diff --git "a/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/load_file.txt" "b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DdFJT4oBgHgl3EQfBSxP/content/tmp_files/load_file.txt" @@ -0,0 +1,1749 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf,len=1748 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='11424v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The canonical model structure on strict ∞-categories can be recovered as a left Bousfield local- ization of this model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show that there ought to be at least three different notions of (∞, ∞)- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 The street nerve as a right Quillen functor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 The two (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=') notions of (∞, ∞)-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 2 ∞-categories and marked ∞-categories 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 ∞-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Marked ∞-categories .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Tensor product of m-marked ∞-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5 The saturated localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 27 4 Comparison with other model structures 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Truncation functors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Comparison with the folk model structure on ∞-Cat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 The folk model structure vs the limit of the π-tower .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 Complicial sets and stratified Street nerve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 42 1 1 Introduction In the present paper, we introduce (in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2) a category ∞-Catm of “m- marked (strict) ∞-categories” for m ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The objects of ∞-Catm are strict ∞-categories, with, similarly to stratified simplicial sets, some arrows be- ing “marked”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The marked arrows are required to be closed under composition, and all identities arrows as well as all arrows of dimension > m are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' These two monoidal structures are meant to respectively be models for the“lax-Gray tensor product” and the “pseudo-Gray tensor product”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The cofibrations are the map that are cofibrations of the canonical model structure between the underlying ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Fibrations between fibrant objects are the “isofibrations” (as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Weak equivalences between fibrant objects are “equivalence of marked ∞- categories”(as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This model structure is a model for strict “(∞, m)-categories” where “invert- ibility” or arrows of dimension > m is taken in a weak sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The existence of this model structure is established in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We also consider two left Bousfield localizations of this model structure: The saturated inductive model structure, studied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5, whose fibrant objects are the ∞-categories in which every arrow which is weakly invertible up to marked arrows is also marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The coinductive model structure, studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2, whose fibrant objects are the ∞-categories in which every coinductively invertible (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='16) arrow is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This second localization is equivalent2 to the canonical model structure on ∞-categories from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=') 2Though not through a Quillen equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is not directly possible simply because this stratified Street nerve is not a right adjoint functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 we show that this functor is indeed a right Quillen functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The first named author has conjectured [17] that more generally this strictification functor should be conservative on weak (∞, m)-categories for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This allows us to state a concrete version of this conjecture here: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The left Quillen functor | |: sSetm → ∞-Catm from Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 reflects weak equivalence between cofibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 The two (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=') notions of (∞, ∞)-categories C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='Schommer-Pries and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='Rezk have independently argued ([16]) that there should be more than one notion of weak (∞, ∞)-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 This allows to produce two different towers: (∞, 0)-Cat π0 ← (∞, 1)-Cat π1 ← (∞, 2)-Cat π2 ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' πn−1 ← (∞, n)-Cat πn ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (∞, 0)-Cat τ0 ← (∞, 1)-Cat τ1 ← (∞, 2)-Cat τ2 ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' τn−1 ← (∞, n)-Cat τn ← .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' and one can take the projective limit of either of these two towers to give a definition of what is an (∞, ∞)-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If one takes the limits of the π-tower then one can see that an arrow that is “coinductively” invertible (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='16) has to be considered invert- ible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' To clearly show that the two are different, one can for example consider the (∞, ∞)-category of cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In the limit of the τ-tower one can define it by taking Xn to be the (∞, n)-categories of cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In this (∞, ∞)-category, every arrow has a dual, so it follows from a result of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Hence, if one were trying to define Xn in the limit of the π-tower, it would be equivalent to an ∞-groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Using our model structure of marked strict ∞-category, we will make these two constructions formal in the context of strict ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' First, we will show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 that our model structure on ∞-Catm for m = ∞ corresponds to the limit of τ-tower as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 4 In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' However, we will also show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3, that the folk model structure is not equivalent to the limit of the π-tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2 ∞-categories and marked ∞-categories 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 ∞-categories A globular set is a presheaf on the globular category G: D0 D1 D2 D3 D4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 ∈ {+, −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We also denote by iǫ k the map Dk → Dn for k < n obtained by composing any string of arrow ending with iǫ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' These and the identity arrows are the only maps in the category G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X is a globular set, one denotes by Xn the set X(Dn) whose elements are called n-arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The map Xn → Xk induced by iǫ k: Dk → Dn is denoted by πǫ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (2) π− k (x#ny) = π− k (x) and π+ k (x#ny) = π+ k (y) whenever the composition is defined and k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (3) πǫ k(x#ny) = πǫ k(x)#nπǫ k(y) whenever the composition is defined and k > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (4) x#nIπ+ n x = x and Iπ− n x#nx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (5) (x#ny)#nz = x#n(y#nz) as soon as one of these is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (6) If k < n (x#ny)#k(z#nw) = (x#kz)#n(y#kw) when the left-hand side is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A morphism of ∞-categories is a map of globular sets commuting with both operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category of ∞-categories is denoted ∞-Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We thus have a morphism in: ∂Dn → Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that ∂D0 = ∅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In b ) is the n-times iterated unity of a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' of b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Moreover, ΣX inherits from X a structure of ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Eventually, for an integer n, we define the ∞-category ΣnX, called the n- suspension of X, as the n-times iterated suspension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Next, we define the notion of polygraphs, first introduced under the name “computads” by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Street in [28] for 2-categories, with the general notion being hinted at in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Here we will exploit that the category of polygraphs identifies with a (non-full) subcategory of ∞-Cat to give a shorter definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We refer to the references above for a more complete introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An n-polygraph is a polygraph whose generators are all of dimension ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='6 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='7 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='8 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Each arrow in a polygraph can be written as an iterated compos- ite of the generators (not necessarily in a unique way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will say that an n-generator appears in an n-arrow if it appears in any such expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='9 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It was first introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Crans in his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' thesis [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We refer to [1] for an introduction to this tensor product close to its original definition, and to [27] for a more modern account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The proof of the existence of this monoidal structure in [27] contains some gaps that have been fixed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It is easy to see from either of these definitions that Dn ⊗ Dm has a unique non-trivial arrow of dimension n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We recall from [3]: 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='10 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X and Y are polygraphs then X ⊗ Y is also a polygraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The generators of X ⊗ Y are the arrow of the form x ⊗ y where x and y are respectively generators of X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Its weak equivalences are a natural class of equivalence of ∞-categories that generalizes the equivalences of ordinary cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It was shown in [3] that this model structure is a monoidal model structure for the Gray tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Marked ∞-categories For the rest of the article, we fix an m ∈ N ∪ {∞} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='11 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' All arrows of dimension strictly superior to m are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If x and y are marked n-arrows and x#ky is defined, then x#ky is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A morphism of m-marked ∞-categories is a morphism between the under- lying ∞-categories that sends marked arrows to marked arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The category of m-marked ∞-categories is denoted ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that if m = ∞, then the second condition of the definition simply disappears, this is the main case we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If X is an ∞-category we denote by X# the m-marked ∞- category (X, X>0) where all arrows of positive dimension are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We denote by X♭ the m-marked ∞-category where only identity arrows and k-arrows for k > m are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='13 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For example X♭ = (X, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='14 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 8 This is easily shown by checking that the right-hand side has the universal property of the colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='15 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='16 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Conversely, this set of arrows contains M and all identities (as no n-dimensional arrows appear in their expression) and is closed under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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 →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Both are obtained by putting different markings on the Gray tensor product from Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For example, the lax-Gray tensor product D1 → D1 is C♭ 1 where C1 is the polygraph C1 = \uf8eb \uf8ec \uf8ed \uf8f6 \uf8f7 \uf8f8 while D1 ∼ D1 is the special m-marked polygraph (C1, D) where D only contains the unique 2 dimension generator of C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' So, unless m = 0 or m = 1, the two tensor products are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' At the derived or homotopy theoretic level, the pseudo-Gray tensor product should correspond to the cartesian product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The formal definition goes as follows 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='17 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that M → N and M ∼ N are not marking on X ⊗Y : they are not stable under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='37 that both make the category of m-marked ∞-categories into a monoidal closed category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order to show this, it is convenient to introduce the following notations: 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='18 Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='20 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X be an ∞-category and M, N two subsets of arrows of X then: M ∪ N = M ∪ N = M ∪ N = M ∪ N Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X, Y be two ∞-categories and M ⊂ X⩾0 and N ⊂ Y⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then: M ⊗ N = M ⊗ N = M ⊗ N = M ⊗ N Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will only show the equality M ⊗ N = M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The evident inclusion M ⊂ M implies M ⊗ N ⊂ M ⊗ N, so it is then enough to show that M ⊗ N ⊂ M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let K be the set of arrows k in X such that k ⊗ n ∈ M ⊗ N for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We need to show that K is closed by identity and composition to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If k = Ix, then k ⊗ n = Ix⊗n ∈ M ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let now k, k′ ∈ K of dimension n such that k#ik′ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Together these induced a map e: � Dn � Di Dn � ⊗Dm → X⊗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' � 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We have proved that k#ik′ ⊗ y ∈ M ⊗ N for all y ∈ N, hence k#ik′ ∈ K and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='22 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X, Y be two ∞-categories, M ⊂ X⩾0 and N ⊂ Y⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then we have M → N = M → N M ∼ N = M ∼ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Given the formula for M → N and M ∼ N from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='18, this is a direct consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='20 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='23 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X, Y, Z be three ∞-categories, M ⊂ X>0, N ⊂ Y>0 and P ⊂ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Then we have (M → N) → P = M → (N → P) (M ∼ N) ∼ P = M ∼ (N ∼ P) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We begin with the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let E: = (M ⊗ Y⩾0 ⊗ Z⩾0) ∪ (X⩾0 ⊗ N ⊗ Z⩾0) ∪ (X⩾0 ⊗ Y⩾0 ⊗ P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='20 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='24 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let X be an ∞-category, M ⊂ X>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The first equality is a straightforward application of the definition of →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the second case, we also use that all arrows of (D0)>0 ⊗ X>0 are identities and so all belong to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='25 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In both cases the forgetful functor to ∞-categories is monoidal and their unit is D♭ 0 = D# 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that D♭ 0 = D# 0 = (D0, ∅) as all arrows of D0 of dimension > 0 are identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the unit, let (X, M) be an m-marked ∞-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For the associativity isomorphism, let (X, M), (Y, N) and (Z, P) be three ∞- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='23 shows that these two marking on X ⊗ Y ⊗ Z, in the lax and the pseudo case, coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='26 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The pseudo and lax-Gray tensor product → and ∼ preserves colimits in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, as ∞-Catm is locally presentable, this immediately implies that both tensor products are closed monoidal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='14 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 The semi-model structure In this section, we will construct a left semi-model structure on the category ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow in ∞-Catm is said to be a cofibration if it has the left lifting property against all trivial fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='28 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='29 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A trivial fibration is a map that has both these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='30 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Indeed, transfinite compositions of pushouts by arrows in Ia only starting from the empty ∞- category exactly give all polygraphs with no markings on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Finally, it was shown in [23] that polygraphs are closed under retract in ∞-Cat, so they constitute all cofibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If f and g are two cofibrations in ∞-Catm then f ˆ → g and f ˆ ∼ g are both cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' By the usual properties of the corner-product, it is enough to check this when f and g are generating cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 13 The forgetful functor ∞-Catm → ∞-Cat is monoidal for both tensor prod- uct and preserves colimits, so it preserves the corner-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='32 Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We define I: = D♯ 1 = (D1, {e1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It is the ∞-category with two objects, e− 0 and e+ 0 and a marked arrow e1: e− 0 → e+ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We denote by j− and j+ the two maps D0 → I corresponding respectively to the two objects e− 0 and e+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This gives a diagram: D0 � D0 \u058c I → D0 Which will play the role of the interval object for our semi-model structure on ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='32, and i is one of the generating cofibrations as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that an arrow in ∞-Catm is an anodyne cofibrations if it has the right lifting property against all naive fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that a cofibration in ∞-Catm is acyclic if it has the lifting property against all naive fibrations between (naively) fibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that a map in ∞-Catm is a fibration if it has the right lifting property against all acyclic cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='34 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It immediately follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 that, as j+ is a cofibration, all maps of the form j+ ˆ ∼ i are cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, all trivial fibrations are also naive fibrations and all anodyne cofibrations are cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='35 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Acyclic cofibrations and fibrations form a cofibrantly gen- erated weak factorization system on ∞-Catm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is a direct application of the results of Section 4 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' All the claim in the proposition follows from this Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='36 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that replacing ˆ ∼ by ˆ → in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 would not change the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' As this applies to both the domain and the co-domain of j+ it follows that j+ ˆ ∼ i = j+ ˆ → i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Also, the reader should not be worried about the use of j+ in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 rather than j− or both j− and j+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='37 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If f is an anodyne (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' acyclic) cofibration and g is a cofibra- tion then f ˆ ∼ g and f ˆ → g are anodyne (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' acyclic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' To get the result for “anodyne cofibrations” it is enough to prove it for the generating anodyne cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let i be one of the generating cofibrations and f = j+ ˆ ∼ i′ be one of the generating anodyne cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We have f ˆ ∼ i = j+ ˆ ∼ (i ˆ ∼ i′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' As i′ ˆ ∼ i is a pushout of generating cofibrations i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' , ik by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 it follows that j+ ˆ ∼ (i ˆ ∼ i′) is a pushout of the j+ ˆ ∼ ik and hence is an anodyne cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If p is a (naive) fibration between fibrant objects then ⟨p/i⟩ is a naive fibration between fibrant objects hence a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The case of → works exactly the same considering the first half of Re- mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27 and the fibrations are as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This immediately follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Because of Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='31 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='37, tensoring by the interval object I of Construc- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='32 is a “strong Quillen functor” in the sense of section 6 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that to apply Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The first ones hold because the generating cofibrations are cofibrations between cofibrant objects and the second because that is how we defined acyclic fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='39 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='33 and whose cofibrations are as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The two classes of equivalence however coincide for arrows that are between fibrant or cofibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='40 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We do not know if ∞-Catm is actually a Quillen model cate- gory or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In the unmarked case, this follows from the fact that all objects are fibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' But that is no longer the case in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We conclude this section with the following lemma that will be useful later: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='41 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will show it is a retract of the map j+ ˆ ∼ in where in is the map ∂Dn → Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In order to achieve this, we will compute j+ ˆ ∼ in more explicitly using the description of D1 ⊗ Dn given in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 of [4] (see proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' #k−1(a ⊗ e+ k−1) π+(a ⊗ eǫ k) = (a ⊗ e− k−1)#k−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' #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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The last two equations are given by proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 of [4], though note that this reference is using a different convention than ours regarding the composition order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is well defined because a ⊗ en is a marked arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Next, we define a map p: I ∼ D♭ n → (Dn+1, {en+1}) by: p(aǫ 0 ⊗ eµ k) = eµ k if k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' p(aǫ 0 ⊗ en) = eǫ n p(a ⊗ eǫ k) = Ieǫ k if k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Finally, the composite p ◦ i send the arrow en+1 to p(a ⊗ en) = en+1 and hence is the identity of Dn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' #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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3 Equations and saturations in an m-marked ∞- category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This is achieved using the notion of “equations” in an ∞-categories introduced by the second named author in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Definitions of equations and saturations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (2) y is a marked arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (3) if n ≤ m, x is an unmarked arrow of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (4) The source of y admits a decomposition: π− n y = ln#n−1(ln−1#n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#1(l1#0x#0r1)#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In particular, x appears only once in π− n y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' (5) x does not appear in the target of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Right equations are defined in the exact same way except the source and target of y are exchanged in the last two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We say that (P, M) is an equation to mean that it is either a left or right equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='2 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Note that specifying the arrows x, y ∈ P is exactly the same as specifying the subpolygraphs ΛP ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' For this reason, we will often also call “equation” the map ΛP → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='3 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='4 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' A left saturation is a special marked polygraph (P, M) with arrows x and y satisfying the conditions of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1 except that x is a marked arrow and the target of y is a marked arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Right saturations are defined in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If P is a saturation, one denotes ΩP the special m-marked polygraph (P, M − {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='5 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='6 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let C be an m-marked ∞-category C and P a left equation (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' right equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' the source of y is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Solutions to an equation P are C are weakly unique if C has the right lifting property against P � ΛP P → Uni(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The equation P has unique solutions in C if the equation P has solutions in C and they are weakly unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' It will be useful to have a “coherent” version of Uni(P), noted Unicoh(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Remarks that in both cases, P � ΛP P → Unicoh(P) is an equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='7 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The morphism j+ ˆ ∼ in: = I ∼ ∂Dn � {1} ∼ Dn → I ∼ Dn is a left equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Indeed, let y be the top dimensional generator of I ∼ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' of [4] allows to give an explicit description of I ∼ Dn, which we recalled in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=')#n−1an and all the ak are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We denote it eq n : ΛEq n → Eq n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='8 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' which we denote sat n : ΩSat n → Sat n 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='9 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We define some left equations which play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In each case, k and n are integers with k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' In all equations above, the domain of the arrow is obtained by removing x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We will temporarily call the objects satisfying this characterization “prefibrant” (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12) and then show in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 that these are exactly the fibrant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='10 Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If a ∈ ΛP, we denote by a its image in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' So in general p(a) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='11 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let a be a (n + 1)-arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' An arrow is invertible if it has an inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='12 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This directly implies that if b and c: a → b are marked, so is b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' This notion is purely temporary: we will show in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='19 that an object is fibrant for the model structure of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='38 if and only if it is prefibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='13 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If C is prefibrant, then equations Eq k,n and Eq k,n have weakly unique solutions in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show the result by a decreasing induction on k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The initialization corresponds to k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let ν: a−1#na → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If b is marked so is x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We now show the weak unicity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let (¯x, ¯y) be another solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We then have a marked arrow: z: ¯x ν−1 −−→ a−1#na#n¯x ¯y−→ a−1 ∗ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The assertion for Eq n,n is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Suppose now the result is true for all k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We start by showing that solutions of Eq k,n and Eq k,n are weakly unique in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let (x, y: a#k−1x → b) be a solution of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let ν: a−1#na → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The unicity of solution of Eq k,n is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We show now that Eq k,n and Eq k,n have solutions in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let (x, y) be a solution of the equation (ν#0s)#kx (a−1#k−1b)#k(ν#k−1t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' y Moreover, we can find such x marked whenever b is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We then have (ν#0s)#kx = (a−1#k−1a#k−1x)#k(ν#k−1t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If b is marked, the arrow x that we produce is also marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' The existence of solution of Eq k,n is proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='14 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' If equations Eq k,n and Eq k,n have solutions in C, then all equations have solutions in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' Let P be a left equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' There is a decomposition of the source of y of the shape π− n y = ln#n−1(ln−1#n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#1(l1#0x#0r1)#1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content='#n−2rn−1)#n−1rn where for each i, li and ri are marked i-arrow in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFJT4oBgHgl3EQfBSxP/content/2301.11424v1.pdf'} +page_content=' We can then use the existence of solutions to Eq k,n and Eq k,n to get two sequences of arrows (xk)0