From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9254 Path: news.gmane.org!.POSTED!not-for-mail From: Jamie Vicary Newsgroups: gmane.science.mathematics.categories Subject: Homotopy hypothesis for contractible operad definitions of weak n-categories Date: Tue, 11 Jul 2017 22:21:01 +0100 Message-ID: Reply-To: Jamie Vicary NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1499862989 8524 195.159.176.226 (12 Jul 2017 12:36:29 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 12 Jul 2017 12:36:29 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Wed Jul 12 14:36:25 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 1dVGsS-0001ur-PY for gsmc-categories@m.gmane.org; Wed, 12 Jul 2017 14:36:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52077) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dVGt7-0006q3-4z; Wed, 12 Jul 2017 09:37:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dVGrV-0002aF-Qu for categories-list@mlist.mta.ca; Wed, 12 Jul 2017 09:35:25 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9254 Archived-At: Hi, Batanin, Leinster and other have presented related definitions of weak n-groupoid in terms of contractible globular operads. I personally find these definitions of "contractible n-groupoids" extremely beautiful. I am interested to learn what evidence we have that the homotopy hypothesis might be true for (at least one of) these definitions. Some good evidence is provided by Peter LeFanu Lumsdaine's [1] proof that a homotopy type gives rise to an infinity-groupoid in the sense of Leinster. There is other work along similar lines. But, as far as I am aware, it remains possible that contractible n-groupoids might in general be weaker structures than homotopy n-types. A fun way to investigate this would be to verify small instances of phenomena associated to the periodic table in contractible n-groupoids. For example, Christoph Dorn has shown me a proof that the Eckmann-Hilton argument holds in a Leinster 2-category; that is, for an object X, and for 2-morphisms f,g:id[X]-->id[X], we have f.g=g.f, thereby establishing one of the first phenomena predicted by the periodic table. Have any higher phenomena from the periodic table been verified? Or, is there other evidence that contractible n-groupoids behave "homotopically" in general? Best wishes, Jamie [1] http://peterlefanulumsdaine.com/research/Lumsdaine-Weak-omega-cats-from-ITT-LMCS.pdf [For admin and other information see: http://www.mta.ca/~cat-dist/ ]