From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3742 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Tensor products (and C*-algebras) Date: Tue, 1 May 2007 08:53:18 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019495 10093 80.91.229.2 (29 Apr 2009 15:38:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:38:15 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Tue May 1 12:12:36 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 01 May 2007 12:12:36 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HityA-0003Au-KA for categories-list@mta.ca; Tue, 01 May 2007 12:08:46 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 1 Original-Lines: 20 Xref: news.gmane.org gmane.science.mathematics.categories:3742 Archived-At: I think it important to note that tensor products have no universal mapping property and no categorical definition. Given an internal hom, there might or might not be a tensor product that provides a left adjoint for it. Given a bifunctor, there might or might not be an internal hom (or two if the bifunctor is not symmetric) right adjoint to it. Another point is that the tensor product on abelian groups (or modules over any commutative right) has two universal mapping properties: It provides left adjoints for the quite obvious (but still not categorically defined) internal hom and also represents the functor that takes A to the functor of bilinear maps out of A x B. In any case, it makes no sense to ask what is the "right" tensor product. The right tensor product will be the one that is appropriate to the job you want to do with it. Michael