regular epimorphism

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

Relevant implications

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

Examples

There are 5 morphisms with this property.

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

Counterexamples

There are 10 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
• presentation of the walking idempotent
• universal morphism

Unknown

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

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