nLab extremal epimorphism

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.)

Related concepts

• monomorphism, epimorphism
• extremal monomorphism, extremal epimorphism
• strong monomorphism, strong epimorphism
• strict monomorphism, strict epimorphism
• regular monomorphism, regular epimorphism
• normal monomorphism, normal epimorphism
• effective monomorphism, effective epimorphism
• split monomorphism, split epimorphism

References

• Francis Borceux, Def. 4.3.2 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

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1. Under a default meaning of image that makes reference to the class $M$ of all monomorphisms. ↩