Re: Literature on Category Theory and Biology

Re: Literature on Category Theory and Biology

[David Ellerman]

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> http://ssrn.com/author=294049

Re: Literature on Category Theory and Biology

[mjhealy]

Regarding Deniz Kural's question about references to papers on category
theory and biology, Tom Caudell and I have been investigating a semantic
theory for neural networks both biological and artificial, cognitive and
non-cognitive.  We are designing and conducting experiments to test and
refine the theory in both neuroscience and cognitive psychology working
with colleagues in those disciplines.  An initial paper on the theory is

M. J. Healy and T. P. Caudell (2006)
Ontologies and Worlds in Category Theory: Implications for Neural Systems,
Axiomathes, vol. 16, nos. 1-2, pp. 165-214.

The only experiment appearing in a publication to date is one on an
artificial neural network application, presented at IJCNN 2005 in Montreal
(the results were presented also at CT06).

Regards,
Mike Healy

Re: Literature on Category Theory and Biology

[Andree Ehresmann]

In answer to Deniz Kural

With Jean-Paul Vanbremeersch we have been developing a model for
biological and neural systems based on category theory called Memory
Evolutive Systems. Since 20 years we have published a series of papers
on this subject, most of which are posted on our Internet site
http://perso.wanadoo.fr/vbm-ehr
Recently we have written a book on this subject:
Memory Evolutive sytems: hierarchy, emergence, cognition"
  due to appear this month in the series "Studies in
Multidisciplinarity" of Elsevier (volume 4).

Sincerely
Andree C. Ehresmann

Re: Literature on Category Theory and Biology

[Wojtowicz, Ralph]

The links:

http://perso.orange.fr/vbm-ehr/

and 

http://alf.nbi.dk/%7Eemmeche/coPubl/97d.NABCE/ExplEmer.html  

and a related paper "Categorical language and hierarchical models for
cell systems" by R. Brown, R. Paton and T. Porter may be of interest.  I
believe that Brown and Porter have other references of this nature.  See
also the recent work by M. Healy on neural networks.

Best wishes,
Ralph Wojtowicz
wojtowicz@metsci.com

Re: Literature on Category Theory and Biology

[Zippie Arzi-Gonczarowski]

Hi,

1.You may want to look at the writings of Robert Rosen (a Google search
will provide the necessary links).

2. For a different direction, and on a far more modest note, you may try
looking at some of my papers on my web page:

http://www.actcom.co.il/typographics/zippie

Of course I will be very intrested to hear your comments about that.

Good luck,

Zippie
-- 

____________________________________________
Zippora Arzi-Gonczarowski, Ph.D.  Aka:Zippie

Typographics, Ltd.
46 Hehalutz St.
Jerusalem 96222, ISRAEL

zippie@actcom.co.il

http://www.actcom.co.il/typographics/zippie

Tel: +972-2-6437819     Fax: +972-2-6434252
_____________________________________________

Re: Literature on Category Theory and Biology

[Colin McLarty]

The first time the term "category theory" (I mean, exactly that term)
appeared in Mathemtical Reviews was in the review of

Rosen, R. [1961]: `A relational theory of  the structural changes
induced in biological systems by alterations in  environment', {\em
Bulletin of Mathematical Biophysics} \bf 23}, pp.~165--71.

The full review reads:

The author uses, among other things, some previous results from his
biologico-mathematical applications of abstract category theory [#B416]
in order to further develop another paper concerning relational biology
[#B418]. Some biological applications are treated; e.g., an
interpretation of the mitotic cycle.

Rosen's earlier related works are reviewed as dealing with "the theory
of categories."

best, Colin

----- Original Message -----
From: Deniz Kural <kural@fas.harvard.edu>
Date: Friday, May 4, 2007 8:55 pm
Subject: categories: Literature on Category Theory and Biology
To: categories@mta.ca

>
>
> Hi,
>
> I was wondering if you are aware of any bibliographies, reviews,
> or papers
> relating category theory to biology - mathematical biology, systems
> biology, computational biology or bioinformatics.
>
> I would be interested in papers relating category theory to areas of
> knowledge representation or other areas of computer science used in
> abovementioned areas.
>
> Please feel free to email me in person if you wish not to
> overburden the
> mailing list.
>
> Regards,
> Deniz Kural
>
>
>

Literature on Category Theory and Biology

[Deniz Kural]

Hi,

I was wondering if you are aware of any bibliographies, reviews, or papers
relating category theory to biology - mathematical biology, systems
biology, computational biology or bioinformatics.

I would be interested in papers relating category theory to areas of
knowledge representation or other areas of computer science used in
abovementioned areas.

Please feel free to email me in person if you wish not to overburden the
mailing list.

Regards,
Deniz Kural