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Let $X$ be a ringed space. The abelian category of (quasi-coherent) $\mathcal{O}_X$-modules does not behave as well as the category of $A$-modules for a commutative ring $A$. The reason is that there are not enough projective objects. In particular, $\mathcal{O}_X$ is not projective, since $Hom(\mathcal{O}_X,-)$ is the global section functor, which is not exact.

I want to fix this by enriching the abelian category over $Sh(X)$, the monoidal abelian category of abelian sheaves on $X$. Remark that $\underline{Hom}(\mathcal{O}_X,-)$ is just the identical functor, thus $\mathcal{O}_X$ should be projective from this point of view. Actually I want that $\mathcal{O}_X$-Mod has all the homological properties as $A$-Mod, except that we talk about them in another topos, namely $Sh(X)$ instead of $Set$.

In particular I want to generalize the following well-known theorem: Let $\mathcal{A}$ be an abelian category, which has a progenerator $P$ (i.e. a finite projective generator), whose coproducts exist. Then $\mathcal{A}$ is equivalent to $Mod_A$ for some (noncommutative) ring $A$, namely $A$ is the endomorphism ring of $P$.

So how do we define an abelian category $\mathcal{A}$ over $Sh(X)$ in reasonable way? I've already written down a possible definition, but I wonder if it is possible to avoid the category of elements of the enriched category, since this makes the whole story nonlocal. For example when you want to write down what a kernel of a morphism should be, or when you want to define the notions of generators or projective objects.

Or should we just say that everthing is a sheaf? A category is a pair $(O,M)$, consisting of a sheaf $O$ on $X$ (objects) and a sheaf $M$ on $X$ (morphisms) together with maps $M \to O \times O$, etc. ...

I'm pretty sure that somebody has worked this out in detail some decades ago, so a reference would be ok.

EDIT [12. 09]: I think the following definition works: A sheabelian ;) category is a pair $(X,A)$, consisting of a topological space $X$ and a presheaf $A$ on $X$, valued in the category of abelian categories, such that the following sheaf condition holds:

1) If $U = \cup_i U_i$ and $f,g : F \to G$ are morphisms in $A(U)$ and $\eta_i$ are natural transformations between $f|_{U_i}, g|_{U_i}$, which are compatible on the $U_i \cap U_j$, then they lift uniquely to a natural transformation between $f$ and $g$.

2) If $U = \cup_i U_i$ and $T_i \in A(U_i)$ are objects, which are isomorphic on $U_i \cap U_j$, such that the cocycle condition is satisfied, then there is an object $T \in A(U)$, which restricts to an object isomorphic to $T_i$, and the isomorphisms here are compatible with each other.

Basically this concept works for every $2$-category. Objects of $(X,A)$ are defined to be objects of $A(X)$, but properties and constructions of these objects are defined locally. For example, $Hom(P,-)$ for an object $P$ is a functor $A(X) \to Sh(X)$. Currently, I'm trying to work out the details for the generalization of the above theorem. Note that if $(X,\mathcal{O}_X)$ is a ringed space, then $O_X$ should be a progenerator of $Mod_{O_X}$.

EDIT [15.09] Ok I think I have just reinvented the notion of a stack. ;)

So my question is: Is there literature about a sort of homological algebra of stacks of abelian categories? Here I'm mainly interested in the site associated to a topological space, and some of the $2$-isomorphisms in the definition of a stack should become identities (since, for example, we have $(F|_V)|_W = F|_W$ for a sheaf $F$ on $X$).

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Martin, if you look at the category of all $\mathbb{A}^1_S$-modules in the big étale or fppf topos over $S$, does this possibly fix anything? –  Harry Gindi Oct 14 '10 at 1:43
    
Also, I'm pretty sure that $Hom(\mathcal{O}_S,-)$ is not the global sections functor. Shouldn't that actually be $Hom(-,\mathcal{O}_S)$? –  Harry Gindi Oct 14 '10 at 1:46
    
1) Sounds good. How is this category exactly defined? 2) No, we are considering module homomorphisms. –  Martin Brandenburg Oct 14 '10 at 8:29
    
Oh, alright, here's why! You have to use the "sheafy" hom functor. That is: $Hom_{\mathcal{O}_X-Mod}(F,-)$ is given as the right adjoint of the "sheafy" tensor product. Then $\mathcal{O}_X$ is projective over itself (in fact free of rank 1). –  Harry Gindi Oct 14 '10 at 22:12
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1 Answer 1

Yes, you do seem to be looking basically at stacks. Note that stacks, and more generally prestacks and fibered categories, can be identified with categories enriched over the self-indexing of the topos of sheaves, or equivalently over the bicategory of spans in the topos of sheaves. There is a bit of a dearth of good expositions of that point of view, but you can try B2.2 in Sketches of an Elephant and the paper "Variation through enrichment" (which is a bit dense and very category-theoretic).

However, it's not true that just working in another topos will resolve the issue of abelian sheaves not having enough projectives, because the proof in Set that module categories have enough projectives is not constructive, and so does not relativize to all topoi. In particular, the fact that free modules are projective relies on the fact that all sets are projective in Set, which is equivalent to the axiom of choice. You don't need the full strength of AC to show that there are enough projectives -- the presentation axiom suffices -- but it's still not true in most/all topoi.

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Thank you for your answer. If $X$ is a ringed space, then the quasi-coherent modules on $X$ should behave like a usual module category over a ring (since there are, by definition, local presentations). I've edited my question above. Where can I find a exposition on category theory or even homological algebra "valued in a topos"? Or does anything works exactly as in the case of Set? Currently I'm trying to work this out for $Sh(X)$, but some things get too complicated... –  Martin Brandenburg Sep 17 '10 at 7:31
    
I'm not sure what you mean by saying that quasi-coherent modules "should" behave like a usual module category. The discussion of indexed and internal categories in Sketches of an Elephant part B is "category theory valued in a topos"; I believe cohomology theory of topoi is going to be in the forthcoming volume 3, but Johnstone's original book Topos theory includes some of it. –  Mike Shulman Sep 17 '10 at 23:57
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