From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7886 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Slick proof that f (x) (g+h) = (f (x) g) + (f (x) h) in a monoidal category with 0, biproducts and duals Date: Sun, 13 Oct 2013 18:05:27 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1381756100 14244 80.91.229.3 (14 Oct 2013 13:08:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 14 Oct 2013 13:08:20 +0000 (UTC) Cc: Categories list To: Jamie Vicary Original-X-From: majordomo@mlist.mta.ca Mon Oct 14 15:08:25 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 1VVhsm-000463-9F for gsmc-categories@m.gmane.org; Mon, 14 Oct 2013 15:08:24 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40745) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VVhqn-0005sN-Ah; Mon, 14 Oct 2013 10:06:21 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VVhqp-00048L-Hh for categories-list@mlist.mta.ca; Mon, 14 Oct 2013 10:06:23 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7886 Archived-At: Dear Jamie, Here is one possible proof. The 2-category of categories with biproducts and zero objects and functors preserving them is biequivalent to the 2-category of CMon-enriched categories with finite coproducts and arbitrary CMon-functors between them (because finite coproducts are absolute in the CMon-enriched world). If C is a category with biproducts, a zero object, and monoidal with left and right duals, then it is in particular monoidal biclosed; whence the tensor product A (x) (-) : C ----> C preserves colimits, so in particular biproducts and the zero object. Via the above biequivalence, it follows that A (x) (-) : C ---> C is a CMon-enriched functor for each A. Thus given a map f : A --->B and objects X,Y in C, we have mappings C(X,Y) ---> C(A(x)X, A(x)Y) ---> C(A(x)X, B(x)Y) in CMon, whose underlying mapping in Set sends g : X ---> Y to f (x) g : A(x)X --> B(x)Y. That these mappings lift to CMon expresses precisely the distributivity you state above. Best, Richard On 11 October 2013 23:24, Jamie Vicary wrote: > 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/ ]