nLab
strong epimorphism

Definition

A strong epimorphism in a category $C$ is an epimorphism which is left orthogonal to any monomorphism in $C$.

Remarks

• If $C$ has equalizers, then any morphism which is left orthogonal to all monomorphisms must automatically be an epimorphism.

• Every regular epimorphism is strong. The converse is true if $C$ is regular.

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