categories: Re: Query about Ab[C]

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

> 
> 
> 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

You can't hope to characterize them: knowing properties of Ab[C] can't
tell you everything about C. For example, if C has a strict terminal object,
then Ab[C] is abelian (because it's degenerate), but that gives you no
information about what else C might contain. If you're looking for a
sufficient condition on C, a canonical one is "Barr-exact" (= effective
regular, in Freyd's terminology): Ab[C] inherits Barr-exactness from C,
and abelian is equivalent to Barr-exact plus additive. Conversely, every
abelian category A is isomorphic to Ab[C] for a suitable Barr-exact C,
namely C = A.

Peter Johnstone