Category

A Category consists of a collection of

• Objects
• Morphisms

where there is a definition for

• Composition
• Identity Morphism

• $M_{ab}$ is a Morphism from $A$ to $B$
• $M_{bc}$ is a Morphism from $B$ to $C$
• $M_{bc}$ $\circ$ $M_{ab}$ is the Composition of the above morphisms that maps $A$ to $C$
  • It is equivalent to applying $M_{ab}$, then applying $M_{bc}$
  • It can be thought of as: $M_{bc}$ follows $M_{ab}$
  • Composition is Associative $\to (f\circ g)\circ h=f\circ (g\circ h)$
• id is the Identity Morphism, which is depicted above to map $A$ to $A$
  • It exists for all Objects, but is intentionally omitted for $B$ and $C$
• Think of a Category as a generalized version of Elements, Functions, and Operations in Set Theory.

Examples of Categories

The Category of Proofs

• Objects: Propositions
• Arrows: Proofs

Set Categories

• Objects: Sets
• Arrows: Functions

Functional Programming Type Categories

• Objects: Data Type
• Arrows: Function

Vector Space Categories

• Objects: Vector Spaces
• Arrows: Linear Transformation