# Filtered colimits are exact in abelian categories

It is well known that filtered colimits commute with finite limits in $\mathsf{Set}$, and hence in every algebraic category - $R\mathsf{Mod}$ in particular. Unless I'm wrong, from the Mitchell embedding theorem one may conclude that filtered colimits commute with finite limits in arbitrary abelian categories.

How can I prove this exactness result from the axioms of abelian categories?

Ideally, I would like a reference to an organized proof.

• The Mitchell embedding theorem lets you conclude nothing of the sort. (Reread the statement!) In fact, there are abelian categories where filltered colimits are not exact. – Zhen Lin Apr 2 '15 at 7:46
• The exactness of filtered colimits is exactly the axiom (AB5) (given by A. Grothendieck in his famous Tōhoku paper). There are a couple of interesting remarks about this axiom: an equivalent form of (AB5), which is worked out magnificently here, and the incompatibility with axiom (AB5*), which gives you many examples of abelian categories which do not satisfy (AB5), $\mathbf{Ab}^{\text op}$ in particular. – Andrea Gagna Apr 2 '15 at 8:35