milewski-programmers-1.jmd

# Category: The Essence of Composition

> 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_.

```mermaid

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?

1. Composition is associative.

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

$$h \circ (g \circ f) = (h \circ g) \circ f = h \circ g \circ f$$

2. 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.

## Challenges

1. 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). 

```julia, results = "hidden", eval = true
id(x) = x
```

2. Implement the composition function in your favorite language.
It takes two functions as arguments and returns a function that is their composition.

```julia, eval = true, results = "hidden"
f(x) = x + 2
g(x) = x^2
h = g ∘ f
```

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

Test one: 

```julia, eval = true
(id ∘ h)(2) == h(2)
```

Test two:

```julia, eval = true
(h ∘ id)(13) == h(13)
```

Test three:

```julia, eval = true
(id ∘ h)(-7) == h(-7)
```

4. 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.

5. 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.

6. 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.