Subject: categories: Re: A challenge (Re: Demystifying the categorical approach)

From: "Wojtowicz, Ralph" <[email-masked]>
Date: Tue, 11 Apr 2006 08:02:38 -0400

> Steve Stevenson wrote:
>As an outsider, let me say something about outsiders in this.
>In a word, "Where's the rosetta stone?"

>From my experience, "Conceptual Mathematics" by Lawvere and Schanuel
can serve as a Rosetta stone for undergraduate students interested
in dynamic systems. I had the opportunity to use "Conceptual
Mathematics" as a reference in an upper-level applied math class
at the University of Dallas. Sample course materials are available at:
http://www.adjoint-functors.net/teaching/notes1.pdf
http://www.adjoint-functors.net/teaching/exam1.pdf
http://www.adjoint-functors.net/teaching/quiz3.pdf
http://www.adjoint-functors.net/teaching/lifeS.pdf
Lawvere's definitions of symbolic dynamics and chaos were of
particular interest to the students and to me.

I found that categorical ideas also provided useful exercises
and organization into an analysis course I taught using Rudin's
"Principles of Mathematical Analysis". The following notes
provide examples: page 8 of
http://www.adjoint-functors.net/teaching/rudin1.pdf
pages 1--7 and 11 of
http://www.adjoint-functors.net/teaching/rudin2.pdf
pages 2--6, 8, 10--12, and 16 of
http://www.adjoint-functors.net/teaching/rudin4.pdf
There are many other opportunities to introduce categorical notions
throughout the course perhaps using material from "Sets for Mathematics"
by Lawvere and Rosebrugh.

As a graduate student, I found that "Conceptual Mathematics" clarified
concepts from dynamic systems theory and provided insights into Lawvere's
categorical dynamics program and into the core literature on dynamic
systems. See http://www.adjoint-functors.net/aipcasys2.pdf and the
references cited.

For applications to computer science, recent papers by Katis, Rosebrugh,
and Walters have applications to concurrency; provide categorical
definitions of transition systems, bisimulation, and related concepts; and
clarify the treatment of those topics in the books by Arnold and by
Clarke, et al. Although not in textbook form, some of these articles are
accessible to undergraduates. Model-checking and automatic software
verification researches get considerable financial support  from U.S.
government sponsors (see http://www.sba.gov/SBIR).

Another Rosetta stone for students interested in computer science
is the literature on sketch-based data models. Papers by Johnson,
Piessens, Rosebrugh, Steegmans, and others are accessible to advanced
undergraduates with faculty guidance. Interoperability of information systems
is also a focus area of the sponsors mentioned above.

In a previous e-mail, Peter Selinger suggested that "[categorical
research with applications to physics] is supported because it is
original, timely, and interesting". Since taking a non-academic
position, I have found that the novelty of category-based proposals
does get the attention of sponsors looking for new ideas (although
I have not submitted any proposals for physics research). In seeking
applications of categories to new areas, we are fortunate to already
have a theory of such great depth. How would thermodynamics be
taught now if Truesdell had been born 100 years earlier?

I will be eager to purchase an undergraduate text on computability
concepts from a categorical perspective. A few years ago I spent
time trying to learn this material from Soare's text without much
success.

Sincerely,
Ralph Wojtowicz
Metron, Inc.
11911 Freedom Drive, Suite 800
Reston, VA  20190-5602
[email-masked]