Definition: A category C consists of
Definition: if domain and codomain of a morphism is same, then it is called endomorphism: .
Definition: a morphism is an isomorphism if it has a two sided inverse, i.e. s.t. and .
If all morphisms in a category are isomorphisms, then that category is called groupoids.
If an endomorphism is also an isomorpihm, it is called automorphism. The set of automorphisms is denoted as , which is a subset of .
Aut(A) is a group for all objects of all categories.
Definitions: A morphism is a monomorphism if for all objects and all morphisms of the category, we have
(3.3) |
In words monomorphisms cannot morph different morphisms into the same one. Similarly, epimorphisms cannot be morphed into the same morphism by different morphisms, with the definition: A morphism is a epimorphism if for all objects and all morphisms of the category, we have
(3.4) |
A morphism being both a monomorphism and epimorphism does not necessarily guarantee that it is also an isomorphism. This is only true in some categories such as Set.