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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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      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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      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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      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 ...
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      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 ...
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      1answer
      147 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 ...
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      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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      1answer
      91 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 ...
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      1answer
      328 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}]}(...
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      1answer
      115 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
      537 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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      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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      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 \...
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      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 $X \times_Z Y$ ...
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      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 ...
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      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,...

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