diff --git "a/1NFST4oBgHgl3EQfWTh0/content/tmp_files/load_file.txt" "b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1NFST4oBgHgl3EQfWTh0/content/tmp_files/load_file.txt" @@ -0,0 +1,2251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf,len=2250 +page_content='Commuting Cohesions David Jaz Myers Mitchell Riley February 1, 2023 Abstract Shulman’s spatial type theory internalizes the modalities of Lawvere’s axiomatic cohesion in a homotopy type theory, enabling many of the constructions from Schreiber’s modal approach to differential cohomology to be carried out synthetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In spatial type theory, every type carries a spatial cohesion among its points and every function is continuous with respect to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But in mathematical practice, objects may be spatial in more than one way at the same time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' a simplicial space has both topological and simplicial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Moreover, many of the constructions of Schreiber’s differential cohomology and Schreiber and Sati’s account of proper equivariant orbifold cohomology require the interplay of multiple sorts of spatiality — differential, equivariant, and simplicial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this paper, we put forward a type theory with “commuting focuses” which allows for types to carry multiple kinds of spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The theory is a relatively painless extension of spatial type theory, and enables us to give a synthetic account of simplicial, differential, equivariant, and other cohesions carried by the same types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We demonstrate the theory by showing that the homotopy type of any differential stack may be computed from a discrete simplicial set derived from the ˇCech nerve of any good cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We also give other examples of multiple cohesions, such as differential equivariant types and supergeometric types, laying the groundwork for a synthetic account of Schreiber and Sati’s proper orbifold cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Contents 1 Introduction 2 2 A Type Theory with Commuting Focuses 6 3 Specializing a Focus 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Detecting Continuity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Equivariant Differential Cohesion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 Supergeometric Cohesion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 32 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13780v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='CT] 31 Jan 2023 A Proof Sketches for Admissible Rules 36 1 Introduction Homotopy type theory is a novel foundation of mathematics which centers the notion of identification of mathe- matical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In homotopy type theory, every mathematical object is of a certain type of mathematical object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and, if x and y are both objects of type X, then we know by virtue of the definition of the type X what it means to identify x with y as elements of the type X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, if x and y were real vector spaces (so that X was the type of real vector spaces), then to identify x with y would be to give a R-linear isomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x and y were smooth manifolds, then to identify them would be to give a diffeomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x and y were mere numbers, then to identify them would be simply to prove them equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And so on, for any type of mathematical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Homotopy theory, in the abstract, is the study of the identifications of mathematical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Homotopy type theory is well suited for synthetic homotopy theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [12, 24, 13, 18] and many others), but to apply these theorems in algebraic topology — where objects are identified by giving continuous deformations of one into the other — requires a modification to the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To emphasize the difference here, compare the higher inductive circle S1, which is the type freely generated by a point with a self-identification, with the topological circle S1 defined as the set of points in the real plane with unit distance from the origin: S1 ≡ {(x,y) : R2 | x2 +y2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The base point of the higher inductive circle S1 has many non-trivial self-identifications, whereas two points of the topological circle may be identified (in a unique way) just when they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The two types are closely related however: the higher inductive circle S1 is the homotopy type of the topological circle S1 obtained by identifying the points of the latter by continuous deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Ordinary homotopy type theory does not have the language to express this relationship, and therefore cannot apply the synthetic theorems concerning the higher inductive circle to topological questions about the topological circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' What is needed is a way to distinguish between types which carry topological structure and discrete types with only homotopical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In his Cantor’s ‘Lauter Einsen’ and Cohesive Toposes [27], Lawvere points out that this distinction between natively cohesive and discrete sets is already present in the writings of Cantor as the distinction between the Menge of mathematical practice and the abstract Kardinalzahlen which arise by abstracting away from the relationships among the points of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the paper, and his subsequent Axiomatic Cohesion [28], Lawvere formalizes this opposition between cohesion and discreteness as an adjoint triple between toposes: Mengen Kardinalen points codiscrete discrete This adjoint triple induces an adjoint pair of idempotent (co)monads on the topos of spaces or Mengen: the left adjoint, ♭, retopologizes a space with the discrete topology, and the right adjoint, ♯, retopologizes it with the codiscrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lawvere notes that in many cases — when the spaces in question are “locally connected” — there will be a fourth adjoint π0 on the left which produces the discrete set of connected components of a space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' this system of adjoint functors characterizes his axiomatic cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But the real power of Lawvere’s axiomatic cohesion is unlocked by Schreiber’s move from 1-toposes whose objects are cohesive sets to ∞-toposes whose objects are cohesive homotopy types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In his Differential Cohomology in a Cohesive ∞-Topos (DCCT) [44], Schreiber shows that Lawvere’s axiomatics, when interpreted in ∞-toposes, give rise to the hexagonal fracture diagrams which characterize differential cohomology — alongside many other observations about the centrality of the defining adjoints of cohesion in higher topology and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' What was the functor π0 that took the connected components of a space becomes, in the ∞-categorical setting, the functor 2 Π∞ which takes the shape (in the sense of Lurie [31, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6]) of a stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All in all, a cohesive ∞-logos has three adjoint endofunctors S ⊣ ♭ ⊣ ♯ where S takes the shape or homotopy type of a higher space considered as a discrete space, ♭ takes its under- lying homotopy type of discrete points, and ♯ takes the underlying homotopy type of points but retopologized codiscretely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Brouwer’s Fixed Point Theorem in Real-Cohesive Homotopy Type Theory [47] (henceforth Real Cohesion), Shulman brings this distinction between cohesive Mengen and discrete Karndinalen to homotopy type theory via his spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory internalizes the ♭ and ♯ modalities from Schreiber’s DCCT which relate discrete (but homotopically interesting) types like S1 and spatial types like S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory also improves upon a previous axiomatization of these modalities in HoTT due to Schreiber and Shulman [45], by replacing axioms with judgemental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Cohesive homotopy type theory is spatial type theory with an additional axiom that implies the local contractibility of the sorts of spaces in question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' from this axiom the further left adjoint S to ♭ may be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Homotopy type theory may be interpreted into any ∞-topos [26, 46], so that a type in homotopy type theory becomes a sheaf of homotopy types externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, if we interpret the topological circle S1 defined as a subset of R2 into the ∞-topos of sheaves on the site of continuous manifolds, it becomes the sheaf (of sets) represented by the external continuous manifold S1, while the higher inductive circle S1 gets interpreted as the constant sheaf at the homotopy type of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the Yoneda lemma, then, any function definable on S1 is necessarily continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since functions f : X → Y in HoTT are defined simply by specifying an element f(x) : Y in the context of a free variable x : X, variation in a free variable confers a liminal sort of continuity: such an expression could be interpreted in a spatial ∞-topos in which case it necessarily defines a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Shulman’s spatial type theory works by introducing the notion of a crisp free variable to get around this liminal continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' An expression in spatial type theory depends on its crisp free variables discontinuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The modalities ♭ and ♯ of spatial type theory represent crisp variables universally on the left and right respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this way, ♭X is the discrete retopologization of the spatial type X, while ♯X is its codiscrete retopologization — a map out of ♭X is a discontinuous map out of X, while a map into ♯X is a discontinuous map into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory is intended to be interpreted into local geometric morphisms γ : E → S of ∞-toposes, those for which γ∗ has a fully faithful right adjoint γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' which gives a geometric morphism f : S → E (with f∗ := γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=') adjoint to γ which acts as the focal point of E as a space over S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The adjoint modalities ♭ and ♯ are interpreted as the adjoint idempotent (co)monad pair γ∗γ∗ and γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='γ∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A crisp free variable is then one which varies over an object of the focal point S : a free variable is crisp when it is in focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is not only one way for mathematical objects to be spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spaces may cohere with smooth, analytic, algebraic, condensed, and simplicial or cubical combinatorial structures — and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Each of these cases would give rise to a particular spatial type theory as the internal language of an appropriate local ∞-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But there are many cases arising in practice where we need not just one axis of spatiality, but many at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, it is a classical theorem that the homotopy type of a manifold may be computed as the realization of a (topologically discrete) simplicial set associated to the ˇCech nerve of a good open cover of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This theorem relates a simplicial set to a continuous space, via an intermediary simplicial space which is both continuous and simplicial at the same time — the ˇCech nerve of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But in spatial homotopy type theory there is only one notion of crisp variable, and therefore just one sort of spatiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For simplicial types, the discrete reflection is the 0-skeleton sk0, while the codiscrete reflection is the 0- coskeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For simplicial spaces, we then have both the (topologically) discrete ♭ and codiscrete ♯, as well as the simplicially 0-skeletal sk0 and 0-coskeletal csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Interestingly, the ˇCech nerve itself arises from these modal- ities: the ˇCech nerve of a map f : X → Y between 0-skeletal types (that is, continuous or differential stacks with no simplicial structure) is its csk0-image, as we will see later in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A simplicial space has both a shape SX and a realization (or colimit) reX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' the first is a topologically discrete simplicial type, while the latter is a 0-skeletal but spatial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' With all these modalities, we can prove the theorem about good covers described above as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3 Another use case for multiple axes of spatiality is Sati and Schreiber’s Proper orbifold cohomology [43], where orbifolds are understood both as having both differential structure (as differential stacks) and global equivariant structure (concerning their singularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In order to get the correct generalized cohomology of orbifolds without relying on ad-hoc constructions based on a global quotient presentation of the orbifold, Sati and Schreiber work with the ∞-topos of global equivariant differential stacks, which is local both over the global equivariant topos and the topos of differential stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Here the differential modalities S, ♭ and ♯ are augmented with the modalities of equivariant cohesion [41]: < , ⊂ , and ≺ , which take the strict quotient, the underlying space as an invariant type, and the Yoneda embedding of the underlying space of a global equivariant type respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Again the modalities play a central role in the theory, with the ordinary Borel cohomology of a global quotient orbifold X �G being the ordinary cohomology of S ⊂ (X �G), while the proper equivariant Bredon cohomology of X �G is the cohomology of ≺ (X �G), twisted by the map to ≺ BG classifying the quotient map ≺ X → ≺ (X �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In these cases, modalities that lie in the same position in their adjoint chain commute with each other, so, for example, ♭ commutes with sk0 and ♯ commutes with csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, there are cases where these modalities are nested, with one spatiality being a refinement of another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This occurs for example in supergeometry as formulated by Schreiber in [44] with the modalities of solid cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The supergeometric focus is given by the even comodal- ity ⇒ (which takes the even part of a superspace) and the rheonomic modality Rh which is given by localizing at the odd line R0|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this paper, we put forward a modification of spatial type theory to allow for multiple axes of spatiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our theory works by allowing for a meet semi-lattice of focuses ♥,♣,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', each with a separate notion of ♥-crisp variable and pair of adjoint (co)modalities ♭♥ and ♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Like spatial type theory, our custom type theory gets us to the coalface of synthetic homotopy theory very efficiently while staying simple enough to be used in an informal style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The presence of multiple notions of crispness forces a more complex context structure than spatial type theory’s separation of the context into a crisp zone and cohesive zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similar to many other modal type theories [30, 22, 21, 9, 38], we annotate each variable with modal information, here, the focuses for which that variable is crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The typing rules for the modalities of each focus then work essentially independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The exception is ♭-elimination, which is upgraded to allow the crispness of the term being eliminated to be maintained in the variable bound by the induction (a ‘crisp’ induction principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Ours is far from the only extension of type theory with multiple modalities, but as we discuss in more detail later, no existing theory has the combination of features that we are looking for: dependent types (ruling out [30]) that may depend on modal variables (ruling out [9]), multiple commuting comodalities (ruling out [47, 11, 38]) each with a with right-adjoint modality (ruling out [33]) and no further left-adjoints (ruling out [22, 21] and [16, §14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In addition to allowing us to formalize the theorem about ˇCech nerves of open covers as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5, our type theory will be able to handle the equivariant differential cohesion used by Sati and Schreiber in their Proper orbifold cohomology [43], as well as the nested focuses of Schreiber’s supergeometric solid cohesion [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This extends the work of Cherubini [17] and the first author [34, 35, 36] of giving synthetic accounts of the constructions of Schreiber [44] and Sati-Schreiber [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Positing an additional focus does not disturb arguments made using existing focuses, so we also expect our theory to be helpful when dipping into simplicial arguments in the course of other reasoning by adding a simplicial focus and making use of the new modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The problem of defining simplicial types in ordinary Book HoTT remains open, and there are now a number of different approaches to constructing simplicial types which each use some extension to the underlying type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this paper, we will axiomatize the 1-simplex ∆[1] as a linear order with distinct top and bottom and use the cohesive modalities to define the ˇCech nerve of a map and the realization or colimit of a simplicial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We believe our approach here would pair nicely with other approaches to simplicial types for the purposes of synthetic (∞,1)-category theory such as [42, 14, 49, 48], where the sk0 modality would take the core of a Rezk type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 1Though we have not looked in detail at how the focuses would work with the Riehl-Shulman simplicial type theory, and in particular how they would interact with the cubes/topes zones of the Riehl-Shulman context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4 Outline of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' After presenting our type theory in §2, we will look at ways to specialize the spatiality of a focus in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, we will observe that in many cases there is a small class of test spaces Gi so that codiscreteness (that is, being ♯-modal) is detected by uniquely lifting against the ♭-counits ♭Gi → Gi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' such Gi will be said to detect continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Externally, the Gi could be any family which generates the logos under colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In practice, the Gi will be test spaces which minimally carry the appropriate spatiality: in the simplicial case, the simplices ∆[n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in the real-cohesive case, the Euclidean spaces Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' for condensed sets, the profinite sets, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In §3, we will also meet a family of axioms which hold for spatialities that are locally contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, continuous manifolds which are built from Euclidean spaces by colimits are locally contractible, while condensed sets which are built from profinite sets by colimits need not be locally contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In general, a space is locally contractible when it has a constant shape in the sense of Lurie [31, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may define a space C to be contractible when any map C → S to a discrete space S is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If the converse holds — a space S is discrete (♭-modal) if every map C → S is constant — then we say that C detects the connectivity of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, R detects the connectivity of continuous ∞-groupoids, and ∆[1] detects the connectivity of simplicial ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If there is a space (or family of spaces) which detects connectivity, then the local geometric morphism p corresponding to the morphism is furthermore strongly locally contractible in that p∗ has a left adjoint p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' which takes the (constant value of the) shape of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the case that p is both local and strongly locally contractible, we say that p is cohesive following Lawvere [28], Schreiber [44], and Shulman [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Nullifying at the family of spaces which detect connectivity gives a modality S which is left adjoint to ♭;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' it may be thought of as taking the homotopy type of a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In §4 we will give example axioms for specializing single focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will review Shulman’s axioms for real cohesion, where the Euclidean spaces Rn detect continuity and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will then see simplicial cohesion in some detail, where the simplices ∆[n] detect continuity and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We give our types simplicial structure by axiomatizing the 1-simplex ∆[1] as a total order with distinct top and bottom elements, following Joyal’s characterization of simplicial sets as the classifying topos for such orders [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We use the csk0 modality to construct ˇCech nerves of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then we will describe the global equivariant cohesion first observed by Rezk [41] and used by Sati and Schreiber in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we will briefly describe axioms for topological focuses such as Johnstone’s topological topos of sequential spaces [25] and the condensed/pyknotic topos of Clausen-Scholze [19] and Barwick-Haine [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' After surveying some of the different sorts of spatiality which types might carry, we turn our attention to multiple focuses in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3, we define what it means for two cohesions to be orthogonal: when the family which detects the connectivity of one is discrete with respect to the other, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We then prove a few lemmas concerning orthogonal cohesions, in particular concerning when it is possible to commute the various modalities past each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we give examples of multiple focuses in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We begin with simplicial real cohesion, which has both a simplicial focus and a real-cohesive focus which are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We prove, in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5, that the shape of any 0-skeletal type M may be computed as the realization of a topologically discrete simplicial type constructed from the ˇCech nerve of any good cover U of M — one for which finite intersections of the Ui are contractible in the sense of being S-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Next, we combine equivariant cohesion with differential cohesion to give the series of modalities used in Sati and Schreiber’s Proper orbifold cohomology [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Happily, no extra axioms are needed to show that the two cohesions are orthogonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we prove this in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we describe the supergeometric or “solid” cohesion of Schreiber’s Differential Cohomology in a Co- hesive ∞-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This extends real cohesion with the odd line R0|1, where the “discrete” comodality of the super- geometric focus takes the even part of a supergeometric space, and the “codiscrete” modality takes a rheonomic reflection of the space, one whose super structure is uniquely determined by its even structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Unlike our other examples where the focuses involved are orthogonal, here the differential focus is included in the supergeometric focus: any discrete space is also purely even, as is any codiscrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We would like to thank Urs Schreiber for his careful reading and extensive comments during 5 the drafting process of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And we would like to thank Hisham Sati for his feedback and words of encour- agement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The authors are grateful for the support of Tamkeen under the NYU Abu Dhabi Research Institute grant CG008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2 A Type Theory with Commuting Focuses The fundamental duality in higher topos theory is between the ∞-topos — a general sort of space — and the ∞- logos — the category of sheaves of homotopy types on such a space [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This duality is perfect: a map of ∞-toposes E → F is defined to be a lex accessible functor Sh∞(F) → Sh∞(E ) between their corresponding ∞-logoses in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This duality between toposes and logoses gives a nice perspective on the distinction between the petite toposes, which are used as generalized spaces in practice, and the gros toposes — or rather, their dual logoses — which are used as categories of spaces, rather than as spaces in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Quite opposite to their names, the petite toposes are “big” spaces, while the gros toposes are “small” spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' it is their dual logoses which are correctly described by the adjectives “petite” and “gros”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the logos is the category of sheaves on the topos, or equivalently the category of ´etale maps into the topos, the “larger” the topos the more constraining the ´etale condition becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For that reason, the gros toposes have qualitatively “smaller” categories of sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On the other hand, the more general the ´etale spaces may be, the “smaller” the base topos must be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In general, the “biggest” logoses, the logoses of spaces, must correspond to the “smallest” toposes: those toposes which are infinitesimal patches around a focal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This point of view is emphasized in Chapter 4 of DCCT [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may therefore, as a first pass, identify logoses of spaces as dual to those toposes E which are local over a focal point F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A geometric morphism p : E → F is local when it admits a left adjoint right inverse f : F → E in the (∞,2)-category of toposes which we call the focal point of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If E is a topological space (that is, if its corresponding logos Sh∞(E ) is the category of sheaves Sh∞(X) on a sober topological space X), then the terminal geometric morphism γ : E → S is local just when X has a focal point: a point f ∈ X whose only open neighborhood is the whole of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, the prime spectrum of a ring A is local if and only if A is a local ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in this case, the focal point is the unique maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On the logos side, this means that the direct image p∗ admits a fully faithful right adjoint p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' (which is f∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All together, this gives an adjoint triple between the corresponding logoses: Sh∞(E ) Sh∞(F) p∗ p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' p∗ Thinking of the objects of Sh∞(E ) as generalized spaces and the objects of Sh∞(F) as mere homotopy types (sheaves on a point), we may see the direct image p∗ as taking the underlying homotopy type of points of a space, while p∗ and p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' are the discrete and codiscrete topologizations of bare homotopy types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This adjoint triple gives rise to an adjoint pair p∗p∗ ⊣ p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='p∗ of a idempotent comonad p∗p∗ and idempotent monad p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='p∗ on the logos Sh∞(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Understood as operations on spaces, these are the discrete and codiscrete retopologizations of a space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Examples of local toposes with focal point F having category of sheaves Sh∞(F) = ∞Grpd the ∞-category of ∞-groupoids include simplicial types ∞Grpd∆op (where discrete is 0-skeletal and codiscrete is 0-coskeletal),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' continuous and differentiable ∞-groupoids2 Sh∞({Rn}) (where discrete means all charts are constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and codis- crete means that any function valued in the set of points is a chart),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' condensed ∞-groupoids (where discrete means discrete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and codiscrete means codiscrete),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' and global equivariant ∞-groupoids ∞GrpdGloop (where discrete means 2These are the gros toposes of C 0 and C ∞ manifolds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6 being a constant presheaf on the global orbit category, and codiscrete means being a presheaf representable by an ordinary ∞-groupoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Shulman [46] has shown that every ∞-logos may be presented by a model of homotopy type theory, allowing reasoning conducted in homotopy type theory to be interpreted in any ∞-logos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this sense, homotopy type theory is to ∞-logoses as set theory is to the 1-logoses of Grothendieck, Lawvere, and Tierney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Brouwer’s Fixed Point Theorem in Real-Cohesive Homotopy Type Theory [47], Shulman also put forward a spatial type theory which may (conjecturally) be interpreted into any local geometric morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Spatial type theory is characterized by including an adjoint pair ♭ ⊣ ♯ of a lex comodality ♭ and lex modality ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These are to be interpreted as p∗p∗ and p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='p∗ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In spatial type theory, any type has a spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The existence of this spatial structure is witnessed by the two opposite ways that we can get rid of it: either we can remove all the spatial relationships between points, using the “discrete” ♭ comodality, or we can trivialize the spatial relations using the “codiscrete” ♯ modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We emphasize that this spatial structure is distinct from the homotopical structure that all types have by virtue of the identifications between their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, the topological circle S1 := {(x,y) : R2 | x2 +y2 = 1} has a spatial structure as a subset of the Euclidean plane (as a sheaf on the site of continuous manifolds, for example), but is a homotopy 0-type (or “set”) without any non-trivial identifications between its points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in particular ΩS1 = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The homotopy type S1 of the circle, however, is spatially discrete but has many non-trivial identifications of its point: in particular ΩS1 = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is not, however, only one way to be spatial in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example a simplicial topological space has both a simplicial structure and a topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This can be witnessed at the level of toposes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If p : E → F admits a focal point f : F → E , then f ∆op : F ∆op → E ∆op is also a focal point of p∆op : E ∆op → F ∆op, where the logos Sh∞(E ∆op) := (Sh∞(E ))∆op is the category of simplicial objects in the logos Sh∞(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But there is another local geometric morphism γ : E ∆op → E where γ∗ sends a simplicial sheaf X• to X0 and γ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' is given by the 0-coskeleton csk0 Sn := Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These two different axes of spatiality on the objects of Sh∞(E )∆op commute, in that the following diagram of adjoints commutes: Sh∞(E )∆op Sh∞(F)∆op Sh∞(E ) Sh∞(F) p∆op ∗ γ∗ γ∗ p∗ In particular, we have that p∗∆op p∗∆op and γ∗γ∗ commute as endofunctors of Sh∞(E )∆op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The former discretely retopologizes a simplicial space, while the latter includes the space of 0-simplices as a 0-skeletal simplicial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Each focus gives an axis along which the objects of the top logos Sh∞(E )∆op may carry spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' When working in, say, simplicial differential spaces, we would like to have access to both the S ⊣ ♭ ⊣ ♯ of real cohesion and the re ⊣ sk0 ⊣ csk0 of simplicial cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Shulman’s spatial type theory offers no way to do this: the ♭ and sk0 comonads have incompatible claims on the notion of ‘crisp’ variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The solution is to allow a separate notion of crispness for each focus we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section, we will describe the rules for a type theory with commuting focuses, generalizing ordinary spatial type theory in the case of a single non-trivial focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will then describe axioms which make these into commuting cohesions, in the sense of cohesive type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To this end, we will fix an commutative idempotent monoid Focus of focuses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we will write the product of the focus ♥ and the focus ♣ as ♥♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This product induces an ordering on the focuses by saying that ♣ ≤ ♥ whenever 7 ♣♥ = ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' With respect to this ordering, the product becomes the meet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we may therefore also think of Focus as a meet semi-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will write the identity element of Focus as ⊤, and note that it is the top focus with respect to the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For most of our purposes in this paper, our commutative idempotent monoid Focus of focuses will be freely generated by a finite set of basic focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Explicitly, we may take Focus = Pf (BasicFocuses)op to be the set of finite subsets of the set of basic focuses with union as the product, and therefore the opposite of the ordering of subsets by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All variables in the context will be annotated with the focus that they are in: x :♥ X ⊢ t : T In general, we will abbreviate the context entry x :⊤ X as x : X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the case that Focus = {♥ ≤ ⊤} is freely generated by one basic focus, we recover the split context used in Shulman’s spatial type theory, where our context x :♥ X, y :⊤ Y ctx corresponds to Shulman’s context x :: X | y : Y ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To describe the typing rules, we will need a couple of auxiliary operations on contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first operation ♥Γ adds a specific focus ♥ to the annotation on every variable in a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So: ♥(·) :≡ · ♥(Γ,x :♣ A) :≡ (♥Γ),x :♥♣ A We also need an operation ♥ \\ Γ that deletes any variables not contained within a given focus ♥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' this is the equivalent of going from ∆ | Γ ctx to ∆ | · ctx in ordinary spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♥\\(·) :≡ · ♥\\(Γ,x :♣ A) :≡ � (♥\\Γ),x :♣ A if ♣ ≤ ♥ ♥\\Γ otherwise We say that a variable x :♣ X is ♥-crisp if ♣ ≤ ♥, and so the ♥-crisp variables are precisely those that survive the ♥\\Γ operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that a term t : T is ♥-crisp if both it and its type T only contain ♥-crisp variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', it is well-formed in context ♥\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We are now ready to describe the rules of the type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' All the usual type formers — Σs, Πs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' — will be included as usual, only referring to variables of the top focus ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the convention that x :⊤ X be written as x : X, these rules look exactly as they do usually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We therefore focus on the new features of type theory with commuting focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Structural Rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' CTX-EMPTY · ctx CTX-EXT ♥\\Γ ⊢ A type Γ,x :♥ A ctx VAR Γ,x :♥ A,Γ′ ctx Γ,x :♥ A,Γ′ ⊢ x : A In prose, these rules read as follows: CTX-EMPTY: The empty context is a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' CTX-EXT: If A is a ♥-crisp type in context Γ, then Γ,x :♥ A ctx is a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' VAR: If x :♥ A appears in a context, then the variable x has type A in that context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Given a context Γ,x :♥ A ctx, it must be the case that A only depends on the variables in Γ which are themselves ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This careful context formation rule is what replaces the division of the context into two zones in Shulman’s spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the conclusion of the variable rule, the type A is well-formed in context Γ,x :♥ A,Γ′ ctx by the admissible DIVIDE-WK rule given below, followed by further weakening with Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 8 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rather than annotating variables, may be tempting to try a floating context separator |♥ for each focus, so that the variables to the left of |♥ are precisely the ♥-crisp ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Such contexts are not sufficiently general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' specifically, the ♭-elimination rule will let us produce a context containing x :♥ A,y :♣ B which clearly cannot be separated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following rules and equations will be made admissible, with the proofs sketched in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' WK Γ,Γ′ ⊢J Γ,x :♥ A,Γ′ ⊢J −−−−−−−−−− SUBST ♥\\Γ ⊢ a : A Γ,x :♥ A,Γ′ ⊢J Γ,Γ′[a/x] ⊢J [a/x] −−−−−−−−−−−−−−−−−−−−− PROMOTE-CTX Γ ctx ♥Γ ctx −−−− PROMOTE Γ ⊢J ♥Γ ⊢J −−−−− DIVIDE-CTX Γ ctx ♥\\Γ ctx −−−−−− DIVIDE-WK ♥\\Γ ⊢J Γ ⊢J −−−−−− ♥(♣Γ) ≡ (♥♣)Γ ♥\\(♣\\Γ) ≡ (♣♥)\\Γ First, we have ordinary weakening by a variable, and a ‘crisp’ substitution similar to that used in spatial type theory, where crisp variables may only be substituted with similarly crisp terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These specialize to the ordinary weakening and substitution rules when used for ♥ ≡ ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PROMOTE-CTX corresponds to the application of the endofunctor ♥ to the context Γ, and PROMOTE to precomposition with the counit morphism ♥Γ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DIVIDE-CTX gives the largest ‘subcontext’ ♥ \\ Γ of Γ such that there is a substitution Γ → ♥(♥ \\ Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The context operation ♥ \\ − thus acts like a left-adjoint to ♥−, although semantically a left-adjoint may not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rules for ♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We now come to the rules for the ♭ comodality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\Γ ⊢ A type Γ ⊢ ♭♥A type ♭-INTRO ♥\\Γ ⊢ M : A Γ ⊢ M♭♥ : ♭♥A ♭-ELIM ♣♥\\Γ ⊢ A type Γ,x :♣ ♭♥A ⊢ C type ♣\\Γ ⊢ M : ♭♥A Γ,u :♣♥ A ⊢ N : C[u♭♥/x] Γ ⊢ (let u♭♥ := M inN) : C[M/x] ♭-BETA ♣♥\\Γ ⊢ A type Γ,x :♣ ♭♥A ⊢ C type ♣♥\\Γ ⊢ K : A Γ,u :♣♥ A ⊢ N : C[u♭♥/x] Γ ⊢ (let u♭♥ := K♭♥ inN) ≡ N[K/u] : C[K♭♥/x] In prose, these rules read as follows: ♭-FORM: If A is a ♥-crisp type, then we may form ♭♥A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-INTRO: If M is a ♥-crisp term of type A, then we may form M♭♥ of type ♭♥A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-ELIM: If C is a type depending on the ♣-crisp variable x :♣ ♭♥A, and M : ♭♥A is a ♣-crisp element of type ♭♥A, then we may assume that M is of the form u♭♥ for a ♣♥-crisp variable u :♣♥ A when defining an 9 element of C[M/x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We write this element as (let u♭♥ := M inN) : C[M/x] where N : C[u♭♥/x] is the element we defined assuming that M was of the form u♭♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The equation ♥\\(♣\\Γ) ≡ (♣♥)\\Γ is necessary here to know that the type ♭♥A is well-formed in context ♣\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-BETA: If M actually is of the form K♭♥ for suitably crisp K, then we simply substitute K in for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The term K must be ♣♥-crisp for both the ♭-INTRO and ♭-ELIM to have been applied, and so its substitution for the ♣♥-crisp variable u is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These rules are stronger than the ones used by Shulman for spatial type theory, even in the case of a single focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We have built in a ♣-crisp induction principle for ♭♥, for any two focuses ♥ and ♣: if the term we are inducting on is already ♣-crisp, then we may maintain that crispness in the new assumption u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we have a single non-trivial focus ♥, as is the case in Shulman’s type theory, then taking ♣ = ♥ in the above expression yields the ‘crisp ♭ induction’ principle of [47, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This induction principle is proven by taking a detour through ♯, but here we choose to build it into the rule directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our elimination rule is in fact also admissible from the less general one that requires the freshly bound variable to only be ♥-crisp, but we choose the more general rule for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rules for ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The rules for ♯ are a little simpler, and in the case of a single focus specialize exactly to the rules of spatial type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥Γ ⊢ A type Γ ⊢ ♯♥A type ♯-INTRO ♥Γ ⊢ M : A Γ ⊢ M♯♥ : ♯♥A ♯-ELIM ♥\\Γ ⊢ N : ♯♥A Γ ⊢ N♯♥ : A ♯-BETA ♥\\Γ ⊢ M : A Γ ⊢ (M♯♥)♯♥ ≡ M : A ♯-ETA Γ ⊢ N : ♯♥A Γ ⊢ N ≡ (N♯♥)♯♥ : ♯♥A In prose, these rules read as follows: ♯-FORM: When forming the type ♯♥A, all variables may be used in A as though they are ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-INTRO: When forming a term M♯♥ : ♯♥A, all variables may be used in M as though they are ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-ELIM: If N is a ♥-crisp element of ♯♥A, we may extract an element N♯♥ : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-BETA: If M is a ♥-crisp element of A, then M♯♥♯♥ ≡ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-ETA: Any term of N : ♯♥A is definitionally equal to N♯♥ ♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As in ordinary spatial type theory, the term N♯♥ may not be well-typed on its own, because it may use non-crisp variables of the context Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is however well-typed underneath the outer (−)♯♥, since the introduction rule allows us to use any variable as though it is ♥-crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Perhaps surprisingly, the shape of the ♯-FORM and ♯-INTRO rules is what builds the left-exactness of ♭ into the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is the case even in ordinary spatial type theory, not a feature that only appears in this multi-focus setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The trick is that the promotion operation ♥Γ distributes over the context extensions in Γ rather than being a ‘stuck’ context former applied to Γ as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Specifically, when using ♯ to derive crisp Id-induction, one applies ♯ to a type x :: A,y :: A, p :: (x = y) | · ⊢ C type, yielding a type | x : A,y : A, p : (x = y) ⊢ ♯C type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Internalized, the former context represents the type (x : ♭A)×(y : ♭A)×♭(x♭ = y♭), but ♯-FORM treats it as identical to ♭((x : A)×(y : A)×(x = y)) when applying adjointness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 10 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In most cases of interest, our commutative idempotent monoid of focuses is freely generated by a finite set of basic focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this situation, it suffices to provide the ♭ and ♯ only for the basic focuses, as the remainder can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The top focus ⊤ (which semantically corresponds to the entire topos we are working in) has both ♭⊤A and ♯⊤A canonically equivalent to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And given focuses ♥ and ♣, it is quickly proven that ♭♥♣ is equivalent to ♭♥♭♣ and similarly for the ♯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Related Type Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Besides the original spatial type theory, there are several other dependent modal type theories that come close to our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ‘adjoint type theory’ perspective [40, 29, 30] was the guiding principle that led to the original spatial type theory of [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Indeed, when instantiated with appropriate mode theory, the framework of [30] reproduces a simply typed version of the theory presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The specific mode theory to be used is a cartesian monoid with a system of commuting, product-preserving endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A dependently typed variant of adjoint type theory is not yet forthcoming, but we expect that our dependent type theory would be an instance of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' An separate line of work on modal type theories is Multimodal Type Theory [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In MTT, every mode morphism µ is reified in the type theory as a positive type former, and each modality modµ must have a left- adjoint-like context operator written �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we do not assume the existence of S, then we are only able to describe ♯ in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Later work [21] describes a multimodal type theory where each mode morphism becomes a (more convenient) negative type former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The semantic requirements are even stronger: the functor corresponding to the modality must be a dependent right-adjoint [11], whose left adjoint is itself a parametric right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is too strict even to capture ♯ without additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In [16, §14], an alternative ‘cohesive type theory’ is presented, using a combination of the above two styles of modal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Rather than working with the endofunctors on the topos of interest, the cohesive setting is kept as an adjoint quadruple Π0 ⊣ Disc ⊣ Γ ⊣ CoDisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A positive type former is used for Disc and negative type formers for Γ and CoDisc, due to the requirements on having one or two left-adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is likely that this could be extended to commuting cohesions, but the interactions of the various context �− operations for the left-adjoints may be difficult to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The type theory with context structure most formally similar to ours is ParamDTT [38, 37], where variables in the context annotated with a modality indicate a variable under that modality directly, not its left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is from this work that we take the left-division notation − \\ Γ for the clearing operation on contexts, which itself has appeared in other guises, for example [39, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ParamDTT uses a fixed ‘mode theory’ with three modalities {¶,id,♯} equipped with a particular composition law, but it is clear that the rules for contexts and basic type formers would work equally well for other sets of modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A version of the cohesive ♭ can be derived from the ‘modal Σ-type’, fixing the second component to be the unit type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There does not appear to be a way to derive the ordinary (negative) rules for ♯ in ParamDTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3 Specializing a Focus A focus gives a specific axis along which a type may be spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In simplicial cohesion, we have a simplicial focus sk0 ⊣ csk0 and in differential cohesion a differential focus ♭ ⊣ ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But what makes the simplicial focus simplicial and the differential focus differential?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section, we will investigate two axioms schemes which can determine the peculiarities of a given focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the next section, we will see these axioms in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, we note that with a single focus, type theory with commuting focuses is the same theory as Shulman’s spatial type theory in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Any of the lemmas and theorems proven in §3, 4, 5, and 6 of Real Cohesion [47] concerning ♭ and ♯ and using no axioms are true also of ♭♥ and ♯♥ for any fixed focus ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorems which do involve the use of axioms are also valid, so long as the crispness used in those axioms is interpreted as ♥-crispness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The rules for ♭♥ and ♯♥ specialize to Shulman’s rules, and therefore his proofs carry through directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 11 Specifically, ♭♥ is a coreflector and ♯♥ is a monadic modality, both are lex, and ♭♥ is (♥-crisply) left-adjoint to ♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since adding a focus only expands the rules of the type theory and does not restrict the application of any of the rules for any of the other focuses, any of the theorems proven in this section for a single focus will apply when working with multiple focuses as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For the rest of this section, we will work within a single focus ♥, and for that reason we will drop the anno- tations by ♥ in our expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, we will write ♭♥ as simply ♭, and we will write x :♥ X as x :: X, following Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Detecting Continuity In this section, we will look at an axiom which ties the liminal sort of “continuity” implied by the crisp variables of the type theory to the concrete continuity of a particular type G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our axiom will take the form of a lifting property characterizing those crisp maps which are ♯-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As we will show in the upcoming Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2, a crisp map is ♯-modal if and only if it lifts crisply (in a sense made precise in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1) against all of the ♭-counits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let c :: A → B and f :: X → Y be crisp maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that c lifts crisply against f if for any crisp square as on the left below, there is a unique crisp filler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A X B Y c f ∀ ∀ ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭(XB) ♭(XA) ♭(Y B) ♭(Y A) ♭(◦ f) ♭(c◦) ♭(◦f) ♭(c◦) ⌟ More formally, we write c ⊥♭ f for the proposition that the square on the right is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A crisp map f :: X → Y is ♯-modal if and only if for all crisp A, (ε : ♭A → A) ⊥♭ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If f is ♯-modal, then since ♯ is lex, it lifts on the right against all ♯-equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any crisp A, the ♭-counit ε : ♭A → A is a ♯-equivalence by [47, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, the square XA X♭A Y A Y ♭A f◦ ε f◦ ε ⌟ is a pullback, and since ♭ preserves crisp pullbacks ([47, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10]), we see that ε ⊥♭ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On the other hand, suppose that f lifts crisply on the right against all ♭-counits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show that f is ♯-modal, it will suffice to show that its ♯-naturality square is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let X → ♯X ×♯Y Y be the gap map of the ♯-naturality square of f, seeking to show that this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It suffices to split the gap map over the naturality square, by the universal property of the pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So, consider the crisp square ♭(♯X ×♯Y Y) X ♯X ×♯Y Y Y snd ε f F k where F(t) :≡ (let t := p♭ in(fst p)♯) is a version of the first projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To check that the square commutes, it suffices by ♭-induction to give, for crisp elements u :: ♯X, y :: Y, and p :: (♯f(u) = y♯), a term of type f(u♯) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But we have crisply that ♯f(u) ≡ ♯f(u♯♯) = (f(u♯))♯ by the definition of ♯f, and composing this path with p we know 12 p′ :: (f(u♯))♯ = y♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the lexness of ♯, we therefore also have p′′ :: ♯(f(u♯) = y), so that the square commutes by p′′♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By hypothesis, there is a unique crisp map k : ♯X ×♯Y Y → X filling this square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The bottom triangle says precisely that k lives over the second projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will turn the top triangle into a proof that k lives over the first projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let (u,y, p) : ♯X ×♯Y Y, seeking to show that k(u,y, p)♯ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This latter type of paths is codiscrete (because ♯X is codiscrete), and so when mapping into it we may assume by that u is of the form x♯, reducing our goal to k(x♯,y, p)♯ = x♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the lexness of ♯, it suffices to give an element of ♯(k(x♯,y, p) = x), and for this it suffices to give an element of k(x♯,y, p) = x under the hypotheses that x, y, and p are crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this case, (x♯,y, p)♭ : ♭(♯X ×♯Y Y), and so we have that k(x♯,y, p) = F((x♯,y, p)♭) by the upper triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But by definition, F((x♯,y, p)♭) ≡ x♯♯ ≡ x, so that we have succeeded in giving the desired identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We have shown that k lives over the naturality square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' now we need to show that it splits the gap map X → ♯X ×♯Y Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To that end, consider the following diagram: ♭X ♭(♯X ×♯Y Y) X X ♯X ×♯Y Y Y gap ♭gap ε ε f f k Showing that the diagram commutes as drawn follows easily by ♭-induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We then have two crisp fillers of the outer square: first we have idX : X → X and k ◦gap : X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the uniqueness of crisp fillers, we conclude that these must be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Knowing that the crisp ♯-modal maps may be characterized by lifting crisply against ♭-counits suggests that we could axiomatize the particular qualities of ♯ by restricting the class of ♭-counits which it suffices to lift against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To that end, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G :: I → Type be a crisp type family indexed by a ♭-modal and inhabited type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that G detects continuity when, for every crisp map f :: X → Y, { f is ♯-modal} �(ε : ♭Gi → Gi) ⊥♭ f for all i :: I � Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Thinking externally, it is straightforward to see that any family Gi which generates a local topos E in question under colimits will detect continuity for the focus given by the terminal map of toposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is because ♭, as a left adjoint, commutes with all colimits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' therefore the problem of lifting against the ♭-counit of any object of E can be reduced to that of lifting against the ♭-counits of the generators Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A crisp type X is ♯-modal if and only if ♭(A → X) → ♭(♭A → X) is an equivalence for all crisp types A, and if G detects continuity then it suffices to check for each Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' When f : X → 1, the square defining crisp lifting is a pullback iff the top map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If a family detects continuity, then it is a separating family for crisp maps in the following precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that G :: I → Type detects continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f :: X → Y be a crisp map for which ♭f : ♭X → ♭Y is an equivalence and for all i :: I, the induced map ♭(Gi → X) → ♭(Gi → Y) given by post-composing with f is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then f is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, note that f is a ♯-equivalence since it is by hypothesis a ♭-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It therefore suffices to show that f is ♯-modal, which by the assumption that G detects continuity means showing that f lifts crisply against all ♭-counits ♭Gi → Gi for i :: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider the following diagram: ♭(XGi) ♭(X♭Gi) ♭(♯XGi) ♭(Y Gi) ♭(Y ♭Gi) ♭(♯Y Gi) ♭(f◦) ♭(f◦) ∼ ∼ ♭(♯ f◦) The square on the left is the one we need to show is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this, it will suffice to show that the middle map ♭(f◦) : ♭(X♭Gi) → ♭(Y ♭Gi) is an equivalence, since the leftmost vertical map is an equivalence by hypothesis and any square with two sides equivalences is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For that, the middle vertical map is equivalent by the adjunction ♭ ⊣ ♯ to the rightmost vertical map ([47, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But the rightmost vertical map is an equivalence because it is post-composition by the equivalence ♯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Detecting Connectivity A focus is said to be cohesive if ♭ has a further left adjoint S which is itself a modality: ♭(SA → X) ≃ ♭(A → ♭X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This adjunction only determines S for crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is better to define S by nullifying a family of objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' then S is determined for all types (of any size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To this end, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G :: I → Type be a crisp type family indexed by a ♭-modal type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that G detects connectivity when, for any crisp type X, {X is ♭-modal} �X is Gi-null for all i :: I � If G detects connectivity, then S is defined to be nullification at the family Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In Real Cohesion [47], the assertion that a given family G detects connectivity is known as Axiom C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In [44, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='48], a single object with this property is said to ‘exhibit the cohesion’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If there is a family G which detects connectivity, then we say that the focus is cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is justified by the following theorem, which we may import directly from Real Cohesion [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that G detects connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then a crisp type is S-modal if and only if it is ♭-modal, and furthermore S is crisply left-adjoint to ♭: ♭(SA → X) → ♭(A → ♭X) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is [47, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4 Examples of Focuses To keep the various operators visually distinct, we will use completely different symbols for each focus we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The rules governing the type formers are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' S ⊣ ♭ ⊣ ♯ denotes real cohesion, where a set of real numbers (possible the Dedekind reals or an axiomatically asserted set of “smooth reals”) detects connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 14 re ⊣ sk0 ⊣ csk0 denotes simplicial cohesion, where the (axiomatically asserted) 1-simplex ∆[1] detects con- nectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' < ⊣ ⊂ ⊣ ≺ denotes global equivariant cohesion, where connectivity is detected by ≺ BG for finite groups G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This notational convention follows Sati and Schreiber [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Various topological toposes exhibit spatial type theory, with ♭ ⊣ ♯ retopologizing types with the discrete and codiscrete topologies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, Johnstone’s topological topos has a focus whose continuity is detected by the walking convergent sequence N∞, which may be constructed as the set of monotone functions N → {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Real Cohesions In Real Cohesion [47], Shulman gives the axiom R♭ which states that a crisp type is ♭-modal if and only if it is RD-null, where RD is the set of Dedekind cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the terminology of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1, this says that RD detects continuous real connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 1 (Continuous Real Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that RD detects continuous real connectivity, and also Shul- man’s Axiom T: For every x : RD, the proposition (x > 0) is ♯-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Though Shulman does not consider this axiom, we may also add the assumption that the family Rn D detects continuous real continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Using this assumption, we may internalize the arguments of Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='33 of [47] to show that the mysterious Axiom T follows from the proposition that if f :: Rn D → RD is crisp and f(x) > 0 for any crisp x :: Rn D, then in fact f(x) > 0 for all (not necessarily crisp) x : Rn D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since, by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='28 of [47] (assuming the crisp LEM or the axiom of countable choice), any crisp Dedekind real is a Cauchy real, we are equivalently asking if a function f : RD → RD is positive on all Cauchy reals, is it always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This seems obvious, but as Shulman notes in Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='34, this obvious statement is not always true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' though assuming countable choice it is likely provable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There are continuous but non-differentiable functions f : RD → RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we want to work in a topos where the types have a smooth structure instead of just a continuous structure, then we must work with a type of smooth reals RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The most common way to axiomatize the type of smooth reals is using the Kock-Lawvere axiom and the other axioms of synthetic differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 of [36] for a list of these axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In any case, if RS is a type of smooth reals, then we will take differential cohesion to mean that RS detects connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 2 (Differential Real Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If RS is a type of smooth reals (say, from synthetic differential geometry), then we assume that RS detects differential connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Simplicial Cohesion There is a well known difficulty in describing simplicial types in ordinary homotopy type theory — the infinite amount of coherence data is difficult to describe formally given the tools of type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This difficulty has led to extensions of type theory such as two-level type theories [5, 8, 4] which augment HoTT with strict equalities which can be use to define simplicial homotopy types satisfying the simplicial identities strictly, bypassing the problematic tower of coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Another approach to avoiding simplicial difficulties is to simply interpret type theory into a topos of simplicial homotopy types, rather than mere homotopy types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is the approach taken by Riehl and Shulman in [42], where they present a type theory that makes every type into a simplicial type and has as primitives the simplices ∆[n], so that a simplex in a type A is a function σ : ∆[n] → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section we will also work with simplicial homotopy types, but by different means to Riehl and Shulman’s type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Instead, we will describe simplicial cohesion, with adjoint modalities re ⊣ sk0 ⊣ csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These are defined semantically as follows: 15 The (“simplicial flat”) 0-skeletal comodality X �→ sk0 X sends a simplicial type to its 0-skeleton: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X2 X1 X0 sk0 �−−−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X0 X0 X0 The (“simplicial sharp”) 0-coskeletal modality X �→ csk0 X sends a simplicial type to its 0-coskeleton: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X2 X1 X0 csk0 �−−−−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X0 ×X0 ×X0 X0 ×X0 X0 The (“simplicial shape”) realization modality X �→ reX sends a simplicial type to its realization (or homotopy colimit), considered as a 0-skeletal simplicial type: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X2 X1 X0 re· �−−−−−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' colimXn colimXn colimXn Because simplicial sets are the classifying (0-)topos for strict intervals (totally ordered sets with distinct top and bottom elements) [50], and since the ∞-topos of simplicial homotopy types is the enveloping ∞-topos3 of simplicial sets [6], we may assume the existence of an interval ∆[1] to make sure that our type theory is interpreted in an ∞-topos equipped with a geometric morphism to S ∆op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may then define the n-simplex ∆[n] to be the n-length increasing sequences in ∆[1], and define an n-simplex in a type X to be a map ∆[n] → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 3 (Simplicial Axioms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We presume that ∆[1] is a total order with distinct top and bottom elements which we call 1 and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Explicitly, this means that we have elements 0, 1 : ∆[1] and a proposition x ≤ y : Prop for x, y : ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This order must satisfy the following axioms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, x ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, y, and z, if x ≤ y and y ≤ z then x ≤ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x and y, if x ≤ y and y ≤ x, then x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, y, either x ≤ y or y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For all x, 0 ≤ x and x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 0 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3The enveloping ∞-topos of a topos is its free (homotopy) cocompletion, fixing existing homotopy colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 16 From these axioms, we may define the other simplices ∆[n] to be the chains of length n in ∆[1]: ∆[n] :≡ {⃗x : ∆[1]n | x1 ≤ x2 ≤ ··· ≤ xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We also assume the following: (Axiom ∆sk0) ∆[1] detects simplicial connectivity: a simplicially crisp type X is 0-skeletal if and only if every map ∆[1] → X is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The family ∆[−] : N → Type detects simplicial continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' (Axiom ∂∆) For i : ∆[1], we have csk0((i = 0)∨(i = 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Each ∆[n] is crisply projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, for a simplicially crisp E : ∆[n] → Type, we have a map sk0((i : ∆[n]) → ∃Ei) → ∃sk0((i : ∆[n]) → Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As there is an obvious map the other way, this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let us quickly set the stage by proving that the n-simplices have trivial geometric realization and the 0-skeleton of the n-simplex ∆[n] is the ordinal [n] ≡ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The order ∆[1] has finite meets and joins, and they distribute over each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Moreover, the inclusion {0,1} �→ ∆[1] is an inclusion of lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that x,y : ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then either x ≤ y or y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the former case, define x∧y :≡ x and x∨y :≡ y, and in the latter case x∧y :≡ y and x∨y :≡ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If both hold, then x = y and the definitions agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x,y,z : ∆[1], then these three may find themselves in any of 6 orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' One may then check that in each of these cases, meets distribute over joins and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For example, supposing that x ≤ y ≤ z, then x ∧ (y ∨ z) = x∧z = x, while (x∧y)∨(x∧z) = x∨x = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The n-simplex ∆[n] is a retract of the n-cube ∆[1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Moreover, the inclusion [n] �→ ∆[n] given by i �→ 0 ≤ ··· ≤ 0 ≤ i times � �� � 1 ≤ ··· ≤ 1 is a retract of the inclusion {0,1}n → ∆[1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Given x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=',xn : ∆[1], define m1 :≡ � i:n xi and let i1 : n be its index, then m2 :≡ � n\\{m1} xi, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Note that m1 ≤ m2 ≤ ··· ≤ mn, so that m : ∆[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, if the xi were already in increasing order, then mi = xi, showing that this is a retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We note that this retract argument works just as well on {0,1}n → [n], if we identify [n] with the subset {⃗x : {0,1}n | x1 ≤ ··· ≤ xn} of increasing sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since it only makes use of the lattice structure of {0,1}n and ∆[1]n, and the inclusion is a lattice homomorphism, we conclude that the necessary squares commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The n-simplex has trivial realization: re∆[n] = ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The realization re∆[n] is a retract of the realization re∆[1]n, and this is contractible since ∆[1] detects simplicial connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The inclusion [n] �→ ∆[n] given by i �→ 0 ≤ ··· ≤ 0 ≤ i times � �� � 1 ≤ ··· ≤ 1 is a sk0-equivalence, showing that sk0 ∆[n] ≃ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will show that [1] �→ ∆[1] is a sk0-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This will imply that [1]n �→ ∆[1]n is a sk0-equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' since [n] �→ ∆[n] is a retract of this, we may conclude that it is a sk0-equivalence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since this inclusion {0,1} �→ ∆[1] is simplicially crisp, to show that it is a sk0-counit it will suffice to show that it is a csk0-equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We therefore need an inverse csk0 ∆[1] → csk0{0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the codomain is 0-coskeletal, it suffices to define this map on ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So let i : ∆[1], seeking csk0{0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Axiom ∂∆, we have csk0((i = 0)∨(i = 1)), and since our goal is 0-coskeletal, we may assume that i = 0 or i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If i = 0, then we send it to 0csk0, if i = 1, then we send it to 1csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show that this map is the inverse of csk0{0,1} → csk0 ∆[1], we may appeal to the fact that identities in a modal type are modal, and so we may remove the csk0 around csk0((i = 0) ∨ (i = 1)) and check that the maps invert each other on these elements, which they clearly do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can also define the type of n-simplices in a simplicially crisp type, and prove a few elementary lemmas concerning the n-simplices of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let X be a simplicially crisp type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then define the type Xn of n-simplices in X as Xn :≡ sk0(∆[n] → X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If f : X → Y is a simplicially crisp map, then it induces a map fn : Xn → Yn by post-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If fn : Xn → Yn is an equivalence for all n, then f is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is a special case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6, noting that X0 ≃ sk0 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then for a crisp y : ∆[n] → Y, we have fib fn(ysk0) ≃ sk0((i : ∆[n]) → fib f (y(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We compute: fib fn(ysk0) ≡ (x : Xn)×(fnx = ysk0) ≡ (x : sk0(∆[n] → X))×let τsk0 := xin(f ◦τ)sk0 = ysk0 ≃ (x : sk0(∆[n] → X))×let τsk0 := xinsk0(f ◦τ = y) ≃ sk0((x : ∆[n] → X)×(f ◦x = y)) ≃ sk0((x : ∆[n] → X)×((i : ∆[n]) → (f(x(i)) = y(i)))) ≃ sk0((i : ∆[n]) → fib f (y(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then (im f)n ≃ im fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We use the projectivity of the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 (im f)n ≡ sk0(∆[n] → (y : Y)×∃fib f (y)) ≃ (y : Yn)×let σsk0 := yinsk0((i : ∆[n]) → ∃fib f (σi)) ≃ (y : Yn)×let σsk0 := yin∃sk0((i : ∆[n]) → fib f (σi)) ≃ (y : Yn)×∃fib fn(y) ≡ im fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4This lemma is in fact equivalent to assuming the projectivity of the simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 18 The definition of the n-simplices that we gave above is simple, but it is not that straightforward to see that it is functorial in the ordinal [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can give an alternative definition of the n-simplices which makes the functoriality evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let Interval denote the category of intervals: totally ordered sets with distinct top and bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The maps of Interval are the monotone functions preserving top and bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let FinOrd+ denote the category of finite inhabited ordinals and order preserving maps between them — the usual “simplex category”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We denote by [n] the ordinal {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will need a standard reformulation of the category of finite ordinals in terms of intervals (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [32, §VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7]) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is a contravariant, fully faithful functor ι : FinOrdop + → Interval sending [n] to [n+1] with top element n+1 and bottom element 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To a map f : [n] → [m], we define ι f : [m+1] → [n+1] by ι f(i) :≡ � min{ j | i ≤ f(j)} n+1 if no such minimum exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Conversely, to a monotone map g : [m+1] → [n+1] preserving top and bottom, we define ι−1g : [n] → [m] by the dual formula (ι−1g)( j) :≡ max{i | g(j) ≤ i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may now define the n-simplices in a way which makes clear their functoriality in the category of finite inhabited ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We define the n-simplex ∆[n] to be ∆[n] :≡ Interval(ι[n],∆[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, ∆ : FinOrd+ → Set gives a functor from finite inhabited ordinals to the category of sets, where ∆(f) : ∆[n] → ∆[m] is given by precomposing with ι f : [m+1] → [n+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Noting that [n] ∼= Interval(ι[n],[1]) by the fully-faithfulness of ι, the inclusion of top and bottom elements [1] �→ ∆[1] induces a natural inclusion [n] �→ ∆[n] by post-composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As we saw in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4, these inclusions are sk0-counits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 The ˇCech Complex The 0-coskeleton modality csk0 is useful for working in simplicial cohesion since it enables us to give an easy construction of the ˇCech complex of a map f : X → Y between 0-skeletal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ˇCech complex of such a map is, externally speaking, the simplicial type formed by repeatedly pulling back f along itself: ˇC(f) :≡ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' X ×Y X ×Y X X ×Y X X Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ˇCech complex ˇC(f) of f is defined to be its csk0-image: ˇC(f) :≡ (y : Y)×csk0((x : X)×(fx = y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 19 We will justify this definition by calculating the type of n-simplices of ˇC(f) when both X and Y are 0-skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let f : X → Y be a simplicially crisp map between 0-skeletal types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ˇC(f)n ≃ X ×Y ···×Y X ≃ (y : Y)×((x : X)×(fx = y))n+1 is the (n+1)-fold pullback of f along itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We calculate: ˇC(f)n :≡ sk0(∆[n] → ˇC(f)) ≡ sk0(∆[n] → (y : Y)×csk0((x : X)×(fx = y))) ≃ sk0((σ : ∆[n] → Y)×((i : ∆[n]) → csk0((x : X)×(fx = σi)))) Since Y is 0-skeletal, any map ∆[n] → Y is constant, so we may continue: ≃ sk0((y : Y)×(∆[n] → csk0((x : X)×(fx = y)))) Since, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4, sk0 ∆[n] = [n], we may use the adjointness of sk0 and csk0 to continue: ≃ sk0((y : Y)×csk0([n] → (x : X)×(fx = y))) Now, we may use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8 of [47] to pass the sk0 into the pair type, and then use that sk0 csk0 = sk0 to continue: ≃ ((u : sk0Y)×let u := ysk0 insk0([n] → (x : X)×(fx = y))) However, all types involved are already 0-skeletal, so we may remove the sk0s: ≃ ((y : Y)×([n] → (x : X)×(fx = y))) ≃ (y : Y)×((x : X)×(fx = y))n+1 This last type is the (n+1)-fold pullback of f along itself, displayed in terms of its diagonal map to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can prove modally that the realization of the ˇCech nerve of a map f : X → Y is the image im f of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This follows from Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 of Real Cohesion [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If A is 0-coskeletal, then reA is a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As a corollary, re(csk0 X) ≃ ∥X∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In [47], this theorem is said to rely on the crisp Law of Excluded Middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However a glance at the proof reveals that this assumption is only used to assume the decidable equality of sk0 ∆[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since we know that sk0 ∆[1] ≃ {0,1} has decidable equality, the proof goes through without assuming crisp LEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For a map f : X → Y, the realization re ˇC(f) of the ˇCech nerve is the realization reim f of the image of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If furthermore Y is 0-skeletal, then re ˇC(f) ≃ im f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We compute: re ˇC(f) ≡ re((y : Y)×csk0 fib f (y)) ≃ re((y : Y)×recsk0 fib f (y)) ≃ re((y : Y)× ��fib f (y) ��) ≡ reim f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now if Y is 0-skeletal, then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='17 of [47], im f is also 0-skeletal, since it is a subtype of a 0-skeletal type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, reim f ≃ im f, so that in total re ˇC(f) ≃ im f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 20 As an application of ˇCech nerves, we can see how to extract coherence data for higher groups from their deloopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If we take the ˇCech nerve of the inclusion ptBG : ∗ → BG of the base point of the delooping of G, we recover a simplicial type whose simplicial identities give coherences for the multiplication of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G be a crisp, 0-skeletal higher group — a 0-skeletal type identified with the loops of a pointed, 0-connected type BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ˇC(ptBG)n ≃ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, d1 : ˇC(ptBG)2 → ˇC(ptBG)1 is the product of the projections d0 and d2 : G2 → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13, we know that ˇC(ptBG)n ≃ (e : BG)×((x : ∗)×(ptBG∗ = e))n+1 ≃ (e : BG)×(ptBG = e)n+1 ≃ (e : BG)×(ptBG = e)×(ptBG = e)n ≃ (ptBG = ptBG)n = Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the second to last step, we contract (e : BG)×(ptBG = e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now, di : ˇC(ptBG)2 → ˇC(ptBG)1 is given by forgetting the ith component of the list (e,(a,b,c)) : (e : BG) × (ptBG = e)n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, d0(e,(a,b,c)) = (e,(b,c)) d1(e,(a,b,c)) = (e,(a,c)) d2(e,(a,b,c)) = (e,(a,b)) Contracting away e and the first element of the pair, we get the three equations d0(ba−1,ca−1) = cb−1 d1(ba−1,ca−1) = ca−1 d2(ba−1,ca−1) = ba−1 and indeed, we have d1(ba−1,ca−1) = d0(ba−1,ca−1)d2(ba−1,ca−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is equivalent, but not quite the same, as the standard presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It amounts to d0(g,h) = hg−1 d1(g,h) = h d2(g,h) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Using the ˇCech nerve, we can extract all the coherence conditions governing a homomorphism of higher groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We first note that the realization of the ˇCech nerve of a group is a delooping of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G be a 0-skeletal higher group with simplicially crisp delooping BG, and let ˇC(G) be the ˇCech nerve of the basepoint inclusion ptBG : ∗ → BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then the projection fst : ˇC(G) → BG is a re-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 of [34], BG is 0-skeletal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Axiom ∆sk0, it is therefore re-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, to show that fst : ˇC(G) → BG is a re-unit, it suffices to show that it is re-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since BG is 0-connected, it suffices to show that the fiber over the base point is re-connected, and this fiber is equivalent to csk0 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This follows by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' recsk0 G is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 21 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let G and H be 0-skeletal higher groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then the type of homomorphisms G → H is equivalent to the type of pointed maps ˇC(G) ·→ ˇC(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Recall that a homomorphism of higher groups is by definition a pointed map between their deloopings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, a homomorphism ϕ : G → H is equivalently a diagram as on the left, while a pointed map between the ˇCech nerves is a diagram as on the right: � � � � � � � � � � � ∗ ∗ BG BH ptBG Bϕ ptBH � � � � � � � � � � � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='≃ � � � � � � � � � � � ∗ ∗ ˇC(G) ˇC(H) ptBG f ptBH � � � � � � � � � � � We are aiming for an equivalence between these two types, which we may present as a one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' to Bϕ : BG → BH and ptBG : ptBH = Bϕ(ptBG) and f : ˇC(G) → ˇC(H) and ptf : pt ˇC(H) = f(pt ˇC(G)) associate the type (□ : (x : ˇC(G)) → (Bϕ(fstx) = fst(fx)))×(ptBϕ ·□(pt ˇC(G)) = fst∗ ptf ) which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' diagrammatically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' is the type of witnesses that the following diagram commutes: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='Bϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='fst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='fst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='To show that this gives a one-to-one correspondence means showing that the types of diagrams ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='Bϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ˇC(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='Bϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='are both contractible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' the left for any homomorphism ϕ and the right for any pointed map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let ϕ be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since by definition ˇC(G) and ˇC(H) were the csk0-factorizations of the basepoint inclusions, there is a unique filler of this square: ∗ ∗ ˇC(H) ˇC(G) BG BH Bϕ ptBG ptBH ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But this is precisely a rearrangement of the diagram on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similarly, if f is a pointed map, then re f : BG → BH makes the diagram on the right commute, and by the universal property of the re-unit this is the unique such map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 Global Equivariant Cohesion In Global Homotopy Theory and Cohesion [41], Rezk shows that the ∞-topos of global equivariant homotopy types is cohesive over the ∞-topos of homotopy types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' While Rezk constructs his site out of all compact Lie groups, we will follow Sati and Schreiber [43] in restricting our attention to the finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The global orbit category Glo is defined to be the full subcategory of homotopy types spanned by the deloopings BG of finite groups G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is a (2,1)-category, and the global equivariant topos is defined to be the ∞-category of homotopy type valued presheaves on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is an adjoint quadruple connecting the global equivariant topos and the topos of homotopy types: S Gloop S colim ∆ Γ ∇ colimX is the colimit of the functor X : Gloop → S , which takes the strict quotient of the global equivariant homotopy type X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ∆S is the inclusion of constant functors: ∆X(BG) :≡ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will refer to such equivariant types as invariant types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ΓX :≡ X(∗) is the evaluation at the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is known as the homotopy quotient of the global equivariant homotopy type X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ∇S is the Yoneda embedding: ∇S(BG) :≡ S (BG,S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This adjoint quadruple gives rise to the cohesive modalities < ⊣ ⊂ ⊣ ≺ of equivariant cohesion: The (“equivariant shape”) strict quotient modality X �→ < X sends a global equivariant type to its strict quotient, considered as an invariant type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The (“equivariant flat”) homotopy quotient modality X �→ ⊂ X sends a global equivariant type to its homotopy quotient, considered as an invariant type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Internally speaking, we say that an equivariantly crisp type is invariant when it is ⊂ modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The (“equivariant sharp”) orbisingular modality X �→ ≺ X sends a global equivariant type to its homotopy quotient, but considered with its natural equivariance via maps from the deloopings of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our axioms for global equivariant cohesion are quite straightforward: Axiom 4 (Global Equivariant Axioms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The type family ≺ B : FinGrp → Type sending a finite group G to ≺ BG detects equivariant continuity and connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The types ≺ BG for finite groups G are the orbi-singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By the definition above, we may recover X(BG) (considered with its natural equivariance) as ≺ ( ≺ BG → X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The family ≺ BG for finite groups G is a large family, but we may reduce it to a small family by noting that the type of finite groups is essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is a useful observation, since it allows us to conclude that < , defined by nullifying all ≺ BG, is an accessible modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Global equivariant cohesion shares a feature with Shulman’s continuous real cohesion: both are definable in the sense that the types which detect continuity and connectivity are definable without axioms in the type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is not the case for simplicial cohesion, which appears to require postulating the 1-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is not clear to us whether there are any general features shared by definable cohesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 23 In Proper Orbifold Cohomology [43], Sati and Schreiber work with equivariant differential cohesion to give an abstract account of the differential cohomology of orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can prove some of their lemmas easily in global equivariant cohesion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' we will return to prove the lemmas relating equivariant and differential cohesion in the upcoming §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following lemma appears as Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='62 in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We have the following equivalences for the generic orbi-singularities ≺ BG: <≺ BG ≃ ∗ ⊂≺ BG ≃ BG ≺≺ BG ≃ ≺ BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first equivalence follows by the assumption that ≺ BG detects equivariant connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The second follows by combining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='22 of [47] to see that ⊂< BG ≃ ⊂ BG with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 of [34] to note that since G is crisply ⊂ modal (as a finite set), BG is as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The third is simply the idempotence of ≺ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following lemma is a slight strengthening of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='65 of [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let X be an equivariantly crisp set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then X is both invariant ( ⊂ modal) and orbi-singular ( ≺ modal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In both cases we will use that ≺ BG detects equivariant continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show that X is invariant, we must show that the ⊂ counit is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6, it suffices to show that the map ⊂ ( ≺ BG → ⊂ X) → ⊂ ( ≺ BG → X) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But ≺ is a lex modality and BG is 0-connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' therefore, ≺ BG is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, ⊂ X is a set since X is, using Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7 of [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, the above map is equivalent to ⊂ ε : ⊂⊂ X → ⊂ X, which is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similarly, to show that X is orbi-singular, it suffices to show that the map ⊂ ( ≺ BG → X) → ⊂ (BG → X) given by pre-composing with the ⊂ counit of ≺ BG is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But again, by the connectivity of ≺ BG and BG, this map is equivalent to the identity ⊂ X → ⊂ X, which is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Miller’s theorem (formerly the Sullivan conjecture) states that the space of maps BG → X with G a finite group and X a finite cell complex is equivalent to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In equivariant modal terms, this says that finite cell complexes (the closure of the class {/0, ∗} under pushout) are ≺ modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It is not likely that this theorem could be proven on purely modal grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, if Miller’s theorem were proven in ordinary HoTT, then the modal statement could be proven in a manner similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 but instead of appealing to the truncatedness of X, appealing to the proof of Miller’s theorem (since any crisp finite cell complex is ⊂ modal as a crisp pushout of ⊂ types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 Topological Toposes Johnstone defined his topological topos in [25] in order to provide a topos of spaces for which the geometric realization of simplicial sets was a geometric morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The problem with using a real-cohesive topos for this purpose (as suggested previously by Lawvere) is the failure of the analytic lesser limited principle of omniscience which says that RD = (−∞,0]∪[0,∞) (see Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7 of [47] and the discussion in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 of ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This failure means that gluing together simplices along their (closed) faces gives the wrong topology on the resulting space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Johnstone remedies this by changing the test space from the real numbers to the walking convergent sequence N∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The walking convergent sequence may be defined internally as the set of monotone functions N → {0,1} (see for example [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, it is rather straightforward to give an internal axiomatization for Johnstone’s topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 24 Axiom 5 (Topological Focus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The topological focus is determined by asserting that N∞ detects topological conti- nuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may also be able to determine condensed homotopy types as in [19] — or rather the similar but more topos-theoretic pyknotic homotopy types of [10] — using a similar axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Define a profinite set to be the limit of a crisp diagram of finite sets indexed by a discrete (♭-modal) partially ordered set with decidable order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may then assert that the family of profinite sets detects condensed continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, we do not know if these axioms are sufficient for proving theorems in these topological toposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 5 Multiple Focuses Now, we turn our attention to generalities on possible relationships between different focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this section, fix two focuses ♥ and ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, we should show that the focuses do indeed commute: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any type B, the map ♯♥♯♣B → ♯♣♯♥B defined by x �→ x♯♥♯♣ ♯♥♯♣ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, the maps ♯♥♯♣B → ♯♥♣B and ♯♣♯♥B → ♯♥♣B defined by x �→ x♯♥♯♣ ♯♥♣ x �→ x♯♣♯♥ ♯♥♣ are also equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first map is well-defined because the use of ♯♥- and ♯♣-introduction means that the assumption x becomes crisp for both ♥ and ♣, so we may apply ♯♥- and then ♯♣-elimination to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may similarly define an inverse by x �→ x♯♣♯♥ ♯♣♯♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These maps are definitional inverses by the computation rules for ♯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The other maps are similarly well defined since being crisp for both ♥ and ♣ means being crisp for focus ♥♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The inverse may be defined in the straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We also note that the ordering of focuses is reflected in the containment of their ♯-modal (and so also ♭-modal) types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♣ ≤ ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then any ♯♣-modal type is ♯♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since ♣ ≤ ♥ is defined to mean ♥♣ ≡ ♣, we know that ♯♥♣A ≡ ♯♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By assumption ♯♣A ≃ A, and chaining this with the commutativity equivalence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 ♯♥A ≃ ♯♥♯♣A ≃ ♯♥♣A ≡ ♯♣A ≃ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Tracing these simple equivalences through, this does indeed give an inverse to the ♯♥-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We can similarly show that the ♭s commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is made simpler through the use of crisp induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let A be an ♥♣-crisp type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ♭♥♭♣A → ♭♣♭♥A defined by u �→ let v♭♥ := uin(let w♭♣ := vin(w♭♥)♭♣) is an equivalence, natural in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In words, the map is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Performing ♭♥-induction on u : ♭♥♭♣A gives an assumption v :♥ ♭♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A second induction on the term v : ♭♣A then gives us an assumption w :♥♣ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This second induction is ‘♥-crisp ♭♣-induction’: the resulting assumption w inherits the ♥-crispness of term v and gains ♣-crispness from the removal of ♭♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we form (w♭♥)♭♣ by applying ♭-introduction twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A map in the other direction is constructed in the same way, and then the proofs these are inverse are immediate by another pair of inductions each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We note also that the ♭-inductions commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭♥-induction and ♭♣-induction commute: let u♭♣ := (let v♭♥ := M inN)inC = let v♭♥ := M in(let u♭♣ := N inC) (when this is well-typed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', v does not occur in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First using uniqueness of ♭♣, we have let u♭♣ := (let v♭♥ := M inN)inC = let w♭♥ := M in(let u♭♣ := (let v♭♥ := w♭♥ inN)inC) ≡ let w♭♥ := M in(let u♭♣ := N[w/v]inC) ≡ let v♭♥ := M in(let u♭♣ := N inC) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Commuting Cohesions Now let’s turn our attention to the relationships between two commuting cohesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are both cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If a ♥♣-crisp type A is ♭♥-modal, then S♣A is still ♭♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We need to produce an inverse to the counit ε♥ : ♭♥S♣A → S♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First construct the composite A s−→ ♭♥A ♭♥η♣ −−−→ ♭♥S♣A where A s−→ ♭♥A is the assumed inverse to ε♥ : ♭♥A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The type ♭♥S♣A is certainly ♭♣-modal by commutativity of ♭♥ and ♭♣: ♭♣♭♥S♣A ≃ ♭♥♭♣S♣A ≃ ♭♥S♣A and therefore the above map factors through i : S♣A → ♭♥S♣A, our purported inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For one direction, consider the naturality square for the counit ε♥: ♭♥S♣A ♭♥♭♥S♣A S♣A ♭♥S♣A ε♥ ♭♥i ε♥ i The map ♭♥i is equal to the comultiplication δ♥ : ♭♥A → ♭♥♭♥A defined by a♭♥ �→ a♭♥♭♥, because both are inverse to the map ♭♥ε♥ : ♭♥♭♥A → ♭♥A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' And so the bottom composite in the square is equal to ε♥ ◦ δ♥, which is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 26 In the other direction, because S♣A is S♣-modal, it suffices to show that the composite A → S♣A → ♭♥S♣A → S♣A is equal to the unit η♣ : A → S♣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this we have the following commutative diagram: A ♭♥A A S♣A ♭♥S♣A S♣A ∼ η♣ ♭♥η♣ ε♥ η♣ i ε♥ The left square commutes by the definition of i, the right square by naturality of ε♥, and the composite along the top is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are both cohesive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then S♥S♣A → S♣S♥A is an equivalence for any ♥♣-crisp type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The map η♣ ◦η♥ : A → S♥A → S♣S♥A factors through S♥S♣A, because S♣S♥A is ♭♣-modal, as a type of the form ♭♣X, and also ♭♥-modal, by the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The map the other way is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To show these are inverses it suffices to show that they become so after precomposition with the composites of the units, because S♣S♥A and S♥S♣A are ♥♣-discrete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' this is immediate by the definition of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We might also hope that, say, ♭♥ and S♣ commute in general, but there is a useful sanity check that shows this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In the bare type theory with no axioms, there is nothing that prevents interpretation in a model where ♭♥ ≡ ♭♣ and S♥ ≡ S♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In ordinary cohesive type theory it is certainly not the case that ♭ and S commute, and so ♭♥S♣ ≃ S♣♭♥ cannot be provable without further assumptions on ♥ and ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A sufficient assumption on our focuses to make ♭♥ and S♣ commute in this way is the following: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that G :♥♣ I → Type and H :♥♣ J → Type detect ♥ and ♣ connectivity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We say that focuses ♥ and ♣ are orthogonal if Gi is ♭♣-modal for all i, and Hj is ♭♥-modal for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our present goal is to show that this indeed makes ♭♥S♣ ≃ S♣♭♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will in fact only use that the Gi are ♭♣-modal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' of course the dual results, flipping ♥ and ♣, require the other half of orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let ♥ and ♣ be cohesive focuses that are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then for any ♣-crisp A, if A is S♥-modal, ♭♣A is still S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our goal is to show that ♭♣A is equivalent to Gi → ♭♣A via precomposition by Gi → 1, for any i : I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We easily check that the type Gi → ♭♣A is ♣-discrete: for any Hj, Hj → (Gi → ♭♣A) ≃ Hj → (Gi → ♭♣A) ≃ Gi → (Hj → ♭♣A) ≃ Gi → ♭♣A because ♭♣A is ♣-discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then, by adjointness of S♣ and ♭♣: (Gi → ♭♣A) ≃ ♭♣(Gi → ♭♣A) ≃ ♭♣(S♣Gi → A) ≃ ♭♣(Gi → A) ≃ ♭♣A 27 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5 (Crisp S♥-induction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are cohesive and orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If B is ♥-discrete and ♣-crisp, then for any ♣-crisp A the map ♭♣(♭♣S♥A → B) → ♭♣(♭♣A → B) given by precomposition by ♭♣η♥ : ♭♣A → ♭♣S♥A is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As a rule, crisp S-induction would be written: ♣\\Γ,x :♣ S♥A ⊢ C ♣\\Γ,x :♣ S♥A ⊢ w : is-♭♥-modal(C) ♣\\Γ ⊢ M : S♥A ♣\\Γ,u :♣ A ⊢ N : C[uS♥/x] Γ ⊢ (let uS♥ := M inN) : C[M/x] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We deploy the usual trick for deriving crisp induction principles: using ♯♣ to move the ♭♣ out of the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Crucially, the previous proposition is what allows us to apply the universal property of S♥ on maps into ♯♣B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭♣(♭♣S♥A → B) ≃ ♭♣(S♥A → ♯♣B) ≃ ♭♣(A → ♯♣B) ≃ ♭♣(♭♣A → B) Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♥ and ♣ are orthogonal and cohesive focuses, then S♥ and ♭♣ commute on ♣-crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This is now straightforward induction on S♥ and ♭♣ in both directions, using crisp S♥-induction when defining ♭♣S♥A → S♥♭♣A and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 when defining S♥♭♣A → ♭♣S♥A to know that ♭♣S♥A is ♥-discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♥ and ♣ are orthogonal and cohesive focuses, then ♭♥ and ♯♣ commute on ♥♣-crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On ♥♣-crisp types, ♭♥♯♣ is right adjoint to ♭♣S♥, and ♯♣♭♥ is right adjoint to S♥♭♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, if ♭♣S♥X ≃ S♥♭♣ for a ♥♣-crisp type X, then also ♭♥♯♣X ≃ ♯♣♭♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Next we investigate a relationship between S and ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Again, this depends on a relationship between the families which detect continuity and connectivity of the two focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♣ is cohesive, and that the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' H :♥♣ I → Type detects ♣-connectivity, and I is ♭♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Hi is ♭♥-modal for all i :♥ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' (In particular, if ♥ and ♣ are orthogonal) Then if X is S♣-modal then ♯♥X is also S♣-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since H detects ♣-connectivity, it suffices to show that ♯♥S♣X is Hi-null for every i : I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since I is ♭♥-modal, we may assume that i :♥ I is ♥-crisp by ♭♥-induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then we can compute: (Hi → ♯♥X) ≃ ♯♥(Hi → ♯♥X) ≃ ♯♥(♭♥Hi → X) ≃ ♯♥(Hi → X) since Hi was assumed ♭♥-modal ≃ ♯♥X since X is S♣-modal Tracing upwards through this series of equivalences shows that the composite is indeed the inclusion of constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 28 On the opposite extreme of orthogonality, we can see that if the Gi which detect the connectivity of ♥ are S♣-connected, then any S♣-modal type is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that ♥ and ♣ are cohesive focuses where G : I → Type detects the connectivity of ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Every Gi is S♣-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Any S♣-modal type is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that Gi is S♣-connected for all i and that X is S♣-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may compute: (1 → X) ≃ (S♣Gi → X) ≃ (Gi → X) In the first equivalence, we use that Gi is S♣-connected, and in the second that X is S♣-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We conclude that X is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Conversely, suppose that any S♣-modal type is S♥-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then in particular S♣Gi is S♥-modal, so that the identity map S♣Gi → S♣Gi factors through S♥S♣Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 we have S♥S♣Gi ≃ S♣S♥Gi ≃ S♣∗ ≃ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, the identity of S♣Gi factors through the point, which means it is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6 Examples with Multiple Focuses In this section, we will see examples with multiple focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In particular, we will see simplicial real cohesion, equivariant differential cohesion, and supergeometric cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 Simplicial Real Cohesion We assume two basic focuses: the real (continuous or differential) focus S ⊣ ♭ ⊣ ♯, and the simplicial focus re ⊣ sk0 ⊣ csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will write R for whichever flavor of real numbers is used in the real cohesive focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will assume both the axioms of real cohesion and simplicial cohesion, as well as the following axiom relating the two focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 6 (Simplicial Real Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that the real focus and simplicial focus are orthogonal — which is to say, R is 0-skeletal and that ∆[1] is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, we assume that S is computed pointwise: for any simplicially crisp type X, the action (ηS)n : Xn → (SX)n of the S-unit of X on n-simplices is itself a S-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Our goal in this section will be to prove that if M is a 0-skeletal type — to be thought of as a “manifold”, having only real-cohesive structure but no simplicial structure — and U is a good cover of M — one for which the finite intersections are S-connected whenever they are inhabited — then the homotopy type SM of M may be constructed as the realization of a discrete simplicial set — namely, the ˇCech nerve of the open cover, with each open replaced by the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let M be a 0-skeletal type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A cover of M consists of a discrete 0-skeletal index set I, and a family U : I → (M → Prop) of subobjects of M so that for every m : M there is merely an i : I with m ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may assemble a cover into a single surjective map c : � i:IUi → M, where � i:I Ui :≡ (i : I)×(m : M)×(m ∈ Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' A cover U : I → (M → Prop) is a good cover if for any n : N and any k : [n] → I, the S-shape of the intersection � i:[n] Uk(i) :≡ (m : M)×((i : [n]) → (m ∈ Uk(i))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' is a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, S(Uk(0) ∩···∩Uk(n)) is contractible whenever there is an element in the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 29 We begin with a few ground-setting lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � i:IUi → M be the associated covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider the projection π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Over a simplicially crisp n-simplex k : ∆[n] → csk0 I, we have fibπn(ksk0) ≃ � i:[n] Uk(isk0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As a corollary, we have that ˇC(c)n ≃ (k : I[n])× � i:[n] Uk(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We compute: fibπn(ksk0) ≡ (x : ˇC(c)n)×(πnx = ksk0) ≃ ((m,z) : (m : M)×((i : I)×(m ∈ Ui))[n])×(πne(m,z) = ksk0) ≃ (m : M)×(K : [n] → I)×(p : (i : [n]) → (m ∈ UK(i)))×(πne(m,i �→ (K(i), p)) = ksk0) Here, e(m,z) is image under the equivalence from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' When all the modal dust settles, we will be left knowing that πne(m,i �→ (K(i), p)) : sk0(∆[n] → csk0 I) is the unique correspondent to K : [n] → I under the sk0 ⊣ csk0 adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, we may contract K away with ksk0 in the above type to get: ≃ (m : M)×((i : [n]) → m ∈ Uk(isk0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For the next lemma, we will need to know that S commutes with csk0 on suitably crisp types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any simplicially crisp type X, we have that Scsk0 X ≃ csk0 SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We know by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 that csk0 SX is S-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We therefore have a map Scsk0 X → csk0 SX given as the unique factor of csk0(−)S : csk0 X → csk0 SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will show that this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since it is crisp, it suffices to show that it is an equivalence on n-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' To that end, we compute: sk0(∆[n] → Scsk0 X) ≃ Ssk0(∆[n] → csk0 X) ≃ Ssk0([n] → X) ≃ sk0(S([n] → X)) ≃ sk0([n] → SX) ≃ sk0(∆[n] → csk0 SX) It remains to show that this is indeed the right equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the first equivalence in the series above is given as the inverse of (−)S n : (csk0 X)n → (Scsk0 X)n, it suffices to check that given a crisp z : ∆[n] → csk0 X, (zsk0)S corresponds under the above equivalences to (csk0(−)S ◦z)sk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, we send (zsk0)S to ((z◦(−)sk0)sk0)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then, we send it to ((z◦(−)sk0)S)sk0, and then to (i �→ (z(isk0)S))sk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we map this to (i �→ ((z(isk0sk0)S))csk0 sk0, which does equal (csk0(−)S ◦z)(i) ≡ (z(i)S)csk0 at i : ∆[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let U : I → (M → Prop) be a simplicially crisp cover, and let c : � i:IUi → M be the covering map itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider the “projection” π : ˇC(c) → csk0 I defined by (m,z) �→ (fstzcsk0)csk0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then U is a good cover if and only if the restriction π : ˇC(c) → imπ is a S-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since csk0 and S commute by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 and I is discrete, csk0 I is also discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' As the subtype of a discrete type, imπ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Therefore, it suffices to show that π : ˇC(c) → imπ induces an equivalence S ˇC(c) ∼−→ imπ if and only if the cover U is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since π is crisp, S ˇC(c) → imπ is an equivalence if and only if it is an equivalence on all n-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On n-simplices, this map (at the top of the following diagram) is equivalent to the map on the bottom of the following diagram: (S ˇC(c))n (imπ)n S(ˇC(c)n) imπn S � (k : I[n])× � i:[n]Uk(i) � (k : I[n])×S �� i:[n]Uk(i) � (k : I[n])×∃� i:[n]Uk(i) The bottom map is an equivalence if and only if the cover is good, and so we conclude the same for the top map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Finally, we can piece these lemmas together for our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let U : I → (M → Prop) be a simplicially crisp good cover of a 0-skeletal type M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Let π : ˇC(U) → csk0 I be the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then reimπ ≃ SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This exhibits the shape SM as the realization of a discrete (simplicial) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the cover is good, we have that imπ ≃ SˇC(U), so that reimπ ≃ reSˇC(U) ≃ Sre ˇC(U) ≃ SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now, since I is a set and csk0 is a lex modality, csk0 I is also a set and so imπ is a set as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Furthermore, since subtypes of discrete types are discrete by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='17 of [47], and csk0 I is discrete since by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 S and csk0 commute on simplicially crisp types, imπ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 Equivariant Differential Cohesion In Proper Orbifold Cohomology, Sati and Schreiber work in equivariant differential cohesion to describe the differ- ential cohomology of orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This cohesion involves both the equivariant focus < ⊣ ⊂ ⊣ ≺ and the differential (real-cohesive) focus S ⊣ ♭ ⊣ ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In this section, we will assume both the axioms of equivariant cohesion and differ- ential real cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will refer to the smooth reals by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Unlike the simplicial real cohesive case, we do not need to add additional axioms to ensure that the equivariant and differential cohesion are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Equivariant and differential cohesion are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The smooth reals R are invariant ( ⊂ modal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any finite group G, ≺ BG is discrete (♭-modal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since the smooth reals are a set, they are invariant by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Similarly, since BG is discrete and hence S-modal (by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9 of [34]), ≺ BG is still S-modal by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 31 The following lemma appears as Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='67 of [43], and is proven quickly with our general lemmas concern- ing orthogonal cohesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Suppose that X is both differentially and equivariantly crisp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Then ⊂ SX ≃ S ⊂ X ⊂ ♭X ≃ ♭ ⊂ ♭X ⊂ ♯X ≃ ♯ ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The first equivalence follows by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The second equivalence follows by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The third follows by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3 Supergeometric Cohesion In his habilitation, Differential Cohomology in a Cohesive ∞-Topos [44], Schreiber describes an increasing tower of adjoint modalities which appear in the setting of supergeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The setting for supergeometric cohesion — called “solid cohesion” in ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' — is sheaves on the opposite of a category of super C ∞-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Schreiber calls these sheaves super formal smooth ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Specifically, the site is (the opposite of) the full subcategory of the category of super commutative real algebras spanned by objects of the form C ∞(Rn)⊗W ⊗ΛRq where W is a Weil algebra — a commutative nilpotent extension of R which is finitely generated as an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The factor C ∞(Rn)⊗W is even graded, while the Grassmannian ΛRq is odd graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' See Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='13 of [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The inclusion of algebras of the form C ∞(Rn) ⊗W has a left and a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The left adjoint is given by projecting out the even subalgebra, and the right adjoint is given by quotienting by the ideal generated by the odd graded elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This gives rise to an adjoint quadruple between the resulting toposes of sheaves and thus an adjoint triple of idempotent adjoint (co)monads on the topos of super formal smooth ∞-groupoids: ⇒ ⊣ ⇝ ⊣ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Of these, ⇒ and Rh are idempotent monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' However, ⇒ does not preserve products, and so does not give an internal modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The action of Rh is easy to define: RhX(C ∞(Rn)⊗W ⊗ΛRq) := X(C ∞(Rn)⊗W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' That is, RhX is defined by evaluating at the even part of the superalgebra in the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We may characterize it internally by localizing at the odd line R0|1, which is the sheaf represented by the free superalgebra on one odd generator ΛR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We turn to the internal story now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The topos of super formal smooth ∞-groupoids also supports the differential real cohesive modalities S ⊢ ♭ ⊢ ♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' These destroy all geometric structure — super and otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For this reason, we will work with the lattice {diff < super < ⊤} of focuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The modalities of the super focus are ⇝ ⊢ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We will refer to ⇝ X as the even part of X, while Rh is known as the rheonomic modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume the following axioms for supergeometric or solid cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Axiom 7 (Solid Cohesion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Solid cohesion uses the focus lattice {diff < super < ⊤}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We use the definition of real superalgebras due to Carchedi and Roytenberg [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume a commutative ring R1|0 satisfying the axioms of synthetic differential geometry (as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 of [36]) known as the smooth reals or the even line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume an R1|0-module R0|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There is furthermore a bilinear multiplication R0|1×R0|1 → R1|0 which sat- isfies a2 = 0 for all a : R0|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Together these axioms imply that R1|1 := R1|0×R0|1 is a R1|0-supercommutative superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume the following odd form of the Kock-Lawvere axiom: For any function f : R0|1 → R0|1 with f(0) = 0, there is a unique r : R1|0 with f(x) = rx for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that R0|1 is ⇝-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that R1|0 detects differential connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' We assume that a type is Rh-modal if and only if it is R0|1-null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' It might seem prudent to instead ask that differential connectivity is detected by the family con- sisting of both R1|0 and R0|1, since we want S to nullify all representables Rn|q, but it suffices to test with R1|0 since R0|1 admits an explicit contraction by its R1|0-module structure (appealing to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='10 of [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2, any ♯-modal type is Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' But also every ♭-modal type is Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If X is S-modal (and, in particular, if X is ♭-modal), then X is Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Since R0|1 is S-connected due to its explicit contraction by the scaling of its module structure, any S-modal type is R0|1-null, and therefore Rh-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' References [1] LICS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Oxford, United Kingdom: Association for Computing Machinery, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ISBN: 978-1-4503-5583- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [2] Andreas Abel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “A Polymorphic Lambda-Calculus with Sized Higher-Order Types”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' June 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://www2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='tcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='de/~abel/diss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [3] Andreas Abel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Polarised subtyping for sized types”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Mathematical Structures in Computer Science 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5 (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 797–822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1017/S0960129508006853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [4] Benedikt Ahrens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The Univalence Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' arXiv: 2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='06275 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='CT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [5] Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Extending homotopy type theory with strict equal- ity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Computer Science Logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Leibniz Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Schloss Dagstuhl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Leibniz-Zent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=', Wadern, 2016, Art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 21, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [6] Mathieu Anel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Enveloping ∞-topoi”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Seminar on Higher Homotopical Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='youtube.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' by Mathieu Anel and Gabriel Catren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Cambridge University Press, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 155–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1017/9781108854429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [8] Danil Annenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Two-Level Type Theory and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' arXiv: 1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='03307 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='LO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [9] Robert Atkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Syntax and Semantics of Quantitative Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' LICS ’18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Oxford, United Kingdom: Association for Computing Machinery, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 56–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ISBN: 978-1-4503-5583-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1145/3209108.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1017/S0960129519000197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [12] Guillaume Brunerie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “On the homotopy groups of spheres in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Universit´e de Nice, 2016.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1184/R1/14555691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [17] Felix Cherubini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Formalizing Cartan Geometry in Modal Homotopy Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Karlsruher Institut f¨ur Technologie, July 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Daniel Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Localization in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Higher Structures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://higher- structures.' metadata={'source': 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10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2178/jsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7803040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [21] Daniel Gratzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Modalities and Parametric Adjoints”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Comput.' metadata={'source': 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Licata and Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Adjoint Logic with a 2-Category of Modes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Logical founda- tions of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 9537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lecture Notes in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Springer, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 219–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1007/978- 3- 319- 27683- 0_16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://dlicata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} 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International Conference on Formal Structures for Computation and Deduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' by Dale Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' FSCD ’17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2017.' 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extensionality, and proof irrelevance in modal type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 221–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1109/LICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='932499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [40] Jason Reed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “A Judgmental Deconstruction of Modal Logic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='cs.' metadata={'source': 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+page_content=' URL: https://faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' edu/~rezk/global-cohesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [42] Emily Riehl and Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “A type theory for synthetic ∞-categories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: High.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 147–224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1007/s42001-017-0005-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [43] Hisham Sati and Urs Schreiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proper Orbifold Cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://ncatlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/ schreiber/files/orbi220627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [44] Urs Schreiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Differential cohomology in a cohesive infinity-topos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://ncatlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/ schreiber/files/dcct170811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [45] Urs Schreiber and Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Quantum Gauge Field Theory in Cohesive Homotopy Type Theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Workshop on Quantum Physics and Logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4204/EPTCS.' metadata={'source': 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+page_content='07004 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='AT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [47] Michael Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Brouwer’s fixed-point theorem in real-cohesive homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Mathe- matical Structures in Computer Science 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 856–941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ISSN: 0960-1295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 1017 / S0960129517000147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [48] Jonathan Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Internal sums for synthetic fibered (∞,1)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2022.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [49] Jonathan Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Two-sided cartesian fibrations of synthetic (∞,1)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='48550/ ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='00938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='00938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' [50] Gavin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Wraith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' “Using the generic interval”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' In: Cahiers de Topologie et G´eom´etrie Diff´erentielle Cat´egoriques 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4 (1993), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 259–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='numdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='org/item/CTGDC_1993__34_4_259_0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 35 A Proof Sketches for Admissible Rules We sketch proofs that the operations on syntax are admissible, demonstrating the interesting cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' those that involve division (CTX-EXT, ♭-FORM/INTRO/ELIM, ♯-ELIM) or promotion (♯-FORM/INTRO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The ♥Γ and ♥\\Γ context operations extend in the obvious way to telescopes Γ′, so that ♥(Γ,Γ′) ≡ (♥Γ),(♥Γ′) ♥\\(Γ,Γ′) ≡ (♥\\Γ),(♥\\Γ′) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='2 (Weakening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Single-variable weakening is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' WK Γ,Γ′ ⊢J Γ,w :♣ W,Γ′ ⊢J −−−−−−−−−−− Moreover, weakening does not change the size of the derivation tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Induction onJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\(Γ,w :♣ W,Γ′) ⊢ A type Γ,w :♣ W,Γ′ ⊢ ♭♥A type There are two subcases: If ♣ ≤ ♥: then ♥\\(Γ,w :♣ W,Γ′) ≡ (♥\\Γ),w :♣ W,(♥\\Γ′), in which case we induct on the type A and reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♣ ̸≤ ♥: then ♥\\(Γ,w :♣ W,Γ′) ≡ (♥\\Γ),(♥\\Γ′) ≡ ♥\\(Γ,Γ′), and so already ♥\\(Γ,w :♣ W,Γ′) ⊢ A type and we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥(Γ,w :♣ W,Γ′) ⊢ A type Γ,w :♣ W,Γ′ ⊢ ♯♥A type By definition ♥(Γ,w :♣ W,Γ′) ≡ ♥Γ,w :♥♣ W,♥Γ′, and so we induct on A (now weakening with a variable of focus ♥♣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' For any two focuses ♥ and ♣, the context ♣\\(♥Γ) is an iterated weakening of ♥(♣\\Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This can be checked variablewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Given a variable x :♠ A in Γ, if it survives to ♥(♣\\Γ) as x :♥♠ A, then we must have ♠ ≤ ♣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Multiplying by ♥, it follows that ♥♠ ≤ ♥♣ ≤ ♣, and so x :♥♠ A also occurs in ♣\\(♥Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The following equations involving contexts and telescopes hold: (♥Γ′)[s/z] ≡ ♥(Γ′[s/z]) (♥\\Γ′)[s/z] ≡ ♥\\(Γ′[s/z]) 36 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='5 (Substitution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' SUBST ♣\\Γ ⊢ s : S Γ,z :♣ S,Γ′ ⊢J Γ,Γ′[s/z] ⊢J [s/z] −−−−−−−−−−−−−−−−−−−− Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Induction onJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Three subcases as usual, for x ∈ Γ, x ≡ z and x ∈ Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The interesting one is x ≡ z: VAR Γ,z :♣ S,Γ′ ⊢ s : S Applying DIVIDE-WK to ♣\\Γ ⊢ s : S gives Γ ⊢ s : S, which can be further weakened to Γ,Γ′ ⊢ s : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\(Γ,z :♣ S,Γ′) ⊢ A type Γ,s :♣ S,Γ′ ⊢ ♭♥A type There are two subcases: If ♣ ≤ ♥: then ♥ \\ (Γ,z :♣ S,Γ′) ≡ (♥ \\ Γ),z :♣ S,(♥ \\ Γ′), in which case we induct, getting (♥ \\ Γ),(♥\\Γ′)[s/z] ⊢ A[s/z] type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' This context is equal to ♥\\(Γ,Γ′[s/z]) ctx, so we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♣ ̸≤ ♥: then ♥ \\ (Γ,z :♣ S,Γ′) ≡ (♥ \\ Γ),(♥ \\ Γ′) ≡ ♥ \\ (Γ,Γ′), and so z does not occur in A, and A ≡ A[s/z], in which case we may reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥(Γ,z :♣ S,Γ′) ⊢ A type Γ,z :♣ S,Γ′ ⊢ ♯♥A type By definition ♥(Γ,s :♣ S,Γ′) ≡ ♥Γ,s :♥♣ S,♥Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Applying PROMOTE to ♣\\Γ ⊢ s : S yields ♥(♣\\Γ) ⊢ s : S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3, this can be weakened to ♣(♥\\Γ) ⊢ s : S, whose context is equal to (♥♣)\\(♥Γ), which is of the correct shape to be substituted into ♥Γ,z :♥♣ S,♥Γ′ ⊢ A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Substitution gives ♥Γ,(♥Γ′)[s/z] ⊢ A[s/z] type, and this context is equal to ♥(Γ,Γ′[s/z]) ctx, so we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='6 (Promote).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' PROMOTE-CTX Γ ctx ♣Γ ctx −−−− PROMOTE Γ ⊢J ♣Γ ⊢J −−−−− Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First, PROMOTE-CTX is by induction on the length of the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider a context Γ,x :♥ A ctx, so that ♥ \\ Γ ⊢ A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Applying PROMOTE to A gives ♣(♥ \\ Γ) ⊢ A type, which can be weakened to (♣♥) \\ (♣Γ) ⊢ A type by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='3, letting us form the context ♣Γ,x :♣♥ A ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' The variable rule is immediate, because modifying the annotation on a variable does not change whether it is usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 37 Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\Γ ⊢ A type Γ ⊢ ♭♥A type Inductively ♣(♥\\Γ) ⊢ A type, which can be weakened to ♥\\(♣Γ) ⊢ A type, and we reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥Γ ⊢ A type Γ ⊢ ♯♥A type Inductively ♣(♥Γ) ⊢ A type, and ♣(♥Γ) ≡ ♥(♣Γ), so we may reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content='7 (Division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' DIVIDE-CTX Γ ctx ♣\\Γ ctx −−−−−− DIVIDE-WK Γ ctx ♣\\Γ ⊢J Γ ⊢J −−−−−−−−−−−− Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' First DIVIDE-CTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Consider a context Γ,x :♥ A ctx, so that ♥\\Γ ⊢ A type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' There are two cases: If ♥ ≤ ♣: Then ♥\\Γ ≡ (♥♣)\\Γ ≡ ♥\\(♣\\Γ), and so ♥\\(♣\\Γ) ⊢ A type is of the right shape to form ♣\\Γ,x :♥ A ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If ♥ ̸≤ ♣: Then ♣\\(Γ,x :♥ A) ≡ ♣\\Γ is well-formed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Now DIVIDE-WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' On terms: Case (variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' If x :♥ A is in context ♣\\Γ, then it must also be in �� and so we may reuse the variable rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♭-FORM ♥\\(♣\\Γ) ⊢ A type ♣\\Γ ⊢ ♭♥A type We know ♥\\(♣\\Γ) ≡ ♣\\(♥\\Γ), and so inductively ♥\\Γ ⊢ A type, and we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Case (promotion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' ♯-FORM ♥(♣\\Γ) ⊢ A type ♣\\Γ ⊢ ♯♥A type ♥(♣ \\ Γ) ⊢ A type may be weakened to ♣ \\ (♥Γ) ⊢ A type (without increasing the size of the derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' Inductively ♥Γ ⊢ A type, and we can reapply the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFST4oBgHgl3EQfWTh0/content/2301.13780v1.pdf'}