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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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      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
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      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
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      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
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      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
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      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
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      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 ...
      4
      votes
      1answer
      92 views

      Coherence theorem for tetracategories, weak $n$-categories

      Is there a coherence theorem/conjecture for tetracategories (weak $4$-categories)? Todd Trimble mentions in his notes on tetracategories that his pasting definitions are essentially unambiguous due ...
      3
      votes
      1answer
      332 views

      Composition map in $\infty$-categories

      Let $\mathcal{C}$ be an $\infty$-category, and let $u:x\rightarrow y$ be an edge. It seems reasonable to say that: The map $Map_{\mathbb{C}[\mathcal{C}]}(a,x)\rightarrow Map_{\mathbb{C}[\mathcal{C}]}(...
      2
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      1answer
      121 views

      On the Group Structure of Morphism Set of a Strict 2-Group

      The standard definition of a strict 2-group says that it is a strict monoidal category in which every morphism is invertible and each object has a strict inverse. Also it is a well known fact that a ...
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      1answer
      540 views

      What does the homotopy coherent nerve do to spaces of enriched functors?

      Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a ...
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      0answers
      145 views

      Comparing cohomology using homotopy fibre

      I have a question, which might be very basic, but I don't know enough topology to answer. Suppose you have a map of topological spaces (or homotopy types) $f : X \to Y$, with homotopy fibre given by ...
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      votes
      0answers
      58 views

      On cofibrations of simplicially enriched categories

      Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells. We have a canonical inclusion functor , $$i: C \...
      13
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      2answers
      500 views

      Describing fiber products in stable $\infty$-categories

      Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...
      7
      votes
      2answers
      281 views

      What is an example of a quasicategory with an outer 4-horn which has no filler?

      A quasicategory has fillers for all inner horns $\Lambda^i[n]$ where $n\geq 2$ and $0<i<n$, but it need not have fillers for $i=0$ (or $i=n)$. In particular, for $n=2$ and $n=3$ there are easy ...
      10
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      0answers
      236 views

      When is Fun(X,C) comonadic over C with respect to the colimit functor?

      Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular,...
      5
      votes
      1answer
      200 views

      Are semisimplicial hypercoverings in a hypercomplete $\infty$-topos effective?

      A hypercovering in an $\infty$-topos $E$ is a simplicial object $U \in {\rm Fun}(\Delta^{\rm op},E)$ such that each map $U_n \to ({\rm cosk}_{n-1} U)_n$ is an effective epimorphism. Theorem 6.5.3.12 ...
      6
      votes
      0answers
      132 views

      2-categorical Yoneda embedding and limits

      Does the 2-categorical Yoneda embedding defined in A.2 Section 5 of `A study in derived algebraic geometry' (Gaitsgory and Rozenblyum, version of 2018-11-14) preserve limits for any $(\infty,2)$-...
      10
      votes
      2answers
      857 views

      Results relying on higher derived algebraic geometry

      Are there any results in number theory or algebraic geometry whose statement does not involve either higher categories or any derived structures but whose most natural (known) proof uses derived $n$-...
      7
      votes
      1answer
      1k views

      Anabelian geometry ~ higher category theory

      Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...
      3
      votes
      1answer
      203 views

      Geometry of 2-arrows

      It is frequently said that one of the contributions of Grothendieck to geometry was to systematically think about the properties of morphisms, as opposed to the properties of spaces themselves (the ...
      4
      votes
      0answers
      200 views

      A compendium of weak factorization systems on $sSet$

      A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying Every morphism $f:x \to y \in \mathcal{C}$ can be factored (...

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