Re: Adjoint Functor Pairs Preserve Limits and Colimits in Categories of Biological Systems, Automata and (M,R)Systems

[Ionel <***>]

Tim:

I missed your previous request / question about adjoint functors.

Not having seen John Baez's site that you mentioned, I'm providing here
a 'minimal response' so that we don't get too technical.
Adjoint Functors were in troduced by Daniel Kan in a much cited
paper back in 1958. A pair of Adjoint Functors, F and G  is defined between
two categories, A and B, F: A---->B and G:B---->A so that they are capable
of universal properties such as preserving limits and colimits. They are
now fundamental concepts/ tools in the theory and categories and functors.
Examples of adjoint functors abound in Topological Algebra, category
theory, Topoi, Homology Theory, Algebraic Geometry. An adjoint pair of
functors is thus formed between a mathematical structured category, such as
Top- the category of topological spaces and homeomorphisms, and the
Category of sets, Set, with mappings between sets:
  F: Top--->Set, G:Set---->Top, where F is the 'forgetful' functor
that forgets the structure of the topological space X and maps the
topological space on its base set S.

To provide an intuitive picture of the adjunction situation Saunders
MacLane in his..."Working Mathematician gives the published example of a
functor that constructs the "minimal realization" of an automaton and its
right adjoint that prescribes the corresponding "behavior" of the automaton
in the category of automata.

We introduced adjoint functors in biological modeling in our BMB paper
published in 1973 entitled "On Adjoint Dynamical Systems." where the
results of nuclear transplant experiments were discussed/ interpreted in
terms of the complex systems analogy between the different developmental
stages of organisms such as newts, from the fertilized egg onwards to
the morula, blastula stages and beyond. In essence, in our model, the pair
of adjoint functors between different stages of nuclear transplantation,
during development of the organism, preserves limits and colimits that are
representing essential functional dynamics in supercategories. Today, I'd
re-write the paper--as Saunders MacLane suggests in Ch.12 of
his "...working mathematician" --in terms of Kan extensions. Incidentally,
this model would also be applicable, albeit in a modifies form, to cloning
experiments; our model has also predicted correctly the results of the
first "successful" cloning that was carried out 25 years later after the
publication of our paper. (Please note that I have left out a great deal of
mathematical detail which is important for a correct understanding of the
concept of adjointness, such as the presence of the pre-requisite natural
transformations / equivalences applicable to the pair of adjoint functors.
Therefore, consulting the ..."working mathematician" chapter on Adjoint
Functors, or any textbook of Category Theory would be important for anyone
who wishes to understand in further depth the 'strength and usefulness of
adjoint functors' for preserving limits and colimits between certain pairs
of categories. There is a great deal more to them...nowadays they have uses
in comparing different types of Logic, for example!). Saunders says they're
ubiquitous in fundamental mathematical problems. As an oversimplified
example he provided the analogy with Adjoint Operators in a Hilbert space,
such as those that are utilized in Quantum Theory.

Regards,

Ionel

I wrote previously in the published article:

     " We suggest that there must exist completion laws-which are
biologically significant, and which rule out improper completions (that is,
completions which would lead to unreal organisms starting from a
"primordial" organism)." [p.11]
and:
  "Then the organismic supercategory representing a "primordial" organism
would play the role of an initial object in the metacategory of all
organismic supercategories and left adjoint functors. To complete this
metacategory with a limit would mean to suppose that there is an organism
which has any other organism as a simple analog. This organism would be the
highest that can possibly develop in a class of realizations of the
corresponding abstract organismic supercategories.
      The above mentioned completion determines the sense of biological
evolution. Even more, it results from this completion that the highest
developed organism would be unique, up to an isomorphism."[p.13]

Tim asked:

>Do these remarks suggest that an organism's development is, in some sense,
anticipatory? Perhaps I am reading too much into this, but it strikes me
that perhaps the organism itself at each developmental stage is also a kind
of model of which particular completions can follow.
Also, if possible, can you speak a bit about "adjoint functors" and their
utility? Maybe the corrected file will make it more clear - it is not a term
I've run across before in my (limited) reading on category theory.


Regards,
Tim>>