# Category Theory/Definition, examples

**Definition (category)**:

A **category** is a class of objects, together with a class of so-called morphisms, each of which have a domain and a target, and a composition of morphisms, such that the following set of axioms hold, if for any two objects and of the subclass of morphisms with domain
and target is denoted :

- Whenever either or , and are disjoint
- For any objects of and any morphisms and , there exists a morphism , called the
**composition**of and - Composition is associative, ie.
- Whenever is an object of , then there exists a unique morphism that acts as an identity both on the left and on the right for the composition of morphisms.

## Exercises edit

- If both and are set mappings such that is injective, prove that is injective.