From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6077 Path: news.gmane.org!not-for-mail From: Micah Blake McCurdy Newsgroups: gmane.science.mathematics.categories Subject: Re: String diagrams, adjunction and autonomous categories. Date: Sun, 29 Aug 2010 23:31:02 -0300 Message-ID: References: Reply-To: Micah Blake McCurdy NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1283185671 16357 80.91.229.12 (30 Aug 2010 16:27:51 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 30 Aug 2010 16:27:51 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Mon Aug 30 18:27:50 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Oq7D4-0006t1-84 for gsmc-categories@m.gmane.org; Mon, 30 Aug 2010 18:27:50 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33679) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Oq7Bv-0006uQ-I4; Mon, 30 Aug 2010 13:26:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Oq7Bt-0001e2-43 for categories-list@mlist.mta.ca; Mon, 30 Aug 2010 13:26:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6077 Archived-At: Hallo! In an autonomous category, the functor B -o _ is "representable" in the sense that it can be written as B* (x) _ for some object B*. If k is the unit for (x), then there are maps "unit" : k ----> B* (x) B and "counit" : B (x) B* ---> k which satisfy triangle identities, usually written graphically as zig-zags which equal identities. The isomorphism you mention in your first paragraph is obtained by appending the unit for A. So, the bends in the wires are the same in both paragraphs. Incidentally, although I'm not sure that I'm familiar with the work of Baez to which you refer, I would imagine that the use of string diagrams to describe units and counits in an autonomous category is considerably older, at least as far back as "Planar Diagrams and Tensor Algebra" by Joyal and Street (available on the website of the latter) from 1988. Cheers, Micah On Sun, Aug 29, 2010 at 2:48 AM, David Leduc wrote: > As shown by Baez, in an autonomous category the isomorphism hom(A (X) B, C) > = hom (B, A -o C), when drawn as a string diagram, is like the bending of > the input wire A to make it an output. > > Now one can also draw string diagrams to represent the zigzag equations > between the adjoint pair of functors _ (X) B and B -o _. > > How does the latter diagram relate to the former one? > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]