An Introduction to Functional Programming

…requires support for commutative diagrams…

A category $\bf C$ is a collection $A, B, …$ of objects together with a set of arrows:

for every objects $A$ and $B$.

Functional programming is concerned with the category types $\bf T$.

Each object ${\tt a}$ of $\bf T$ represents a type of variables, and each arrow ${\tt f : a \rightarrow b}$ in ${\rm Hom}({\tt a, b})$ is a program with input of type ${\tt a}$ and output of type ${\tt b}$.

The set of programs ${\bf T}(\tt{a, b})$ from $\tt a$ to $\tt b$ is itself a type, denoted by ${\tt a \rightarrow b}$. This special property of ${\bf T}$ is characteristic of cartesian categories.