Re: Applications for Category Theory

[categories <cat-dist@mta.ca>]

Date: Sat, 23 Aug 1997 13:41:12 -0400 (EDT)
From: F W Lawvere <wlawvere@acsu.buffalo.edu>


This is a partial reply to the inquiry by Dan Yoder of Tazent Systems.

To create a more engineering-friendly mathematics
has been one of the goals of category theory
(at least for me since 1959).That of course doesn't
prevent some people who know a little about it to
claim that it can be enjoyed as "mysticism" and 
that applied mathematics has no place in it.The
fact is that this goal is taking several workers
several years of work to achieve; but it is in
sight.

It must be emphasized that "reading about" algebra
will never suffice to understand its applications.
Indeed no mathematical science can be "comprehensible
to someone without the formal framework".At least
a few conscious acts on the part of the individual 
to learn by participating in the actual scientific
reasoning are necessary.

As an attempt to provide he interested reader with
the materials for doing just that,(1) Steve Schanuel
and I prepared a text, CONCEPTUAL MATHEMATICS
which will be available from Cambridge University 
Press after September 2. It is based on a course
we gave for freshmen at Buffalo several times in the 
early 90's, and aims to provide the reader having no
previous advanced mathematics with a non-watered-down
grasp of some of the basic concepts and examples of
categories. We tried to do this without shrinking 
from correct proofs or precise definitions (as too
many books do on the basis of the absurd theory that 
actual understanding would be incompatible with intuition).
In 1987 I prepared for those having a basic
understanding of categories,(2) an introduction to the
method used in nearly all mathematical applications of
categories, namely the systematic use of categories of
actions (="presheaves"or A-sets) and natural maps 
(=homogeneous or equivariant or intertwining or.. maps)
between them as the first approximation to modelling
any category of situations. This text was written
with computer science specifically in mind, and was
published as the second section of my paper
"Qualitative distinctions between toposes of
generalized graphs" in volume 92 of the
American Mathematical Society's series
CONTEMPORARY MATHEMATICS

I would much appreciate to learn opinions
on the two questions:  a) Is (1) sufficient
background for the student who undertakes
a serious study of (2) ?   and b) Are the
applications alluded to in (2) sufficiently
suggestive to those who want to use that
method ?

Further examples of the kind in (2) are in
my "Kinship and mathematical categories"
which will appear in a volume edited by
Jackendoff and Wynn in memory of John
Macnamara ( who worked to apply categorical 
logic to psychology). Although that paper
is directed to a problem in anthropology,
computer scientists will quickly recognize
the kinship with concurrency and other
problems of interest to them.

Of course there are many writings by other
authors with much the same purpose, but I 
take this opportunity to suggest that (1)
followed by (2) may be an  approximation to a
reasonable course for self-study.

Bill Lawvere