In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism.
The two conditions have been studied in the categories of groups, rings, modules, and topological spaces.
The terms "hopfian" and "cohopfian" have arisen since the 1960s, and are said to be in honor of Heinz Hopf and his use of the concept of the hopfian group in his work on fundamental groups of surfaces.
By injectivity, f factors through the identity map IA on A, yielding a morphism g such that gf=IA.
The morphisms in the category of rings with unity are required to preserve the identity, that is, to send 1 to 1.