From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/932 Path: news.gmane.org!not-for-mail From: john baez Newsgroups: gmane.science.mathematics.categories Subject: Re: query Date: Tue, 17 Nov 1998 20:12:45 -0800 (PST) Message-ID: <199811180412.UAA15669@charity.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017346 28377 80.91.229.2 (29 Apr 2009 15:02:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:02:26 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Wed Nov 18 12:51:08 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id KAA23834 for categories-list; Wed, 18 Nov 1998 10:53:47 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.4 PL24 PGP3 *ALPHA*] Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 42 Xref: news.gmane.org gmane.science.mathematics.categories:932 Archived-At: Jim Stasheff writes: > The language of higher category theory in more than analogous > to homotopy theory, especially in the cellular version. What > is the appropriate reference for a non-categorical reader? > Have the A_\infty categories of Smirnov or Fukaya been treated > in the categorical literature?? Unfortunately the truly appropriate reference has not yet been written, because the equivalence between weak infinity-groupoids and homotopy types has not yet worked out in full detail, at least not in the cellular version. The *dream* of translating all of homotopy theory into higher category theory is outlined in: John Baez and James Dolan, Categorification, to appear in Proceedings Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997, eds. Ezra Getzler and Mikhail Kapranov, preprint available as math.QA/9802029. This also has lots of references to different places where various bits of the dream have been realized. Basically the dream consists of working out the following correspondence: HIGHER CATEGORY THEORY HOMOTOPY THEORY omega-groupoids homotopy types n-groupoids homotopy n-types k-tuply groupal omega-groupoids homotopy types of k-fold loop spaces k-tuply groupal n-groupoids homotopy n-types of k-fold loop spaces k-tuply monoidal omega-groupoids homotopy types of E_k spaces k-tuply monoidal n-groupoids homotopy n-types of E_k spaces stable omega-groupoids homotopy types of infinite loop spaces stable n-groupoids homotopy n-types of infinite loop spaces Z-groupoids homotopy types of spectra How do A_infinity categories fit in? As far as I can tell they should correspond to omega-categories where all j-morphisms are invertible for j > 1.