From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2817 Path: news.gmane.org!not-for-mail From: "John Baez" Newsgroups: gmane.science.mathematics.categories Subject: Re: lax crossed modules Date: Wed, 21 Sep 2005 11:42:55 -0700 (PDT) Message-ID: <200509211842.j8LIgt423375@math-cl-n03.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 1241018922 5937 80.91.229.2 (29 Apr 2009 15:28:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:28:42 +0000 (UTC) To: categories@mta.ca (categories) Original-X-From: rrosebru@mta.ca Thu Sep 22 17:31:06 2005 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 22 Sep 2005 17:31:06 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EIXaG-0007Ko-Sg for categories-list@mta.ca; Thu, 22 Sep 2005 17:22:20 -0300 X-Mailer: ELM [version 2.5 PL6] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 27 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:2817 Archived-At: Urs Schreiber wrote: > [...] weak (coherent) structure 2-groups (I guess these > are essentially "the same" as lax crossed modules?) [...] Since not everyone will understand this remark by my esteemed coauthor, let me elaborate. There's a nice way to weaken the concept of crossed module. A crossed module is just another way of looking at a group object in Cat - otherwise known as a "categorical group" or "strict 2-group". But, starting with the concept of group object in Cat, one can then weaken the usual group axioms to natural isomorphisms and impose suitable coherence laws, obtaining the notion of "gr-category" or "coherent 2-group". One could then backtrack and formulate this concept so that it resembles the concept of crossed module as closely as possible. I guess this would deserve to be called a "weak crossed module" or something like that. All this stuff except the last paragraph is well-known and summarized here: John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-Groups, Theory and Applications of Categories 12 (2004), 423-491. http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html One might also seek a "lax" version of the concept of crossed module, where "lax" is taken in the Australian sense of replacing equations by morphisms rather than isomorphisms - "lax" as opposed to "pseudo". If I were forced to do this, I'd try to do it by laxifying the concept of group object in Cat. But, I don't see which way all the 2-arrows should point. Best, jb