# Category

## Definition

A category  is the following data:

• Objects: A collection  of objects.
• Morphisms: For any objects , a collection  of morphisms. Every element in  is termed a morphism from  (i.e., with source or domain ) to  (i.e., with target or co-domain ). The morphism sets for different pairs of objects are disjoint. Note that  is also written as . The collection  is sometimes also denoted  or simply .
• Identity morphism: For every object , a distinguished morphism . This is called the identity morphism of .
• Composition rule: For , a map, called composition of morphisms, from  to . This map is denoted by .

satisfying the following compatibility conditions:

• Associativity of composition: For , with , we have .
• Identity behaves as an identity: For , with , we have .

The collections of objects and morphisms need not be sets. If the collection of objects is a set, and the collection of morphisms between any two objects is a set, the category is termed a small category. If the collection of morphisms between any two objects is a set, the category is termed a locally small category.