From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3145 Path: news.gmane.org!not-for-mail From: Robin Houston Newsgroups: gmane.science.mathematics.categories Subject: Products in a compact closed category Date: Thu, 23 Mar 2006 12:17:05 +0000 Message-ID: <20060323121705.GA26650@rpc142.cs.man.ac.uk> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019119 7423 80.91.229.2 (29 Apr 2009 15:31:59 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:31:59 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Mar 23 23:15:08 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 23 Mar 2006 23:15:08 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1FMce8-0001kK-06 for categories-list@mta.ca; Thu, 23 Mar 2006 23:07:28 -0400 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 91 Original-Lines: 20 Xref: news.gmane.org gmane.science.mathematics.categories:3145 Archived-At: Dear categorists, I recently proved that products (or coproducts) in a compact closed category are necessarily biproducts, and I'm wondering whether this is a known theorem. I can't find any reference to it in the literature, but the proof is not hugely complicated and it would not surprise me to learn that someone noticed it before this week! (More precisely, I can prove that given a monoidal category that has (finite) sums and products, if the tensor distributes over the sums on one side and the products on the other -- e.g. for every object A, -*A preserves sums and A*- preserves products -- then the products and coproducts are both really biproducts.) Any references or recollections will be much appreciated. Yours, Robin