In Mark Hovey's article Model category structures on chain complexes of sheaves (arXiv:math/9909024) a model structure on the category $Ch(A)$ of unbounded chain complexes for a Grothendieck abelian category $A$ is constructed. This is called the injective structure and has the derived category of $A$ as its homotopy category.

Can this injective model be given the structure of a simplicial model category? How is the simplicial enrichment given?

The Dold-Kan correspondence identifies non-negative chain complexes with with simplicial objects in $A$ and there is an evident simplicial enrichment but the question is for unbounded.


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It is instructive to consider a simpler case, namely that of chain complexes over a ring. Hovey deals with this example in great detail in his book Model Categories. In particular, on page 114 Hovey states that $Ch(R)$ is not a simplicial model category. In algebraic situations like this, it's not really the right question to ask a model category to be simplicial. Instead, one should ask it to be a dg-model category, a.k.a. a $Ch(\mathbb{Z})$-model category. This does hold for $Ch(R)$ with the injective model structure, and should probably hold in the generality of your question as well, though I haven't checked it. A good reference is chapter 4 of Hovey's book, where he defines the notion of a $\mathscr{C}$-model category $\mathscr{M}$ where $\mathscr{C}$ is a model category. This is a generalization of the notion of a simplicial model category (when $\mathscr{C}$ is simplicial sets), i.e. the coherence between $\mathscr{M}$ and $\mathscr{C}$ is analogous to Quillen's SM7 axiom, though Hovey was the first to realize the necessity of the unit axiom for general $\mathscr{C}$.

Note that the model categories $Ch(A)$ in your question are certainly Quillen equivalent to simplicial model categories, e.g. by Dan Dugger's work (because they are combinatorial model categories). Dugger's paper is called "Combinatorial model categories have presentations"

EDIT: Because you're in an additive setting, you actually have more machinery at your disposal than I realized. If you wish to compare an enrichment over $\mathscr{C}$ with an enrichment over $\mathscr{D}$ (e.g. comparing dg-model categories with simplicial model categories), then a great resource is the paper Enriched Model Categories by Dugger and Shipley, especially their notion of Quillen adjoint module. I only just found this paper today. Their machinery can be used to say when two enrichments are quasi-equivalent, i.e. equivalent up to homotopy. Now, monoids in sSet and Ch(k) are not so different (thanks to Dold-Kan) so you may be able to use this machinery to beef up the answer I gave from "Quillen equivalent to a simplicial model category" to something stronger. For example, their Proposition 5.2 and their section 9 let you get your hands on what's going on a bit more than the fully general results of Dugger I mentioned previously. On the other hand, if what I wrote before is enough for whatever application you had in mind then feel free to ignore this.

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    $\begingroup$ Hi David, I don't agree with you when you say that asking whether the category of chain complexes is simplicial is not the right questions. Beign simplicial is a very powerful property since it allows an easy computation of mapping spaces, and you don't get that with other enrichments, such as the one you indicate. Besides, beign equivalent to a simplicial model is a much weaker property which is not that useful for computing mapping spaces either, since the equivalence itself may be as complicated as the construction of a framing in your original category. $\endgroup$ Dec 2, 2013 at 23:32
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    $\begingroup$ Hi Fernando. Yes, I made the edit because I wanted to give the OP extra machinery to do more than just know that his model category was Quillen equivalent to a simplicial one. I also felt my answer was a bit unsatisfactory but I hope the current one is closer to helping with whatever application the OP has in mind. Do you happen to know of a model category which is both a simplicial model category and a dg-model category? It seems like most places I've read about this get one property or the other, but not both. $\endgroup$ Dec 3, 2013 at 0:34
  • $\begingroup$ David, concerning your question, I don't know. The fact that the Eilenberg-Zilber equivalence is not an isomorphism suggests me that maybe the answer is no, except for things like the empty model category or so. $\endgroup$ Dec 3, 2013 at 9:34
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    $\begingroup$ @FernandoMuro: If you have a Ch_{≥0}(Z)-enriched model category, can't you compute the simplicial mapping space as the Dold-Kan functor applied to the derived enriched hom? I'd expect that for a cofibrant object X the cosimplicial object N(Z[Δ^n])⊗X is a cosimplicial resolution of X, which would enable us to compute the simplicial mapping space as described above. $\endgroup$ Sep 24, 2014 at 12:09
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    $\begingroup$ @DmitriPavlov I think it's even true in the unbounded case (after truncation), see the proofs of Prop. 5.6.7 and Corollary 5.6.10 in Hovey's book. $\endgroup$ Sep 24, 2014 at 13:21

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