categories: Re: Query about Ab[C]

[Peter Freyd <pjf@saul.cis.upenn.edu>]

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.