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      7
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      2answers
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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 ...
      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
      416 views

      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 ...
      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 ...
      3
      votes
      0answers
      136 views

      Why is the Straightening functor the analogue of the Grothendieck construction?

      In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between peseudo functors into the category of ...
      4
      votes
      2answers
      200 views

      Proof that a quasicategory is equivalent to the homotopy coherent nerve of a simplicial category

      On the abstract of a paper by Emily Riehl and Dominic Verity, it is stated that Every quasicategory arises as a the homotopy coherent nerve of a simplicial category up to equivalence. Where can ...
      6
      votes
      1answer
      409 views

      Need help understanding comment in Higher Topos Theory

      I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1. Lemma 2.4.4.1. Let $p : \mathcal{C} \rightarrow \mathcal{...
      2
      votes
      1answer
      81 views

      Factorization of a map from a contractible Kan complex through a Kan complex

      Suppose we are given a contractible Kan complex $S$ and a map of simplicial set $f : S \rightarrow T$. Under what conditions can we say that $f$ factors through the largest Kan complex $Z$ contained ...
      5
      votes
      1answer
      316 views

      Remark 2.4.1.4 Higher Topos Theory

      In HTT, given a inner fibration $p : X \rightarrow S$ of simplicial, an edge $f : x \rightarrow y$ of the simplicial set $X$ is said to be a $p$-Cartesian if the induced map $$ X_{/f} \rightarrow ...
      9
      votes
      1answer
      540 views

      Higher Topos Theory Theorem 2.2.5.3

      The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem. We have a trivial Kan ...
      7
      votes
      0answers
      171 views

      Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?

      In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$ Joyal's isofibration theorem says precisely An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...
      6
      votes
      2answers
      645 views

      Theorem 2.1.2.2 Higher Topos Theory

      At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
      11
      votes
      0answers
      204 views

      Higher homotopical information in racks and quandles

      A quandle is defined to be a set $Q$ with two binary operations $\star,\bar\star\colon\ Q\times Q\to Q$ for which the following axioms hold. Q1. a $\star$ a = a Q2. (a $\star$ b) $\bar\star$ b = (a $...
      3
      votes
      1answer
      138 views

      Cellularity of anodyne extensions?

      Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts) If not, is there a known counterexample? Similarly, does ...
      9
      votes
      0answers
      254 views

      Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

      I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one. Suppose we are given a (strict) pullback square ...

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