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