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      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 ...
      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}]...
      1
      vote
      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 ...
      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 (...
      5
      votes
      1answer
      273 views

      Are there universal homological functors?

      There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable? That is, for each small abelian ...
      5
      votes
      1answer
      305 views

      If the Tate construction vanishes for all trivial $G$-actions, then does it vanish for all $G$-actions?

      Let $\mathcal{C}$ be a semiadditive $\infty$-category, complete and cocomplete, and let $G$ be a finite group. Then for any $X \in Fun(BG,\mathcal{C})$, there is a norm map $N_X: X_G \to X^G$. For ...
      3
      votes
      0answers
      103 views

      Descent for the cotangent complex along faithfully flat SCRs

      By Theorem 3.1 of Bhatt-Morrow-Scholze II (https://arxiv.org/pdf/1802.03261.pdf), we know that for $R$ a commutative ring, $\wedge^{i}L_{(-)/R}$ satisfies descent for faithfully flat maps $A \...
      6
      votes
      0answers
      142 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 ...
      0
      votes
      1answer
      84 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 ...
      5
      votes
      1answer
      388 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 ...
      26
      votes
      1answer
      810 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 ...
      5
      votes
      1answer
      171 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 ...
      4
      votes
      1answer
      206 views

      A finite Whitehead Theorem for $\infty$-topos

      Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces ...
      7
      votes
      2answers
      413 views

      What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

      A paradox: Goodwillie calculus considers only finitary functors. $TC$ isn't finitary. Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem. (...
      4
      votes
      0answers
      103 views

      Minimal infinity categories in the Segal space picture

      There is a well-known notion of a minimal Kan complex (see Goerss/Jardine's book) which is generalized to a minimal quasi-category in these notes by Joyal www.math.uchicago.edu/~may/IMA/Joyal.pdf (...

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