nLab extremal epimorphism

Extremal morphisms

Extremal morphisms


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


  • If CC has all equalizers, then the assumption that ee is an epimorphism is redundant in the definition – in this case any morphism admitting no non-trivial factorizations through monomorphisms is automatically epic.

  • Any strong epimorphism is extremal. The converse is true if CC has all pullbacks.

  • Any regular epimorphism is strong, and hence extremal. The converse is true if CC is regular.

  • An image factorization1 of a morphism ff is, by definition, a factorization f=mef = m \circ e where mm is a monomorphism and ee is an extremal epimorphism.

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 August 26, 2021 at 18:59:33. See the history of this page for a list of all contributions to it.