From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6095 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Makkai's suggestion Date: Fri, 3 Sep 2010 12:04:55 +0800 Message-ID: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1283556502 31556 80.91.229.12 (3 Sep 2010 23:28:22 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 3 Sep 2010 23:28:22 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Sat Sep 04 01:28:20 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OrfgC-00061M-JH for gsmc-categories@m.gmane.org; Sat, 04 Sep 2010 01:28:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47284) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Orff5-0001IM-R3; Fri, 03 Sep 2010 20:27:11 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Orff3-0007f3-HG for categories-list@mlist.mta.ca; Fri, 03 Sep 2010 20:27:09 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6095 Archived-At: Greg writes: > Of course, after following such a path of least resistance, a journeyman > categoryist might look at a variety of alternatives and consider their > trade-offs. In this effort, the very quality you point out is of great > utility in developing a genuine understanding of the design space. The > initial presentation, this path of least resistance formulation, however, > ought to have a precise sense in which it is *initial*, like an initial > algebra. Leinster's refinement of Batanin's approach defines weak infinity-categories as algebras of an "initial globular operad with contractions". Here "globular" means we're doing infinity-categories in the obvious way, where given two n-morphisms f,g: x -> y we can talk about (n+1)-morphisms from f to g. "Algebra of a globular operad with contractions" means we can compose these n-morphisms in all the pictorially obvious ways, and every pictorially plausible law holds *up to a higher morphism*. "Initial" means we're doing this in exactly the right way: for example, there aren't any *extra* ways of composing morphisms, and we're not sticking in *too many* of these higher morphisms. I am sure people will eventually come up with better ways to do infinity-category theory. Eventually most math majors will learn it in college (unless our current civilization collapses in less than, say, 150 years). But the approaches we've got right now are already pretty good. Learn 'em! Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]