Category Theory

A lot of this is sourced from Category Theory for Programming

The power of category theory arises from abstraction: by boiling down constructions to their essence, analogous situations can be formally identified using category theory.

In essence, a simple collection which can be thought of as a graph. Three components

1. A collection of objects (nodes)
2. A collection of morphisms (edges).
   • If $f$ is a morphism with source C and target B, we write $f:C\to B$
3. A notion of composition of morphisms.
   • If we have $g:A\to B$ and $f:B\to C$, they can be composed resulting in a morphism $f\circ g:A\to C$
   • Composition of morphisms needs to be associative. Typically applied right to left

Categories

A category $\mathcal{C}$ consists of

• A collection of objects, denoted by ${\mathcal{C}}_0$
• For any given $X,Y\in {\mathcal{C}}_0$, a collection of morphisms from $X$ to $Y$ denoted by ${\text{hom}}_{\mathcal{C}}(X,Y)$ or $\mathcal{C}(X,Y)$ or even $X\to Y$
• An identity morphism for each object $X\in {\mathcal{C}}_0$ such that ${\text{Id}}_X\in {\text{hom}}_{\mathcal{C}}(X,X)$
• A binary infix composition operator $\circ$ that takes $\text{hom}(Y,Z)\to \text{hom}(Y,Z)\to \text{hom}(X,Z)$ (this can be thought of as $X\to Y\wedge Y\to Z⟹X\to Z$)

See also: functional programming, Homotopy Theory