# Non standard (?) model category structure on (co)chain complexes.

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.

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.

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## 1 Answer

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 object is projective if and only if it is injective (injectivity (resp. projectivity) is defined with respect to inflations (resp. deflations)). If you forget about the existence of (finite) limits and colimits, any Frobenius category satisfies all the axioms of a Quillen model category: the cofibrations (resp. fibrations) are the inflations (resp. deflations), while the weak equivalences are the maps which factor through some projective-injective object. Therefore, any Frobenius category which has finite limits as well as finite colimits is a (stable) closed model category in the sense of Quillen.

Now, given an additive category $A$, the category of chain complexes $C^\sharp(A)$ (where $\sharp=\varnothing$ for unbounded chain complexes, $\sharp=b$ for bounded chain complexes, etc) is a Frobenius category: inflations (resp. deflations) are the degreewise split monomorphisms (resp. split epimorphisms), and projective-injective objects are the contractible chain complexes. If $A$ has finite limits as well as finite colimits, this shows that the category $C^\sharp(A)$ is a stable closed model category whose cofibrations (fibrations) are the degreewise split monomorphisms (resp. epimorphisms), and whose weak equivalences are the chain homotopy equivalences.

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Thank you, Denis-Charles Cisinski: any reference for this result? –  a.r. Oct 19 '10 at 12:05
For the fact that any Frobenius category defines a model category, you may find a proof with Proposition 4.19 and Example 5.3 in a paper of mine: "Catégories dérivables", Bull. Soc. Math. France 138 (2010), 317-393 (but this is an old folklore result; I think the first instance of this appeared in the paper of A. Heller: "The loop-space functor in homological algebra", Trans. Amer. Math. Soc. 96 (1960), 382–394). The fact that C(A) is a Frobenius category for any additive category A is a nice exercise. –  Denis-Charles Cisinski Oct 20 '10 at 14:23
Merci encore, Denis. Je crois qu'on s'est rencontré à Barcelone il y a peut-être deux ans, n'est-ce pas? Amitiés. –  a.r. Oct 20 '10 at 19:01
En effet. A bientôt! –  Denis-Charles Cisinski Oct 20 '10 at 21:49