split monomorphism

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

Relevant implications

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

Examples

There are 2 morphisms with this property.

• identity map of a group
• identity map of a set

Counterexamples

There are 13 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
• 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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