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 :

  1. Whenever either or , and are disjoint
  2. For any objects of and any morphisms and , there exists a morphism , called the composition of and
  3. Composition is associative, ie.
  4. 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

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