Subject: categories: Re: Regular embedding

From: [email-masked] (Ross Street)
Date: Fri, 22 May 1998 14:05:30 +1000

The proof Michael Barr had in mind at the time of writing his 1986 JPAA
paper on representations of categories, and has now clarified, is truly
beautiful.  I may have misrepresented what the problem was with presenting
the transfinite induction part of the proof in my course.  It is a question
of time rather than difficulty.  I am not known amongst my colleagues as
one who shrinks from teaching or avoiding hard proofs.  The point is that I
have a course consisting of 24 lectures to a group including a (bright) 4th
year student (who had hardly heard of categories before the course), a
graduate students in number theory, physics and computer science, my own 5
PhD students, and a professional category theorist.  I am trying to keep it
interesting for all. The students (even the ones who have done a formal
set-theory course) have not really used ordinals; so I would need to
sacrifice some category theory to provide that background. I am still
undecided about this; perhaps, the transfinite induction and the embedding
theorem can be an appendix to the course.

Perhaps the most novel aspect of the course, considering my 2-cell
background, is that I have given 18 lectures so far without introducing
natural transformations. I'm about to talk about adjunctions without them
too.

I must admit that the course content has evolved on the run. But looking
back, I think it has some unity of purpose.  I headed straight for the
definition of regular category as having a terminal object, pullbacks,
strong epi - monic factorizations, and stability of strong epis under
pulling back along arbitrary arrows.  We developed the method of
generalised elements and constructed the Poset-enriched category of
relations in a regular category.  We proved relations with right adjoints
are graphs of arrows.  We proved strong epis are regular. Equivalence
relations and (Barr-)exact categories were examined.

[One of my (intended?) questions to Michael Barr was whether he knew of a
diagram lemma which could be proved significantly quicker using the
embedding theorem than by the generalised-element/relations technology.]

I proved in detail that an exact additive category is abelian (as defined
in Freyd's book "Abelian Categories" - beginning of Chapter 2). Not only is
that a deep theorem of category theory (in my opinion - despite not needing
a transfinite argument) but it is a microcosm of categorical ideas that
have proved useful in other contexts.  We proved the Five Lemma, Snake
Lemma using relations in the abelian category.

So far I have set 15 exercises: some quite challenging. One was to prove
Cat is not regular (I gave a bit of a hint).

Now I am discussing 2-sided discrete fibrations (just for categories) in
depth.  These are being advertised as the "relations" of category theory.
In fact, this means we do have presheaf categories disguised as  DFib(A,1);
so some natural transformations are there hiding there.

I have this weekend to decide where we go next!

But I think there is enough meat in all this to deal with sceptical colleagues.

When it comes to a game of "my subject's harder than yours, nya, nya", I
would also argue that the concepts category theory has to offer are just as
hard to really master as difficult theorems.  This is why many
mathematicians calculate hard to do what we can do with our concepts.

Regards,
Ross

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From: Michael Barr <[email-masked]>
Date: Fri, 22 May 1998 11:25:25 -0400 (EDT)

I certainly welcome Ross' clarification of his problem.  One possibility
would be to give the proof except for the existence of sufficient regular
functors and either refer the students to the paper or (better) write it
out carefully and distribute it.  One thing I have given only a little
thought to is whether it can be done using a maximal principle argument.
The point is that it is not a question of extending a map to a larger and
larger subobject, but of building larger and larger objects and not as
subobjects of something already given.  This makes it different from the
proof, say, that divisible abelian groups are injective (from which the
existence of sufficient injectives in any module category follows easily).
I think a maximal principle argument goes down a lot more easily than one
based overtly on ordinals or transfinite induction.

Ross, it sounds like a beautiful course and if you have notes, I would
like to see them. 

Michael