Categories

Categories contain:

A collection of objects
A collection of morphisms (also called arrows) connecting objects denoted by $f : S \to T$ , where $f$ is the morphism, $S$ is the source, and $T$ is the target
- Note: $f : A \to B$ and $g : A \to B$ DOES NOT IMPLY $f = g$
- Formally this can also be expressed as a relation between a collection of objects and a collection of morphisms
- Morphisms have a notion of composition, that being if $f : A \to B$ , $g : B \to C$ , then $g \circ f : A \to C$

There are three rules for categories:

Associativity: For morphisms $a$ , $b$ , and $c$ , $(a \circ b) \circ c = a \circ (b \circ c)$
Closed composition: If for morphisms $a$ and $b$ , $a \circ b$ exists, then there must be morphism $c = a \circ b$
Identity morphisms: For every object $A$ in a category, there must be an identity morphism $id_{A} : A \to A$

@craisin's notes