# Exactness of filtered colimits

Are filtered colimits exact in all abelian categories?

In Set, filtered colimits commute with finite limits. The proof carries over to categories sufficiently like Set (i.e. where you can chase elements round diagrams), in particular A-Mod where A is a commutative ring. This implies that filtered colimits are exact in A-Mod.

I am aware of a vague principle that things that are true in A-Mod are true for all abelian categories, but I have never seen a precise statement of this principle so I am not sure if it applies in this case.

• In case you are still curious about your "vague principle," you might be thinking of the Freyd-Mitchell Embedding Theorem: en.wikipedia.org/wiki/Mitchell's_embedding_theorem – Justin Curry Nov 14 '12 at 15:34
• See also Steve Lack's similar question (as yet not answered successfully) mathoverflow.net/questions/57099/… – ziggurism Mar 21 '13 at 4:26
• Filtered colimits commute with finite limits in any algebraic category (for a definition and characterization of 'algebraic categories' see for instance Borceux, Vol II, Thm 3.9.1). Thus filtered colimits are exact in any cocomplete algebraic category. – Exterior Feb 25 '15 at 0:48

A counterexample which is non-trivial is given in Chapter 6 of Neeman's book Triangulated Categories. The category in question is the full subcategory of additive functors Cat(S^{op}, Ab) where S satisfies some hypotheses (e.g. is an essentially small triangulated category) and we take those functors which are product preserving for small enough products (so as contravariant functors S-> Ab they send sufficiently small coproducts to products). This category is complete and cocomplete but has neither exact filtered limits or exact filtered colimits. More precisely it satisfies [AB4] and [AB4*] but neither [AB5] nor [AB5*].