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There is a projective model structure on the category of (pre)sheaves with value in any reasonnable model category (e.g. simplicial sets, complexes of abelian groups, commutative k-dg-algebras, where …

This is well known, but formulated in a slightly different way. Recall that a Frobenius category is an exact category which has enough injectives as well as enough projectives, and such that an objec …

I complete the answer of Charles Rezk by some precise references: you may find the answer to your questions in my book Les préfaisceaux commes modèles des types d'homotopie, Astérisque 308 (2006). In …

If, given any fixed cofibrant object $A$, there is a funtor $map(A,-)$ from $M$ to simplicial sets which preserves weak equivalences between fibrant objects and commutes with homotopy limits up to can …

I suspect that you already have one, but here is a proof. I will assume that $M$ and $N$ are combinatorial and that $M$ is left proper (otherwise, I don't think that the literature contains a general …

If $V$ is a reasonnable model category (i.e. combinatorial, etc), and $C$ a small category endowed with a Grothendieck topology $\tau$, there are $\tau$-local model structures on the category $Fun(C^{ …

If we look at the proof that I know of in the case this is true - see below - we need in fact that, for any small $1$-category $I$, $h_\infty(C^I)$ has small (co)limits and that they can be computed …

Quillen's original proof (in Homotopical Algebra, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that th …

For any elementary topos $T$, there is a model category structure on $T$, whose cofibrations are the monomorphisms, and whose weak equivalences are the maps $X\to Y$ such that, either $Y$ is empty, ei …

The axiomatic way to present this is Heller's theory of homotopy theories, which is the same as Grothendieck's theory of derivators (see the references in the links below). As any model category defin …

Let $A$ be a small category, and $W$ an $A$-localizer. Then we say that $W$ is regular if any presheaf $X$ over $A$ is canonically the homotopy colimit of the representable presheaves above $X$; see D …

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 …

The answer to question 1) is no. However, the functor $$N Elts_A:Psh(A)\to sSet$$ commutes with colimits and is a left Quillen equivalence whenever $A$ is a test category. We may transfer the model st …

Homotopy limits in any model category always coincide with limits in the associated $(\infty,1)$-category. To see this, you need to know the following (classical) facts: 1) given a cofibrant object $ …

The fact that G is a test category falls in large class of examples. Grothendieck proved that a small category $A$ is a local test category if and only if there exists a presheaf $I$ on $A$ which is a …