Let $\cal{A}$ be an abelian category with enough projectives and $\mathbf{C}_+ (\cal{A})$ the category of bounded below chain complexes.

Since Quillen (Homotopical algebra, 1.2, examples B), there is a well-known "standard" model category structure on $\mathbf{C}_+ (\cal{A})$ taking as weak equivalences the maps inducing isomorphisms on homology, as fibrations the degree-wise epimorphisms in $\cal{A}$ and with cofibrations maps $i$ which are injective and such that $\mathrm{cok}\ i$ is a complex having a projective object of $\cal{A}$ in each degree.

More recently, Hovey (Model categories), proved an analogous result for the category of *unbounded* chain complexes, but with ${\cal A} = R$-modules, $R$ a ring (but cofibrations are not so easy to characterise). Finally, it's folklore (at least, I don't know if it is published somewhere) that the same holds for $\cal{A}$ an abelian category with a projective generator -the fact that allows the small object argument to work, as Eric Wofsey points out in his answer to [this MO question][1].

>I'm interested in the following variant of this problem: is there a model structure on $\mathbf{C}_+ (R)$ taking as weak equivalences the **homotopy** equivalences?

If it's true, I think this should be easy to verify: just taking a look to the classical proof and seing if you can change "**homology** equivalences" everywhere by "**homotopy** equivalences". I'm willingly going to do it, but, prior to start, I would like to know if it is already done, much as in the case of topological spaces where, together with the "standard" (Quillen too) model structure with **weak homotopy** equivalences as weak equivalences, there is the Strom model structure (The homotopy category is a homotopy category), with **homotopy** equivalences as weak equivalences.



  [1]: https://mathoverflow.net/questions/141/model-category-structures-on-categories-of-complexes-in-abelian-categories