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The following might be very well known for people who works with model categories, but I do not find the answer.

Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain complexes of $A$-modules (complexes where differential rises the degree and which are zero in negative degree). There is the injective model structure on $\mathbf{Ch}_+(A)$ which has quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations and epimorphisms with injective kernel as fibrations. Let $K^\bullet$ be a complex, then its injective resolution $I^\bullet (K^\bullet)$ turn out to be its fibrant replacement.

Let $X$ be a topological space and consider $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$, the category of positive complexes of sheaves of $A$-modules on $X$. My question is the following:

Question: Is there an analogue model structure in $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$ to the injective model structure in $\mathbf{Ch}_+(A)$? More concretely, if we consider quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations, and morphisms having the right lifting property with respect to injective quasi-isomorphisms, do they define a model structure? If $\mathcal{K^\bullet}$ is a complex of sheaves, will its injective resolution $I^\bullet (\mathcal{K}^\bullet)$ be its fibrant replacement?

Thank you very much

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    $\begingroup$ I don't have time to write a full answer just now, but I think the answer is probably yes, and that proof should use cotorsion pairs. I recommend you look at the work of Hovey and Gillespie. Gillespie has a paper handling categories of chain complexes of quasi-coherent sheaves over a scheme, and perhaps some mention is made there of the case you care about. I believe there are two cotorsion pairs in these settings, one for the projective model structure and one for the injective. $\endgroup$
    – David White
    Aug 30, 2015 at 22:22
  • $\begingroup$ I think that, in all cases, cofibrations should be cochain maps which are mono in strictly positive degrees (not necessarily in degree $0$), so that the $t$-structure truncation functor from the unbounded category is a left Quillen functor. $\endgroup$
    – Fernando Muro
    Aug 31, 2015 at 0:03

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David White's comment led me to adequate references to answer this question. Thank you very much

Question: Is there an analogue model structure in $\mathbf{Ch}_+(\mathbf{Shv}_X(A))$ to the injective model structure on $\mathbf{Ch}_+(A)$? More concretely, if we consider quasi-isomorphisms as weak equivalences, monomorphisms as cofibrations, and morphisms having the right lifting property with respect to injective quasi-isomorphisms, do they define a model structure?

Answer: Yes, there is such model structure. The above mentioned three clasess do form a model structure. Find the result in a letter to A. Grothendieck from Joyal in Théorème 2 (p.10)

Question: If $\mathcal{K}^\bullet$ is a complex of sheaves, will its injective resolution $I^\bullet(\mathcal{K}^\bullet)$ be its fibrant replacement?

Answer: Yes, it will. The same argument that shows that for a chain complex of modules its fibrant replacement is a complex made of injective modules holds for sheaves. Find the argument for modules (although the case of projective resolutins) in this paper,

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    $\begingroup$ Great! Glad you got it resolved! $\endgroup$
    – David White
    Sep 6, 2015 at 15:56

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