From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6160 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: Evil in bicategories Date: Mon, 13 Sep 2010 23:28:35 -0700 Message-ID: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1284504970 3152 80.91.229.12 (14 Sep 2010 22:56:10 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 14 Sep 2010 22:56:10 +0000 (UTC) Cc: categories , David Leduc ,droberts@maths.adelaide.edu.au To: Toby Bartels Original-X-From: majordomo@mlist.mta.ca Wed Sep 15 00:56:08 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 1OveQ4-00029l-5x for gsmc-categories@m.gmane.org; Wed, 15 Sep 2010 00:56:08 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:32923) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OveOg-00046F-80; Tue, 14 Sep 2010 19:54:42 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OveOZ-00025z-9z for categories-list@mlist.mta.ca; Tue, 14 Sep 2010 19:54:35 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6160 Archived-At: On Mon, Sep 13, 2010 at 11:08 PM, Michael Shulman wrote: > This is also true of Batanin's definition, which takes as basic > underlying data a globular set....potentially including (I believe) > pretty much all definitions of higher category. I wrote this without thinking hard enough; sorry. Of course, one also has to consider the extra structure placed on the underlying data. Extra structure of the "horn-filling" variety, as in Joyal's and Street's definitions, consists of conditional assertions that certain dependent types are inhabited, which is certainly "non-evil." I would expect that all the "non-algebraic" definitions could be dealt with similarly; for instance, the Simpson-Tamsamani definition involves also the assertion that certain maps of (n-1)-categories are equivalences, which should itself be a "non-evil" assertion based again on inhabitation of certain dependent types. But it would be tricky to write all of that out carefully. For Batanin-type definitions, it is going to depend on what operad you pick; for instance strict omega-categories are definitely "evil." But I would guess that if you use a "CW operad" which is built up freely, as an operad, by attaching operations of successively higher dimension whose boundaries are composite operations of lower dimension (which is how I usually think of an operad for weak higher categories), then its algebras should also be definable in a "non-evil" way. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]