Re: Adjoint Functor Pairs Preserve Limits and Colimits in Categories of Biological Systems, Automata and (M,R)Systems; John Baez site on Adoint Functors

[Ionel <***>]

Tim:

I'm glad you found the Pareigis's book on Categories and Functors:
it is expensive at $77, used compared with Mitchell's. Do you think you can
find the one that I translated back in 1971, published in 1973:
N. Popescu: "Abelian Categories with Applications to Rings and
Modules., Academic Press: N.Y. and London. Let's see if one can still
obtain it?

Regards,
Ionel

>
>> -----Original Message-----
>> From: ROSEN Forum [mailto:*** Behalf Of Ionel
>> Sent: Saturday, June 05, 2004 3:27 PM
>> To: ***
>> Subject: Re: Adjoint Functor Pairs Preserve Limits and Colimits in
>> Categories of Biological Systems, Automata and (M,R)Systems; John Baez
>> site on Adoint Functors
>>
>>
>> Tim:
>>
>> I did look at John Baez 's presentation of adjoint functors: it has
>> a number of useful and good things, but it's a bit hard to follow as it
>> spreads over several weeks--it is very diluted. The good thing is it has
>> quite a few examples. The bad thing is the basic Adjointness theorems
seem
>> to be missing; not showing the importance of preservation of limits and
>> colimits is a basic limitation; also I did not see any examples of
>> adjointness in N-categories, unless I missed one of his several weeks
>> presentations in this multiple-week series on adjointness. Maclane's book
>> does much better on that side of important theorems on adjointness.
>> Another useful source on adjointness and category theory is Barry
>> Mithcell's "The theory of Categories." in Academic Press-- really nice
and
>> elegantly written for an introduction, but now dated and long out
>> of print.
>> Also there is a later book by Pariegis, but also out-of-print and
>> practically impossible to find: I'd like to also have Pareigis's book but
>> I don't really know where to find it short of conatacting the author
>> himself.
>>
>> Regards,
>>
>> Ionel
>>On Sun, 6 Jun 2004 12:05:38 -0400, Tim Gwinn <***> wrote:

>Hi Ionel,
>
>Yes, Baez's probably isn't the best...but it is free. :)
>
>I do have a copy of Mitchell. I think it would be a good book as a
classroom
>text, but not the best self-learning book. Colimits were only barely
>mentioned, as I recall. I'll have to give the chapters on adjoint functors
a
>read now that you've given me some more understanding of them.
>
>If anyone's interested, there are several used copies available, both of
>Mitchell's "Theory of Categories" and Bodo Pareigis' "Categories and
>Functors". See http:://used.addall.com.
>
>Regards,
>Tim
>>
>>
>>
>> On Tue, 1 Jun 2004 07:58:48 -0400, Ionel <***> wrote:
>>
>> >Tim:
>>
>> >
>> >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>>