Re: Abelian-topos (AT) categories

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

Date: Thu, 6 Nov 1997 10:39:35 GMT
From: Michael Barr <barr@triples.math.mcgill.ca>

In reply to Steve Vickers' post, I have a few comments.  First off,
it was not Lubkin-Heron-Freyd-Mitchell who proved the full embedding
theorem.  The first three proved only a faithful functor into set
(and subject to some smallness condition), while Mitchell showed
the full embedding theorem.  And not merely into a Grothendieck abelian
category, but one with a small projective generator.  The analogue would
be an embedding into a set-valued functor category.  Unfortunately,
a beautiful observation of Makkai's shows that that is impossible.  
Makkai pointed out that under a full embedding that preserves finite
limits, finite sums and epis the boolean algebra of complemented
subobjects, which is classified by maps into 1 + 1, would have to
be preserved.  But in a functor category that lattice is complete
and atomic (that is, completely distributive), so that fact, which
is not true for toposes in general, becomes a necessary condition
for the existence of an embedding.  (Is it sufficient?)  There is, of
course, a full embedding theorem for small exact categories, but that
is a lot less than a topos.  But is true that additive + exact = abelian,
so maybe that is also a good analogy.

Michael