diff --git "a/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/2301.11424v1.pdf.txt" "b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/2301.11424v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/2301.11424v1.pdf.txt" @@ -0,0 +1,2967 @@ +arXiv:2301.11424v1 [math.CT] 26 Jan 2023 +An inductive model structure for strict +∞-categories +Simon Henry and Felix Loubaton +Abstract +We construct a left semi-model category of “marked strict ∞-categories” +for which the fibrant objects are those whose marked arrows satisfy nat- +ural closure properties and are weakly invertible. The canonical model +structure on strict ∞-categories can be recovered as a left Bousfield local- +ization of this model structure. We show that an appropriate extension +of the Street nerve to the marked setting produces a Quillen adjunction +between our model category and the Verity model structure for complicial +sets, generalizing previous results by the second named author. Finally, +we use this model structure to study, in the setting of strict ∞-categories, +the idea that there are several non-equivalent notions of weak (∞, ∞)- +categories - depending on what tower of (∞, n)-categories is used. We +show that there ought to be at least three different notions of (∞, ∞)- +categories. +Contents +1 +Introduction +2 +1.1 +The street nerve as a right Quillen functor . . . . . . . . . . . . . +3 +1.2 +The two (?) notions of (∞, ∞)-categories +. . . . . . . . . . . . . +3 +2 +∞-categories and marked ∞-categories +5 +2.1 +∞-categories +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.2 +Marked ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3 +Tensor product of m-marked ∞-categories . . . . . . . . . . . . . +9 +2.4 +The semi-model structure . . . . . . . . . . . . . . . . . . . . . . +12 +3 +Equations and saturations in an m-marked ∞-category. +17 +3.1 +Definitions of equations and saturations . . . . . . . . . . . . . . +18 +3.2 +Characterization of fibrant objects . . . . . . . . . . . . . . . . . +20 +3.3 +Isofibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.4 +Equivalences +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +3.5 +The saturated localization. +. . . . . . . . . . . . . . . . . . . . . +27 +4 +Comparison with other model structures +29 +4.1 +Truncation functors +. . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.2 +Comparison with the folk model structure on ∞-Cat . . . . . . . +36 +4.3 +The folk model structure vs the limit of the π-tower +. . . . . . . +38 +4.4 +Complicial sets and stratified Street nerve . . . . . . . . . . . . . +42 +1 + +1 +Introduction +In the present paper, we introduce (in Section 2.2) a category ∞-Catm of “m- +marked (strict) ∞-categories” for m ∈ N ∪ {∞}. The objects of ∞-Catm are +strict ∞-categories, with, similarly to stratified simplicial sets, some arrows be- +ing “marked”. The marked arrows are required to be closed under composition, +and all identities arrows as well as all arrows of dimension > m are marked. This +category ∞-Catm is equipped with two monoidal closed structures denoted → +and +∼ that are both the Gray-Crans tensor product on the underlying strict +∞-categories but act differently on markings. These two monoidal structures +are meant to respectively be models for the“lax-Gray tensor product” and the +“pseudo-Gray tensor product”. +Our main result is the construction of a model structure1 on ∞-Catm similar +to the canonical (or “Folk”) model structure on strict ∞-category from [19]: +1.1 Theorem. There is a combinatorial left semi-model structure on the cate- +gory ∞-Catm of m-marked ∞-categories such that: +• This model structure is monoidal for both tensor products +∼ and → (from +Section 2.3). +• The cofibrations are the map that are cofibrations of the canonical model +structure between the underlying ∞-categories. +• The fibrant objects are the marked ∞-categories in which all marked arrows +admit marked weak inverses, and in which if there is a marked arrow a → b +then a is marked if and only if b is marked. +• Fibrations between fibrant objects are the “isofibrations” (as defined in +Section 3.3). +• Weak equivalences between fibrant objects are “equivalence of marked ∞- +categories”(as defined in Section 3.4). +This model structure is a model for strict “(∞, m)-categories” where “invert- +ibility” or arrows of dimension > m is taken in a weak sense. The existence of +this model structure is established in Section 2.4, but some of its properties, in +particular, the characterization of fibrant objects and fibrations between fibrant +objects will only be established in Section 3. +We also consider two left Bousfield localizations of this model structure: +• The saturated inductive model structure, studied in Section 3.5, whose +fibrant objects are the ∞-categories in which every arrow which is weakly +invertible up to marked arrows is also marked. +• The coinductive model structure, studied in Section 4.2, whose fibrant +objects are the ∞-categories in which every coinductively invertible (see +Definition 4.16) arrow is marked. +This second localization is equivalent2 to the canonical model structure on +∞-categories from [19]. +The motivations to introduce this model structure come from two different +lines of investigations that we will explain separately below: +1We use the term “model category” as a generic name for all sorts of model categories +(Quillen model categories, semi-model categories, weak model categories, etc...) +2Though not through a Quillen equivalence. +2 + +1.1 +The street nerve as a right Quillen functor +In [20], the second named author has shown that the Street nerve of a strict +∞-category can be made into a complicial set by defining the “thin” simplexes +as being those whose top dimensional arrows are weakly invertible. +From there, it is natural to ask whether this stratified version of the Street +nerve, also preserves fibrations, and hence is a morphism of categories of fibrant +objects (and this will be shown in the present paper as Proposition 4.51). +In fact, more generally, one could ask if it is possible to make this version +of the Street nerve into a right Quillen functor (for the Verity model structure +on complicial sets from [30]). This is not directly possible simply because this +stratified Street nerve is not a right adjoint functor. The solution to this problem +is to work with marking on both sides: The usual Street nerve from strict ∞- +categories to simplicial sets is a right adjoint functor, and one can extend it to a +right adjoint functor from marked ∞-categories to “marked” simplicial sets (or +rather stratified simplicial sets to follow the terminology of [30]). In Section 4.4 +we show that this functor is indeed a right Quillen functor. +This right Quillen functor from marked ∞-categories to stratified simplicial +sets is meant to be a model for the forgetful functor from strict (∞, ∞)-categories +to weak (∞, ∞)-categories. In particular, the corresponding left Quillen functor +from stratified simplicial sets to marked ∞-categories is a model for the more +mysterious “strictification functor”, sending weak ∞-categories to strict ∞- +categories. +At the level of ∞-groupoids, this strictification functor corresponds essen- +tially to (non-abelian) homology, through the equivalence between strict ∞- +groupoids and crossed chain complexes ([7]) which is well-known to be a conser- +vative functor by Whitehead’s theorem for homology. The first named author +has conjectured [17] that more generally this strictification functor should be +conservative on weak (∞, m)-categories for all m. +This allows us to state a +concrete version of this conjecture here: +1.2 Conjecture. The left Quillen functor | |: sSetm → ∞-Catm from Sec- +tion 4.4 reflects weak equivalence between cofibrant objects. +1.2 +The two (?) notions of (∞, ∞)-categories +C.Schommer-Pries and C.Rezk have independently argued ([16]) that there +should be more than one notion of weak (∞, ∞)-categories. More precisely, +they both arrive at the conclusion that even if one accepts (which seems to +be a clear consensus nowadays) that there is only one notion of weak (∞, n)- +categories for finite n, there are at least two different ways to build a notion of +(∞, ∞)-categories out of it. +Before we go into further details, we should say that the following discussion +is mostly informal and speculative and most of it has not been formalized in +any models - in fact, one motivation for the present paper is to formalize some +of it in the context of strict ∞-categories. +First, let us go over the argument put forward by Rezk and Schommer- +Pries, or at least how we understand it: The forgetful (or inclusion) functor +from (∞, n)-categories to (∞, n + 1)-categories is supposed to have both a left +adjoint πn, which freely adds inverses to all (n + 1)-arrows and a right adjoint +τn which remove all non-invertible (n + 1)-arrows. +3 + +This allows to produce two different towers: +(∞, 0)-Cat +π0 +← (∞, 1)-Cat +π1 +← (∞, 2)-Cat +π2 +← . . . +πn−1 +← (∞, n)-Cat +πn +← . . . +(∞, 0)-Cat +τ0 +← (∞, 1)-Cat +τ1 +← (∞, 2)-Cat +τ2 +← . . . +τn−1 +← (∞, n)-Cat +τn +← . . . +and one can take the projective limit of either of these two towers to give a +definition of what is an (∞, ∞)-category. +If one takes the limits of the π-tower then one can see that an arrow that +is “coinductively” invertible (see Definition 4.16) has to be considered invert- +ible. To be precise, we mean that if F: X → Y is a morphism in the limit of +the π-tower which admits an inverse up to a coinductively invertible natural +transformation then F is an equivalence. +The situation in the limit of the τ-tower however is fairly different: Given an +(∞, ∞)-category in this sense, it corresponds to a collection of (∞, n)-categories +Xn such that Xn ≃ τnXn+1, and an n-arrow corresponds to an n-arrow of Xn +(or of Xk for k > n). In this setting one has an intrinsic notion of equivalence: +an n-arrow is said to be an equivalence if it belongs to Xn−1 (equivalently if it +is invertible in the (∞, n)-category Xn). In this setting, coinductively invertible +arrows do not have to be invertible if none of the higher cells witnessing the +coinductive invertibility are not themselves invertible. +To clearly show that the two are different, one can for example consider the +(∞, ∞)-category of cobordisms. In the limit of the τ-tower one can define it by +taking Xn to be the (∞, n)-categories of cobordisms. In this (∞, ∞)-category, +every arrow has a dual, so it follows from a result of E.Cheng (see [9]) that every +arrow in the cobordisms (∞, ∞)-category is coinductively invertible, although +there are many non-invertible n-arrows in Xn for all n. +Hence, if one were +trying to define Xn in the limit of the π-tower, it would be equivalent to an +∞-groupoid. +Using our model structure of marked strict ∞-category, we will make these +two constructions formal in the context of strict ∞-category. This is of course +only meant to be a toy model for the case of weak ∞-categories, but it is already +interesting, and it will show that the picture above while correct, needs to be +refined a little. +First, we will show in Section 4.1 that our model structure on ∞-Catm +for m = ∞ corresponds to the limit of τ-tower as above. More precisely, we +will show that it is Quillen equivalent to an appropriate homotopy limit of the +∞-Catm for m < ∞ using the τn functor as transition functors. +The notion of homotopy limit of a tower of model structure we are using has +been introduced in [6], and we will use their construction of the homotopy limit. +Here there is a small gap we should disclaim: [6] only develops the theory of such +limits for Quillen model categories and not semi-model categories, and we will +apply their construction to our left semi-model categories directly. In order for +our argument to be complete despite this, we will prove that the construction +from [6] does yield to a left semi-model category, but we will not reprove that +it corresponds to a homotopy limit as in [6, Theorem 5.1]. However, it should +be noted that in practice, the argument of [6] seems to carry over to our setting +with almost no changes, so this gap is not really a concern. +4 + +In Section 4.2 we will show that the folk model structure is equivalent to the +left Bousfield localization of our model structure which corresponds to turning +all coinductively invertible arrows into equivalences. +However, we will also show in Section 4.3, that the folk model structure is +not equivalent to the limit of the π-tower. It is unclear if the limit of the tower +of πn corresponds to further localization of our model structure, or is something +entirely different, but we find that the argument we will give in Section 4.3 to +distinguish between the folk model structure and the limit of π-tower shows +that this limit is exhibiting behaviors that are not really expected from a notion +of (∞, ∞)-categories, or at least are not typical of any known model of ∞- +categories. +Coming back to the world of weak (∞, ∞)-categories, this suggests that +the two most interesting notions of weak (∞, ∞)-categories are the limit of +τn tower, which corresponds to an “inductive” notion of equivalences, and its +localization that turn the coinductive equivalence into equivalences, but this +localization should be different from the limit of the πn-tower which might not +be an interesting notion of (∞, ∞)-categories. What we mean here is that we are +not aware of any attempt of giving a concrete definition of (∞, ∞)-categories +that seems to produce something that could be equivalent to this limit. All +definitions we have seen can be reasonably conjectured to be equivalent to either +the limit of the τn tower or to its “coinductive” localization. +2 +∞-categories and marked ∞-categories +2.1 +∞-categories +A globular set is a presheaf on the globular category G: +D0 +D1 +D2 +D3 +D4 . . . +i+ +0 +i− +0 +i+ +1 +i− +1 +i+ +2 +i− +2 +i+ +3 +i− +3 +With the relations iǫ +ni+ +n−1 = iǫ +ni− +n−1 for all n > 0 and ǫ ∈ {+, −}. We also +denote by iǫ +k the map Dk → Dn for k < n obtained by composing any string of +arrow ending with iǫ +k. These and the identity arrows are the only maps in the +category G. +If X is a globular set, one denotes by Xn the set X(Dn) whose elements are +called n-arrows. The map Xn → Xk induced by iǫ +k: Dk → Dn is denoted by πǫ +k. +2.1 Definition. An ∞-category is a globular set X together with operations +of compositions +Xn ×Xk Xn → Xn +(0 ≤ k < n) +which associates to two n-arrows (x, y) verifying π+ +k (x) = π− +k (y), one n-arrow +x#ky, as well as identities +Xn → Xn+1 +associating to an n-arrow x, an (n + 1)-arrow Ix, and satisfying the following +axioms: +5 + +(1) ∀x ∈ Xn, πǫ +n(Ix) = x. +(2) π− +k (x#ny) = π− +k (x) and π+ +k (x#ny) = π+ +k (y) whenever the composition is +defined and k ⩽ n. +(3) πǫ +k(x#ny) = πǫ +k(x)#nπǫ +k(y) whenever the composition is defined and k > +n. +(4) x#nIπ+ +n x = x and Iπ− +n x#nx = x. +(5) (x#ny)#nz = x#n(y#nz) as soon as one of these is defined. +(6) If k < n +(x#ny)#k(z#nw) = (x#kz)#n(y#kw) +when the left-hand side is defined. +A morphism of ∞-categories is a map of globular sets commuting with both +operations. The category of ∞-categories is denoted ∞-Cat. +2.2 Definition. An (n + 1)-arrow c in an ∞-category is said to be trivial, or +an identity arrow, if there exists an n-cell d such that c = Id. +2.3 Example. By abuse of notation, we also denote Dn the ∞-category that +admits for any k < n only two k-non-trivial arrows, denoted e− +k and e+ +k , and a +single non-trivial n-arrow, denoted en verifying : +π− +l (eǫ +k) = e− +l +π+ +l (eǫ +k) = e+ +l +for l ≤ k < n +π− +l (en) = e− +l +π+ +l (en) = e+ +l +for l ≤ n +The ∞-category ∂Dn is obtained from Dn by removing the n-arrow en. We +thus have a morphism +in: ∂Dn → Dn. +Note that ∂D0 = ∅ +2.4 Definition. If X is an ∞-category, we define the globular set ΣX, called +the suspension of X, by the formula +(ΣX)0 = {a, b} +(ΣX)n+1: = Xn ∪ {Ina, Inb} +where In +a (resp. In +b ) is the n-times iterated unity of a (resp. of b). Moreover, +ΣX inherits from X a structure of ∞-category. +Eventually, for an integer n, we define the ∞-category ΣnX, called the n- +suspension of X, as the n-times iterated suspension of X. +Next, we define the notion of polygraphs, first introduced under the name +“computads” by R. Street in [28] for 2-categories, with the general notion being +hinted at in [29]. As far as we know the first formal introduction of polygraphs +in the literature is in [25] and independently in [8], where the name “polygraphs” +was introduced. Here we will exploit that the category of polygraphs identifies +with a (non-full) subcategory of ∞-Cat to give a shorter definition. We refer +to the references above for a more complete introduction. +6 + +2.5 Definition. +• We say that an ∞-category X is a polygraph if it can be constructed +from the empty ∞-category by freely adding arrows with specified source +and target. That is if X can be obtained as a transfinite composition +∅ = X0 → X1 → · · · → Xi → Colim Xi = X where for each i, the map +Xi → Xi+1 is a pushout of Y × ∂Dn → Y × Dn+1. +• An arrow of a polygraph is said to be a generator if it is one of the arrows +that has been freely added at some stage. +• A morphism of ∞-categories between two polygraphs is said to be a mor- +phism of polygraphs or a polygraphic morphism if it sends each generator +to a generator. +• An n-polygraph is a polygraph whose generators are all of dimension ⩽ n. +2.6 Remark. Generators of a polygraph can be shown to be exactly the arrows +that cannot be written as a composite in a non-trivial way, so the notion of +generator does not depend on the choice of the presentation of X, and any +isomorphism between polygraphs is automatically polygraphic. +2.7 Example. The only n-polygraphs for n < 0 is the empty ∞-category, the +category of 0-polygraphs is equivalent to the category of sets and corresponds +to discrete ∞-categories, the category of 1-polygraphs (and polygraphic mor- +phisms between them) is equivalent to the category of directed graphs, and they +corresponds to categories that are free on a graph. +We will sometimes distinguish between a polygraph seen as an object of +the category of polygraphs and polygraphic morphisms, and the corresponding +∞-category, which we call the free ∞-category on the polygraph. +2.8 Remark. Each arrow in a polygraph can be written as an iterated compos- +ite of the generators (not necessarily in a unique way). For an n-arrow f, the set +of generators of dimension n that appear in such an expression (and even the +number of times they appear) is the same for all such expressions. We will say +that an n-generator appears in an n-arrow if it appears in any such expression. +2.9 Construction. The category ∞-Cat admits a closed monoidal structure, +called the Gray tensor product or Crans-Gray tensor product, which we denote +as +∞-Cat × ∞-Cat +→ +∞-Cat +X, Y +�→ +X ⊗ Y +Its explicit construction is very involved and we will assume the reader is already +familiar with it. It was first introduced by S. Crans in his Ph.D. thesis [10]. +We refer to [1] for an introduction to this tensor product close to its original +definition, and to [27] for a more modern account. The proof of the existence of +this monoidal structure in [27] contains some gaps that have been fixed in [4]. +It is easy to see from either of these definitions that Dn ⊗ Dm has a unique +non-trivial arrow of dimension n + m. If f and g are respectively an n-arrow +of X and an m-arrow of Y , which corresponds to morphisms f: Dn → X and +g: Dm → Y , we denote by f ⊗ g the m + n arrow of X ⊗ Y obtained as the +image of this non-trivial (n+m)-arrow by the functor f ⊗g: Dn ⊗Dm → X ⊗Y . +We recall from [3]: +7 + +2.10 Proposition. If X and Y are polygraphs then X ⊗ Y is also a polygraph. +The generators of X ⊗ Y are the arrow of the form x ⊗ y where x and y are +respectively generators of X and Y . +Finally, we recall from [19] that ∞-Cat carries a model structure, called the +folk model structure in which every object is fibrant and where the generating +cofibrations are the ∂Dn → Dn. Its weak equivalences are a natural class of +equivalence of ∞-categories that generalizes the equivalences of ordinary cate- +gories. It was shown in [23] that the cofibrant objects are exactly the polygraphs +and it also follows from this that the cofibrations between cofibrant objects are +the polygraphic inclusions. It was shown in [3] that this model structure is a +monoidal model structure for the Gray tensor product. +2.2 +Marked ∞-categories +For the rest of the article, we fix an m ∈ N ∪ {∞} +2.11 Definition. An m-marked ∞-category is an ∞-category X, together with +a set M ⊂ � +k>0 X(k) of arrows of positive dimension called marked arrows such +that: +• All identity arrows Ix are marked. +• All arrows of dimension strictly superior to m are marked. +• If x and y are marked n-arrows and x#ky is defined, then x#ky is marked. +A morphism of m-marked ∞-categories is a morphism between the under- +lying ∞-categories that sends marked arrows to marked arrows. The category +of m-marked ∞-categories is denoted ∞-Catm. +Note that if m = ∞, then the second condition of the definition simply +disappears, this is the main case we are interested in. +2.12 Example. If X is an ∞-category we denote by X# the m-marked ∞- +category (X, X>0) where all arrows of positive dimension are marked. We denote +by X♭ the m-marked ∞-category where only identity arrows and k-arrows for +k > m are marked. +2.13 Construction. If X is an ∞-category and M ⊂ � +k>0 Xk is a set of +arrows of X, we denote by M the smallest set of arrows such that M ⊂ M +and (X, M) is an m-marked ∞-category. That is M is the reunion of the set of +arrows of dimension strictly superior to m and the set of all n-arrows that can +be written as iterated composites of n-arrows in M and arrows of the form Ix +for x an (n − 1)-arrow. For example X♭ = (X, ∅). +2.14 Construction. The category of m-marked ∞-categories has all colimits, +and they are easily described in terms of colimits of ∞-category and of Con- +struction 2.13: if (Xi, Mi)i∈I is a diagram of m-marked ∞-category indexed by +a category I then: +Colim +i∈I (Xi, Mi) = +� +Colim +i∈I +Xi, ∪ifi(Mi) +� +where fi denotes the canonical map fi: Xi → Colimi∈I Xi and fi(Mi) is simply +the set of arrows of the form fi(x) for x ∈ Mi. +8 + +This is easily shown by checking that the right-hand side has the universal +property of the colimit. +2.15 Definition. A special m-marked polygraph is an m-marked ∞-category of +the form (X, M) where X is free on a polygraph and M only contains generators +of X. +2.16 Proposition. If (X, M) is a special m-marked polygraph, then an n-arrow +f is in M if and only if n > m or if all the generating n-arrows that appear in +f are in M. +Proof. An arrow satisfying this condition is a composite of marked n-arrows and +identities of lower dimensional arrows, so it has to be in M. Conversely, this +set of arrows contains M and all identities (as no n-dimensional arrows appear +in their expression) and is closed under composition. +2.3 +Tensor product of m-marked ∞-categories +In this section we construct two monoidal closed structures on the category of +m-marked ∞-categories, respectively called the pseudo-Gray tensor product +∼ +and the lax-Gray tensor product +→. +Both are obtained by putting different +markings on the Gray tensor product from Construction 2.9. For example, the +lax-Gray tensor product D1 → D1 is C♭ +1 where C1 is the polygraph +C1 = + + + +• +• +• +• + + + +while D1 ∼ D1 is the special m-marked polygraph (C1, D) where D only contains +the unique 2 dimension generator of C1. So, unless m = 0 or m = 1, the two +tensor products are distinct. At the derived or homotopy theoretic level, the +pseudo-Gray tensor product should correspond to the cartesian product. +The formal definition goes as follows +2.17 Construction. Given two m-marked ∞-categories (X, M) and (Y, N) we +define two sets of arrows in X ⊗ Y : +• M → N is the set of arrows of the form x ⊗ y ∈ X ⊗ Y where either x ∈ M +or y ∈ N. +• M ∼ N contains all arrows in M → N together with all arrows of the form +x ⊗ y with x and y both of dimension > 0. +Note that M → N and M ∼ N are not marking on X ⊗Y : they are not stable +under composition. So we define: +(X, M) → (Y, N) = (X ⊗ Y, M → N) +(X, M) ∼ (Y, N) = (X ⊗ Y, M +∼ N) +We will show in Lemma 2.37 that both make the category of m-marked +∞-categories into a monoidal closed category. +In order to show this, it is convenient to introduce the following notations: +9 + +2.18 Notation. For A and B subsets of arrows in ∞-categories, we denote by +A ⊗ B the set of arrows of the form a ⊗ b ∈ X ⊗ Y for a ∈ A and b ∈ B. For +X and ∞-category, we denote by X⩾0 the set of all arrows of X and by X>0 +the set of all arrows of dimension > 0. We can hence, for (X, M) and (Y, N) to +m-marked ∞-category rewrite the definitions above as: +M → N += +(M ⊗ Y⩾0) ∪ (X⩾0 ⊗ N) +M +∼ N += +(M → N) ∪ (X>0 ⊗ Y>0) += +(M ⊗ Y⩾0) ∪ (X⩾0 ⊗ N) ∪ (X>0 ⊗ Y>0) +By definition of the Gray tensor product, we have the following result: +2.19 Lemma. Let X and Y be two ∞-categories, then +X⩾0 ⊗ Y⩾0 = (X ⊗ Y )⩾0 +X>0 ⊗ Y⩾0 ∪ X⩾0 ⊗ Y>0 = (X ⊗ Y )>0 . +That is X ⊗ Y is generated under composition by arrows of the form x ⊗ y, +and the arrows of dimension > 0 of X ⊗ Y are generated under compositions +by arrows of the form x ⊗ y with x or y of dimension > 0 +2.20 Lemma. Let X be an ∞-category and M, N two subsets of arrows of X +then: +M ∪ N = M ∪ N = M ∪ N = M ∪ N +Proof. This is straightforward. +2.21 Lemma. Let X, Y be two ∞-categories and M ⊂ X⩾0 and N ⊂ Y⩾0. +Then: +M ⊗ N = M ⊗ N = M ⊗ N = M ⊗ N +Proof. We will only show the equality M ⊗ N = M ⊗ N. The equality M ⊗ N = +M ⊗ N is proved in the exact same way and the last equality follows immedi- +ately by applying the result to M and N. We will also only proves the results +for m = ∞, the case of a general m follows immediately as it marks all arrow of +dimension > m on each side of these equalities. The evident inclusion M ⊂ M +implies M ⊗ N ⊂ M ⊗ N, so it is then enough to show that M ⊗ N ⊂ M ⊗ N. +Let K be the set of arrows k in X such that k ⊗ n ∈ M ⊗ N for all n ∈ N. We +need to show that K is closed by identity and composition to finish the proof. +If k = Ix, then k ⊗ n = Ix⊗n ∈ M ⊗ N. Let now k, k′ ∈ K of dimension n such +that k#ik′ is defined. They are encoded by a map Dn +� +Di Dn → X and let +y ∈ N be an arrow of dimension m of Y , encoded by a map Dm → Y . +Together these induced a map e: +� +Dn +� +Di Dn +� +⊗Dm → X⊗Y . +� +Dn +� +Di Dn +� +⊗ +Dm is a polygraph of dimension m + n with only two generating arrows of +maximal dimensions that are sent to k ⊗ y and k′ ⊗ y, which are by hypothesis +in M ⊗ N. +Now the arrow corresponding to (k#ik′) ⊗ y in +� +Dn +� +Di Dn +� +⊗ Dm is in +M ⊗ N as all the top dimensional generators that appear in it are in M ⊗ N. +We have proved that k#ik′ ⊗ y ∈ M ⊗ N for all y ∈ N, hence k#ik′ ∈ K and +this concludes the proof. +10 + +2.22 Lemma. Let X, Y be two ∞-categories, M ⊂ X⩾0 and N ⊂ Y⩾0. Then +we have +M → N += +M → N +M +∼ N += +M +∼ N. +Proof. Given the formula for M → N and M +∼ N from Notation 2.18, this is a +direct consequence of Lemma 2.20 and Lemma 2.21. +2.23 Lemma. Let X, Y, Z be three ∞-categories, M ⊂ X>0, N ⊂ Y>0 and +P ⊂ Z>0. Then we have +(M → N) → P += +M → (N → P) +(M +∼ N) ∼ P += +M +∼ (N +∼ P) +Proof. We begin with the first equality. Let +E: = (M ⊗ Y⩾0 ⊗ Z⩾0) ∪ (X⩾0 ⊗ N ⊗ Z⩾0) ∪ (X⩾0 ⊗ Y⩾0 ⊗ P) . +The lemmas 2.19, 2.20 and 2.21 implies the following equalities: +E += +M ⊗ Y⩾0 ⊗ Z⩾0 ∪ X⩾0 ⊗ (N ⊗ Z⩾0 ∪ Y⩾0 ⊗ P) += +M ⊗ (Y ⊗ Z)⩾0 ∪ X⩾0 ⊗ (N → P) += +M → (N → P) +A very similar computation also shows that E = (M → N) → P, which concludes +the proof of the first equality. +For the second equality, we define +F: = (X⩾0 ⊗ Y>0 ⊗ Z>0) ∪ (X>0 ⊗ Y⩾0 ⊗ Z>0) ∪ (X>0 ⊗ Y>0 ⊗ Z⩾0) +The second equality of Lemma 2.19 implies that: +F = Xk⩾0 ⊗ Y>0 ⊗ Z>0 ∪ X>0 ⊗ (Y ⊗ Z)>0 +and then that +E ∪ F += +M ⊗ (Y ⊗ Z)⩾0 ∪ X⩾0 ⊗ (N +∼ P) ∪ X>0 ⊗ (Y ⊗ Z)>0 += +M +∼ (N +∼ P) +and here again, a similar computation shows E ∪ F = (M +∼ N) ∼ P, which +concludes the proof. +2.24 Lemma. Let X be an ∞-category, M ⊂ X>0. +Then the empty set, +considered as a subset of the ∞-category D0, verifies (up to the identifications +D0 ⊗ X ≃ X ⊗ D0 ≃ X): +∅ → M = M → ∅ = M +∅ ∼ M = M +∼ ∅ = M +Proof. The first equality is a straightforward application of the definition of →. +For the second case, we also use that all arrows of (D0)>0 ⊗ X>0 are identities +and so all belong to M. +11 + +2.25 Proposition. Both the lax-Gray tensor product +→ and the pseudo-Gray +tensor product +∼ as defined above are monoidal structures on the category of +m-marked ∞-categories. In both cases the forgetful functor to ∞-categories is +monoidal and their unit is D♭ +0 = D# +0 . +Proof. Note that D♭ +0 = D# +0 = (D0, ∅) as all arrows of D0 of dimension > 0 are +identities. +The proposition exactly says that the structural map (associativity and unit +isomorphism) of the Gray tensor product of ∞-categories preserves the marking +we specified on the tensor product. +For the unit, let (X, M) be an m-marked ∞-category. The Lemmas 2.21 +and 2.24 imply that +(X, M) → (D0, ∅) += +(X ⊗ D0, M → ∅) += +(X, M) +(X, M) ∼ (D0, ∅) += +(X ⊗ D0, M +∼ ∅) += +(X, M) +and +(D0, ∅) → (X, M) += +(D0 ⊗ X, ∅ → M) += +(X, M) +(D0, ∅) ∼ (X, M) += +(D0 ⊗ X, ∅ ∼ M) += +(X, M). +For the associativity isomorphism, let (X, M), (Y, N) and (Z, P) be three ∞- +categories. Lemma 2.21 implies that +� +(X, M) → (Y, N) +� +→ (Z, P) += +(X ⊗ Y ⊗ Z, (M → N) → P) +� +(X, M) ∼ (Y, N) +� +→ (Z, P) += +(X ⊗ Y ⊗ Z, (M +∼ N) → P) +and +(X, M) → � +(Y, N) → (Z, P) +� += +(X ⊗ Y ⊗ Z, M → (N → P)) +(X, M) ∼ � +(Y, N) → (Z, P) +� += +(X ⊗ Y ⊗ Z, M +∼ (N → P)). +Lemma 2.23 shows that these two marking on X ⊗ Y ⊗ Z, in the lax and +the pseudo case, coincide. +2.26 Proposition. The pseudo and lax-Gray tensor product → and ∼ preserves +colimits in each variable. +In particular, as ∞-Catm is locally presentable, this immediately implies +that both tensor products are closed monoidal structures. +Proof. It follows from the fact that the Gray tensor product ⊗ preserves colimits +in each variables, the description of colimits of m-marked ∞-category given in +Construction 2.14 and Lemma 2.21. +2.4 +The semi-model structure +In this section, we will construct a left semi-model structure on the category +∞-Catm. +2.27 Definition. We define the set I = Im ∪ Ia to be our set of generating +cofibrations in ∞-Catm where: +Ia = {in: ∂Dn → Dn, |n ⩾ 0} +12 + +Im = {Dn → (Dn, {en}) +, n ⩾ 0} +An arrow in ∞-Catm is said to be a trivial fibration if it has the right lifting +property against all arrows in I. An arrow in ∞-Catm is said to be a cofibration +if it has the left lifting property against all trivial fibration. +2.28 Remark. It immediately follows from the small object argument that +every arrow can be factored into a cofibration followed by a trivial fibration +and that all cofibrations are retracts of transfinite compositions of pushouts of +arrows in I. +2.29 Remark. An arrow π: X → Y has the right lifting property against all +arrows in Ia if its image by the forgetful functor to ∞-Cat is a trivial fibration, +that is if for every pair of parallel n-arrows u, v in X, the map HomX(u, v) → +HomY (π(u), π(v)) is surjective. +π has the right lifting property against all arrows in Im if and only for every +arrow f ∈ X such that π(f) is marked in Y , f is marked in X. A trivial fibration +is a map that has both these properties. +2.30 Remark. The cofibrant objects of ∞-Catm are exactly the m-marked ∞- +categories whose underlying ∞-category is free on a polygraph, with any possible +marking on them (not just the special markings of 2.15). Indeed, transfinite +compositions of pushouts by arrows in Ia only starting from the empty ∞- +category exactly give all polygraphs with no markings on them. Pushouts by +Im are simply changing the marking and can make any arrow marked, so by also +taking pushouts by arrows in Im one obtains all polygraphs with any possible +marking on them. Finally, it was shown in [23] that polygraphs are closed under +retract in ∞-Cat, so they constitute all cofibrant objects. +The pushout-product, or corner-product (sometimes also called Leibniz prod- +uct) f ˆ +→ g and f ˆ +∼ g is defined as usual: if f: X → Y and g: A → B are two +arrows in ∞-Catm, then f ˆ +→ g is the canonical arrow: +X → B +� +X →A +Y +→ A → Y +→ B +and f ˆ +∼ g is the canonical arrow +X ∼ B +� +X ∼ A +Y +∼ A → Y +∼ B +We refer to the appendix of [18] for the general theory of pushout products and +their formal properties. +2.31 Proposition. If f and g are two cofibrations in ∞-Catm then f ˆ +→ g and +f ˆ +∼ g are both cofibrations. +Proof. By the usual properties of the corner-product, it is enough to check this +when f and g are generating cofibrations. If f and g are both in Ia, then f → g +has no marked arrows in either its domain or codomain and coincides with the +corner-product f ˆ⊗ g in ∞-Cat, which has been shown to be a cofibration in +[3]. f +∼ g is the same except that some arrows are marked, but we can always +add these marking by taking additional pushouts by arrows in Im, so it is again +a cofibration. +13 + +The forgetful functor ∞-Catm → ∞-Cat is monoidal for both tensor prod- +uct and preserves colimits, so it preserves the corner-product. In particular, if +either f or g is in Im then it is sent to isomorphisms by this forgetful functor +and hence f ˆ +→ g and f ˆ +∼ g induces isomorphisms between their underlying ∞- +categories. Now, if f: (X, N) → (X, M) is a morphism in ∞-Catm that induces +an isomorphism on underlying ∞-categories, then it is a pushout of arrows in +Im: one simply needs to take such pushout to make all arrows in M marked. +2.32 Construction. We define I: = D♯ +1 = (D1, {e1}). It is the ∞-category with +two objects, e− +0 and e+ +0 and a marked arrow e1: e− +0 → e+ +1 . We denote by j− and +j+ the two maps D0 → I corresponding respectively to the two objects e− +0 and +e+ +0 . This gives a diagram: +D0 +� +D0 ֌ I → D0 +Which will play the role of the interval object for our semi-model structure on +∞-Catm. +We will take as a set of “generating anodyne cofibrations” (also called a +“pseudo-generating set of trivial cofibrations”) the set of maps of the form j+ ˆ +∼ i +where i is a generating cofibration, more precisely: +2.33 Definition. +• We say that an arrow in ∞-Catm is a naive fibration if it has the right +lifting property against all arrows of the form j+ ˆ +∼ i, where j+: D0 → I is +as in Construction 2.32, and i is one of the generating cofibrations as in +Definition 2.27. +• We say that an arrow in ∞-Catm is an anodyne cofibrations if it has the +right lifting property against all naive fibrations. +• We say that a cofibration in ∞-Catm is acyclic if it has the lifting property +against all naive fibrations between (naively) fibrant objects. +• We say that a map in ∞-Catm is a fibration if it has the right lifting +property against all acyclic cofibrations. +As before, it immediately follows from the small object argument that every +arrow factors as an anodyne cofibration followed by a naive fibration, and all +anodyne cofibration are retracts of transfinite compositions of pushouts of the +“generating anodyne cofibrations”. +2.34 Remark. It immediately follows from Proposition 2.31 that, as j+ is a +cofibration, all maps of the form j+ ˆ +∼ i are cofibrations. In particular, all trivial +fibrations are also naive fibrations and all anodyne cofibrations are cofibrations. +2.35 Proposition. Acyclic cofibrations and fibrations form a cofibrantly gen- +erated weak factorization system on ∞-Catm. An object is “naively fibrant” if +and only if it is fibrant and more generally an arrow between fibrant objects is +a fibration if and only if it is a naive fibration. +Proof. This is a direct application of the results of Section 4 of [15]. Starting +from the premodel structure on ∞-Catm whose weak factorization systems are +(cofibrations, trivial fibrations) and (anodyne cofibrations, naive fibrations), we +obtain the one with (cofibrations, trivial fibrations) and (acyclic cofibrations, +fibrations) as its “left saturation” L(∞-Catm) in the sense of Theorem 4.1 of +[15]. All the claim in the proposition follows from this Theorem 4.1. +14 + +2.36 Remark. Note that replacing ˆ +∼ by ˆ +→ in 2.33 would not change the +definition. +Indeed, if X = Y ♯ is an m-marked ∞-category whose arrows of +dimension > 0 are all marked then for any m-marked ∞-category Z one has +X +∼ Z = X → Z. As this applies to both the domain and the co-domain of j+ +it follows that j+ ˆ +∼ i = j+ ˆ +→ i. +Also, the reader should not be worried about the use of j+ in Definition 2.33 +rather than j− or both j− and j+. While putting j− or both j− and j+ instead +of j+ would change the definition of naive fibrations and anodyne cofibrations, +this does not affect the definition of (naive) fibrations between fibrant objects, +hence the acyclic cofibrations and fibrations would not be changed. Indeed, once +the existence of a (monoidal) model structure is established, it follows that j− +is acyclic by 2-out-of-3, and hence all the maps j− ˆ +∼ i = j− ˆ +→ i are also acyclic +cofibrations. +2.37 Lemma. If f is an anodyne (resp. acyclic) cofibration and g is a cofibra- +tion then f ˆ +∼ g and f ˆ +→ g are anodyne (resp. acyclic). +Proof. To get the result for “anodyne cofibrations” it is enough to prove it for +the generating anodyne cofibrations. Let i be one of the generating cofibrations +and f = j+ ˆ +∼ i′ be one of the generating anodyne cofibrations. We have f ˆ +∼ i = +j+ ˆ +∼ (i ˆ +∼ i′). +As i′ ˆ +∼ i is a pushout of generating cofibrations i1, . . . , ik by +Proposition 2.31 it follows that j+ ˆ +∼ (i ˆ +∼ i′) is a pushout of the j+ ˆ +∼ ik and +hence is an anodyne cofibration. +The result for acyclic cofibrations follows from formal properties of the +pushout product: it follows that if i is a cofibration and p is a naive fibra- +tion then the (right) pullback exponential ⟨p/i⟩ is a naive fibration. If p is a +(naive) fibration between fibrant objects then ⟨p/i⟩ is a naive fibration between +fibrant objects hence a fibration. It follows that if i a acyclic cofibration and +j is a cofibration then i ˆ +∼ j is an acyclic cofibration as it is a cofibration by +Definition 2.27 and if p is a fibration between fibrant objects then i ˆ +∼ j has the +right lifting property against p because j has the left lifting property against +⟨p/i⟩. +The case of +→ works exactly the same considering the first half of Re- +mark 2.36. +2.38 Theorem. The category ∞-Catm of m-marked ∞-category admits a left +semi-model structure, called the inductive model structure, in which the cofi- +brations and trivial fibrations are as in Definition 2.27 and the fibrations are as +in Definition 2.33. +Proof. This immediately follows from Theorem 6.12 of [15]. Because of Propo- +sition 2.31 and Lemma 2.37, tensoring by the interval object I of Construc- +tion 2.32 is a “strong Quillen functor” in the sense of section 6 of [15]. Note that +to apply Theorem 6.12 one needs to observe that ∞-Catm, with the (cofibra- +tions, trivial fibrations) and (acyclic cofibrations, fibrations) weak factorization +systems, is both “right saturated” and “left saturated” that is, that a fibration +that has the right lifting property against all cofibrations between cofibrant ob- +jects is a trivial fibration and that a cofibration that has the left lifting property +against all fibrations between fibrant objects is a trivial cofibration. The first +ones hold because the generating cofibrations are cofibrations between cofibrant +objects and the second because that is how we defined acyclic fibrations. +15 + +2.39 Remark. The proof of Theorem 2.38 above also shows that ∞-Catm +also admits a right semi-model category structure whose fibrations and trivial +cofibrations are the fibrations and acyclic cofibrations of Definition 2.33 and +whose cofibrations are as in Definition 2.27. +This however does not clearly make ∞-Catm into a Quillen model struc- +ture but rather into a “two-sided model category” as in Section 5 of [15]. We +refer to Section 5 of [15] for what this means more precisely, but in short, +the problem is that the left and right semi-model categories have different +classes of weak equivalences. The two classes of equivalence however coincide +for arrows that are between fibrant or cofibrant objects. Another way to talk +about this difference is that left and the right semi-model categories are Quillen +equivalent and have the same homotopy category, but define different functors +∞-Catm → Ho(∞-Catm). The two functors agree on objects that are either +fibrant or cofibrant but differ on general objects: one sends an object X to its +cofibrant replacement while the other sends it to a fibrant replacement, and we +do not know if these are always homotopy equivalent when X is neither fibrant +nor cofibrant itself. +2.40 Remark. We do not know if ∞-Catm is actually a Quillen model cate- +gory or not. In the unmarked case, this follows from the fact that all objects +are fibrant. But that is no longer the case in this situation. In terms of the +“two-sided model structure” mentioned in the previous remark, the question is +whether ∞-Catm satisfies one of the equivalent conditions of Proposition 5.3 of +[15]. +We conclude this section with the following lemma that will be useful later: +2.41 Lemma. The map +i+ +n : D♭ +n → (Dn+1, {en+1}) +where en+1 is the unique non-identity arrow of Dn+1, is an anodyne cofibration. +Proof. We will show it is a retract of the map j+ ˆ +∼ in where in is the map +∂Dn → Dn. +In order to achieve this, we will compute j+ ˆ +∼ in more explicitly using the +description of D1 ⊗ Dn given in appendix B.1 of [4] (see proposition B.1.4): As +a polygraph, the generating arrows of D1 ⊗ Dn are the: +a− +0 ⊗ eǫ +k +a+ +0 ⊗ eǫ +k +a ⊗ eǫ +k +where the arrows of D1 have been denoted “a” instead of “e” to distinguish +them, and ǫ is either + or −, k ⩽ n and e+ +n = e− +n . Their source and target are +given as follows: +π−(a− +0 ⊗ eǫ +k) = a− +0 ⊗ e− +k−1 +π+(a− +0 ⊗ eǫ +k) = a− +0 ⊗ e+ +k−1 +π−(a+ +0 ⊗ eǫ +k) = a+ +0 ⊗ e− +k−1 +π+(a+ +0 ⊗ eǫ +k) = a+ +0 ⊗ e+ +k−1 +π−(a ⊗ eǫ +k) = (a− +0 ⊗ eǫ +k)#0(a ⊗ e+ +0 )#1 . . . #k−1(a ⊗ e+ +k−1) +π+(a ⊗ eǫ +k) = (a ⊗ e− +k−1)#k−1 . . . #1(a ⊗ e− +0 )#0(a+ +0 ⊗ eǫ +k) +16 + +We did not put parenthesis in the expression above, to keep them shorter, the +default convention is to do the composition #i in order of increasing values of i. +The last two equations are given by proposition B.1.4 of [4], though note that +this reference is using a different convention than ours regarding the composition +order. Note that the object we are interested in is I +∼ D♭ +n which is the same +polygraph endowed with the special marking where all the arrows a ⊗ eǫ +k are +marked. +We then realize (Dn+1, {en+1}) as a retract of I +∼ D♭ +n+1 as follows: We +call i: (Dn+1, {en+1}) → I +∼ D♭ +n the unique morphism sending en+1 to a ⊗ en. +This is well defined because a ⊗ en is a marked arrow. Next, we define a map +p: I ∼ D♭ +n → (Dn+1, {en+1}) by: +p(aǫ +0 ⊗ eµ +k) = eµ +k if k < n. +p(aǫ +0 ⊗ en) = eǫ +n +p(a ⊗ eǫ +k) = Ieǫ +k if k < n. +p(a ⊗ en) = en+1 +In order to check that this is well defined, we first need to check that this +definition is compatible with the source and target given above, which follow +from an immediate calculation. Then we need to show that this is compatible +with the marking, which is the case as both Ieǫ +k and en+1 are marked. +Finally, the composite p ◦ i send the arrow en+1 to p(a ⊗ en) = en+1 and +hence is the identity of Dn+1. +To conclude the proof, we just have to observe that the maps f and i defined +above send the domain of i+ +n and of j+ ˆ +∼ in to each other. +The domain of j+ ˆ +∼ in is the sub-polygraph of I +∼ D♭ +n which contains all +the generators except a− +0 ⊗ en and a ⊗ en, while the domain of i+ +n contains all +generators of Dn+1 except en+1 and e− +n . +In order to check that the map i is compatible with these sub-polygraphs, +it is enough to check that i(e+ +n ) is in the domain of j+ ˆ +∼ in, to see this, we +compute: +i(e+ +n ) = π+i(en+1) = π+(a ⊗ en) = (a ⊗ e− +n−1)#n−1 . . . #1(a ⊗ e− +0 )#0(a+ +0 ⊗ en) +and we observe that this expression involves neither a− +0 ⊗ en nor a ⊗ en, hence +it does belong to the domain of j+ ˆ +∼ in. +In order to check that the map p is compatible with these sub-polygraphs, we +need to check the image by p of all the generators of I ∼ D♭ +n except a− +0 ⊗ en and +a⊗en. These are given by the formulas p(aǫ +0⊗eµ +k) = eµ +k if k < n, p(a+ +0 ⊗en) = e+ +n +and p(a ⊗ eǫ +k) = Ieǫ +k, which all indeed belong to the image of i+ +n . +3 +Equations and saturations in an m-marked ∞- +category. +The general goal of this section is to arrive at a better description of the fibrant +objects and fibrations between fibrant objects of the model structure of Theo- +rem 2.38. This is achieved using the notion of “equations” in an ∞-categories +introduced by the second named author in [20]. We will recall the basic theory of +equations, in a slightly different language and introduce an analog of equations +to deal with the markings, which we call saturations. +17 + +3.1 +Definitions of equations and saturations +3.1 Definition. A left equation is a special m-marked polygraph (P, M) with +two arrows x, y ∈ P such that: +(1) y is the unique arrow of dimension n + 1 and P contains no arrows of +dimension > n + 1. +(2) y is a marked arrow. +(3) if n ≤ m, x is an unmarked arrow of P. +(4) The source of y admits a decomposition: +π− +n y = ln#n−1(ln−1#n−2...#1(l1#0x#0r1)#1...#n−2rn−1)#n−1rn +where for each i, li and ri are marked i-arrow in P, with ln and rn not +containing x. In particular, x appears only once in π− +n y. +(5) x does not appear in the target of y. +Right equations are defined in the exact same way except the source and +target of y are exchanged in the last two conditions. +We say that (P, M) is an equation to mean that it is either a left or right +equation. If P, with its arrows x and y as in the definition, is an equation one +denotes by ΛP the sub-polygraphs of P that contains all arrow except x and y. +3.2 Remark. Note that specifying the arrows x, y ∈ P is exactly the same as +specifying the subpolygraphs ΛP ⊂ P. For this reason, we will often also call +“equation” the map ΛP → P. +We say that an equation ΛP → P has solutions in C ∈ ∞-Catm if C has the +right lifting property against ΛP → P and we say that a morphism f: C → D +lifts solutions of the equation if it has the right lifting property against the map +ΛP → P. +3.3 Remark. The name “equation” comes from the idea that we are looking for +an element x such that a certain composite of x with other arrows is isomorphic +to another given arrow. From this point of view, a map ΛP → X corresponds +to such an equation in X, and an extension P → X corresponds to a solution +of the equation, or rather the image of x is the solution and y represents the +isomorphism witnessing that x is a solution. +3.4 Definition. A left saturation is a special marked polygraph (P, M) with +arrows x and y satisfying the conditions of Definition 3.1 except that x is a +marked arrow and the target of y is a marked arrow. Right saturations are +defined in the same way. +If P is a saturation, one denotes ΩP the special +m-marked polygraph (P, M − {x}). +3.5 Definition. If P is an equation, we define the m-marked ∞-category +Uni(P) which is the colimit of the following diagram: +∂Dn +D♯ +n +P � +ΛP P +Uni(P) +x∐x′ +z +⌟ +18 + +A map Uni(P) → X corresponds to a map ΛP → X, which is an equation in +X, together with two solutions P → X, given by pairs (x, y) and (x′, y′), and a +marked arrow z: x → x′ which express that the two solutions are isomorphic. +3.6 Definition. Let C be an m-marked ∞-category C and P a left equation +(resp. right equation). +The equation P has solutions in C if for all morphisms ΛP → C, there +exists a lifting (x, y): P → C such that x is sent on a marked arrow whenever +the target of y is (resp. the source of y is). +Solutions to an equation P are C are weakly unique if C has the right lifting +property against P � +ΛP P → Uni(P). +The equation P has unique solutions in C if the equation P has solutions in +C and they are weakly unique. +It will be useful to have a “coherent” version of Uni(P), noted Unicoh(P). +If P is a left equation, P � +ΛP P → Unicoh(P) is obtained as the following +sequence of pushout: +∂Dn +Dn +P � +ΛP P +��� +Unicoh(P) +∂Dn+1 +Dn+1 +x∐x′ +z +⌟ +s[x/z]#ny′∐y +⌟ +where s is the source of y. Conversely, if P is a right equation, P � +ΛP P → +Unicoh(P) is obtained as the following sequence of pushout: +∂Dn +Dn +P � +ΛP P +• +Unicoh(P) +∂Dn+1 +Dn+1 +x∐x′ +z +⌟ +y#nt[x/z]∐y′ +⌟ +where t is the source of y. Remarks that in both cases, P � +ΛP P → Unicoh(P) is +an equation. By definition, if C is an m-marked ∞-category such that Unicoh(P) +has a solution in C, then C as the right lifting property against P � +ΛP P → +Uni(P). +3.7 Example. Let n be a non-negative integer. The morphism +j+ ˆ +∼ in: = I ∼ ∂Dn +� +{1} ∼ Dn → I +∼ Dn +is a left equation. Indeed, let y be the top dimensional generator of I +∼ Dn. If +we denote by x the top dimensional arrow of {0} ∼ Dn, and for 0 < k ≤ n, by +ak the image of the top dimensional k-generator of I +∼ Dk−1 by the morphism +I +∼ δ− +k−1: I +∼ Dk−1 → I +∼ Dn, +19 + +Section B.1. of [4] allows to give an explicit description of I +∼ Dn, which we +recalled in the proof of Lemma 2.41. Using this description, we see that if we +name y = a ⊗ en and x = a− +0 ⊗ en the two arrows of I ∼ Dn that are not in the +image of j+ ˆ +∼ in, then we have a decomposition of the source of y of the form: +(((x#0a0)#1a2)...)#n−1an +and all the ak are marked. We denote it +eq +• +• +• +• +n : ΛEq +• +• +• +• +n → Eq +• +• +• +• +n . +3.8 Example. Similarly, the morphism +j+ ˆ +∼ sn: I +∼ Dn +� +{1} ∼ (Dn, {en}) → I +∼ (Dn, {en}) +where sn is the “identity” map Dn → (Dn, {en}) is a left saturation. which +we denote +sat +• +• +• +• +n : ΩSat +• +• +• +• +n → Sat +• +• +• +• +n +3.9 Definition. We define some left equations which play an important role. +In each case, k and n are integers with k ⩽ n. +• eq +• +• +• +k,n : ΛEq +• +• +• +k,n → Eq +• +• +• +k,n , whose target is generated by x and b of di- +mension n, a a marked arrow of dimension k and y: (a#k−1x) ⇒ b. +• eq +• +• +• +k,n : ΛEq +• +• +• +k,n → Eq +• +• +• +k,n , whose target is generated by x and b of di- +mension n, a a marked arrow of dimension k and y: (x#k−1a) ⇒ b. +In all equations above, the domain of the arrow is obtained by removing x +and y. Also, in each case, we have not listed all the constraints of the source +and target that are necessary to make sense of the definition of y. For example, +in Eq +• +• +• +k,n , we have the relation π+ +k−1(a) = π− +k−1(x) for the composition a#k−1x +to exists, and the relations πǫ +n−1(b) = πǫ +n−1(a#k−1x), as b needs to be parallel +to a#k−1x for y to exists. +3.2 +Characterization of fibrant objects +In this section, we will give a simple characterization of the fibrant objects of +the model structure introduced in Theorem 2.38. We will temporarily call the +objects satisfying this characterization “prefibrant” (Definition 3.12) and then +show in Proposition 3.19 that these are exactly the fibrant objects. +3.10 Notation. Suppose given an equation P and a lifting problem of the form: +ΛP +C +P +D +p +Given a a generator of P, we will denote its image in D also by a. If a ∈ ΛP, +we denote by a its image in C. So in general p(a) = a. If the dotted diagonal +lift exists, or in the process of constructing such a lift, the image of x, y ∈ P in +C are also denoted x and y, and we hence also have p(x) = x and p(y) = y. +20 + +3.11 Definition. Let a be a (n + 1)-arrow. An inverse for a is an arrow a−1 +such that there exist two marked arrows: +ǫ: a#na−1 → I +ν: a−1#na → I. +An arrow is invertible if it has an inverse. +3.12 Definition. An m-marked ∞-category C is prefibrant if +(1) marked arrows are invertible and their inverses are marked, +(2) whenever a and c: a → b are marked, so is b . +This directly implies that if b and c: a → b are marked, so is b. +This notion is purely temporary: we will show in Proposition 3.19 that an +object is fibrant for the model structure of Theorem 2.38 if and only if it is +prefibrant. +3.13 Proposition. If C is prefibrant, then equations Eq +• +• +• +k,n and Eq +• +• +• +k,n have +weakly unique solutions in C. +Proof. We show the result by a decreasing induction on k ≤ n. The initialization +corresponds to k = n. +In this case, the data of a morphism ΛEq +• +• +• +n,n → C +corresponds to two n-arrows a and b sharing the same source and such that a is +marked. Let ν: a−1#na → I. If we define x: = a−1#nb and y: ψ#nb: a#nx → b, +the couple (x, y) is a solution of Eq +• +• +• +n,n . If b is marked so is x. We now show +the weak unicity of the solution. Let (¯x, ¯y) be another solution. We then have +a marked arrow: +z: ¯x +ν−1 +−−→ a−1#na#n¯x +¯y−→ a−1 ∗ b. +The assertion for Eq +• +• +• +n,n is similar. +Suppose now the result is true for all k + 1. +We start by showing that +solutions of Eq +• +• +• +k,n and Eq +• +• +• +k,n are weakly unique in C. The data of a morphism +ΛEq +• +• +• +k,n → C corresponds to an n-arrow x: s → t, a k-invertible arrow a, and an +arrow b: a#k−1s → a#kt. Let (x, y: a#k−1x → b) be a solution of this equation. +Let ν: a−1#na → I. The arrow x is then also a solution of Eq +• +• +• +k+1,n: +(ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t) +(a−1#k−1b)#k(ν#k−1t) +(a−1#k−1y)#k(ν#k−1t) +and so is weakly unique. The unicity of solution of Eq +• +• +• +k,n is proved similarly. +We show now that Eq +• +• +• +k,n and Eq +• +• +• +k,n have solutions in C. Let (x, y) be a +solution of the equation +(ν#0s)#kx +(a−1#k−1b)#k(ν#k−1t). +y +Moreover, we can find such x marked whenever b is. We then have +(ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t). +21 + +By weakly unicity of solution of Eq +• +• +• +k+1,n, we then have a marked arrow +z: a−1#k−1a#k−1x → a−1#k−1b. +But a#k−1x and b are solutions of an equation Eq +• +• +• +k,n , and so there exist a +marked arrow +˜y: a#k−1x → b. +If b is marked, the arrow x that we produce is also marked. The existence of +solution of Eq +• +• +• +k,n is proved similarly. +3.14 Lemma. If equations Eq +• +• +• +k,n +and Eq +• +• +• +k,n +have solutions in C, then all +equations have solutions in C. +Proof. Let P be a left equation. There is a decomposition of the source of y of +the shape +π− +n y = ln#n−1(ln−1#n−2...#1(l1#0x#0r1)#1...#n−2rn−1)#n−1rn +where for each i, li and ri are marked i-arrow in P. We can then use the existence +of solutions to Eq +• +• +• +k,n and Eq +• +• +• +k,n to get two sequences of arrows (xk)0 0) of Defini- +tion 3.25. Indeed, given a weak lifting diagram: +∂Dn +X +Dn +(Dn+1, {en}) +Dn +Y +p +The solid part of the diagram corresponds to a pair of parallel (n − 1)-arrows +(a, b) in X, together with an n-arrow c: p(a) → p(b) in Y , the top dotted mor- +phism gives us an arrow ˜c: a → b, while the bottom dotted morphism corre- +sponds to a marked (n + 1)-arrow e: p(˜c) → c, so this lifting condition corre- +sponds exactly to the third point of Definition 3.25 ( with the second point +corresponding to the case n = 0). +3.5 +The saturated localization. +Proposition 3.19 produces a characterization of fibrant objects of the model +structure of Theorem 2.38: a marked ∞-categories is fibrant if the marked +arrows have inverses and if an arrow isomorphic to a marked arrow is marked. +A careful reader might have noticed however that this is not sufficient to +show that the marked arrows are exactly the arrows that have inverses in the +sense of Definition 3.11. +3.27 Example. Let C be a category, seen as an ∞-category with no non- +identity arrows of dimension > 1. We endow C with the marking C♭, where +only the identity arrows are marked. +With this marking C is fibrant, indeed, it satisfies all the conditions of +Proposition 3.19. +But if the category C has non-identity invertible arrows, +these would be arrows that have inverses in the sense of Definition 3.11 without +being marked. +In this section, we “fix” this problem by introducing a Bousfield localization +in which the fibrant objects have these properties. +3.28 Definition. A marked ∞-category C is said to satisfy the 2-out-of-6 +property if given three composable n-arrow f,g and h such that f#n−1g and +g#n−1h are marked, then f, g and h are marked. +3.29 Remark. If C is a fibrant m-marked ∞-category. +Then the relation +f ∼ g defined by ∃c: f → g a marked (n + 1)-arrow, is an equivalence relation +27 + +on n-arrow. Indeed it is reflexive and transitive as identities are marked and +composites of marked arrows are marked, and it is symmetric as marked arrows +have inverses. +This equivalence relation is moreover compatible with all composition op- +erations, so that one can define a “homotopy n-category” hnC, which is an +n-category whose k arrows for k < n are these of C and its n-arrows are equiva- +lence classes for this relations. We will use in particular that given two parallel +n − 2 arrows u, v in C we have a category hnC(u, v) whose objects are n − 1 +arrows u → v and whose morphisms are equivalence classes of n arrows between +them. +3.30 Lemma. For an m-marked ∞-category C the following conditions are +equivalent: +(1) An arrow in C is marked if and only if it has an inverse in the sense of +Definition 3.11. +(2) C is fibrant in the model structure of Theorem 2.38 and satisfies the 2- +out-of-6 property. +Proof. We first consider C an m-marked ∞-category which satisfies (1), and we +check it is fibrant using Proposition 3.19. The first condition of Proposition 3.19 +is immediate, we check the second condition : if c: a → b is marked and b is +marked, then considering b−1 an inverse of b, the marked arrow connecting +ǫ: b−1#nb → I and ν: b#nb−1 → I, we can simply compose: +b−1#na +b−1#nc +→ +b−1#nb +ǫ→ I +a#nb−1 c#nb−1 +→ +b#nb−1 +ǫ→ I +This shows that b−1 is also an inverse for a, and hence if all arrows with an +inverse are marked a is marked as well. Note that if it is a which is marked in +the first place then one can consider an inverse c−1: b → a and apply the same +argument. +Next, we show that C satisfies 2-out-of-6. +For this, we can rely on Re- +mark 3.29. An n arrow has an inverse in the sense of Definition 3.11 if and +only if it is an isomorphism in the category hnC(u, v) where u and v are its +(n − 2)-dimensional source and target. +Our assumption is then that an n- +arrow is marked if and only if its equivalence class is invertible in the category +hnC(u, v). The fact that marked arrows satisfy 2-out-of-6 then follows from the +fact that isomorphism in a category satisfies the 2-out-of-6 condition. +Conversely, assuming that C satisfies condition (2) we have that marked +arrows have inverses because C is fibrant and Proposition 3.19. If an arrow a +has an inverse a−1 then both a#n−1a−1 and a−1#n−1a are marked because +they are equivalence to identities, and it follows from the 2-out-of-6 condition +that a (and a−1) is marked. +3.31 Theorem. The model structure of Theorem 2.38 admits a Bousfield lo- +calization (as a left semi-model structure) in which the fibrant objects are the +marked ∞-categories which satisfy the equivalent conditions of Lemma 3.30. +We call this model structure the saturated inductive model structure. +28 + +As a Bousfield localization, this model structure has the same cofibrations +and the same fibration between fibrant objects as the model structure from +Theorem 2.38. +Proof. The key point here is that the 2-out-of-6 condition for a marked ∞- +category corresponds to the lifting property against certain cofibrations. +For each n we consider the polygraphs Xn generated by three composable n +arrow +Dn +� +Dn−1 +Dn +� +Dn−1 +Dn +Where each pushout uses the target maps on the left and the source map on +the right. We call f, g and h the three n-dimensional generators of Xn. We +consider the map sn: +sn: +� +Xn, {f#n−1g, g#n−1h} +� +→ +� +Xn, {f, g, h} +� +which is the identity of Xn (with two different markings). sn is a cofibration, +and a marked m-category has the right lifting property against all the sn if and +only if it satisfies the 2-out-of-6 property. +We hence take the left Bousfield localization of the model structure of The- +orem 2.38 at the set sn. The theory of left Bousfield localization for left semi- +model structure can be found in [5] or [15]. +Given that the maps in sn are already cofibration between cofibrant objects, +the fibrant objects of the left Bousfield localization are the fibrant objects that +have the right lifting property against the maps sn and all their iterated cylinder +maps ∇ksn, where if i: A → B is a cofibration, ∇i is the cofibration B � +A B → +IAB for some relative cylinder object. However, in the case of the map sn, given +that it only changes the marking, the pushout B � +A B is just B, hence it is +already a cylinder object, and the map ∇sn is an isomorphism. It hence follows +that an object is fibrant in the localization if it is fibrant and has the lifting +property against all the sn, i.e. has the 2-out-of-6 property. This concludes the +proof. +4 +Comparison with other model structures +4.1 +Truncation functors +4.1 Definition. Let m < p ≤ ∞. There is a functor: +πm: +∞-Catm+1 +→ +∞-Catm +(X, M) +�→ +(X, M). +that marks every arrow of dimension m + 1, an obvious inclusion functor: +ιm+1: +∞-Catm +→ +∞-Catm+1 +(X, M) +�→ +(X, M) +and eventually, a functor: +τm: +∞-Catm+1 +→ +∞-Catm +(X, M) +�→ +(τ(X), M) +29 + +where τ(X) is the sub ∞-category of X whose arrows of dimension strictly su- +perior to m are the ones in M. As M is assumed to be closed under composition +and contains the identities, τ(X) is indeed an ∞-category. +These functors fit in following adjunctions: +πm ⊣ ιm+1 ⊣ τm. +4.2 Notation. For m < p, we also denote by ιp: ∞-Catp → ∞-Catm (resp. +πp: ∞-Catm → ∞-Catp, resp. τp∞-Catm → ∞-Catp) the iterate composite +of ιk (resp. πk, resp. τk) when k range over [p, m]. +Moreover, because ιp is the inclusion of a full subcategory, we will of- +ten identify X and ιpX in our notation. +In the same way, for a morphism +f ∈ Hom(X, τm(Y )), the corresponding morphism in Hom(ιpX, Y ) will also be +denoted f. +4.3 Proposition. For m < p, the adjoint pairs (πm ⊣ ιp) and (ιp ⊣ τm) are +Quillen pairs. +Proof. The functors πm and ιp obviously preserve cofibrations and anodyne +cofibrations. +As mentioned in the introduction, we can consider the two towers of model +structures: +∞-Cat0 π0 +← ∞-Cat1 π1 +← ∞-Cat2 π2 +← . . . +πn−1 +← ∞-Catn πn +← . . . +∞-Cat0 τ0 +← ∞-Cat1 τ1 +← ∞-Cat2 τ2 +← . . . +τn−1 +← ∞-Catn τn +← . . . +and take the projective limit of either tower to get a definition of strict (∞, ∞)- +categories. +Our goal in this section is to show that the inductive model structure on +∞-Cat∞ is equivalent to the limit of the second tower (with τ functors). Here +by projective limit we mean a homotopy theoretic limit of these towers, that is a +homotopy limit of the corresponding tower of (∞, 1)-categories. Such projective +limits of model structures have been studied in [6] and [12] and we will use the +construction from these papers. +4.4 Remark. It should be noted that the results from [6] and [12] are only +proved for Quillen model structures, so they do not immediately apply to the +left semi-model structures that we are using here. The proof from these two +papers easily adapts to the setting of left semi-model structures with very few +modifications, so it should be safe to assume these results can be applied here +as well. Though to avoid relying on this, we will give an independent proof that +the model structure we use as a model of this projective limits exists and state +our main theorem as an equivalence with this model structure. The only aspect +that still relies on applying the results of [6] or [12] to semi-model structures is +in order to interpret our results as saying something about homotopy limits of +towers. +4.5 Definition. A category with weak equivalences is a couple (C, W) where C +is a category and W is a class of map C satisfying the two-out-of-three property. +We define the homotopy category of (C, W) as ho(C, W): = C[W −1]. +30 + +4.6 Definition. We define the category with weak equivalences LimLaxn∈N ∞-Catn, +whose object are sequences X• = {(Xn, fn)}n∈N where Xn ∈ ∞-Catn and +fn: Xn → τnXn+1, and whose weak equivalences are pointwise equivalences. By +adjunction, objects are in bijection with sequences +X0 +f0 +−→ X1 +f1 +−→ . . . +fn−1 +−−−→ Xn +fn +−→ . . . +where each Xn ∈ ∞-Catn. +The category with weak equivalences limn∈N ∞-Catn, is the full sub-category +of LimLaxn∈N ∞-Catn composed of objects {(Xn, fn)}i∈N where for all n, fn: Xn → +τiXn+1 is a weak equivalence of the model structure on ∞-Catn. Weak equiv- +alences are pointwise equivalences. +4.7 Proposition. There exist a model structure on LimLaxn∈N ∞-Catn,, called +the lax-limit structure, where fibrations and weak equivalences are pointwise, and +cofibrations are morphisms h: X• → Y• such that h0: X0 → Y0 is a cofibration +in ∞-Cat0, and for all n, the dotted morphism in the following diagrams is a +cofibration in ∞-Cati+1: +Xn +Xn+1 +Yn +Yn +� +Xn Xn+1 +Yn+1 +Proof. First let us notice that LimLaxn∈N ∞-Catn, can be identified with the +full subcategory of the functors X: N → ∞-Cat∞ such that Xn ∈ ∞-Catn. +There is a model structure on such functors, where fibrations and weak +equivalences are pointwise: the projective (or Reedy) model structure. +The +cofibrations of this model structure are as described in the proposition and this +model structure restricts to LimLaxn∈N ∞-Catn. +4.8 Definition. We have an adjunction +LimLaxn∈N ∞-Catn +∞-Cat∞ +c +τ +⊣ +where the left adjoint sends a sequence X• to its colimit: +c(X•): = Colim +n∈N Xn, +and the right adjoint sends a ∞-marked ∞-category X on the sequence +τ0(X) → · · · → τn(X) → . . . +4.9 Proposition. This adjunction induces a Quillen adjunction between the +lax-limit model structure and the inductive model structure where the left adjoint +preserves weak equivalences and fibrant objects. +31 + +Proof. The left adjoint c clearly preserves cofibrations. Secondly, because the +model structure on ∞-Cat∞ is ω-combinatorial, its fibrant objects are closed +under ω-filtered colimits, and because its factorization systems can be obtained +as ω-accessible functors its weak equivalences are closed under ω-filtered colimit +(this is shown for Quillen model structure as Proposition 7.3 of [11] and for left +semi-model structure as Proposition 7.7 of [13]). This implies that the functor +c preserves cofibrations and weak equivalences (as it is a filtered colimit) and +hence it preserves acyclic cofibrations. +4.10 Proposition. There is a left Bousfield localization of the model structure +on LimLaxn∈N ∞-Catn, called the limit structure, where X• is fibrant if and +only if it is fibrant in the lax-limit model structure and if for all integer n, +fn: Xn → τnXn+1 is a weak equivalence. Moreover, weak equivalences between +fibrant objects are pointwise equivalences. +According to our claim (see Remark 4.4) that the results of [6] or [12] can be +applied to left semi-model structures, the ∞-category obtained as the localiza- +tion of this Bousfield localization is equivalent to the limit of the ∞-categories +obtained as the localization of the ∞-Catn (with the τn functors as transitions). +We need to introduce certain constructions before proving the theorem: +4.11 Construction. Let k be any integer. We define LimLaxi∈k,k+1(∞-Cati, τi) +to be the category whose objects are triplet (X, X′, f: X → τk(X′)) where X and +X′ are respectively k-marked and (k + 1)-marked ∞-categories. By adjunction, +these objects are in bijection with sequences: +X +f−→ X′ +where X and X′ are respectively k-marked and (k + 1)-marked ∞-categories. +There is an adjunction +LimLaxi∈k,k+1(∞-Cati, τi) +LimLaxi∈N(∞-Cati, τi) +U +r +⊣ +where the left adjoint U sends X → Y to the sequence +∅ → ... → ∅ → X +f−→ Y → Y → · · · → Y → . . . +while the right adjoint sends X• to +Xk +f−→ Xk+1. +4.12 Remark. Given i: A ֌ B a cofibration between cofibrant objects in a +(possibly left semi-) model category, we call the (or a) homotopy codiagonal +of i the cofibration B � +A B ֌ IAB where IAB is some choice of a relative +cylinder object for this cofibration. Given that this homotopy codiagonal is +itself a cofibration between cofibrant objects this construction can be iterated. +When constructing a left Bousfield localization at a set S of cofibration between +cofibrant objects, the fibrant objects of the localization are exactly the objects +that have the right lifting property against all arrows in S as well as all their +iterated homotopy codiagonal. +This is a fairly standard result on Bousfield +localization, which is proved for weak model structure (in particular for left +semi-model structure) in [15] (See the proof of Theorem 7.3 and Remark 7.6). +32 + +4.13 Construction. Given A ֌ B a cofibration between cofibrant objects +in ∞-Catk, we can see it as a cofibrant object of LimLaxi∈k,k+1(∞-Cati, τi). +Given a choice of a relative cylinder object IAB for A → B, we have a cofibration +in LimLaxi∈k,k+1(∞-Cati, τi) given by the square: +A +B +B +IAB +A key observation for the proof below is that there is a way to choose a +homotopy codiagonal for this map that is also of this form. +Indeed to construct such a codiagonal map, one needs to construct a (cofi- +bration, weak equivalence) of a map (in the vertical direction): +B � +A B +IAB � +B IAB +B +IAB +One can observe that the horizontal map B � +A B ֌ IAB � +B IAB is already +a relative cylinder object for A ֌ B, so that one can first factorize the leftmost +map +B � +A B +IAB � +B IAB +IAB � +B IAB +B +IAB +∼ +One then forms the pushout P: +B � +A B +IAB � +B IAB +IAB � +B IAB +P +B +IAB +⌜ +And the map P → IAB can then be factored in a cofibration followed by a +weak equivalence. +33 + +B � +A B +IAB � +B IAB +IAB � +B IAB +P +W +B +IAB +⌜ +∼ +Which gives a relative cylinder object, and hence a homotopy codiagonal for +our map of the form: +B � +A B +IAB � +B IAB +IAB � +B IAB +W +But one can see that the object W we constructed above is itself a relative +cylinder object for the map B � +A B → IAB � +B IAB and hence this homotopy +codiagonal is again of the desired form. +Proof of Proposition 4.10. As in Construction 4.13, given a cofibration A ֌ +B in ∞-Catk we consider the cofibration {U(A → B) → U(B → IAB)} in +LimLaxi∈N(∞-Cati, τi). We call Ik the set of cofibrations obtained for A ֌ B +a generating cofibration of ∞-Catk. We claim that a fibrant object (Xi, fi) +of LimLaxi∈N(∞-Cati, τi) has the right lifting property against all maps in Ik +if and only if the map fk: Xk → τkXk+1 is a weak equivalence, and that this +also implies that (Xi, fi) has the right lifting property against all maps of the +form {U(A → B) → U(B → IAB)} when A ֌ B is an arbitrary cofibration in +∞-Catk. +For this, we will use the criterion that in any model category a morphism +between fibrant objects f: X → Y is a weak equivalence if and only if for every +generating cofibration A → B, there is, in the category of arrows, a lifting in all +diagram of shape: +(A → B) +(X +f−→ Y ) +(B → IAB) +where IAB is a relative cylinder object for the cofibration A → B. +This is +proved for weak model categories in Appendix A.2 of [14], see Theorem A.2.6 +and Remark A.2.7. +Now, an object (Xi, fi) of the lax-limit has the right lifting property against +morphisms of Ik if and only if (Xk → Xk+1) has the right lifting property against +(A → B) → (B → IAB). +This last condition is, by adjunction, equivalent +to asking that fk: Xk → τkXk+1 has the right lifting property against (A → +B) → (B → IAB), which is, accorded to the criterium, equivalent to ask that +fk: Xk → τkXk+1 is a weak equivalence. And conversely, if fk: Xk → τkXk+1 is +34 + +an equivalence, then it has the right lifting property against (A → B) → (B → +IAB) for any cofibration A ֌ B and any relative cylinder object. +We then defined the limit model structure as the left Bousfield localization +of the lax-limit model structure by all set Ik (for all values of k). The existence +of this localization is asserted by theorem 7.3 of [15]. +It remains to show that the fibrant object of this localization are the fibrant +object satisfying this condition that fk: Xk → τkXk+1 is a weak equivalence. As +discussed in Remark 4.12, the fibrant object of this localization are the objects +that are fibrant in the lax-limit model structure on which have the left lifting +property against all maps in Ik and all their iterated homotopy codiagonal, but +by Construction 4.13, all these iterated homotopy codiagonals are of the form +U(A → B) → U(B → IAB) and hence, we the discussion above show that their +fibrant objects are exactly the objects such that the map fk: Xk → τkXk+1 is +an equivalence as claimed in the proposition. +4.14 Proposition. The adjunction of Definition 4.8 is a Quillen adjunction +between the limit model structure of Proposition 4.10 and the inductive model +structure. +Proof. We need to show that the adjunction of Proposition 4.14 passes to the +localization, which means that all morphisms of I are sent to trivial cofibrations +in ∞-Cat∞. This is immediate because +U(A → B) → U(B → IAB) +c +�−→ +B → IAB. +4.15 Theorem. The Quillen adjunction between the limit model structure of +Proposition 4.10 and the inductive model structure is a Quillen equivalence. +This induces an equivalence of categories: +ho lim +n∈N ∞-Catn ∼= ho ∞-Cat∞. +Proof. Because the left adjoint preserves all weak equivalences and fibrant ob- +jects, we have to show that for every fibrant ∞-marked ∞-category X, and for +every cofibrant and fibrant sequence X• we have two weak equivalences: +cτX → X +and +X• → τcX•. +The first one is immediate because +X ∼= Colim +n∈N τnX. +Let X• be a cofibrant and fibrant object of the limit model structure. Be- +cause X• and τcX• are fibrant, the second comparison morphism is a weak +equivalence, if and only if for all of k, Xk → Colimn∈N τk(Xn) is a weak equiv- +alence. In order to show this, consider the following diagram: +X0 +... +Xk +Xk +... +Xk +... +X0 +... +Xk +τk(Xk+1) +... +τk(Xn) +... +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +∼ +35 + +where all the vertical morphisms are weak equivalence. Because the left adjoint +c preserves weak equivalence, this induces a weak equivalence: +Xk +∼ +−→ Colim +n∈N τk(Xn) ∼= Colim +n∈N τk(Xn). +4.2 +Comparison with the folk model structure on ∞-Cat +Following [19, Definition 6], we can also give a coinductive definition of invert- +ibility of arrows in an ∞-category. The notion is called “weakly invertible” in +[19]. Explicitly, we define by coinduction: +4.16 Definition. We say that an n-arrow f: a → b in a (marked) ∞-category +is coinductively invertible if there exist g: b → a and two coinductively invertible +(n + 1)-arrows α: f#n−1g → 1a and β: g#n−1f → 1b. +Of course, this implies that there are also (n+1)-arrows α′: 1a → f#n−1g and +β′: 1b → g#n−1f, and then (n + 2)-arrows α#nα′ → 11a , 1f#n−1g → α′#nα, +. . . then followed by several (n + 3)-arrows and so on up to infinity. +4.17 Lemma. Let X be a ∞-category, and M the set of coinductively invertible +arrows. The set M satisfies the two following properties: +(1) M = M. +(2) For all c: a → b in M, a ∈ M ⇔ b ∈ M. +Proof. The first point is the third and the fourth point of example 1.1.9 of [20], +and the second one is a consequence of proposition 1.1.10 of loc cit. +4.18 Proposition. If X is a fibrant m-marked ∞-category, all marked arrows +in X are coinductively invertible in the underlying ∞-category. +Proof. This is a direct consequence of Lemma 3.17. +4.19 Proposition. Let X be an ∞-category and M the set of coinductively in- +vertible arrows. The marked ∞-category (X, M) is then fibrant in the inductive +model structure. +Proof. We show that (X, M) verifies the conditions of Proposition 3.19. +By +definition, marked arrows of (X, M) have inverses in the sense of definition +3.11, and the first condition is fulfilled. +The second condition is implied by +Lemma 4.17. +4.20 Definition. Let G1 be the ∞-category obtained in the factorization of +D1 → D0 in a cofibration k1: D1 → G1 followed by a trivial fibration t1: G1 → +D1 of the folk model structure. +We then define Gn: = Σn−1G1 and kn: = +Σn−1k1: Dn → Gn, tn: = Σn−1t1: Gn → Dn−1. Let us recall that the definition +of the functor Σn−1 is given in Definition 2.4. As the suspension preserves triv- +ial fibrations and cofibrations, the pair (kn, tn) is a factorization of Dn → Dn−1 +into a cofibration followed by a trivial fibration. +36 + +4.21 Proposition. Let X be a ∞-category, and f an n-arrow of X. There +exist a lifting in the following diagram : +Dn +X +Gn +f +if and only if f is weakly invertible. +Proof. This is a reformulation of lemma 18 of [19]. +4.22 Definition. The coinductive model structure is the left Bousfield local- +ization of the model structure on ∞-Cat∞ by the set of morphisms: +{(Gn, ∅) → Dn−1, n ∈ N∗} +4.23 Remark. Remark that if we define ˜ +Gn: = πn−1(Gn, ∅), the sequence +(Gn, ∅) +pn +−→ ˜ +Gn +˜ +kn +−→ Dn−1 +is a factorization in a cofibration followed by a trivial fibration in the inductive +model structure. Using the terminology of [15], we will say that the cofibration +pn represents the morphism (Gn, ∅) → Dn−1. As we can see in the construction +of the left Bousfield localization provides in the proof of the theorem 7.3 of +op cit, a marked ∞-category X is fibrant in the coinductive model structure +if and only if X is fibrant in the inductive model structure and has the right +lifting property against morphisms kn and iterated homotopy codiagonal of kn +for all n > 0. +4.24 Proposition. Let X be a fibrant ∞-marked ∞-category in the inductive +model structure. Then X is fibrant in the coinductive model structure if and only +if marked arrows are exactly the coinductively invertible arrows of the underlying +∞-category. +Proof. Suppose first that X is fibrant in the coinductive model structure and +let f be a coinductively invertible arrow of the underlying ∞-category. +By +proposition 4.21, this corresponds to a morphism f: (Gn, ∅) → X. As remarked +in 4.23, X as the right lifting property against kn, which implies that f can +be lifted by πn−1Gn. That shows that f is marked. Moreover, the proposition +4.18 states that all marked arrows are coinductively invertible. This shows that +marked arrows exactly correspond to coinductively invertible ones. +For the other direction, suppose that X is a marked ∞-category, fibrant +is the inductive model structure, whose marked arrows are the coinductively +invertible ones. We want to show that X is fibrant in the coinductive model +structure. Accorded to Proposition 4.19, X is fibrant is the nonlocalized model +structure. We then have to show for all integers n > 0, X has the left lifting +property against kn and iterated homotopy codiagonal of kn. Remarks now that, +as +˜ +Gn +� +(Gn,∅) ˜ +Gn = +˜ +Gn, all the iterated homotopy codiagonals are identities. +To conclude, it is enough to show that X has the left lifting property against +morphisms kn for n > 0, which is obvious by assumption. +37 + +4.25 Theorem. The subcategory of fibrant objects of the coinductive model +structure on ∞-Cat∞ is isomorphic to ∞-Cat. Moreover, a morphism between +fibrant ∞-marked ∞-categories is a weak equivalence if and only if the corre- +sponding morphism in ∞-Cat is a weak equivalence of the folk model structure. +We then have +hocoind(∞-Cat∞) ∼= hofolk(∞-Cat). +Proof. Let Fib(∞-Cat∞) be the subcategory of fibrant objects of the coinduc- +tive model structure on ∞-Cat∞. We define φ: Fib(∞-Cat∞) → ∞-Cat to be +the functor that forgets the marking. Proposition 4.24 implies that this functor +is an equivalence of category. +Eventually, a morphism f: X → Y between fibrant ∞-marked ∞-categories +is a weak equivalence if and only if, every diagram in the category of arrows of +shape: +(∂Dn → Dn) +(X +f−→ Y ) +(Dn → ˜Gn+1) +admit a lifting. This is equivalent to asking that every diagram in the category +of arrows of ∞-Cat of shape +(∂Dn → Dn) +(X +φ(f) +−−−→ Y ) +(Dn → Gn+1) +admit a lifting, which is equivalent to ask that φ(f) is a weak equivalence. +Note that if m < ∞, then every m-marked ∞-category which is fibrant for +the saturated inductive model structure is also fibrant for the coinductive model +structure, hence when restricting the previous theorem to m-marked objects for +m < ∞, we no longer need to move to the coinductive model structure and we +directly obtain the following: +4.26 Corollary. If m < ∞, the full subcategory of fibrant objects of the satu- +rated inductive model structure on ∞-Catm is isomorphic to the subcategory of +∞-Cat composed of ∞-category whose arrow of dimension strictly superior to +n are coinductively invertible. Moreover, a morphism between fibrant m-marked +∞-categories is a weak equivalence if and only if the corresponding morphism +in ∞-Cat is a weak equivalence of the folk model structure. +4.3 +The folk model structure vs the limit of the π-tower +In this section, we will compare the folk model structure with the limits of the +tower of π functor as considered in Section 4.1. We will show that they are not +equivalent by building a morphism that is not an equivalence of the folk model +structure, but become invertible in the limit of the π-tower. It seems unlikely +38 + +that the limit of the π-tower is actually equivalent to any localization of the +inductive model structure, though we have not been able to give an argument +general enough to show this. +More precisely, we will show: +4.27 Proposition. There exist a morphism f between cofibrant ∞-marked ∞- +category such that +(1) f is not a weak equivalence of the coinductive model structure on ∞- +marked ∞-categories defined in Definition 4.16, +(2) for all integer n, πnf is a weak equivalence of the saturated inductive model +structure on n-marked ∞-categories defined in Theorem 3.31. +As an immediate consequence, we get: +4.28 Corollary. The (∞, 1)-functor from the (∞, 1)-categories associated to +the folk model structure to the limit of the (∞, 1)-categories associated to the +saturated inductive model structure for the n-marked defined in Theorem 3.31, +and induced for all n by the left Quillen functor πn: ∞-Cat∞ → ∞-Catn, is +not an equivalence. +4.29 Construction. Let E1 denote the following 2-polygraphs: +b +b +a +a +f +Ib +Ib +and En: = Σn−1E1. +Let us recall that the definition of the functor Σn−1 is +given in Definition 2.4. When writing Dn → En, we will always consider the +morphism representing the n-arrow Σn−1f. We define by induction a sequence +of polygraphs (Pn)n∈N. We set P0: = D1 and Pn as the pushout: +� +(Pn)n+1 Dn+1 +Pn +� +(Pn)n+1 En+1 +Pn+1 +⌟ +Informally, taking a pushout along Dn+1 → En means freely adding a left +and a right inverse to an arrow f (except there is no marking yet) and so Pn+1 +is constructed by freely adding left and right inverses to all (n + 1)-arrows of +Pn. +When writing D1 → Pn, we will always consider the morphisms representing +the 1-arrow P0 → Pn. Finally, for n ∈ N ∪ {∞} we define Cn and Dn as the +following pushouts: +� +k 0, gn is a (n + k)-generator and g0 = g, +(2) for n > 0, gn appears in the decomposition of the source of gn+1. +Proof. We show this result by coinduction on k. Suppose the result is true for +all (k + 1)-arrows, and let f: a → b be a coinductively invertible k-arrow, and +g a k-generator appearing in the decomposition of f. There exists a k-arrow +f ′: b → a and a coinductively invertible (n + 1)-arrow α: f#k−1f ′ → Ia. As g is +a k-generator appearing in the decomposition f#k−1f ′ (which is the source of +α), we can find a (k + 1)-generator β appearing in the decomposition of α and +such that g is in the decomposition of the source of β. As α is coinductively +invertible, one can continue this process coinductively starting from β to build +a sequence of generators (βn)n∈N satisfying the desired property. We then set +g0: = g, and gn: = βn−1. This sequence also satisfies the desired property. +4.31 Corollary. The ∞-categories C∞ and D∞ have no coinductively invertible +arrow except identities. +Proof. We will show this assertion for C∞, the proof for D∞ is essentially the +same. We proceed by contradiction: let f be a non-identity coinductively in- +vertible k-arrow of C∞. As f is not an identity there should be at least one +k-generator g appearing in its decomposition. As C∞ is a polygraph one can +apply the previous lemma and obtain a sequence (gm)m∈N of generators of C∞. +Eventually shifting the sequence one can freely assume that g0 is of dimension +> 1. The generators of C∞ are obtained by gluing the generators of Pn for all +n at the unique generator of D1, so this g0 will have to be in one of the Pn, it +then follows by induction that all the gm are in the same Pn, but this leads to +a contradiction as the dimension of the generator of Pn is bounded above. +4.32 Corollary. The marked ∞-categories C♭ +∞ and D♭ +∞ are fibrant in the coin- +ductive model structure. +Proof. It is immediate that C♭ +∞ and D♭ +∞ are prefibrant (as in Definition 3.12) +and hence they are fibrant in the indutive model structure. Hence by Propo- +sition 4.24 we only need to check that all their coinductively invertible arrows +are marked, but by the previous corollary, only their identity arrows are coin- +ductively invertible, which concludes the proof. +40 + +4.33 Lemma. The morphism C∞ → D∞ is not a weak equivalence of the +coinductive model structure. +Proof. As both C∞ and D∞ are fibrant in the coinductive model structure, +which is a Bousfield localization of the inductive model structure, this map is a +coinductive equivalence if and only if it is an inductive equivalence. Hence one +can test whether it is an equivalence using Definition 3.25 and Proposition 3.26, +but this map fails to satisfy condition (1) of Definition 3.25, as the 1-arrow of +C∞ corresponding to the vertical map D1 → C∞ is not marked and send to an +identity arrow (hence marked) in D∞. +Let us now show the second point, namely that for any integer n, πnC∞ → +πnD∞ is a weak equivalence. +4.34 Lemma. For all n > 0, the map πn+1En+1 → πnEn+1 is a weak equiv- +alence in saturated inductive model structure for n-marked ∞-category (Theo- +rem 3.31) . +Proof. One should first note that this map is an isomorphism of the underlying +∞-categories and only corresponds to marking all the n-arrows. In particular, it +is a cofibration. Moreover, πn+1En+1 is cofibrant as its underlying ∞-category +is a polygraph. Using the characterization of fibrant objects in the saturated +inductive model structure (see Lemma 3.30 and Theorem 3.31), one easily sees +that fibrant objects have the left lifting property against πn+1En+1 → πnEn+1. +As lifts against these morphisms are unique if they exist, we deduce that any +fibration between fibrant objects has the left lifting property against them. It +follows that this map is an acyclic cofibration and hence a weak equivalence. +4.35 Lemma. For all n, πnPn → D0 is a weak equivalence of the saturated +inductive model structure. +Proof. We define ˜Pn+1 as the pushouts: +� +(Pn)n+1 Dn+1 +Pn +� +(Pn)n+1 πnEn+1 +˜Pn+1 +⌟ +As weak equivalences between cofibrant objects are stable by pushouts, the +previous lemma and the fact that (Dn+1, {en+1}) → πnEn+1 is an acyclic cofi- +bration, imply that all arrows labeled by ∼ in the following diagrams are weak +equivalences: +� +(Pn)n+1 Dn+1 +Pn +� +(Pn)n+1 Dn+1 +Pn+1 +� +(Pn)n+1 πn+1En+1 +πn+1Pn+1 +� +(Pn)n+1(Dn+1, {en}) +πnPn +� +(Pn)n+1 πnEn+1 +˜Pn +� +(Pn)n+1 πnEn+1 +˜Pn +∼ +∼ +∼ +∼ +⌟ +⌟ +⌟ +⌟ +41 + +By two out of three, πn+1Pn+1 → D0 is a weak equivalence if and only if +˜Pn+1 → D0 is, and so if and only if πnPn → D0 is. It remains to show the case +n = 0 which is obvious. +4.36 Lemma. For all n, the induced morphism πnC∞ → πnD∞ is a weak equiv- +alence of the saturated inductive model structure on n-marked ∞-categories. +Proof. Using the last lemma and as weak equivalences between cofibrant objects +are stable by pushout, we have a diagram where all arrows labeled by ∼ are +weak equivalences +� +k∈N πnD1 +� +k∈N πnPk +(� +k0 Xn of simplex of positive dimension called thin simplexes +that includes all degenerate simplexes. +A morphism of m-stratified simplicial sets is a morphism between the under- +lying simplicial sets that sends thin simplexes to thin simplexes. The category +of m-stratified simplicial sets is denoted Strat. +The join is an important operation for simplicial sets, which is defined on +representable by the formula +∆[n] ⋆ ∆[m]: = ∆[n + m + 1] +and extended by colimits to any pair of simplicial set +X ⋆ Y : = +Colim +([n],[m])∈∆×∆ +� +Xn×Ym +∆[n] ⋆ ∆[m]. +See for example [22] Definition 1.2.8.1 and below. We now defined it for stratified +simplicial sets as follows: +42 + +4.38 Definition. If (X, M) and (Y, N) are two stratified simplicial sets, we +define M ⋆ N as the set of simplices of X ⋆ Y of the form x ⋆ y where either x +or y is thin. We then define +(X, M) ⋆ (Y, N): = (X ⋆ Y, M ⋆ N). +4.39 Definition. We define several marking on ∆[n]: +(1) ∆[n]t. The top n-simplex is thin. +(2) ∆k[n]. All simplices that include {k − 1, k, k + 1} ∩ [n] are thin. +(3) (∆k[n])′. All simplices that include {k − 1, k, k + 1} ∩ [n], together with +the (k − 1)-face and the (k + 1) face are thin. +(4) (∆k[n])′′. All simplices that include {k − 1, k, k + 1} ∩ [n], together with +the (k − 1)-face, the k-face and the (k + 1) face are thin. +(5) ∆[3]eq. All simplices of dimension strictly higher than 2, together with +[0, 2] and [1, 3] are thin. +(6) ∆[n]♯. All simplices are thin. +4.40 Definition ([24, Definition 1.19]). An elementary anodyne extension is +one of the following: +(1) The complicial horn inclusions are the regular extensions +Λk[n] → ∆k[n], n ≥ 1, n ≥ k ≥ 0. +(2) The complicial thinness extensions: +(∆k[n])′ → (∆k[n])′′, n ≥ 2, n ≥ k ≥ 0. +(3) The saturation extensions: +∆[n] ⋆ ∆[3]eq → ∆[n] ⋆ ∆[3]♯, n ≥ −1. +(4) The m-triviality extensions: +∆[n] → ∆[n]t, n > m +4.41 Remark. In the case where m = ∞, there is no m-triviality extension. +4.42 Definition. A (saturated) m-complicial set is a marked simplicial set +having the right lifting property against all elementary anodyne extensions. +As demonstrated in [21], m-complicial sets are a model for (∞, m)-categories. +For example, 0-complicial sets and 1-complicial sets are essentially the same +as Kan complexes and quasicategories respectively. The word saturated refers +to the fact that (as in [24]) we have included the “saturation extension” as +part of our elementary anodyne extensions. These are not always included and +play a role similar to the saturated localization of the inductive model structure +considered in Section 3.5. See also [26] for a more general discussion of saturation +for complicial sets. +43 + +4.43 Theorem (Verity [30], Riehl [26], Ozornova-Rovelli [24]). There is a model +structure on Strat where cofibrations are all monomorphisms, and acyclic cofi- +brations are generated by elementary anodyne extension. Fibrant objects of this +structure are the (saturated) m-complicials sets. We denote Stratm the category +Strat endowed with this model structure. +We will use the join to define the adjunction between stratified simplicial +sets and marked ∞-categories. +4.44 Definition. Let C and D be two marked ∞-categories. The joint of C +and D, noted C ⋆ D, is the colimit of the following diagram: +A → {0} → B � A → {1} → B +A → D1 → B +A � B +A ⋆ B +⌟ +As noted in proposition 3.3.11 of [2] at the level of ∞-categories, this is the +usual join of ω-categories, as defined in paragraph 6.30 of [4]. This operation is +then associative. +4.45 Proposition. Let X → Y be a cofibration and K → L an acyclic cofibra- +tion. Morphisms +K ⋆ Y +� +X⋆K +L ⋆ X → L ⋆ Y +and +Y ⋆ K +� +K⋆X +X ⋆ L → Y ⋆ L +are acyclic cofibrations. +Proof. Consider the following diagram: +L � Y +K → ∂D1 → Y ∪ L → ∂D1 → X +K → D1 → Y ∪ L → D1 → X +L � Y +L → ∂D1 → Y +L → D1 → Y +Taking colimit of the lines, this induces a comparison morphism: +K ⋆ Y +� +X⋆K +L ⋆ X → L ⋆ Y. +Proposition 7.5 of [3] implies that this morphism is a cofibration. Lemma 2.37 +implies that vertical morphisms of the previous diagram are weak equivalence. +Furthermore, these colimits are homotopy colimits, the comparison morphism +is then a weak equivalence, and then an acyclic cofibration. We proceed analo- +gously for the second morphism. +4.46 Definition. The terminal category 1 has a monoid structure for this +operation. The multiplication 1 ⋆ 1 → 1 is the unique morphism to the terminal +∞-category. +By the universal property of the category ∆, this induces a cosimplicial +object |−|: ∆ → ∞-Cat∞ where +|∆[n]|: = 1 ⋆ 1 ⋆ ... ⋆ 1. +44 + +The ω-category |∆[n]| is traditionally called the nthoriental. We denote |−|: Sset → +∞-Cat∞ the extension by colimits of this cosimplicial object. For all n, |∆[n]| +is an n-polygraph that admits only one n-generator. If M in a marking for K, +we denote |M| the set of arrows obtained as composition: +Dn → ∆[n] +|v| +−→ K +where the left morphism corresponds to the top cell of the nth orientals, and +the right morphism is in M. We can now extend the realization to stratified +simplicial sets: +|−|: +Strat +→ +∞-Catm +(K, M) +�→ +(|K|, |M|) +This functor is cocontinuous, and induces an adjunction: +Strat +∞-Catm +| | +N +⊣ +The right adjoint is called the stratified Street nerve. By construction, if K and +L are two stratified simplicial sets, we have |K ⋆ L| = |K| ⋆ |L|. +4.47 Remark. In the case m = ∞, this adjunction model the forgetful functor +from strict ∞-categories to weak ∞-categories (given by the stratified Street +nerve N). +The left adjoint corresponds to the “strictification functor” that +sends a weak ∞-category to a strict ∞-category in a universal way. +4.48 Proposition. The stratified nerve preserves fibrant objects. +Proof. Suppose first that m < ∞ and let (X, M) be a fibrant m-marked ∞- +category for the saturated inductive model structure. +According to Corol- +lary 4.26, M consist of coinductively invertible arrows of X, and N((X, M)) +is equal to the stratified simplicial set associated to the Street nerve of X de- +fines in [20, D´efinition 5.2.1]. Theorem 5.2.12 of op.cit. then imply that the +stratified Street nerve sends fibrant objects of the saturated inductive model +structure on ∞-Catm to an m-complicial sets. +Now, let C be a fibrant ∞-marked ∞-category for the saturated inductive +model structure. As the stratified nerve preserves directed colimits, there is an +isomorphism +N(C) ∼= Colim +n∈N N(τnC) +For all n, τnC is fibrant for the saturated inductive model structure for n- +marked ∞-categories, and N(τnC) is then a fibrant of the model structure for n- +complicial sets. As the model structure for ∞-complicial sets is ω-combinatorial, +fibrant objects are stable by directed colimits, and N(C) is fibrant. +4.49 Lemma. The realization functor sends complicial horn inclusion to acyclic +cofibration of the saturated inductive model structure for m-marked ∞-categories. +Proof. The complicial horn inclusion Λ1[2] → ∆[2]1 corresponds to the following +inclusion of marked ∞-categories: +• +• +• +• +• +• +∼ +45 + +which obviously is an equation. The two complicial horn inclusions Λ0[2] → +∆0[2] and Λ2[2] → ∆2[2] are respectively equal to eq +• +• +• +1,1 +and eq +• +• +• +1,1 . +The +realization functor commutes with the join. Furthermore, we can see that for +all 0 < k < n, we have: +∆k[n] = ∆[k − 2] ⋆ ∆1[2] ⋆ ∆[n − k − 2] +and Λk[n] is the sub-object: +∂∆[k − 2] ⋆ ∆1[2] ⋆ ∆[n − k − 2] +∪ +∆[k − 2] ⋆ Λ1[2] ⋆ ∆[n − k − 2] +∪ +∆[k − 2] ⋆ ∆1[2] ⋆ ∂∆[n − k − 2]. +Proposition 4.45 then implies that Λk[n] → ∆k[n] is an equation. We proceed +analogously for the case k = 0 and k = n. +4.50 Theorem. The strictification functor and the stratified Street nerve form +a Quillen adjunction between the model structure for m-complicial sets and the +inductive model structure on ∞-Catm. +Proof. Because of Lemma 4.49, it just remains to show that complicial thinness +extensions, saturation extensions, and m-triviality extensions are sent to acyclic +cofibrations. Let i be such a morphism. According to Proposition 4.48, any +fibrant object of the saturated inductive model structure has the right lifting +property against |i|. As |i| is an identity on the underlying ∞-category, lifts +against it are unique if there exist. This implies that any morphism between +fibrant objects has the right lifting property against |i|, and this morphism is +then an acyclic cofibration. This concludes the proof. +Finally, we can use this to generalize the results from [20]: The stratified +Street nerve: +N: ∞-Cat → sSetm +introduced in [20], is exactly the stratified Street nerve N of the present paper +combined with the fully faithful inclusion ∞-Cat ⊂ ∞-Catm constructed in +Section 4.2, which makes all coinductively invertible arrow marked. Hence: +4.51 Proposition. Let f: X → Y a fibration (resp. a trivial fibration, resp. +weak equivalence) of the canonical model structure on ∞-Cat, then its stratified +Street nerve N(f): N(X) → N(Y ) is a fibration (resp. a trivial fibration, resp. +a weak equivalence) in the Verity model structure on sSetm. +The main result of [20] corresponds to the special case of preservation of +fibrant objects. +Note, that in particular the proposition shows that the stratified Street nerve +from [20], while not being a right Quillen functor, is still a morphism of Brown +categories of fibrant objects, and so it does defines a limit preserving functor on +the corresponding associated ∞-categories. +Proof. As the stratified Street nerve N: ∞-Catm → sSetm is a right Quillen +functor, it preserves fibrations and trivial fibrations, as well as weak equivalences +between fibrant objects. 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