# Extremal morphisms

## Definition

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

Despite the terminology, it is not necessarily true that an extremal epimorphism is in fact an epimorphism, so sometimes this is required as well (and dually for extremal monomorphisms). On the other, this does often follow, as in the first remark below.

## Remarks

• If $C$ has all equalizers, then any extremal epimorphism must actually be an epimorphism.

• 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 factorization 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.)

Revised on August 21, 2015 22:52:13 by Toby Bartels (98.16.168.238)