Subject: categories: RE: cracks and pots 93

From: "Marta Bunge" <[email-masked]>
Date: Sun, 02 Apr 2006 10:03:51 -0400

Dear Eduardo,

I promised to comment on your posting partly since you praise me in having
started this discussion but also since you touch on something very relevant
to the questions that I have been asking.
>
>In recent years J.Baez and his followers have been occupying more and more
>space in the categorical community (this fact is at the starting point of
>the present debate).

You refer to the list of meetings where John Baez has played a prominent
role as speaker in recent years -- and I forgot to mention Coimbra 1999
(School on Category Theory and Applications). I do not recall of anybody,
with the exception perhaps also of Steve Awodey, to have been given so much
attention in recent years.

I was perhaps misled by "theological considerations", namely the Templeton
Foundation and its efforts in trying to mix up science and religion. This is
unfortunately true enough, but it has nothing to do with our present
discussion. This is the conclusion that I have reached, and it is my
obligation to say so publicly.

You offer some interesting explanation for the state of affairs in category
theory meetings.
>
>I think this is so because they have some interesting category theory to
>show, but they are occupying more space than their mathematics deserves
>because they bring a refreshing air to a community until now dominated by
>an old guard that has not shown signs of necessary evolution, and that has
>not being able to attract very good and talented young mathematicians to
>the community. There is now not other exiting body of developments within
>the community. The old guard is being pushed out (prone or supine ?), but,
>alas, not by better mathematicians.

A general comment is that, whereas some of it may be true, there is much
that should be justified, or questioned. For instance, that "they have sone
interesting category theory to show", that "they bring a refreshing air",
that ours is "a community until now dominated by an old guard that has shown
no signs of necessary evolution", that our community "has not been able to
attract very good and talented mathematicians", at present there is "no
other exiting body of developments within the community".

Would you care to be more explicit? At the request of others, and to the
risk of attracting a lot of criticism, I have done so and have been grossly
misunderstood. If I did not request it, others will do so. That way, it
would be easier to respond to the above.

Let us assume for the moment that the is a rift in our category theory
community -- the "old guard" versus the "new guard". We could even put names
to represent each one -- it may seem obvious to many that, whereas Marta
Bunge "must have been influenced by Bill Lawvere (and others)", Eduardo
Dubuc "must be influenced by Andre Joyal (and others)". I am still
speculating. This imaginary discussion I want to refute, and I am sure that
you would want to do the same, or maybe not. But, is there such a rift and,
if so, why take one side against the other?

I believe that *all* progressive forces within category theory (no matter
where they come from) should join efforts rather than split over issues that
very few people understand or care about. I am a great admirer of both Bill
Lawvere and Andre Joyal (and of many others, eg Peter Freyd, Ross Street,
Ieke Moerdijk, to name a few -- of course, beginning with Grothendieck)
because, even if they may have different philosophies of mathematics (in
particular of category theory), they all bring in novel ideas derived from
great experience and insight.

>The old guard is being pushed out (prone or supine ?), but,
>alas, not by better mathematicians.

This too, is cryptic, and too sweep a generalization. I believe that there
is poor mathematics no matter which area of category theory, but there is
also good mathematics both sides of the imaginary (?)  rift. Instead of
making divisions of the sort "the young against the old", which can be used
to justify almost anything, let us make no divisions and try to preserve
excellence and promise in whatever is being done, whether applied or not,
whether fashionable or not, and filter out (as editors of journals,
organizers of conferences) the sort of mathematics that is sure to discredit
us. Who will then be the judges? Obviously good mathematicians, not biased,
completely accountable for their choices, not short-sighted, independent,
very well informed, anxious to preserve high standards. Is there anybody
left after so many requirements? Well, Peter Johnstone is one such person,
but there are several others, of course and it is not for me to make such a
list. Once identified by general consensus (here is the hard part!), let
them consistently make all the decisions (for a certain period of time, at
least). That way, it will be unlikely that the same people be invited over
and over again, and also unlikely that bad papers will be accepted to be
presented at meetings or published in our journals.

Sammy Eilenberg and Saunders MacLane may have represented different
tendencies within category theory, and most of us feel having been
influenced by one rather than by the other (some by both). Yet, there was
never any open rift between them, which was wonderful.

I decided to work on category theory after a brilliant lecture by Eilenberg
at Haverford College in (I think) 1963. I thought -- this is what I want to
do! And there (at Penn, where I was a grad student) was Peter Freyd giving a
course in Algebraic Topology, and later one in Abelian Categories.The sheer
beauty and depth of these new ideas (new then) was overwhelming, and I
decided to ask Peter to let me become his student. As I said publicly at a
recent celebration of Peter's 65th Birthday in Philadephia, this was to
determine the rest of my (I think, most interesting) life. I met Bil Lawvere
later, at a congress in Jerusalem in 1964. His new ideas were in a different
direction, but equally fascinating, and I learnt a lot from him the
following year at the Forschunsinstitut fur Matematik at the E.T.H. in
Zurich, where I also met Anders Kock, Jim Lambek, and Fritz Ulmer, among
others.  I met "the others" (Myles Tierney, Jon Beck, Jean Benabou, Saunders
MacLane, and others) at the first (1966) Oberwolfach meeting, where I was
allowed to present the results of my thesis at the very end of it. I count
as my advisors both Freyd and Lawvere (made official in the Genealogy
Project). It was the confluence of ideas and people working in harmony that
was so wonderful in those days. Why should they be gone forever?  Let us
work together to bring them back, if possible.

Something good has to come out of this discussion, or else it may have been
something worse than just a waste of time. What do you say, Eduardo?

Best wishes,
Marta