From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5529 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: forms Date: Thu, 14 Jan 2010 09:15:17 -0800 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: ger.gmane.org 1263525019 28502 80.91.229.12 (15 Jan 2010 03:10:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Jan 2010 03:10:19 +0000 (UTC) To: Al Vilcius , categories@mta.ca Original-X-From: categories@mta.ca Fri Jan 15 04:10:12 2010 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 4.50) id 1NVcZf-0001JW-TV for gsmc-categories@m.gmane.org; Fri, 15 Jan 2010 04:10:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NVbzv-0007Zl-8Z for categories-list@mta.ca; Thu, 14 Jan 2010 22:33:15 -0400 Content-Disposition: inline In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5529 Archived-At: Al Vilcius wrote: >to which category do multilinear maps belong? >X tensor Y ---> Z linear in k-mod >X cartesian Y ---> Z bilinear >X ---> Y hom Z linear in k-mod >where do the middle arrows live? The slickest answer is perhaps that the middle arrows live in the *multicategory* k-mod: X, Y ---> Z bilinear in k-mod The problem with this is that multicategories are little bit more complicated than categories. http://ncatlab.org/nlab/show/multicategory http://en.wikipedia.org/wiki/Multicategory Because k-mod is a monoidal category (aka tensor category), that is it has a well-behaved operation tensor, we can use mere linear maps X tensor Y ---> Z instead. http://ncatlab.org/nlab/show/monoidal+category http://unapologetic.wordpress.com/2007/06/28/monoidal-categories/ http://en.wikipedia.org/wiki/Monoidal_category And because k-mod is a closed category, that is it has a well-behaved operation hom, we can use mere linear maps X ---> Y hom Z instead. http://en.wikipedia.org/wiki/Closed_category Since these two operations tensor and hom are compatible, in that they correspond to the same multicategory structure, k-mod is in fact a *closed monoidal category*. http://ncatlab.org/nlab/show/closed+monoidal+category http://en.wikipedia.org/wiki/Closed_monoidal_category Category theorists usually turn a bilinear map into a linear map in one or the other of these ways, to avoid multicategories. >>From a structural perpsective, all three perspectives are equivalent, so it really doesn't matter which way you look at them. But multicategories are there if you want them. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]