From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3555 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: groupoids versus homotopy 1-types Date: Wed, 27 Dec 2006 10:53:40 -0800 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019375 9249 80.91.229.2 (29 Apr 2009 15:36:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:36:15 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Thu Dec 28 17:01:49 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 28 Dec 2006 17:01:49 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1H02Bc-0005aP-LH for categories-list@mta.ca; Thu, 28 Dec 2006 16:49:12 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 57 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:3555 Archived-At: 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