Subject: categories: Re: A question on: <newsgroup sci.math>

From: Ronnie Brown <[email-masked]>
Date: Wed, 21 Jun 2000 15:23:33 +0100

I have a paper with Loday on calculating the third homotopy group of a
suspension of a K(G,1)

51. (with J.-L. LODAY), ``Van Kampen theorems for diagrams of spaces'',
{\em Topology} 26
(1987) 311-334. 

in terms of a new tensor product of non abelian groups. It would not be a
sensible task to rewrite the paper without category theory (considered as a
unifying principle, as a mode for efficient calculation in certain
algebraic structures,  and as a supplier of new algebraic structures). 

To go further back (and higher, of course) Grothendieck's extension of the
Riemann-Roch Theorem uses category theory explicitly. In the 1950's people
were trying to give algebraic proofs of this theorem, when AG came up with
an algebraic proof of a vast generalisation. Then there is the categorical
background to the proof of the Weil conjectures ....

But the question is misplaced - in the early part of the 20th century, I
expect some would ask if set theory was really necessary for a confirmed
problem solver! The history of maths shows that maths greatest contribution
to science, culture and technology has been in terms of expressive power,
to give a language for intuitions which enables exact description,
calculation, deduction. (Exam question: discuss the last statement, with an
emphasis on particular examples!) It also allows for the *formulation* of
new problems, which perhaps cause old interests to lapse as people perceive
there are more exciting things to do. It is a narrow view to regard
`important maths' as necessarily that which solves well known problems, and
so leave evaluation as akin to a sports league table: how old is the
problem? who has worked on it? etc, etc. The progress of maths is much more
complicated and interesting than that! It won't help the public image of
maths if it is seen that mathematicians believe the most important aspect
of their subject is a (to the public) bizarre set of problems which seem to
interest no one else. 

However, it is important that these questions be asked, together with
questions on modes for evaluating `good maths'. As AG remarked in a letter,
maths was held back for centuries for lack of the `trivial' concept of
zero! For more discussion, look at 

http://www.bangor.ac.uk/ma/CPM/cdbooklet/knots-m.html
                                         knots2.html

So let the debate be broadened and continued!

Ronnie Brown

(will someone please forward this to the newsgroup?)

PS I am trying to trace a combinatorics problem solved in the 1950s using
categories of paths, and which was at the time held up as the sort category
theory could not do! Maybe my memory is failing! But the question put on
the newsgroup is an old war horse! 

-------- Original Message --------
Subject: categories: A question on: <newsgroup sci.math>
Date: Mon, 19 Jun 2000 17:13:05 +1000
From: Ross Street <[email-masked]>
To: "[email-masked]" <[email-masked]>

I thought the CatNet would be interested in the following question
which appeared on <newsgroup sci.math>. The letter was pointed out
to me by a colleague at Macquarie.

--Ross

*************************************************
From: [email-masked] (Bill Taylor)
Newsgroups: sci.math
Subject: Query about Category Theory.
Date: 18 Jun 2000 05:36:30 GMT
Organization: Department of Mathematics and Statistics, University of
Canterbury, Christchurch, NewZealand

Does anyone know of any cases where Category Theory (morphisms, functors &
all that) has helped solve an unsolved problem?

That is, a problem in some other branch of math, posed without reference to
categorical ideas, and previously unsolved, that was first solved via
Category T.

I realize that CT provides a unifying framework for many seemingly
disparate
ideas in math, and that is a fine thing of course; but I was just wondering
if it had this problem-solving capability.

TIA for any helpful responses.

----------------------------------------------------------------------------
--- Bill Taylor        [email-masked]