Category Theory/Definition, examples

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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


  1. If is injective, prove that is injective.