Embedding abelian categories

Let A be an abelian category. In the stacks project this means that A has a set of objects, and that

• A is a pre-additive category with a zero object and direct sums, i.e., an additive category,
• A has all kernels and cokernels (and hence all finite limits and all finite colimits), and
• Coim(f) = Im(f) for all morphisms f in A

Martin Olsson pointed out that there is a simple direct argument which proves that in such a category any epimorphism (called a surjection in the following) is a universal epimorphism, see Lemma Tag 05PK. Using this fact we obtain a site C whose underlying category is simply A and where a covering is the same thing as a single surjective morphism. Then the Yoneda functor gives a fully faithful, exact functor

A —> Ab(C), X —> h_X

into the category of abelian sheaves, see Lemma Tag 05PN. Combining this with results on abelian sheaves one obtains a proof of Mitchell’s embedding theorem for abelian categories, see Remark Tag 05PR.

I like the argument phrased in this way because I already know about sites, sheaves, etc. It in some sense explains to me (and hopefully an additional handful of readers here) why the embedding theorem should be true. Moreover, I want to make the point that for all applications I can imagine the embedding into the category of abelian sheaves on a site is sufficient.

2 thoughts on “Embedding abelian categories”

1. Johan, is there an example of an interesting or useful fact about abelian categories for which someone might invoke Mitchell’s result and there isn’t an intuitive and straightforward proof via the trick of “chasing members” as in MacLane’s “Categories for the Working Mathematician”? I can’t recall ever encountering a situation where I’d want to use Mitchell’s result.

• Brian,

I have spoken (years ago) with students who struggled with “chasing members” arguments but were very happy with “chasing elements” arguments. Even if it is less intellectually honest, it is still nice to have a formal result which justifies for these students that the “chasing elements” arguments can always be refined to proofs in any Abelian category. Of course that is just my opinion ðŸ™‚

Best regards,
Jason