split epimorphism

A morphism $e : A \to B$ is a split epimorphism if there is a morphism $m : B \to A$ with $e \circ m = \id_B$.

Relevant implications

• isomorphism implies split epimorphism
• monomorphism andsplit epimorphism implies isomorphism
• split epimorphism implies regular epimorphism

Examples

There are 4 morphisms with this property.

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

Counterexamples

There are 11 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
• reduction modulo p
• universal morphism

Unknown

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

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