Re: Abelian-topos (AT) categories

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

Date: Mon, 3 Nov 1997 10:00:59 -0500 (EST)
From: Peter Freyd <pjf@saul.cis.upenn.edu>

I've had trouble with starting a reply to Vaughan's most recent
posting, the one headed, "Abelian-topos (AT) categories". It makes the
wide gap that's come into being all too painful.

Vaughan wrote,

  I don't see much discussion of closed structure for abelian
  categories.  Why is this?  Am I just reading the wrong stuff, or is
  the closed structure of abelian categories boring, or what?

And that was after he had written,

 I only found out what abelian categories were a week ago.

For a long time Category Theory existed (say, in the Mathematical
Reviews) as a subset of Homological Algebra -- which is a way of
saying that category theory was abelian category theory. (I can't
remember a new result in the theory of abelian categories in the last
quarter-century. I do remember, alas, a bunch of new announcements of
such results.)

I have trouble with Vaughan's phrase "the closed structure" on an
abelian category. There can be many. The category of finite-
dimensional complex representations of a compact group has a
distinguished symmetric monoidal closed structure. If they are viewed
just as categories then for any two compact groups with the same
number of conjugacy classes the categories are isomorphic. If viewed
just as monoidal closed categories then the necessary and sufficient
condition that they be isomorphic is that the groups have isomorphic
character tables. On the other hand, if viewed as a _symmetric_
monoidal closed categories, one can recover the group from the
category. If you want a specific example consider the two non-abelian
groups of order eight, the dihedral and the quaternian. In each case
the plain category of representations is the 5-fold cartesian power of
the category of finite-dimensional complex vector spaces. As monoidal
closed categories they are isomorphic (but not isomorphic with the
5-fold cartesian power of the closed monoidal category of finite-
dimensional complex vector spaces). As symmetric monoidal closed
categories they are different.

Anyway, there's a whole body of material. A lot of it is now viewed as
standard in a number of (non-categorical) subjects and as for all
successfull branches of category theory the theory of abelian
categories is no longer considered to be a branch of category theory.

Back to pratt cats: if one wants to axiomatize those categories that
are products of abelian cats and topoi and not worry about the
intervening families the axioms can be made quite simple. After
saying that it's a regular category with a coterminator contained in
its terminator, I'd start with the  P-E-l-r-/\  structure as in my 
last post, define  TX  as the image of  rX  and prove it to be the
correflection of X into the full subcategory of type-T objects. 
0xX -> X  is easily seen to be the correflection of  X  into the full
subcategory of type-A objects. Then the axiom that these two 
correflections yield a coproduct decomposition for each object allows
one to prove quickly that the category is the cartesian product of the
two correfletive subcategories. The type-T objects clearly form a
topos. All that's needed now is a couple of axioms to make the type-A 
objects abelian.