CatDat

extremal monomorphism

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

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

Relevant implications

• epimorphism andextremal monomorphism implies isomorphism
• extremal monomorphism implies monomorphism
• extremal monomorphism () implies regular monomorphism
• extremal monomorphism () implies strong monomorphism
• monomorphism () implies extremal monomorphism
• strong monomorphism implies extremal monomorphism

Examples

There are 8 morphisms with this property.

• Baer-Specker relations
• embedding of A3 into S3
• example of a non-strong extremal monomorphism
• handle of the universal fork
• identity map of a group
• identity map of a set
• inclusion of positive numbers
• map from the empty set

Counterexamples

There are 7 morphisms without this property.

• embedding of integer into rational numbers
• map into the singleton set
• multiplication with 2
• presentation of the walking idempotent
• reduction modulo p
• universal morphism
• universal split epimorphism

Unknown

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

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