# Category theory

Category theory can be helpful in understanding Haskell's type system. There exists a "Haskell category", of which the objects are Haskell types, and the morphisms from types a to b are Haskell functions of type a -> b.

The Haskell wikibooks has an introduction to Category theory, written specifically with Haskell programmers in mind.

## Definition of a category

A category $\mathcal{C}$consists of two collections:

Ob$(\mathcal{C})$, the objects of $\mathcal{C}$

Ar$(\mathcal{C})$, the arrows of $\mathcal{C}$ (which are not the same as Arrows defined in GHC)

Each arrow $f$ in Ar$(\mathcal{C})$ has a domain, dom $f$, and a codomain, cod $f$, each chosen from Ob$(\mathcal{C})$. The notation $f\colon A \to B$ means $f$ is an arrow with domain $A$ and codomain $B$. Further, there is a function $\circ$ called composition, such that $g \circ f$ is defined only when the codomain of $f$ is the domain of $g$, and in this case, $g \circ f$ has the domain of $f$ and the codomain of $g$.

In symbols, if $f\colon A \to B$ and $g\colon B \to C$, then $g \circ f \colon A \to C$.

Also, for each object $A$, there is an arrow $\mathrm{id}_A\colon A \to A$, (often simply denoted as $1$ or $\mathrm{id}$, when there is no chance of confusion).

### Axioms

The following axioms must hold for $\mathcal{C}$ to be a category:

1. If $f\colon A \to B$ then $f \circ \mathrm{id}_A = \mathrm{id}_B\circ f = f$ (left and right identity)
2. If $f\colon A \to B$ and $g \colon B \to C$ and $h \colon C \to D$, then $h \circ (g \circ f) = (h \circ g) \circ f$ (associativity)

### Examples of categories

• Set, the category of sets and set functions.
• Mon, the category of monoids and monoid morphisms.
• Monoids are themselves one-object categories.
• Grp, the category of groups and group morphisms.
• Rng, the category of rings and ring morphisms.
• Grph, the category of graphs and graph morphisms.
• Top, the category of topological spaces and continuous maps.
• Preord, the category of preorders and order preserving maps.
• CPO, the category of complete partial orders and continuous functions.
• Cat, the category of categories and functors.
• the category of data types and functions on data structures
• the category of functions and data flows (~ data flow diagram)
• the category of stateful objects and dependencies (~ object diagram)
• the category of values and value constructors
• the category of states and messages (~ state diagram)

## Categorical programming

Catamorphisms and related concepts, categorical approach to functional programming, categorical programming. Many materials cited here refer to category theory, so as an introduction to this discipline see the #See also section.