Date: Tue, 4 Nov 1997 13:34:49 -0400 (AST)
Subject: Pratt slices 

Date: Tue, 4 Nov 1997 16:55:54 +0000
From: Dr. P.T. Johnstone <P.T.Johnstone@dpmms.cam.ac.uk>

Just a thought about Vaughan's original question: the class of toposes
(and that of pretoposes) is stable under slicing, as are all the
`exactness properties' that they share with abelian categories. Slices
of abelian categories aren't abelian; but, thanks to Aurelio Carboni, 
we know how to characterize them. So we ought surely to be looking for
a common generalization, not of toposes and abelian categories, but of
(pre)toposes and affine categories in Aurelio's sense.

How about it, Peter?

Peter J.


Date: Wed, 5 Nov 1997 17:34:21 -0400 (AST)
Subject: Re: Pratt slices 

Date: Tue, 04 Nov 1997 11:15:46 -0800
From: Vaughan R. Pratt <pratt@cs.Stanford.EDU>


>From: Dr. P.T. Johnstone <P.T.Johnstone@dpmms.cam.ac.uk>
>So we ought surely to be looking for
>a common generalization, not of toposes and abelian categories, but of
>(pre)toposes and affine categories in Aurelio's sense.

In that context let me rephrase my question about adding in Set\op as,
is the common universal Horn theory of Set, Ab, and Set\op, along with
their slices, finitely axiomatizable?  Or (with or without the slices)
is this nice link between Set and Ab confined to the geometric
(discrete) half of mathematics?

Vaughan