From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5767 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Q about_monoidal_functors? Date: Sat, 8 May 2010 22:54:55 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1273443086 27724 80.91.229.12 (9 May 2010 22:11:26 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 9 May 2010 22:11:26 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Mon May 10 00:11:24 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OBEia-0007yu-NV for gsmc-categories@m.gmane.org; Mon, 10 May 2010 00:11:24 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OBEG8-0000hP-J7 for categories-list@mta.ca; Sun, 09 May 2010 18:42:00 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5767 Archived-At: Andre Joyal wrote in part: >We should use a similar terminology for spaces and maps. >E-n space <--> E-n map >Also for (higher) categories and functors. >monoidal category <---> monoidal functor >braided monoidal category <----> braided monoidal functor >2-braided monoidal category <--> 2-braided monoidal functor >3-braided monoidal category <--> 3-braided monoidal functor >...... >symmetric monoidal category <--> symmetric monoidal functor I agree, one should say "symmetric monoidal functor"; if nothing else, that indicates that the source and target are symmetric (not merely braided) monoidal categories. I only put "BMF" in my table to show a particular pattern. Depending on how you write down the definitions, that a braided monoidal functor between symmetric monoidal categories is the same thing as a symmetric monoidal functor between them is either an utter triviality or a deep and interesting theorem; but in either case, we need the words to state it. (I do agree with John about preferring "k-tuply monoidal", but I'll let him make that argument.) >A (n+1)-braided monoidal n-category is symmetric by >the stabilisation hypothesis. >I believe that a (n+1)-braided monoidal functor >between (n+1)-braided monoidal n-categories is symmetric. I think that you mean to say (which is even stronger) that an n-braided monoidal functor between SM n-categories is symmetric. More generally, a k-braided monoidal l-transfor between SM n-categories is symmetric as long as k + l is greater than or equal to n. (A 0-transfor is a functor, a 1-transfor is a natural transformation, etc. This numbering is due to Sjoerd Crans; feel free to argue that it's off.) More generally yet, a k-braided monoidal l-transfor between m-braided monoidal n-categories is m-braided, as long as k + l >= n, regardless of the value of m (although we need m >= k for the antecedent to make sense). >Is this part of the official stabilisation hypothesis? I don't know what's official, but I'll claim the conjecture above as mine if nobody else has written it down yet. (^_^) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]