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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.

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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/… –  Joe Hannon Mar 21 '13 at 4:26
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2 Answers

up vote 12 down vote accepted

Here's a dumb counterexample. If C is an abelian category, so is Cop. In Cop, filtered colimits are filtered limits in C. And, of course, there are many examples of abelian categories (such as abelian groups) where filtered limits aren't exact.

Of course, your question is really: when is an abelian category C sufficiently close to Set, so that we can ratchet up the fact that filtered colimits are exact in Set to a proof for C.

Any category of sheaves of abelian groups on a space (or on a Grothendieck topos) will have exact filtered colimits, for instance.

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OK thanks. What is the easiest way to see that filtered colimits are exact for sheaves of abelian groups? Look at stalks? Is there any sort of simple sufficient condition that would include this example? –  Martin Orr Oct 24 '09 at 10:45
    
Yes, it is apparent by looking at stalks. Or, you can do it in two steps: (1) presheaves of abelian groups have this property, and (2) prove it for sheaves, using that the sheafification commutes with all colimits and is exact. –  Charles Rezk Oct 24 '09 at 19:22
    
@Martin: An exercise with a hands-on hint to do this is 1.6.K in the notes at math216.wordpress.com (Jan 15 2013 version; presumably all later versions, and most earlier versions). –  Ravi Vakil Feb 16 '13 at 0:56
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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*].

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