Notes (CTFP 01/31): Category

1 Category: The Essence of Composition

category: A structure consisting of objects and arrows between objects, such that arrows compose. That is, if a category has objects A, B and C along with arrows A -> B, B -> C, then it also has the arrow A -> C.

1.1 Arrows as Functions.

Arrows are also called morphisms, which comes from the Greek “morphe” meaning form.

Notation:

1.2 Properties of Composition

  1. Composition of arrows is associative, so that

    h . (g . f) = (h . g) . f = h . g . f

    This makes sense with the definition of a category. Suppose a category contains objects A, B, C, D and arrows:

    f :: A -> B
    g :: B -> C
    h :: C -> D

    Then by definition it contains:

    (f . g) :: A -> C
    (h . g) :: B -> D
    (h . g . f) :: A -> D

    Associativity just means that it doesn’t matter what order we compose arrows:

    h . (g . f) == (h . g) . f == h . g . f
  2. There is an identity arrow A -> A for every object A, such that

    f :: A -> B
    g :: B -> A
    id_a :: A -> A
    
    f . id_a = f
    id_a . g = g

One thing that confuses me is that it seems like this implies that in some cases that there can’t be multiple arrows from one object to another.

Suppose a category Cat has objects A, B and arrows

f' :: A -> B
f  :: A -> B
g  :: B -> A
id_a :: A -> A
id_b :: B -> B

If Cat is a category, then the compositions of g . f and g . f' must be arrows in the category:

g . f :: A -> A
g . f' :: A -> A

But the only arrow of type A -> A is id_a, so

g . f = g . f' = id_a

And equivalently:

f . g = f' . g = id_b

Which implies:

f' = f' . id_a = f' . g . f = id_b . f = f

So, either Cat is not a category, or f' == f.

cf. Haskell Wikibook page on Category Theory

I think the important thing to keep in mind here is that when looking at a possible structure to see if its a category, the structure is constant. There aren’t any implicit elements of Cat that could also satisfy the property that e.g (g . f) is in Cat. We actually have to look in Cat to find if something fits the properties. If not, it’s not a category.

When I first approached this topic I confused the statements:

This really confused me for a bit, until I realized that when I was doing the latter I was just trying to build a superset of C that actually was a category. This is apparently called a “free category”.

1. 4 Challenges

  1. I = (\x. x) in the lambda calculus.
  2. compose = (\x y z. x (y z))

  3. compose I F X = (\x y z. x (y z)) I F X = I (F X) = F X compose F I X = (\x y z. x (y z)) F I X = F (I X) = F X

  4. No, links are not composable.
  5. Friendships are not composable.
  6. When the edges are composable, and every node has a self-directed edge.