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      Questions tagged [higher-category-theory]

      For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

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      6
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      0answers
      98 views

      The model category structure on $\mathbf{TMon}$

      I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer. I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...
      3
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      1answer
      135 views

      Proving a Kan-like condition for functors to model categories?

      I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
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      votes
      0answers
      47 views

      Concerning the definition of a 2-crossed module

      Question: Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...
      2
      votes
      1answer
      200 views

      Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

      Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
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      1answer
      149 views

      Existence of pointwise Kan extensions in $\infty$-categories

      This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant ...
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      97 views

      Classifying Objects for Fibrations Defined by a Lifting Property

      I have been studying weak factorization systems for their use in model categories. I am trying to use these to abstract away from a common phenomenon underlying fibrations. In brief, it seems as ...
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      votes
      0answers
      79 views

      Simplicial sets of categories as models for $(\infty,1)$-categories [duplicate]

      Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise. In my understanding, there are several models for $(\infty,...
      4
      votes
      0answers
      112 views

      Derived weight filtration on motivic Galois representations

      Thanks to modern techniques (such as the pro-etale site), we can now understand etale (co)homology of varieties and motives as "genuinely" derived (e.g. DG) Galois-equivariant objects. I'm looking for ...
      6
      votes
      3answers
      342 views

      Definition of $E_n$-modules for an $E_n$-algebra

      The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more ...
      7
      votes
      1answer
      388 views

      Categorical Significance of Fibrations

      It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
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      91 views

      Tensor schemes “with relations”

      In "The geometry of tensor calculus I," Joyal and Street introduce the notion of a tensor scheme, a set of abstract objects, together with formal morphisms between words in this set. They then prove ...
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      0answers
      98 views

      CoCartesian vs. locally CoCartesian fibrations

      Say $\pi: C\to J$ is an inner fibration of $\infty$-categories. Then "morally", $\pi$ corresponds to a diagram indexed by $J$ in the "category of categories with correspondences", and if $\pi$ is ...
      7
      votes
      1answer
      408 views

      Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie

      I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...
      6
      votes
      1answer
      157 views

      Tricategorical coherence

      Why does coherence begin to matter at the tricategorical level? It is well known that every weak $2$-category is equivalent to a strict $2$-category, with the equivalence essentially given by (unless ...
      15
      votes
      1answer
      521 views

      Natural examples of $(\infty,n)$-categories for large $n$

      In Higher Topos Theory, Lurie argues that the coherence diagrams for fully weak $n$-categories with $n>2$ are 'so complicated as to be essentially unusable', before mentioning that several dramatic ...

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