extremal epimorphism

Extremal morphisms

Extremal morphisms


An extremal epimorphism (also sometimes called a cover) in a category CC is a morphism ee such that if e=mge = m \circ g where mm is a monomorphism, then mm is an isomorphism.

The dual notion is an extremal monomorphism: a morphism mm such that if m=gem = g \circ e where ee is an epimorphism, then ee is an isomorphism.

Despite the terminology, it is not necessarily true that an extremal epimorphism is in fact an epimorphism, so sometimes this is required as well (and dually for extremal monomorphisms). On the other, this does often follow, as in the first remark below.


Of course, the dual properties are all true of extremal monomorphisms. (See coequalizer, monomorphism, strong monomorphism, pushout, regular monomorphism, coregular category?, coimage factorization?, epimorphism.)

  1. Under a default meaning of image that makes reference to the class MM of all monomorphisms.

Last revised on June 15, 2017 at 08:29:36. See the history of this page for a list of all contributions to it.