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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, combine Corollary 9.4.1 and Theorem 7.11 of arXiv:1410.5675, for example. This topic is also examined more explicitly in the work of Chu and Haugseng, arXiv:1707.08049. Corollary 5.1.13 shows tha …

The most obvious approach is to consider simplicial $\def\Ai{{\sf A}_∞}\Ai$-categories, where $\Ai$ denotes a nonsymmetric operad in simplicial sets that is weakly equivalent to the terminal operad, i …

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 …

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, …

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 …

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 …

Yes because the model structure on Fun(Δ^op, sSet) is a cartesian model structure. This follows from the fact that the Joyal model structure is cartesian, and §10, §11 in Rezk's paper “A model for the …

One rich source of examples is that every combinatorial symmetric monoidal model category gives rise to such a category V. In particular, this covers all the examples in the main post, including cdga …

The derived tensor product of dg-categories was explored by Toën, see his article The homotopy theory of dg-categories and derived Morita theory, in particular, Section 4, where Toën explains how to …

Categories (co)fibered in groupoids are used to define stacks (stacks in groupoids, to be precise), so any introduction to stacks will do. For stacks on smooth manifolds, see, for example, Ieke Moerd …

Working in the stable ∞-category of morphisms, whose objects are morphisms and morphisms are commutative squares with a choice of homotopy, the statement $δ≃α$ up to some contractile space of homotop …

The case when $M$ is a smooth manifold follows from the smooth Oka principle. See there for an expository account of the argument and references to additional sources. Indeed, the left side of (*) is …

This is closely related to the Baez–Dolan stabilization hypothesis. There are numerous proofs of this statement. One line of reasoning is to establish a general 1-category statement first: given a sym …

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 …