From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9266 Path: news.gmane.org!.POSTED!not-for-mail From: RONALD BROWN Newsgroups: gmane.science.mathematics.categories Subject: Re: Homotopy hypothesis for contractible operad definitions of weak n-categories Date: Sat, 15 Jul 2017 21:59:49 +0100 (BST) Message-ID: Reply-To: RONALD BROWN NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1500230178 27964 195.159.176.226 (16 Jul 2017 18:36:18 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 16 Jul 2017 18:36:18 +0000 (UTC) Cc: categories@mta.ca To: Timothy Porter , camell.kachour@gmail.com Original-X-From: majordomo@mlist.mta.ca Sun Jul 16 20:36:12 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1dWoOi-0006TR-9z for gsmc-categories@m.gmane.org; Sun, 16 Jul 2017 20:36:04 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52935) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dWoPQ-0004ft-IU; Sun, 16 Jul 2017 15:36:48 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dWoNk-0004B1-Ua for categories-list@mlist.mta.ca; Sun, 16 Jul 2017 15:35:04 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9266 Archived-At: Dear All,=C2=A0 Loday's model is for pointed spaces, and Grothendieck was critical of this = in a letter to me in 1983, of which I have quoted part in the Indag Paper o= n my preprint page. =C2=A0I did not worry about this in the 1980s since the= immediate consequences were quite novel. For example, Ellis and Steiner so= lved the old problem of the critical group for (n+1)-ads, and the nonabelia= n tensor product of groups has been well developed by group theorists (see = www.groupoids.org.uk/nonabtens.html).=C2=A0 What has not been looked at is an input of crossed modules over groupoids, = instead of over groups, and considering first the work of Ellis-Steiner in = that light.=C2=A0(crossed n-cubes of groupoids?) We know from examples that strict 2-fold groupoids are more complicated tha= n homotopy 2-types, see my preprint =C2=A0arXiv:0903.2627v2; and the van Ka= mpen theorem with Loday has not so far been given a version with many base = points, unlike the version in the book Nonabelian Algebraic Topology.=C2=A0 The philosophy given in the Indag Paper has relatively recently been put i= n this form, and so no part of it was discussed with Grothendieck, except t= he idea that n-fold groupoids model homotopy n-types, which, as said above,= is not quite correct, though he thought it "absolutely beautiful". At that= time, 1985, he was starting to write "Recollte et Semaille", a task which= seemed to lead him away from mathematics. =20 The work with Loday shows in many explicit examples how low dimensional ide= ntifications in topology can give rise to high dimensional homotopy invaria= nts, and there are explicit and precise calculations using the higher van K= ampen theorems. Such calculation is not the only aim, but it does give a u= seful test.=20 Best=20 Ronnie ----Original message---- >From : t.porter.maths@gmail.com Date : 15/07/2017 - 07:35 (GMTDT) To : camell.kachour@gmail.com Cc : categories@mta.ca Subject : categories: Re: Homotopy hypothesis for contractible operad defin= itions of weak n-categories Dear All, Can I ask why Loday's cat^n groups are not mentioned? (They have been now.) I know they are not globular, but by spreading out the `weakness' of the higher groupoid structures the axioms end up being strict (and very simple as they are really just abstractions of classical commutator identities). Surely they deserve to be used as a reference point to compare some of the other candidates. Loday's models work for *all *n-types for finite n. (I do not know how to handle general homotopy types using any similar methodology.) Tim [For admin and other information see: http://www.mta.ca/~cat-dist/ ]