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.
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 factorization^{1} 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.)
Last revised on August 26, 2021 at 18:59:33. See the history of this page for a list of all contributions to it.