From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5756 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Q. about monoidal functors Date: Fri, 7 May 2010 12:48:42 -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 1273275905 22081 80.91.229.12 (7 May 2010 23:45:05 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 7 May 2010 23:45:05 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Sat May 08 01:45:03 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 1OAXE6-00061f-Qm for gsmc-categories@m.gmane.org; Sat, 08 May 2010 01:45:03 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OAWlO-0001d8-QK for categories-list@mta.ca; Fri, 07 May 2010 20:15:23 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5756 Archived-At: Fred E.J. Linton wrote in part: >Steve Lack wrote: >>Such a T is called a symmetric monoidal functor. >Thanks for helping dispel my illusion that all monoidal >functors might necessarily be thus symmetric :-) : Something like this is true, however. First, every monoidal natural transformation is symmetric monoidal (assuming that it goes between symmetric monoidal functors at all). Also, there is the concept of braided monoidal categories that lies between monoidal categories and symmetric monoidal categories. And every braided monoidal functor is symmetric monoidal (assuming that it goes between symmetric monoidal categories at all). Each of these facts is trivial by itself; for example, the definition of symmetric monoidal functor that you wrote down makes sense for a functor between braided monoidal categories; it is simply the definition of braided monoidal functor, and there is nothing more to add when the braiding is symmetric. But the entire pattern is interesting: PC -- PF -- PNT -- ENT MC -- MF -- MNT -- ENT BMC -- BMF -- MNT -- ENT SMC -- BMF -- MNT -- ENT SMC -- BMF -- MNT -- ENT SMC -- BMF -- MNT -- ENT (etc) (To fit this all on the screen, I have used initialisms: "Categories", "Functors", "Natural transformations", "Equality of", "Pointed", "Monoidal", "Braided", "Symmetric".) The thing to notice is that each column stabilises one row earlier than the column before it. The columns stabilise because there is nothing more to write down. * John Baez, Some definitions everyone should know. http://math.ucr.edu/home/baez/qg-winter2001/definitions.pdf (This discusses strong monoidal functors between weak monoidal categories, but it is easy enough to generalise to lax monoidal functors or to specialise to strict monoidal categories.) It's possible that the columns stabilise only through our ignorance (as once we were ignorant that BMC were there between MC and SMC). However, there is a general theory of k-tuply monoidal n-categories which confirms the pattern, although some of that is still conjecture. * nLab, k-tuply monoidal n-categories http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]