Re: A well kept secret?

[Andrew Stacey <andrew.stacey@math.ntnu.no>]

This discussion has been very interesting.  I have a couple of comments and
a request, but first a little background.  I've only recently truly
encountered category theory - I describe myself as a differential topologist
and as yet see no reason to change that description, but I've increasingly
needed to use at least the language of category theory to express some of the
things that I come across in algebraic and differential topology and this has
led to me learning some category theory at last.

However, I sometimes feel as though I've stumbled into a party by mistake and
can't find the way out.  I'm quite enjoying being at the party, I ought to
say, but every now and then I sit down in a corner and wonder how I got in,
and also suspect that I missed the Big Announcement at the beginning that said
what the party was for.

All this discussion about a "well kept secret" has gone a bit over my head.
I'm not sure what the secret is!  My forays into the categorical landscape
have been two-fold: understanding operations in cohomology theories and
understanding smooth spaces.  The first, paradoxically, relates to trying to
un-categorify something ("decategorify" now has a mathematical meaning and
I don't intend that); namely, the previous description of what we wanted to
understand was extremely categorical and we wanted a much more "hands on"
description, but that actually just led us from one categorical description to
another (our own journey was quite tortuous, I should say).  The second foray
wouldn't have happened if those I'd been talking to hadn't already been
speaking in categories - I had to learn the language just to join the
conversation.

So when you all talk of a "well kept secret" and something that "went wrong in
the 60s" (didn't everything?), please remember that some of us weren't even
born in the 60s, let alone thinking about mathematics, so haven't a clue
what's going on.  And, as I've tried to say above, I'm an outsider but one
with a favourable view of category theory so if it's hard for me to figure out
what the fuss is about, I'm not surprised that it's hard for anyone further
out.

Let me make these remarks a little more concrete with a request (or
a challenge if you prefer).  In my department, the colloquium is called
"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've
been writing too many scripts lately!).  I'm giving this talk in January.  My
original plan was to say something nice and differential, with lots of fun
pictures of manifolds deforming or knots unknotting, or something like that.
However, the discussion here has set me to thinking about saying something
instead about category theory.  It is a pearl of mathematics, it does have
a certain beauty, there's certainly a lot that can be said, even to a fairly
applied audience as we tend to have here (it is the Norwegian university of
Science and Technology, after all), even without talking about programming
(about which I know nothing).

But for such a talk, I need a story.  I don't mean a historical one (I'm not
much of a mathematical historian anyway), I mean a mathematical one.  I want
some simple problem that category theory solves in an elegant fashion.  It
would be nice if there was one that used category theory in a surprising way;
beyond the idea that categories are places in which things happen (so perhaps
about small categories rather than large ones).

I'm not trying to get anyone to write my talk for me!  It's just that as
someone who only recently engaged with category theory then I'm aware
- painfully aware - that I often miss the point.  But to counter that, then as
  someone who only recently engaged with category theory then I can still
remember fairly vividly why I like it and what convinced me that it was worth
thinking about (and learning about), which will hopefully give the talk
a little more omph.

Thanks in advance for your suggestions,

Andrew Stacey

PS I just remembered something else I was going to mention.  Someone else
mentioned MathOverflow.  Well, there was a question about what was missing
from undergraduate mathematics.  I said "category theory".  It currently lies
5th in the list (out of 28, my other suggestion "how to write with chalk so it
doesn't squeak" is 12th).  More interesting than it's placing is the vast
number of comments that followed, mostly saying that too much "abstract
nonsense" would be off-putting to students.  You can read it at:

http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]