Topos-abelian categories that are neither

[categories <cat-dist@mta.ca>]

Date: Mon, 27 Oct 1997 10:19:51 -0800
From: Vaughan Pratt <pratt@cs.stanford.edu>

Toposes and abelian categories are striking for the number of
elementary properties they have in common (monic+epi = iso, mono/epi is
a unique factorization system, etc. etc.) and the paucity of their
common models, namely just the final category.

This observation prompts the following questions.

1.  Are there any other pairs of large and/or useful classes of
categories whose respective theories have so much in common yet whose
models have so little in common?

2.  What can be said of the class of those categories having all the
elementary properties common to toposes and abelian categories?  In
particular does it contain anything other than toposes and abelian
categories?  And if so, does this outcome change when the language is
extended to say second order logic?

To the extent that both toposes and abelian categories share much
pleasant structure, the models of the intersection of their theories,
for a suitable choice of language, would seem to be a nice class in its
own right.

Vaughan Pratt