CatDat

isomorphism

A morphism $f : A \to B$ is an isomorphism if it has an inverse, i.e. a morphism $f^{-1} : B \to A$ satisfying $f \circ f^{-1} = \id_B$ and $f^{-1} \circ f = \id_A$.

• Dual property: isomorphism (self-dual)
• Related properties: epimorphism, monomorphism
• nLab Link

Relevant implications

• epimorphism andmonomorphism () implies isomorphism
• epimorphism andstrong monomorphism implies isomorphism
• epimorphism () implies isomorphism
• isomorphism implies effective epimorphism andsplit epimorphism
• isomorphism implies effective monomorphism andsplit monomorphism
• isomorphism () implies normal epimorphism
• isomorphism () implies normal monomorphism
• monomorphism andstrong epimorphism implies isomorphism
• monomorphism () implies isomorphism

Examples

There are 2 morphisms with this property.

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

Counterexamples

There are 11 morphisms without this property.

• Baer-Specker relations
• embedding of A3 into S3
• embedding of integer into rational numbers
• handle of the universal fork
• 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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