ONT Re: Extension x Comprehension = Information
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Last time we looked at an ordinary function f : X -> Y,
and we glommed onto a single fiber of f, considered in
one of two ways: (1) a set of ordered pairs F c X x Y
such that <x, y> in F if and only f(x) = y, or else (2)
a subset of X, horrifically asciified as f^(-1)(y) c X.
o-----------------------------o-----------------------------o
| Source Domain | Target Codomain |
o-----------------------------o-----------------------------o
| |
| x |
| x · |
| x · · |
| x · · · |
| x · · · · |
| x · · · · y |
| x · · · · |
| x · · · |
| x · · |
| x · |
| x |
| |
| |
| Functional Fiber |
o-----------------------------------------------------------o
Any function f : X -> Y, which is, after all, exactly the same as
the relation f c X x Y, can be treated as an assortment of fibers
of the first sort. So it is easy to grasp the elementary fact of
category theory that any function whatsoever can be factored into
an epic (surjective, 'onto') and a monic (injective, 'one to one')
sequence of composed functions, as illustrated here for one fiber:
o-------------------o-------------------o-------------------o
| Source Domain | Medial Domain | Target Domain |
o-------------------o---------------------------------------o
| |
| x |
| x · |
| x · · |
| x · · · |
| x · · · · |
| x · · · · m · · · · · · · · > y |
| x · · · · |
| x · · · |
| x · · |
| x · |
| x |
| |
| |
| Functional Fiber Factors |
o-----------------------------------------------------------o
In sum, an arbitrary f : X -> Y can always be factored into
a pair of functions of the types g : X -> M and h : M -> Y,
where g is surjective, h is injective, and f = h o g, here
using the "left-composition" convention according to which
the composition h o g is defined by (h o g)(x) = h(g(x)).
To finish off the topic of factoring functions for now,
I will give the commutative diagram and the additional
bit of explanation that I gave once before, to wit:
We began with the trusim from category theory, at least,
the sorts of "concrete categories" of sets and functions
that will be most salient in the minds of most everybody:
That an arbitrary arrow factors into a couple of pieces,
an epic on which a monic ensues, 'Iliad' and 'Odyssey',
if you will, and if you catch my drift, and whether
you will or not, 'tis true.
| f
| arbitrary
| X o-------------->o Y
| \ ^
| \ /
| g \ / h
| surjective \ / injective
| "epic" \ / "monic"
| \ /
| v /
| o
| M
Now, there's a catch here -- there's always a catch, the way I see it --
leastwise, once we begin to think so systematically as to be working
inside any sort of category at all, instead of merely picking up on
this or that isolated instance of an arbalistrary functional arrow,
then this ostensibly trivial truism becomes contingent on the list
of a "suitable transitional object" (STO), like M in our example,
and of the "requisite intermedi-arrows" (RIA's), like g and h,
explicitly listed within the formal category in question.
Otherwise, "you just cannot get there from here" is the
only thing that answers to your desire for mediation.
http://suo.ieee.org/ontology/msg00032.html
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤