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The answer is: no there isn't such a thing. Here is a rough argument (a full proof would deserve a little more care). Using the main result of S. Schwede, The stable homotopy category is rigid, Annals …

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 …

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 …

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 …

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 …

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 …

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 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 …

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 …

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 …

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 …

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 both cofibrations and weak equivalences are stable under filtered colimits, then so are trivial cofibrations. This happens for instance if $\mathcal{M}$ is a presheaf category on an elegant Reedy c …