Re: Relational "Space" - Ulanowicz works

[Tim Gwinn <***>]

> -----Original Message-----
> From: ROSEN Forum [mailto:*** Behalf Of John
> Kineman
> Sent: Friday, March 19, 2004 6:06 PM
> To: ***
> Subject: Re: Relational "Space" - Ulanowicz works
>
>
> Hi Tim,
> Your last suggestion is very intriguing to me. I got the book you
> recommended earlier on category theory (Lawvere & Schanuel - Conceptual
> Mathematics) and am trying to learn - but its tough going.  But here's a
> speculation you probably won't like, but maybe will stimulate some
> conversation.
>
> Suppose we take the criteria for a category which is A -f->B-g->C with all
> the possible associative and identify relations (e.g., A-h->C)
> indicated by
> arrows (can't draw it here). That supposedly is what defines a category
> along with the identity and associative rules in terms of functors
> connecting A,B,C;  like h o (g o f) =  (h o g) o f, and so forth.
> (the idea
> is much simpler than the notation, I think - its three things in a tight
> relationship, with formal rules about how they transform into each other).


As I understand it, a category is more simply defined as a collection of
objects plus a collection of morphisms that map between the objects in the
category. The identity mapping in a category is a morphism, not a functor.
The associative and identity rules you mention apply to morphisms in the
category. The functorial relations would be for providing inter-relations
among categories: for mapping objects to objects and morphisms to morphisms,
between categories (although, it is also possible that both categories can
be the same).

Unfortunately, that book focuses heavily on morphisms, rather than functors;
but I don't know of another book to recommend. I have an old one I bought
used that was referenced by Rosen ("Theory of Categories", Mitchell, 1965)
which focuses more on functors, but besides being out of print, it is one of
those textbooks of the kind that made one queasy when they handed them out
the first day of math class in high school :)

I realize now in my remarks in the previous post I mentioned "flow arrows
replaced by functorial relations", when I should have said "flow arrows
replaced by morphisms". My bad.


> Now suppose a simple relationship between A and B (A-f->B) with
> the inverse
> relationship (B-g->A), along with the identity arrows for A and B defines
> an RR modeling relation.


The general modeling relation, as I understand it, would be the objects and
morphisms (entailment structures) in one category encoded into the objects
and morphisms in another category by a covariant functor. The decoding would
correspondingly be performed by a contravariant functor.


> We can, in the way I have tended to argue the
> case, consider that as an explanation for complexity or picture of a
> complex relationship, where the inverse does not necessarily
> commute (f not
> equal to g), but f and g can be similar enough such that some functors
> based on them could be made to commute, as in a classical reduction).


If f and g are meant to represent encoding and decoding arrows, then I think
I'd disagree. The inability to create a commuting modeling relation might be
due to many reasons: it might be due to complexity, or it might be due to a
bad model, or it might be due to our inability at the moment to be creative
enough to construct the right encoding/decodings for a particular model, or
so on. A failure to commute isn't very definitive. I'd rather go with
Rosen's notion that a complexity is demonstrated by having a successfully
commuting modeling relation with a noncomputable model.


> So then we have that a morphism between two things plus the inverse
> morphism and the identities (which are RR's implication and causality
> identities) defines a modeling relation, and a basic complexity;
> and that a
> category defines an organism! The A, B, C for example could be:
> A = organismic subsystems, the "parts"
> B = organism/behavior as the organic "whole"
> C = environmental context as the next larger domain within which the
> organism derives meaning.
>
> For an organism to exist it must in fact have functors operating between
> these three components as a minimum, so I'm thinking that the
> definition of
> a category tells us what can exist (and persist) as a logical relational
> structure.
>
> An organism is thus a closed set of these complex relations which
> meets the
> formal rules of a category. That would provide a clear distinction between
> complexity and organismic life, the later being a special organization of
> the former.
>


I'm not exactly sure what you are proposing, but it sounds interesting. Are
elements in A,B,C all objects of one category with morphisms between these
elements? How would these elements be related?

Regards,
Tim