From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3757 Path: news.gmane.org!not-for-mail From: "David Ellerman" Newsgroups: gmane.science.mathematics.categories Subject: Re: Literature on Category Theory and Biology Date: Sat, 5 May 2007 09:01:45 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019504 10164 80.91.229.2 (29 Apr 2009 15:38:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:38:24 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Sat May 5 21:26:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 05 May 2007 21:26:31 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HkUYY-0007Nx-2C for categories-list@mta.ca; Sat, 05 May 2007 21:24:54 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 17 Original-Lines: 58 Xref: news.gmane.org gmane.science.mathematics.categories:3757 Archived-At: A new "heteromorphic" treatment of adjoint functors provides applications to biology such as an abstract characterization of selectionist (as opposed to instructionist) mechanisms as in Darwin's evolutionary theory, the selectionist theory of the immune system, and neural darwinism (e.g., Edelman's and Changeux's work). Heteromorphisms, e.g., the injection of a set of generators into the free group on the set, can be formally treated in category theory using bifunctors Het:X^op x A--> Set analogous to the usual Hom:X^op x X-->Set. When the heteromorphisms from objects in a category X to objects in a category A can represented in each of the categories, then the functors giving the representing objects are a pair of adjoint functors and the representations give a pair of natural isomorphisms: Hom_A(Fx,a) = Het(x,a) = Hom-_X(x,Ga). The usual treatment of adjoints leaves out the middle term. And all adjoint functors can be shown to arise in this manner (up to isomorphism). The applications were not available in the usual treatment of adjoints where the heteromorphisms were not explicit. The applications are outlined in a paper just published in Axiomathes (2007) 17: 19-39. A reprint can be retreived from my website: http://www.ellerman.org/Davids-Stuff/Maths/Adjoints-Axiomathes-Reprint.pdf . A rather long (and impenetrable) treatment of the math was in the recent "What is Category Theory" collection of papers (2006: Polimetrica). A short straightforward treatment of the math is available on the ArXiv: http://arxiv.org/abs/0704.2207v1 . Other applications of category theory to biology have been made by Robert Rosen (as mentioned by several posts) and by Andree Ehresmann. Best, David __________________ David Ellerman Visiting Scholar University of California at Riverside Email: david@ellerman.org Webpage: www.ellerman.org View my research on my SSRN Author page: http://ssrn.com/author=294049