# Category theory

Source: https://plumeria.dev/docs/category



A category is a mathematical structure consisting of objects and morphisms.
If there are objects and morphisms, and the following two conditions are met, it is a category:

* Identity morphism

* Associativity

## First Condition (Identity morphism) [#first-condition-identity-morphism]

`create` is a monoid because it has a single object that satisfies the Identity morphism.

* `create -> create`

* `keyname -> keyname`

* `css-property -> css-property`

$A \circ A = A$

$A \oplus I = A$

Where the first expresses idempotency and the second expresses identity.

## Second Condition (Associativity) [#second-condition-associativity]

The law of associativity states that the result is the same regardless of the order in which the expressions are calculated.

Given `[A, B, C]`:

* `(A + B) + C`
* `A + (B + C)`

result in the same result, thus satisfying the compositional condition of the associative law.

$(A \oplus B) \oplus C = A \oplus (B \oplus C)$

## Composition of Monoids [#composition-of-monoids]

When this is assigned to styling,

{`[styles.A, styles.B, styles.C]`}

Then, the monoid `create` is a category with objects in composition.\
This is proof that a styling system closed to static analysis satisfies the axioms of category theory.

Ultimately, it always converges to a single result.\
$parse(A \oplus B \oplus C) \to \text{"classA classB classC"}$