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To find a counterexample, we should choose $C$ to have a lot of fibrant objects, but few cofibrant objects. So let $D$ be a combinatorial model category and let $C$ be the model category structure on …

You can think of the pushout of two maps f : A → B, g : A → C in Set as computing the disjoint union of B and C with an identification f(a) = g(a) for each element a of A. We could imagine forming th …

Yes. The isomorphism in $\mathrm{Ho}(\mathcal{M})$ is represented by a morphism in $\mathcal{M}$ from a cofibrant replacement for $A$ to a fibrant replacement for $B$. The "converse to the Whitehead …

I'm not 100% sure, but I think the answer is that you should choose a cellular model for PSh(C) (the category of presheaves of sets on C), which is a set S of monomorphisms in PSh(C) such that every m …

I came across the nlab page for the axiom COSHEP (category of sets has enough projectives) which seems to be just what's needed to obtain a model category structure, as usually understood, on Set with …

The general statement is that if $V$ is a monoidal model category (I will assume the unit object of $V$ is cofibrant, so that there are no funny extra axioms related to the unit) and $M$ is a $V$-mode …

A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the stru …

A somewhat general statement along these lines would be as follows. Suppose $\mathcal{D}$ is a cartesian closed locally presentable category and $d : \Delta \to \mathcal{D}$ is a cosimplicial object w …

That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left …

Question 1: The model category $\mathcal{C}$ should be left proper, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper. …

Funny, we were just discussing this at dinner last night. Anyways, see Corollary B.13 of Morava K-theories and localization by Hovey and Strickland for some stable presentable ∞-categories with no no …

When $\mathcal{M}$ and $\mathcal{N}$ are combinatorial, this class $\mathcal{C}$ is indeed the left class of a weak factorization system on the category of all left adjoints from $\mathcal{M}$ to $\ma …

Model categories are 1-categorical presentations of (∞,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there …