From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6090 Path: news.gmane.org!not-for-mail From: Greg Meredith Newsgroups: gmane.science.mathematics.categories Subject: Re: Makkai's suggestion Date: Wed, 1 Sep 2010 14:42:16 -0700 Message-ID: References: Reply-To: Greg Meredith NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1283432942 9639 80.91.229.12 (2 Sep 2010 13:09:02 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 2 Sep 2010 13:09:02 +0000 (UTC) Cc: categories To: John Baez Original-X-From: majordomo@mlist.mta.ca Thu Sep 02 15:09:00 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 1Or9XG-0001W3-Ns for gsmc-categories@m.gmane.org; Thu, 02 Sep 2010 15:08:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:56635) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Or9To-00031b-Mb; Thu, 02 Sep 2010 10:05:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Or9Tg-000330-2y for categories-list@mlist.mta.ca; Thu, 02 Sep 2010 10:05:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6090 Archived-At: Dear John, i have always taken this situation 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). to be a pretty clear critique of category theory's basic formulation. This may be too idealistic, but i've always felt that an ideally robust formulation would admit a meta-theory that makes n-categories "just fall out". The fact that they are so hard to formulate suggests that the basic design of the original presentation misses something crucial. i confess that taken together with the fact that it is exceptionally hard to get categorical composition to line up with parallel composition (in the sense of concurrent computation) in a manner that respects Curry-Howard, has really made me evaluate category theory as still very much a work in progress. Personally, i have wondered if there is a presentation that takes monad as the fundamental building block. i think this might not be too much of a stretch goal, actually, as monad as polymorphic comprehension is now well-established. Best wishes, --greg On Mon, Aug 30, 2010 at 9:34 PM, John Baez wrote: > 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/ ]