# nLab extremal epimorphism

Extremal morphisms

category theory

# Extremal morphisms

## Definition

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

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

## Remarks

• If $C$ has all equalizers, then the assumption that $e$ 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 $C$ has all pullbacks.

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

• An image factorization1 of a morphism $f$ is, by definition, a factorization $f = m \circ e$ where $m$ is a monomorphism and $e$ 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 $M$ of all monomorphisms.