From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5528 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Re: forms Date: Thu, 14 Jan 2010 16:19:59 +0100 Message-ID: References: Reply-To: Andrej Bauer NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1263525012 28486 80.91.229.12 (15 Jan 2010 03:10:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Jan 2010 03:10:12 +0000 (UTC) To: Al Vilcius , categories@mta.ca Original-X-From: categories@mta.ca Fri Jan 15 04:10:05 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 1NVcZX-0001GU-MW for gsmc-categories@m.gmane.org; Fri, 15 Jan 2010 04:10:03 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NVbzJ-0007YY-HN for categories-list@mta.ca; Thu, 14 Jan 2010 22:32:37 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5528 Archived-At: On Wed, Jan 13, 2010 at 6:18 PM, Al Vilcius wrote: > Dear Categorists, > Please help me relieve my confusion on a simple question: > > to which category do multilinear maps belong? > > For example, in vector spaces (k-mod) > there are the usual bijective correspondences: > > X tensor Y =C2=A0---> =C2=A0Z =C2=A0 =C2=A0 linear in k-mod > ____________________________________ > > X cartesian Y =C2=A0---> =C2=A0 Z =C2=A0 =C2=A0 bilinear > ____________________________________ > > X =C2=A0---> =C2=A0Y hom Z =C2=A0 =C2=A0 linear in k-mod > > where do the middle arrows live? I always understood the correspondence between the first and the second line as saying "don't talk about bilinear maps on products--talk about linear maps on tensor products instead". But if you twisted my arm (and I did not exectute a proper defense) I would cook up the following: Take the category whose objects are tuples of vector spaces and a morphism (X_1, ..., X_n) -> (Y_1, ..., Y_m) is an m-tuple (f_1, f_2, ..., f_m) of multi-linear maps f_i : X_1 \times ... \times X_n -> Y_i and composition is composition. Doesn't that work? So the answer to your question is: the cartesian sign in the second row is a mirage. But that's something we can easily figure out: it makes no sense to talk about a bilinear map unless we are told how its domain is decomposed into a product (there can be many ways). With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]