From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/456 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Applications for Category Theory Date: Mon, 25 Aug 1997 11:07:42 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016977 25852 80.91.229.2 (29 Apr 2009 14:56:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:56:17 +0000 (UTC) To: categories Original-X-From: cat-dist Mon Aug 25 11:08:43 1997 Original-Received: by mailserv.mta.ca; id AA07350; Mon, 25 Aug 1997 11:07:42 -0300 Original-Lines: 80 Xref: news.gmane.org gmane.science.mathematics.categories:456 Archived-At: Date: Sat, 23 Aug 1997 13:41:12 -0400 (EDT) From: F W Lawvere 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