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.

- Is this possible in full generality? (Unbounded complexes, no restrictions on X)
- 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?
- Given monoids R,S,T in Der X do we get the usual adjunctions in two variables between their categories of bimodules?
- 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.