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From: "Dr. P.T. Johnstone"
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Subject: Re: Query about Ab[C]
Date: Thu, 14 Dec 2000 16:25:39 +0000 (GMT)
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In-Reply-To: <200012140419.eBE4JSF19580@transbay.net> from "Bill Rowan" at Dec 13, 2000 08:19:28 PM
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>
>
> 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