# 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`

. Various other Haskell structures can be used to make it a Cartesian closed category.

## Contents

## Defintion of a category

A category consists of two collections:

Ob, the objects of

Ar, the arrows of (which are not the same as Arrows defined in GHC)

Each arrow in Ar has a domain, dom , and a codomain, cod , each chosen from Ob. The notation means is an arrow with domain and codomain . Further, there is a function called composition, such that is defined only when the codomain of is the domain of , and in this case, has the domain of and the codomain of .

In symbols, if and , then .

Also, for each object , there is an arrow , (often simply denoted as or , when there is no chance of confusion).

### Axioms

The following axioms must hold for to be a category:

- If then (left and right identity)
- If and and , then (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)

### Further definitions

With examples in Haskell at:

## 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.

- Erik Meijer, Maarten Fokkinga, Ross Paterson: Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire. See also related documents (in the CiteSeer page). Understanding the article does not require knowledge of category theory—the paper is self-contained with regard to understanding catamorphisms, anamorphisms and other related concepts.
- Varmo Vene and Tarmo Uustalu: Functional Programming with Apomorphisms / Corecursion
- Varmo Vene: Categorical Programming with Inductive and Coinductive Types. The book gives Haskell examples to illustrate the deep categorical theory topic.
- Tatsuya Hagino: A Categorical Programming Language
- Charity, a categorical programming language implementation.
- Deeply uncurried products, as categorists might like them article mentions a conjecture: relatedness to Combinatory logic

## See also

- Michael Barr and Charles Wells have a paper that presents category theory from a computer-science perspective, assuming no prior knowledge of categories.
- Michael Barr and Charles Wells: Toposes, Triples and Theories. The online, freely available book is both an introductory and a detailed description of category theory. It also contains a category-theoretical description of the concept of
*monad*(but calling it a*triple*instead of*monad*). - A Gentle Introduction to Category Theory - the calculational approach written by Maarten M Fokkinga.
- Wikipedia has a good collection of category-theory articles, although, as is typical of Wikipedia articles, they are rather dense.