ONT Notes On Categories
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NOC. Note 1
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Here I will document some of the computational approaches to
category theory that I starting working on back in the 1980's,
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 that 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 will have to revert to the convention that I was accustomed to using
then, 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:
(1_x)F = 1_(xF), for all x in Obj(C).
(f o g)F = fF o 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" like this,
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
Jon Awbrey
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