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



Injective model structure on sheaves of bounded complexes of $A$-modules
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](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.ps) 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](http://www.math.fsu.edu/~gdungan/MC10.04.07.pdf),


