From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4656 Path: news.gmane.org!not-for-mail From: "Mike Stay" Newsgroups: gmane.science.mathematics.categories Subject: Re: Matrices Category Date: Tue, 14 Oct 2008 10:43:30 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020085 14245 80.91.229.2 (29 Apr 2009 15:48:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:48:05 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Oct 15 08:45:46 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 15 Oct 2008 08:45:46 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kq4i7-0004yv-Hu for categories-list@mta.ca; Wed, 15 Oct 2008 08:38:39 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 33 Xref: news.gmane.org gmane.science.mathematics.categories:4656 Archived-At: On Mon, Oct 13, 2008 at 9:34 AM, Hugo Macedo wrote: > Hello > > I'm trying to study the Category of Matrices but I found almost nothing. Do > you know where > I can find information about them? I think you may have more success searching for information about the category Vect_K of vector spaces and K-linear transformations between them, where K is the field of interest (usually the reals K=R or the complex numbers K=C). > More specifically can we consider the tensor product as the product > bi-functor? In Set, the cartesian product is different from the coproduct, and the product satisfies hom(A x B, C) is isomorphic to hom(A, C^B) making Set into a cartesian closed category, a special kind of symmetric monoidal closed category; but this is not true in Vect. The product and coproduct are the same in Vect, namely the "direct sum", while the tensor product is what makes Vect into a symmetric monoidal closed category: hom(A tensor B, C) = hom(A, B -o C) where -o is linear implication. Vect also happens to be a compact closed category, which means that B -o C is isomorphic to B* tensor C. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com