Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets.
Which extra data do we have to specify on the topos X such that we can recover (up to some notion of equivalence) the essential structure of the underlying site? For example, every representable copresheaf F = Hom(A, _) has a category of modules attached to it, which is a kind of extra data. [I know I am a bit unprecise here with what I mean by essential, so making this precise could also be part of the answer to my question.]
One could also rephrase this question as follows: Given a topos X that satisfies Giraud's axioms, one can extract a site such that X is the Grothendieck topos over this site. Which extra data do we need to impose on X such that we can recover X as a Grothendieck topos over a site that is (the dual) to commutative monoids in a symmetric monoidal category.
When I write down this question, I have the following example in mind: Let C be the category of abelian groups. Then X is the topos of presheaves on the category of affine schemes, which gives rise to algebraic geometry. X possesses a commutative ring object, namely the affine line A1 and one has the stack of categories of quasi-coherent sheaves over objects of X. Taking the idea of (Grothendieck) topoi seriously, one should be able to forget about C and just consider the topos X (i.e. without a fixed base site). Of course, one has to remember (at least) A1. This allows to recover the stack of categories of quasi-coherent modules.
Added for clarification:
But what if C is not the category of abelian groups? In this case, X = CoPSh(Comm(C)) also carries a stack QCoh of categories of quasi-coherent modules as follows: Let F be an object of X, i.e. F is a covariant functor from Comm(C) to Set. An object M of the category QCoh(F) maps a morphism a: Hom(A, _) -> F to an A-module M(a) in C together with natural isomorphisms M(b) = M(a) ⊗A B for all morphisms a -> b in Comm(C). [It is here where the category C itself comes in.]
What is the minimal amount of data we need on X so that X is equivalent to sheaves on a site S such that the dual of S is of the form Comm(C') with C' giving rise to a somewhat equivalent stack of categories of quasi-coherent modules.