Re: Famous unsolved problems in ordinary category theory

["Ronnie Brown" <ronnie.profbrown@btinternet.com>]

In reply to Hasse Riemann's question (see below):

I remember being asked this kind of question at a Topology conference in
Baku in 1987.  It is worth discussing the background to this, as someone who
has never gone for a `famous problem', but found myself trying to develop
some mathematics to express some basic intuitions.

Saul Ulam remarked to me in 1964 at my first international conference
(Syracuse, Sicily) that a young person may feel the most ambitious thing to
do is to tackle a famous problem; but this may distract that person from
developing the mathematics most appropriate to them. It was interesting that
this remark came from someone as good as Ulam!

G.-C. Rota writes in `Indiscrete thoughts' (1997):
What can you prove with exterior algebra that you cannot prove without it?"
Whenever you hear this question raised about some new piece of mathematics,
be assured that you are likely to be in the presence of something important.
In my time, I have heard it repeated for random variables, Laurent Schwartz'
theory of distributions, ideles and Grothendieck's schemes, to mention only
a few. A proper retort might be: "You are right. There is nothing in
yesterday's mathematics that could not also be proved without it. Exterior
algebra is not meant to prove old facts, it is meant to disclose a new
world. Disclosing new worlds is as worthwhile a mathematical enterprise as
proving old conjectures. "

It is like the old military question:  do you make a frontal attack; or find
a way of rendering the obstacle obsolete?

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!

Of course you do not find such questions where everyone is looking! It could
be interesting to investigate how such questions arise, perhaps by pushing a
point of view as far as it will go, or seeing a new analogy.

"If at first, the idea is not absurd, then there is no hope for it." Albert
Einstein
It could be interesting to investigate historically:

 if (let us suppose) category theory has advanced without a fund of famous
open problems, how then has it advanced?

One aim of mathematics is understanding, making difficult things easy,
seeing why something is true. Thus improved exposition is an important part
of the progress of mathematics (even if this is ignored by Research
Assessment Exercises). R. Bott said to me (1958) that Grothendieck was
prepared to work very hard to make something tautological. By contrast, a
famous algebraic topologist replied to a question of mine about his graduate
text by asking: `Is the function not continuous?' He never gave me a proof!
And I never found it! (Actually the function was not well defined, but that
I could fix!)
Grothendieck wrote to me in 1982: `The introduction of the cipher 0 or the
group concept was general nonsense too, and mathematics was more or less
stagnating for thousands of years because nobody was around to take such
childish steps ...'. See also

http://www.bangor.ac.uk/~mas010/Grothendieck-speculation.html

The point I am trying to make is that the question on `open problems' raises
issues on the nature of,  on professionalism in, and so on the methodology
of, mathematics. It is a good question to start with.

Hope that helps.

Ronnie Brown























----- Original Message -----
From: "Hasse Riemann" <rafaelb77@hotmail.com>
To: "Category mailing list" <categories@mta.ca>
Sent: Tuesday, June 02, 2009 5:31 PM
Subject: categories: Famous unsolved problems in ordinary category theory





Hello categorists

I don't know what to make of the silence to my question.
This is the easiest question i have. I can't believe it is so difficult.
It is not like i am asking you to solve the problems.

There must be some important open problems in ordinary category theory.
There are plenty of them in the theory of algebras and
in representation theory, so there should be more of them in category
theory.

Especially if you broaden the boundaries a bit of what ordinary category
theory is.
Take for instance:
model categories,
categorical logic,
categorical quantization,
topos theory-locales-sheaves.
But i had originally pure category theory in mind.

Best regards
Rafael Borowiecki


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