Editing Category theory (section)

==Definition of a category==


A category <math>\mathcal{C}</math>consists of two collections:

Ob<math>(\mathcal{C})</math>, the objects of <math>\mathcal{C}</math>

Ar<math>(\mathcal{C})</math>, the arrows of <math>\mathcal{C}</math>
(which are not the same as [[Arrow]]s defined in [[GHC]])

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

In symbols, if <math>f\colon A \to B</math> and <math>g\colon B \to
C</math>, then <math>g \circ f \colon A \to C</math>. 

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

===Axioms===
The following axioms must hold for <math>\mathcal{C}</math> to be a category:

#If <math>f\colon A \to B</math> then <math>f \circ \mathrm{id}_A = \mathrm{id}_B\circ f = f</math> (left and right identity) 
#If <math>f\colon A \to B</math> and <math>g \colon B \to C</math> and <math>h \colon C \to D</math>, then <math>h \circ (g \circ f) = (h
\circ g) \circ f</math> (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.

* [[Hask]]
* 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:
* [[Category theory/Functor]]
* [[Category theory/Natural transformation]]
* [[Category theory/Monads]]