extremal epimorphism


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

The dual notion is an extremal monomorphism.


  • If CC has equalizers, then any morphism with the property above must automatically be an epimorphism.

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

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

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

Revised on May 23, 2012 02:39:38 by Mike Shulman (