From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3570 Path: news.gmane.org!not-for-mail From: "Ronnie Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: groupoids versus homotopy 1-types Date: Fri, 5 Jan 2007 17:06:06 -0000 Message-ID: <5634.01768818057$1241019384@news.gmane.org> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019383 9298 80.91.229.2 (29 Apr 2009 15:36:23 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:23 +0000 (UTC) To: "categories" Original-X-From: rrosebru@mta.ca Fri Jan 5 13:06:44 2007 -0400 X-Keywords: X-UID: 62 Original-Lines: 79 Xref: news.gmane.org gmane.science.mathematics.categories:3570 Archived-At: I don't have a specific reference in 2-category language but the following should be relevant: The notion of a homotopy theory for groupoids was set up in my 1968 topology book now revised and republished as Topology and Groupoids. In particular \pi_1: spaces \to groupoids preserves homotopies. Fibrations were introduced in an exercise, and developed in later editions. See also Philip Higgins' Categories and Groupoids, (1971) now available as a TAC reprint. The nerve and classifying space of a groupoid are in Graeme Segal's `Classifying spaces and spectral sequences' (IHES) utilising Grothendieck's nerve of a category. These preserve homotopy. The fact that for a CW-complex X, [X,BG] \cong [\pi_1 X, G] is also well known: P.Olum Ann Math 1958? There is also relevant material in Gabriel-Zisman's book, but I do not have it with me. People should also look at 2 papers on groupoids by P A Smith in the Annals, 1951. Hope that helps. Ronnie www.bangor.ac.uk/r.brown ----- Original Message ----- From: "John Baez" To: "categories" Sent: Wednesday, December 27, 2006 6:53 PM Subject: categories: groupoids versus homotopy 1-types > Dear Categorists - > > The following claim should be well-known (or false), > but I don't know a reference: > > Let Gpd be the 2-category consisting of > > groupoids > functors > natural transformations > > and let 1Type be the 2-category consisting of > > homotopy 1-types > continuous maps > homotopy classes of homotopies > > where for present purposes "homotopy 1-types" means "CW complexes with > vanishing higher homotopy groups regardless of the choice of basepoint". > > Claim: Gpd and 1Type are equivalent (or "biequivalent", > in older terminology). > > In fact I bet there is an explicit pseudo-adjunction between them, > with the "fundamental groupoid" 2-functor going one way and the > "Eilenberg-Mac Lane space" 2-functor going the other way. > > Does anyone know for sure? Know a reference? > > Best, > jb > > > > > > > > -- > Internal Virus Database is out-of-date. > Checked by AVG Free Edition. > Version: 7.1.409 / Virus Database: 268.15.6/565 - Release Date: 02/12/2006 >