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Recently I have heard somewhere that any (edit: small) abelian category can be expressed as the colimit of categories of projective modules over some rings. The remark was that this is "basically just idempotent completion". I have been trying to figure out myself how this works but got stuck. Maybe somebody knows how to do this or where I can find a good reference.

My idea so far was to consider the rings $R=End(A,A)$ for any object $A$ in the abelian category $\mathcal A$. Then I can consider the functor $h_A:\mathcal A\to R\text{-}mod$ which maps $X$ to the $R$ module $Hom(A,X)$, which gives by Yoneda an embedding $\mathcal A\to (R\text{-}mod)^\mathcal A$. $A$ itself and its coproducts are then precisely the free $R$-modules.

If I take the idempotent completion of the category of free $R$-modules, I get the category of projective $R$-modules.

But this all doesn't really seem to go anywhere. Am I on the right way?

Also this statement seems so strong that it is odd that I can't find a reference. If I have any funtorial property for projective $R$-modules which commutes with colimits (like so many things in K-theory) I automatically have it for any abelian category. Maybe this means that the statement should be weaker?

edited Jan 31 at 13:22

asked Jan 31 at 12:16

Simon Markett
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If your abelian category $\mathcal{A}$ is small, then it is the category of projective $\mathcal{A}$-modules, where $\mathcal{A}$-modules are contravariant additive functors from $\mathcal{A}$ to abelian groups. – Fernando Muro Jan 31 at 12:33

@Fernando Muro: And how do modules over categories relate to modules over rings? What do you mean by projective? Do you have a reference? (Also: is your comment basically an answer with gaps to be filled, or rather just a remark?) – Simon Markett Jan 31 at 15:37

[I hope this comment makes sense...I'm really tired right now, so no promises.] Once you're in $R$-mod, but before taking idempotent completion, why can't you just use the fact that every $R$-modules is a colimit of a chain of projectives (it's projective resolution)? As for Fernando's comment, I read about this once in a paper of Hovey (though there are probably references which are more algebraic): hopf.math.purdue.edu/Hovey-Lockridge/ssrs.pdf – David White Jan 31 at 19:44

Any abelian category as filtered colimit of categories of projective modules

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