Bimorphism

bijective morphism in a category

One of the categoric-theoretical generalizations of the concept of a bijective mapping between sets. A morphism $u$ in a category $C$ is said to be a bimorphism if it is both a monomorphism and an epimorphism in $C$. A product of bimorphisms is a bimorphism, i.e. the bimorphisms form a subcategory containing all isomorphisms. In the category of sets and the category of groups every bimorphism is an isomorphism. However, the categories of rings, topological spaces, or Abelian groups without torsion contain bimorphisms that are not isomorphisms.