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The ∞-categorical limits (respectively colimits) are given by the right (respectively left) adjoint of the constant diagram functor $$C→C^I,$$ where $I$ is the indexing category and $C$ is the ∞-categ …

Yes. As shown in the paper The enriched Thomason model structure on 2-categories, the category of (strict) 2-categories can be equipped with a model structure that makes it Quillen equivalent to the …

It is easy to construct counterexamples already for the full subcategory $\{[0],[1]\}⊂Δ$. The category $\cal C$ can be constructed by applying the nerve functor to hom-objects of a category $D$ enric …

The statement is false in its current form: there are left Quillen functors between stable model categories that do not preserve finite limits. However, since ∞-categories are mentioned, presumably wh …

The answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the singu …

An obvious necessary condition for $f$ to be a categorical equivalence is that $f$ is weakly equivalent to a (co)cartesian fibration of quasicategories, i.e., $f$ is an analogue of a Street fibration …

The first part of the question was previously asked and answered here: Reference for homotopy colimit = total complex. The second part can be easily reduced to the first part by rectifying homotopy co …

Consider the canonical functor $H$ from the homotopy category of homotopy coherent descent data in the ∞-category of coherent sheaves to the category of descent data in the derived category of coheren …

The first question has a negative answer, given by the simplicial set $\def\Exi{{\sf Ex}^{\sf\infty}}X=\Exi Y$, where $Y$ is a simplicial set generated by vertices $a,b,b',c,c',d,d'$, 1-simplices $ab, …

As shown by Dwyer–Kan (Function complexes in homotopical algebra, Proposition 4.8), for a simplicial model category $A$, the simplicial category $A^\circ$ is Dwyer–Kan equivalent to the simplicial cat …

As pointed out to me by George Raptis, a detailed treatment of Brown representability for $(n,1)$-categories ($1≤n≤∞$), stable or not, is now available in Hoang Kim Nguyen, George Raptis, Christoph S …

More precisely, let A,B be two simplicially enriched model categories, is it true that every adjoint pair of ∞-functors between A,B comes from a Quillen pair? Assuming the model categories are combi …

Here is what André Joyal wrote in an email to me: No, I have not discovered the model structure for quasi-categories in the 1980's. I became interested in quasi-categories (without the name) around 1 …

An account of derived functors between ∞-categories equipped with weak equivalences and fibrations can be found in Section 7.5 of Cisinski's Higher Categories and Homotopical Algebra. This setting is …

A good place to start is the following characterization of nerves of ordinary groupoids: A category is a groupoid if and only if its nerve is a Kan complex. Filling the outer horns is precisely what …