Notes On Categories

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Post       : Notes On Categories : 1
URL        : http://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/
Posted     : February 22, 2013 at 11:00 am
Author     : Jon Awbrey
Categories : Category Theory, Computation, Graph Theory, Logic, Mathematics, Relation Theory, Type Theory

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Continued from “Notes On Categories” (14 Jul 2003)
• Inquiry List  : http://stderr.org/pipermail/inquiry/2003-July/000637.html
• Ontology List : http://suo.ieee.org/ontology/msg04908.html

Here are some notes on a computational approach to category theory
that I started working on back in the 1980s, all of which work as
yet remains in the “Schubert Category” of unfinished symphonies.

It helps me a little bit to write the names of categories in the plural,
so as not to confuse them with individuals.  It also helps if I treat the
arrows of Arr(C) as the primary entities in the category C, recovering the
objects of Obj(C) as secondary entities by collecting all the entities that
appear in s(f) = Source(f) and t(f) = Target(f) as one ranges over all of the
arrows f in Arr(C).

The last time that I tried to do “categories by computer”,
I was using data structures that had the following shapes:

   Category C o
             /|\
            / | \
          ... | ...
              |
      Arrow f o
             / \
            s   t
           /     \
     s(f) o       o t(f)

A functor, then, is something I picture like this:

              Functor F o
                      . | .
                    .   |   .
                  .     |     .
                .       |       .
   Category C o         o         o Category D = CF
              |       ./ \.       |
              |     . /   \ .     |
              |   .  /     \  .   |
              | .   /       \   . |
      Arrow f o    o         o    o Arrow fF
             / \ .   .     .   . / \
            /  .\      . .      /.  \
           s .   t     . .     s   . t
          /.      \  .     .  /      .\
         o         o         o         o
         x         y         xF        yF

This is a rough sketch of the actual data structures
that I used to represent a functor F as a “matching”
between the parallel items of categories C and D.

NB. I have reverted to the convention I was accustomed
to use at the time, where all operators are applied on
the right of their arguments.

What the picture says is that the functor F : C → CF
takes each arrow f in C to an arrow fF in CF, and each
object x in C to an object xF in CF, in such a manner
that (fs)F = (fF)s and (ft)F = (fF)t.  To be a functor,
F must satisfy the following two systems of equations:

(1x)F      =   1(xF),     for all x in Obj(C).

(f ∘ g)F   =   fF ∘ gF,   for all composable f, g in Arr(C).

That was just how I kept track of things on the computer.

It is, of course, more usual to draw a functor square
in the following manner, where we get one such picture
for each object x and arrow f in C.

             F
      x o-------->o xF
        |         |
        |         |
      f |         | fF
        |         |
        v         v
      y o-------->o yF
             F

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