epimorphism

A morphism $e : A \to B$ is an epimorphism if it is right-cancellative, i.e. if $f \circ e = g \circ e$ for two morphisms $f,g : B \rightrightarrows T$, then $f = g$. In many concrete categories appearing in practice, these are are a bit harder to understand than monomorphisms; in particular, epimorphisms usually are not to be confused with surjective structure-preserving maps. Stronger types of epimorphisms (such as regular epimorphisms) are often much better understood.

Relevant implications

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

Examples

There are 9 morphisms with this property.

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

Counterexamples

There are 6 morphisms without this property.

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

Unknown

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

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