From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6082 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: Makkai's suggestion Date: Tue, 31 Aug 2010 12:34:46 +0800 Message-ID: References: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1283299370 7205 80.91.229.12 (1 Sep 2010 00:02:50 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 1 Sep 2010 00:02:50 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Sep 01 02:02:49 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Oqamu-0006DQ-H3 for gsmc-categories@m.gmane.org; Wed, 01 Sep 2010 02:02:48 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52729) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Oqam0-00036J-NT; Tue, 31 Aug 2010 21:01:52 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Oqalx-0004nQ-2J for categories-list@mlist.mta.ca; Tue, 31 Aug 2010 21:01:49 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6082 Archived-At: David wrote: > http://ncatlab.org/johnbaez/show/Towards+Higher+Categories > > Thank you for the reference. But I don't know where to start. Start by reading the above book together with Cheng and Lauda's "Higher categories: an illustrated guidebook": http://www.cheng.staff.shef.ac.uk/guidebook/ and Leinster's "A Survey of Definitions of n-Category": http://arxiv.org/abs/math/0107188 Then try Lurie's "Higher Topos Theory": http://arxiv.org/abs/math/0608040 They're all free online! Expect to spend a decade on this stuff. Or, wait two decades for people to polish it up, and then spend half a decade learning the basics and half a decade learning what people have done in the next two decades. That may be more efficient. Is there a definitive definition of omega-categories somewhere in the > literature or is it still unknown? There are *several* definitions that are almost surely "right" and likely to be studied for many years hence. There is no particular reason to expect that one definition will be best for all applications - but there's a lot of reason to expect that all the "right" definitions will be shown to be equivalent (in a rather subtle sense). > Can it be stated in elementary terms (I mean in terms of object, arrows, > ... without references to simplicial sets or topology) ? > You should learn to love simplicial sets - they're way too important to avoid! If for some reason you're allergic to simplicial sets, you might like Batanin's definition of omega-categories. But then you need to like operads. You could state it without operads, but then it becomes quite long. The book by Cheng and Lauda takes various definitions and makes them less scary by illustrating how they work with lots of pictures. > In the definition of a bicategory, one could replace the coherence > axioms by the statement that all diagrams built from the canonical > ismorphisms commute. Can it be generalized to n=3, ... , omega. > You could say that's the basic idea behind Batanin's definition. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]