Famous unsolved problems in ordinary category theory

[Paul Taylor <pt09@PaulTaylor.EU>]

Rafael Borowiecki, under the alias Hasse Riemann, asked,
> Are there any famous unsolved problems in category theory?

Ronnie Brown's posting in response to this is a classic, and
deserves to be printed out and pinned up in every graduate
student's office!   I particularly like the military analogy
with the choice between a frontal assault and making the
obstacle obsolete.   The following point is especially important:

> I was early seduced (see my first two papers) by the idea of
> looking for questions satisfying 3 criteria:
> 1) no-one had previously asked it;
> 2) the question was technically easy to answer;
> 3) the answer was important.
> **** Usually it has been 2) which failed! ****

I sent (a version of) the following reply to "categories" when
Rafael first asked the question, but then asked Bob to withdraw
it as I thought I could write it better.  I put off doing so
because other topics were under discussion,  but by his posting
Ronnie has obliged me to send it, since otherwise I would just
be a chicken.

So here goes:

Do I hear taunts of "do you have a Fields Medal?"?

These are a bit like those of "do you have a girlfriend?".

Well, no, I admit it. I don't.   I have a boyfriend (Richard), and
some of you have met him.   If you bear with me, you will see that
this is not a completely frivolous answer, even though it is a
personal one.

My point is that there are analogies between being a gay man and
being a conceptual--constructive mathematician:

- They both involve long periods of self-doubt and pretence in the
   face of real and perceived discrimation.  This is very much still
   real in the mathematical case, as evidenced by that fact that
   categorists and consructivists are largely to be found in
   computer science departments, excluded from mathematics in case
   they might corrupt the youth.

- The result of this is a significantly delayed adolescence --
   I have met gay men going through adolescence in the 50s or 70s.

- Finally, there is pride in being who you are, and the recognition
   of "Honi soit qui mal y pense" -- that it is the people who think
   ill of it that have the problem.  In the words of a song from
   "La Cage aux Folles" that is known as the "sweet potato song",
   "I yam what I yam!".

Before I came out as a categorist, I pretended to be interested
in difficult puzzles,   I was in the British team in the
International Mathematical Olympiad in 1979, but didn't do very
well.  I started a magazine called QARCH, whose total output
in 30 years amounts to less than one of my papers now.

I was taught as an undergraduate by the Hungarian analyst and
graph theorist Bela Bollobas.  He set problems for first year
students problems that took three weeks to solve, if at all.
(Bela is a mathematician of considerable stature -- so great that
it took me five years to notice that he is 10cm shorter than me --
and I remember him with great affection, in case he gets to read this.)

However, I hope that Bela (along with Andrej Bauer, Imre Leader and
Dorette Pronk, who help organise IMO things in Slovenia, Britain and
Canada nowadays), will forgive me if I say that there is something
fundamentally unsatisfying about IMO problems.   Once you have the
solution, that is it.  They are like crosswords or jigsaws or sudoku.

After that I had my delayed adolescence (with an unsuitable
boyfriend).  I studied continuous posets instead of algebraic
ones and categories instead of posets, just to show that I could.
Somebody should have told me to get a proper job as a programmer,
but they didn't have the guts to say it to me.  (If graduate
students ask me for advice nowadays, I do tell them to get proper
jobs, and not surprisingly they (mis)interpret this personally.)

Long after this, the first paper on Abstract Stone Duality was
published on my 40th birthday, more or less.   According to
G H Hardy's depressing "Mathematician's Apology", and to the rules
for getting a Fields Medal,  I was officially finished as a
mathematician.  But it is pretty clear that I have been doing
my best mathematics during my fifth decade.  On the other hand,
all of those gratuitously difficult problems had gone into the mix.

Before I return to the question.  please refer to number 6 in
     en.wikipedia.org/wiki/Hilbert's_problems
which asks for the axiomatisation of physics.  Even in this most
famous collection of gratuitously difficult problems, we find a
conceptual question.

The first of Hilbert's problems is called the "continuum hypothesis",
but is about smashing the continuum into dust.   Elsewhere, he
said "no-one shall expell us from Cantor's Paradise", but I regard
it as a dystopia.  I dream of some eventual escape, returning to the
Euclidean paradise.  There we would actually talk about lines,
circles, compact subsets or whatever, instead of families of subsets
or arcane algebra (or, indeed, category theory).   I am looking for
a language for mathematics that would look like "set theory" (as
mathematicians, not set theorists, perceive it) but would yields
computable continua instead of dust.

More categorically, I believe that there is some notion of category
that is very similar to an elementary topos, but in which all
morphisms are continuous (in particular Scott continuous with
respect to an intrinsic order).

I also believe that these ideas are applicable to other subjects.
When I have made the appropriate tools, I hope to be able to understand
algebraic geometry, which was a complete mystery to me as a student.
I am in princple capable of doing this, BECAUSE I am a categorist,
by following the analogy between frames and rings.

One version of this problem that I still cannot solve is a question
that Eugenio Moggi asked me in April 1993, although I forget the
exact words.   We wanted a class of monos (I said they should be
the equalisers targetted at power of Sigma) that was closed under
composition and application of the Sigma^2 functor (ie taking the
exponential Sigma^(-) twice).

Another is how to embed the category of locales in a CCC WITHOUT
using illegitimate presheaves (Vickers and Townsend) or the axiom
of collection (Heckmann).  When I wrote the original version of
this posting a couple of weeks back, I thought I could solve this
one.  I am still hopeful, but it turns out to be a powerful question,
cf Ronnie's (2) above.

Notice that I give the principal formulation of the question in
vague language, not as a Diophantine equation.  The more specific
the question, the more likely it is to have been the WRONG one.
Asking an impertinent question is the best way of getting a
pertinent answer.

This still involves very difficult problems and hundreds of journal
pages of formal proofs.  But for me the problems serve the concepts
rather than the other way round.  This is the essence of what it is
to be a conceptual mathematician.   Ronnie Brown has told you a
different story of his own, but with the same message.  Many
other experienced categorists (including the ones in higher
dimensions, which Rafael excluded from his original question,
for some reason) would do likewise.

What about Fields Medals?   People will get them, using my work,
two or three generations down the line.

Paul Taylor





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