Re: Literature on Category Theory and Biology

["David Ellerman" <david@ellerman.org>]

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



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