Subject: categories: Query about Ab[C]

From: Bill Rowan <[email-masked]>
Date: Wed, 13 Dec 2000 20:19:28 -0800 (PST)

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

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From: Michael Barr <[email-masked]>
Date: Thu, 14 Dec 2000 12:39:57 -0500 (EST)

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.

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From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Thu, 14 Dec 2000 16:25:39 +0000 (GMT)

> those categories C for which Ab[C] is abelian.

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