regular monomorphism

A morphism $m : A \to B$ is a regular monomorphism if it is the equalizer of a pair of morphisms $f,g : B \rightrightarrows C$. In many categories appearing in practice, this is the same as an embedding. This property is strongly related to other types of monomorphisms by the implications below; see also this overview.

Relevant implications

• effective monomorphism implies regular monomorphism
• extremal monomorphism () implies regular monomorphism
• monomorphism () implies regular monomorphism
• normal monomorphism implies regular monomorphism
• regular monomorphism () implies normal monomorphism
• regular monomorphism implies strict monomorphism
• split monomorphism implies regular monomorphism

Examples

There are 5 morphisms with this property.

• embedding of A3 into S3
• handle of the universal fork
• identity map of a group
• identity map of a set
• map from the empty set

Counterexamples

There are 10 morphisms without this property.

• Baer-Specker relations
• embedding of integer into rational numbers
• example of a non-strong extremal monomorphism
• inclusion of positive 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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