Bill Rowan asks if there is a simple characterization of those categories C for which Ab[C] is abelian. I doubt if there can be a useful necessary and sufficient condition. A sufficient condition can be found on page 91 of Cats and Alligators, to wit, that the category be effective regular (where "effective" means that every equivalence relation is effective,i.e. it appears as a pullback of a map against itself). Note that the conclusion (abelian) is self-dual but the condition (effective regular) is not. I'm pessimistic about a useful necessary and sufficient condition because of the following: Let C be a category with cartesian squares (needed to define abelian-group-object) such that Ab[C] is abelian. Let C' be a full subcategory closed under cartesian squaring that contains the image of the forgetful functor from Ab[C] back to C. Then Ab[C'] = Ab[C]. An example of the sort of pathological categories to be found among such C' is the category of all groups in which the commutator subgroup is a product of a finite number of simple groups each of which was described prior to 30 June 1973.

I suspect that it is too much to hope for a characterization. But it is sufficient that either C or C^op be exact. For C to be exact, it is required that it have finite limits, coequalizers of equivalence relations, that regular epis be stable under pullback and that every equivalence relation is the kernel pair of its coequalizer. On Wed, 13 Dec 2000, Bill Rowan wrote:

Hi all,

Ab[C] is just my notation for the category of abelian group objects in the category C. I was wondering if there is a simple characterization of those categories C for which Ab[C] is abelian.

Bill Rowan