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 \text{id}_A: A \to A