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Let $k$ be a commutative ring and let $\mathrm{Ch}(k)$ be the category of non-negatively graded chain complexes of $k$-modules. We endow it with the projective model structure. Weak equivalences are q …

This property holds actually for right derivable categories in the sense of: MR2729017 Reviewed Cisinski, Denis-Charles Catégories dérivables. (French) [Derivable categories] Bull. Soc. Math. France 1 …

Cofibrant CDGAs are retracts of cellular ones. A cellular cofibrant CDGA is a free commutative graded algebra on a (possibly transfinite) sequence of generators $x_1,x_2,\dots$ such that $d(x_i)$ only …

That's not the Moerdijk model structure. In the Moerdijk model structure, weak equivalences and fibrations are created by the diagonal simplicial set construction. Your model structure resembles the B …

The answer is positive when the target is the category of simplicial sets. You can find a proof in Chaper VIII of Goerss-Jardine's book. Then it follows for presented combinatorial model categories in …

I found it in Hirschhorn's book. It's in 11.6 Diagrams in a cofibrantly generated model category. The answer is always, provided K is cofibrantly generated (Theorem 11.6.1). For him (as well as for Ho …

You can consider DGAs, which are DG-categories with only one object. The polynomial ring $k[x]$ concentrated in degree $0$ (with trivial differential) is cofibrant, since it is free as a (graded) $k$- …

Rodríguez González, Beatriz(E-CSIC-IM) Simplicial descent categories. (English summary) J. Pure Appl. Algebra 216 (2012), no. 4, 775–788

I take 'dg-injective' as fibrant in the injective model structure on complexes of right dg-$R$-modules, whose weak equivalences are the quasi-isomorphisms and whose cofibrations are the levelwise inje …

1) Mikhail, mapping cones etc, are defined for arbitrary maps. The problem is that they are not homotopy invariant unless your model category is left proper. Therefore, in general you must take cofibr …

Yes, because the projection functor from the model category to the homotopy category preserves coproducts of cofibrant objects. That is actually the way of showing that the homotopy category has copro …

Let $\tilde C$ be the left Bousfield localization of $C$. As categories $C=\tilde C$, but $\tilde C$ has more weak equivalences. In particular, the identity functor induces a functor $\varphi\colon \o …

Yes, because both statements are equivalent to the existence of an exact triangle of the form $$W\longrightarrow X\oplus Y\oplus Z\longrightarrow V\longrightarrow\Sigma W.$$ The arrows are the same …

I like the following example because it is very close to the origins of homotopy theory (and also because I worked on it at the beginning of my career): proper homotopy theory. Objects are topological …

1) I assume your three model structures have the same weak equivalences, correct me if I'm wrong. Let $\mathcal C$ be a model category and $I$ a small category, e.g. $I=\bullet\leftarrow \bullet\right …