Definition. Category [005K]

A category is a collection of objects with a collection of morphisms with the following properties

1. Each morphism has a domain and codomain . We write .
2. Each object has an identity morphism .
3. For any morphisms there exists a composite morphism
4. For any , both and are equal to .
5. For any composable triple of morphisms , , , the composition is associative, e.g . We will denote it by

We will often call morphisms arrows.