From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7883 Path: news.gmane.org!not-for-mail From: Jamie Vicary Newsgroups: gmane.science.mathematics.categories Subject: Slick proof that f (x) (g+h) = (f (x) g) + (f (x) h) in a monoidal category with 0, biproducts and duals Date: Fri, 11 Oct 2013 13:24:00 +0100 Message-ID: Reply-To: Jamie Vicary NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1381534274 28518 80.91.229.3 (11 Oct 2013 23:31:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 11 Oct 2013 23:31:14 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Sat Oct 12 01:31:16 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1VUmAt-0005sm-NF for gsmc-categories@m.gmane.org; Sat, 12 Oct 2013 01:31:15 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38055) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VUm99-00057k-Mo; Fri, 11 Oct 2013 20:29:27 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VUm9A-0000cG-AN for categories-list@mlist.mta.ca; Fri, 11 Oct 2013 20:29:28 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7883 Archived-At: Hi, In a category with a zero object and biproducts we obtain a unique enrichment in commutative monoids, which we will write as +. If the category is also monoidal with left and right duals for objects, then the tensor product distributes over +, in the sense that f (x) (g+h) = (f (x) g) + (f (x) h) for all morphisms f,g,h with g and h in the same hom-set. I have a proof of this but it is a bit clunky, and rather long. Can anyone give a beautiful one? Best wishes, Jamie. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]