CatDat

extremal epimorphism

A morphism $e : A \to B$ is an extremal epimorphism if it is an epimorphism and whenever $e = m \circ g$ is a factorization in which $m$ is a monomorphism, then $m$ is an isomorphism. The condition that $e$ is an epimorphism follows from the factorization property when the category has equalizers, but in general, we need to explicitly demand it.
By the implications below, extremal epimorphisms are closely related to strong epimorphisms: every strong epimorphism is extremal, and the converse holds when pullbacks exist. See also this overview.

• Dual property: extremal monomorphism
• Related properties: epimorphism, regular epimorphism, strong epimorphism
• nLab Link

Relevant implications

• epimorphism () implies extremal epimorphism
• extremal epimorphism andmonomorphism implies isomorphism
• extremal epimorphism implies epimorphism
• extremal epimorphism () implies regular epimorphism
• extremal epimorphism () implies strong epimorphism
• strong epimorphism implies extremal epimorphism

Examples

There are 6 morphisms with this property.

• identity map of a group
• identity map of a set
• map into the singleton set
• presentation of the walking idempotent
• reduction modulo p
• universal split epimorphism

Counterexamples

There are 9 morphisms without this property.

• Baer-Specker relations
• embedding of A3 into S3
• embedding of integer into rational numbers
• example of a non-strong extremal monomorphism
• handle of the universal fork
• inclusion of positive numbers
• map from the empty set
• multiplication with 2
• universal morphism

Unknown

There are 0 morphisms for which the database has no information on whether they satisfy this property.

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