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Yes, there is such a model structure, at least if we are willing to pick a category of simply connected spaces that has limits and colimits. The modern proof simply applies the existence theorem for l …

One can do even better than a Quillen adjunction: Theorem 3.8 in A model structure for Grothendieck fibrations establishes a Quillen equivalence between the projective model structure on presheaves of …

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 …

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 …

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 …

To answer the question as it is stated: $U$ is an object in a locally presentable category, therefore $U$ is a small object, hence the corepresentable functor of $U$ preserves $α$-filtered colimits fo …

The diagram $G$ is a projectively cofibrant diagram because nondegenerate simplices split off as a coproduct summand in every degree and every simplicial level is a coproduct of (enriched) representab …

There are many connections between Lie groupoids and homotopical algebra. In recent years, a particularly prominent connection is to the theory of simplicial presheaves, originally developed in the c …

Before someone answers I add a related less general question. Does there exist a cofibrant replacement (not necessarily functorial) of the point ∗∈sSetCop with the property that ev0Q(∗)≅∗∈SetCop ? N …

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 …

This depends on whether one insists on derived functors landing in the original model category (as is the case with modern approaches of Hinich and Dwyer–Hirschhorn–Kan–Smith), or in its homotopy cate …

$\def\Cnec{{\frak C}^{\rm nec}} \def\Exi{{\rm Ex}^∞} \def\N{{\rm N}} \def\sCat{{\sf sCat}} \def\sSet{{\sf sSet}_{\sf Joyal}}$ As indicated in the answer A combinatorial approximation functor sSet->qCa …

If we write down the lifting square for an arbitrary cofibration $f\colon A→B$ and the unit map $η\colon X→RLX$ (with the bottom map being $b\colon B→RLX$), and then use the adjunction to pass to the …