Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} \longrightarrow \mathbf{Mod}_R$ in a category of $R$-modules, for some ring $R$.

Now, $V$ being exact is the same as saying that it preserves all *finite* limits and colimits.

I would be glad to know if Mitchell's embedding theorem could be improved in order to have that $V$ preserves also:

(a) arbitrary products, and (b) filtered colimits.

Or, alternatively,

(c) injective objects.

Or, which conditions on the abelian category ${\cal A}$ would guarantee (a) and (b), or (c)? Are there any results in these directions?

The reason behind my question is the following: I realized that my answer to my previous question vanishing theorems is wrong: sheaf cohomology is not defined uniquely in terms of exact sequences, so the fact that $V$ is exact doesn't guarantee that $H^n(X; {\cal F}) = H^n(X; V({\cal F}))$ as I claimed. But, if I had (a) and (b), I could say that $V$ preserves Godement resolutions. And if I had (c), $V$ would preserve injective resolutions.

experience(which you lack) and seeing students use Mitchell's theorem only to realize later that it doesn't help much, that's all. I also know a thing or two about useful ways to work with sheaf cohomology; the given motivation is not leading in a good direction for that. I don't see any big deal about offering precise advice based on experience in a comment box. I wrote "as far as I know" to make it clear that I am open to hearing examples to the contrary. I never wrote that I have an "important opinion"; don't put words in my mouth. Lighten up, dude. $\endgroup$ – BCnrd Jul 17 '10 at 20:103more comments