What is Category Theory?

It actively discourages looking inside objects. In category theory, an object is just a vague thing where the only thing you know about them is how they relate to other objects.

What Is a Category?

A category consists of objects and arrows that go between them. Key to categories is there ability to compose.

What Is a Morphism?

The arrows in the following diagram can be thought of as functions. They are also known as *morphisms*.

graph LR; A --> B; B --> C; A --> C;

In a category, if there is an arrow going from $A$ to $B$ and an arrow from $B$ to $C$, there must be a direct arrow from $A$ to $C$. This arrow from $A$ to $C$ is their composition.

\[ f(A) = B \]

\[ g(B) = C \]

\[ (g \circ f)(A) = C \]

What are the properties of composition in categories?

Composition is associative.

Example: Given three composable morphisms, $f$, $g$, and $h$, parentheses are not needed to compose them. This can be demonstrated below:

\[ h \circ (g \circ f) = (h \circ g) \circ f = h \circ g \circ f \]

For every object $A$, there exists an arrow (morphism) which is a

*unit of composition*.

A "unit of composition" is that whenever there is an arrow starting or ending at $A$, it gives back the same arrow.

Example: The unit arrow (or morphism) for object $A$ is denoted as $\textbf{id}_{A}$ which can be read out as the *identity* on $A$. If there is a morphism $f$ that goes from $A$ to $B$, then

\[ f \circ \textbf{id}_{A} = f \]

and

\[ \textbf{id}_{B} \circ f = f \]

In practice, the identity arrow is implemented as an identity function which returns its argument.

Implement, as best as you can, the identity function in your favorite language (or the second favorite, if your favorite language happens to be Haskell).

id(x) = x

Implement the composition function in your favorite language.

It takes two functions as arguments and returns a function that is their composition.

f(x) = x + 2 g(x) = x^2 h = g ∘ f

Write a program that tries to test that your composition function respects identity.

Test one:

(id ∘ h)(2) == h(2)

true

Test two:

(h ∘ id)(13) == h(13)

true

Test three:

(id ∘ h)(-7) == h(-7)

true

Is the world-wide web a category in any sense? Are links morphisms?

I would say that the world-wide web is a category. Web pages would function as objects and links would be morphisms that go between them. I think links are valid as morphisms as they form arrows to and from objects.

Is Facebook a category, with people as objects and friendships as morphisms?

I would say that Facebook would not be a category as the objects and friendships are not composable. Say if you have friend $A$ who is friends with $B$ and $B$ is friends with $C$, if you wanted to get to friend $C$ from knowing friend $A$, there is not a direct connection between the two.

When is a directed graph a category?

A directed graph is a category when you can come up with associative compositions at any position on the graph.

© Jacob Zelko. Last modified: April 24, 2023. Website built with Franklin.jl and the Julia programming language.