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There are many motivations, but the short answer is that many desirable properties are only available in the world of $\infty$-categories. This is a wonderful miracle. This is particularly visible whe …

I give different arguments for this kind of things here. In short, the main case is the one of the terminal presheaf: one proves by cofinality arguments that the colimit of the Yoneda embedding $S\to\ …

The six functor formalism applies to $D$-modules, but you need to extend the theory to possibly non-smooth schemes. For this, we see that hypersheaves on the site of pairs $(X,Z)$, with $Z$ a closed s …

There are several ways to interpret the homotopy hypothesis. Strictly speaking, Grothendieck's homotopy hypothesis is not a theorem yet: Grothendieck stated it in a very precise way in the very first …

This fails already with the category of abelian groups. If the dg-nerve of the dg-category of chain complexes of abelian groups were presentable, then the associated triangulated category would be wel …

It seems to me that the answer is yes. Here is a sketchy argument. Let us fix some notations. I will write $i(X)$ for the maximal Kan subcomplex of a quasi-category $X$ and $Hom(A,B)$ for the interna …

If your model structures are assumed to have small limits or colimits, the answer to the question of the title is: always. For any model category $M$ and any small category $C$, inverting levelwise we …

Here is a (tautological) proof in the setting of quasi-categories. Let $A$ be a quasi-category. In ordinary category theory, one can describe the category of presheaves of sets over small category $C$ …

No it is not. But there is a replacement for this. There is another model of $\infty$-categories: marked simplicial sets. If $X$ is a simplicial set with a set of marked $1$-simplices $S$, the fibrant …

This is a long comment. Simon's answer is correct, but there are other ways around the difficulty of forming the right internal Hom. If $C$ is a full simplicial subcategory of a simplicial model categ …

The model structure for complete Segal spaces is not right proper. To see this, one can first prove that the model structure for quasi-categories is not right proper: for instance, the map $\delta^1_2 …

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the in …