Injective model structure on sheaves of bounded complexes of $A$-modules 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
 A: 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,
