From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6225 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re: The omega-functor omega-category Date: Sat, 25 Sep 2010 13:22:45 +0200 Message-ID: References: Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1285451717 15908 80.91.229.12 (25 Sep 2010 21:55:17 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 25 Sep 2010 21:55:17 +0000 (UTC) Cc: David Leduc , categories To: Ross Street Original-X-From: majordomo@mlist.mta.ca Sat Sep 25 23:55:15 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Ozci8-0002JQ-1D for gsmc-categories@m.gmane.org; Sat, 25 Sep 2010 23:55:12 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39101) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OzchH-0004se-M7; Sat, 25 Sep 2010 18:54:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OzchE-0000RG-Ln for categories-list@mlist.mta.ca; Sat, 25 Sep 2010 18:54:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6225 Archived-At: Dear Ross, concerning the internal hom of strict omega-categories: >>> Given two strict omega-categories C and D, how do you define the >>> strict omega-category of omega-functors between C and D? >> >> There is the Crans-Gray tensor product on StrOmegaCat that makes it >> biclosed monoidal. > > I think David was asking about the simpler cartesian closed structure on > omega-Cat. This is constructed in > > =A0 =A0 =A0 =A0The algebra of oriented simplexes, J. Pure Appl. Algebra 4= 9 (1987) > 283-335 You know all this, but for the record I say the following: The cartesian closed structure has an internal hom that is a restriction of the internal hom wrt the Gray structure. Usually the one of the Gray structure is the one of interest. It is the one closer to the full oo-category theoretic notion (the one with no strictness constraints whatsoever). The Crans-Gray tensor product with its property that G^k otimes G^l is k+l-dimensional is the fix in the globular model for what in the simplicial model is automatic, namely that Delta^k x Delta^l is k+l-dimensional. That this is automatic for the cartesian product in simplicial sets but requires more work for globular sets is one of the reasons why simplicial models for oo-categrories are more highly-developed than globular ones: they are easier. A good brief introduction to this is on the first few pages of Sjoerd Crans= ' A tensor product for Gray-categories http://www.emis.de/journals/TAC/volumes/1999/n2/5-02abs.html (After that introduction the article goes on to refine the globular Gray tensor product to the case of _weak_ (or rather: semi-strict) 3-categories.) But of course for the purposes of David's application (which I don't know about) the strict version of the internal hom might be sufficient. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]