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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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      A few questions while reading Higher Topos Theory

      I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help. First, in Lemma 2.2.3.6, while ...
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      127 views

      Does Ex^∞ send homotopy inverse limits of ∞-categories to homotopy inverse limits of spaces

      Kan's functor $\operatorname{Ex}^{\infty}$ plays the role of total localization or groupoid completion in the theory of $\infty$-categories; specifically, it can be viewed as the left-adjoint of the ...
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      363 views

      Visualization and new geometry in higher stacks (soft question)

      I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...
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      votes
      1answer
      263 views

      Definition A.3.1.5 of Higher Topos Theory

      I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...
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      votes
      1answer
      81 views

      A characterization of maps that are homotopic relative to $A$ over $S$

      Let $$\begin{array}{ccccccccc} A & \rightarrow & X \\ i\downarrow & & \downarrow p \\ B & \xrightarrow{v} & S \end{array} $$ be a commutative diagram of simplicial sets, with ...
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      1answer
      164 views

      Weighted (co)limits as adjunctions

      It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\...
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      67 views

      Natural transformation of $A_\infty$-functors lifted from homology

      Suppose you have two $A_\infty$-functors $\mathcal{F,G}:\mathcal{A}\longrightarrow \mathcal{B}$ which descend to $F,G:A \longrightarrow B$ in homology (here $A=H^0(\mathcal{A})$ and same for $\mathcal{...
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      votes
      1answer
      359 views

      Understanding the proof of Proposition 1.2.13.8 in Lurie's Higher Topos Theory

      In the first chapter of Lurie's HTT, Proposition 1.2.13.8 shows that the functors out of over (infinity)-categories reflect infinity-colimits. For the life of me I cannot follow the proof. Can ...
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      votes
      1answer
      699 views

      Deligne's doubt about Voevodsky's Univalent Foundations

      In a recent lecture at the Vladimir Voevodsky's Memorial Conference, Deligne wondered whether "the axioms used are strong enough for the intended purpose." (See his remark from roughly minute 40 in ...
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      votes
      3answers
      784 views

      Higher $\infty$-categories

      Is there a reason we consider $\infty$-categories to be the $\omega^{th}$ step in the 2-internalization inside Cat (or enrichment over Cat if you prefer)* process made invertible above some finite ...
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      votes
      1answer
      186 views

      Smash product and the integers in a Grothendieck $(\infty, 1)$-topos

      Let $\mathcal{H}$ be a Grothendieck $(\infty,1)$-topos. According to this page in nlab, for any $X \in \mathcal{H}$, the suspension object $\Sigma X$ is homotopy equivalent to the smash product $B \...
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      votes
      1answer
      106 views

      Inverting a suspension object in a stable monoidal category

      Suppose we are given a cocomplete closed symmetric monoidal stable $(\infty,1)$-category $\mathcal{C}$ with suspension $\Sigma$, and let $X \in \mathcal{C}$ be dualizable. I'd like to create a new ...
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      votes
      1answer
      148 views

      Homotopy groups of Diffeomorphisms of punctured d-dim ball

      Let $\mathbb{D}_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}_n^d)$ that fix the punctures. In ...
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      votes
      0answers
      103 views

      Infinity categorical analogue of 2-dimensional monad theory

      I'm wondering whether there is an infinity categorical analogue to the results of Two-dimensional monad theory. For the most part, I'm interested in the relation between strict functors of infinity ...
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      68 views

      Compact Generation of Co-Module Categories

      Let $\mathcal{C}$ be a compactly generated stable $\infty$-category, linear over a field of characteristic $0$ (i.e., so that it is in particular a dg-category). Let $A$ be a co-monad acting on $\...

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