nLab
extremal epimorphism

Definition

An extremal epimorphism (also called a cover) in a category $C$ is an epimorphism $e$ such that if $e=mg$ where $m$ is a monomorphism, then $m$ is an isomorphism.

The dual notion is an extremal monomorphism.

Remarks

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

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

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

• An image factorization of a morphism $f$ is, by definition, a factorization $f=me$ where $m$ is a monomorphism and $e$ is an extremal epimorphism.