From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5850 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: algebraic models of homotopy types, crossed modules and the free loop space Date: Tue, 25 May 2010 14:48:46 +0100 Message-ID: Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1274810075 7584 80.91.229.12 (25 May 2010 17:54:35 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 25 May 2010 17:54:35 +0000 (UTC) To: "categories@mta.ca" Original-X-From: categories@mta.ca Tue May 25 19:54:33 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OGyKi-0002Hy-9w for gsmc-categories@m.gmane.org; Tue, 25 May 2010 19:54:28 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OGy7B-000417-G1 for categories-list@mta.ca; Tue, 25 May 2010 14:40:29 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5850 Archived-At: There has been discussion on directions for higher category theory and relations with homotopy theory, and I explained that what I have been trying to do might be better described as `higher dimensional group theory'. As an example I mention the following: In March, Niranjan Ramachandram wrote to me as follows: ----------------------------------------------- Permit me to ask a question about the free loop space LX = Maps (S1, X) of a nice connected space X. The path components of LX correspond to conjugacy classes in the fundamental group of X. Is there a good place where the fundamental group of the various path components of LX are calculated? I am particularly interested in the components corresponding to non-trivial conjugacy classes of the fundamental group of X. Perhaps the more natural question is how to describe the fundamental (and higher homotopy) groupoid of LX in terms of the homotopy invariants of X. Is there such an elegant description? ---------------------------------------------------- This led me to write an answer to say that if X is the classifying space of a crossed module of groups, M \to P, then a crossed module of groupoids L(M \to P) can be written down explicitly and which describes completely the homotopy 2-type of L(M \to P); this is rather better than just describing the fundamental groupoid, or even just the various first and second homotopy groups! The paper has been through several revisions and corrections, and the current version is available on arxiv.org/abs/1003.5617 . I hope the next version will include some computer calculations by a colleague! The result is a special case of the description of the weak homotopy type of (BC)^Y when C is a crossed complex and Y is a CW-complex, in terms of CRS(\Pi Y_*,C), using the internal hom for crossed complexes, and \Pi Y* is the fundamental crossed complex of Y with its skeletal filtration, which involves relative homotopy groups, and so relies on a 1991 paper by myself and Higgins on `the classifying space of a crossed complex'. It is interesting to tease out for which circumstances various proposed models of homotopy types are `useful', which again depends on one's aims. The above also generalises the case X=BG is the classifying space of a group, discussed in my 1987 groupoid survey (Bull LMS) for (BG)^Y for a general Y. Ronnie Brown [For admin and other information see: http://www.mta.ca/~cat-dist/ ]