From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8342 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference search: new categories by replacing morphisms with diagrams Date: Fri, 26 Sep 2014 05:52:27 +1000 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 7.3 \(1878.6\)) Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1411726409 25632 80.91.229.3 (26 Sep 2014 10:13:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 26 Sep 2014 10:13:29 +0000 (UTC) Cc: Tom Hirschowitz , "categories@mta.ca" To: Jason Erbele Original-X-From: majordomo@mlist.mta.ca Fri Sep 26 12:13:23 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XXSWg-0007eJ-Nq for gsmc-categories@m.gmane.org; Fri, 26 Sep 2014 12:13:22 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50015) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XXSWN-0005BW-V1; Fri, 26 Sep 2014 07:13:03 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XXSWN-0007St-LM for categories-list@mlist.mta.ca; Fri, 26 Sep 2014 07:13:03 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8342 Archived-At: Dear Jason, I=92m travelling at the moment, and can=92t look up details, but similar = things have been done in the past, in particular by Bob Walters. If you start with a = monoidal category,=20 then you can define a bicategory with the same objects, and in which a = morphism=20 from A to B consists of an object X and a morphism A=97> X @ B, where @ = denotes=20 the tensor product. There is a dual version in which the original category in which = morphisms have the form A@X->B. Your version is a combination of both of these. (Once again, you get a = bicategory rather=20 than a category, unless for some reason + is strictly associative.) Another closely related notion is that of Elgot automaton, studied by = Walters and various collaborators. Regards, Steve Lack. On 25 Sep 2014, at 4:49 am, Jason Erbele wrote: > Ah, it figures I would leave something out in my first post here. My > apologies to all of you that were scratching your heads over the > missing rules for composition. At least five of you have responded > with essentially the same question, so this will be sort of a blanket > response. >=20 > Composition of (f,g,h): A --> B through X and (f',g',h'): B --> C > through Y will go through the biproduct X (+) Y with (f'f, [f'g, g'], > [h, h'(f+gh)]^T): A --> C. That is, the first component composes in > the usual way, the second component is a row matrix, and the third > component is a column matrix. In the original category, the various > morphisms in the composed triple correspond to A --> B --> C, > directly; X --> B --> C and Y --> C; and A --> X and A --> B --> Y, > where this A --> B is f+gh rather than f. >=20 > Identity morphisms in the new category are those with f=3Did and the > zero object for X, which uniquely determines g and h. The embedding I > mentioned of the original category into the new category is a functor, > after all. >=20 > Best, > Jason [For admin and other information see: http://www.mta.ca/~cat-dist/ ]