Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should be triangulated.


1. Is this possible in full generality? (Unbounded complexes, no restrictions on X)
2. Consider a sheaf of rings R or equivalently a ring of sheaves. This gives us two things: An abelian category of left R modules that we can derive; let's call this one Der R; A monoid R in Der X whose category of modules we denote dMod R. Is Der R = dMod R'? Ff not: how do they relate?
3. Given monoids R,S,T in Der X do we get the usual adjunctions in two variables between their categories of bimodules?
4. Given rings R,S,T in Sh X do we get the usual adjunctions in two variables between their derived categories of bimodules?


Now for the question: What is the right setting to do this? As i understand it, there's no suitable model structure that gives 4 in full generality.



Closed monoidal structure on the derived category of sheaves
I think the natural context to answer this and related questions is that of symmetric monoidal stable $(\infty,1)$-categories. In that context Lurie's DAG I-III give all the necessary foundations to straightforwardly generalize all of the usual abelian story you refer to to the derived category. There is such an object for sheaves on X (unbounded complexes thereof), and we can talk about associative algebra objects therein, their stable module categories, bimodules etc. and everything works just as you'd expect. In particular the homotopy categories of all these will be what you expect. (If you're looking for a worked-out example of playing these games in a similar context, I can selfishly suggest [this](http://arxiv.org/abs/0805.0157).)


