An extremal epimorphism (also sometimes called a cover) in a category is an epimorphism such that if where is a monomorphism, then is an isomorphism.
The dual notion is an extremal monomorphism: a monomorphism such that if where is an epimorphism, then is an isomorphism.
If has all equalizers, then the assumption that 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 has all pullbacks.
Any regular epimorphism is strong, and hence extremal. The converse is true if is regular.
An image factorization1 of a morphism is, by definition, a factorization where is a monomorphism and 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.