CatDat

monomorphism

A morphism $f : A \to B$ is a monomorphism if it is left-cancellative, i.e. if $f \circ g = f \circ h$ for two morphisms $g,h : T \rightrightarrows A$, then $g = h$. In many concrete categories appearing in practice, these are injective structure-preserving maps.

• Dual property: epimorphism
• Related properties: isomorphism, regular monomorphism
• nLab Link

Relevant implications

• epimorphism andmonomorphism () implies isomorphism
• monomorphism andstrong epimorphism implies isomorphism
• monomorphism () implies isomorphism
• monomorphism () implies regular monomorphism
• monomorphism () implies strong monomorphism
• regular monomorphism implies monomorphism
• strict monomorphism implies monomorphism
• strong monomorphism implies monomorphism

Examples

There are 9 morphisms with this property.

• Baer-Specker relations
• embedding of A3 into S3
• embedding of integer into rational numbers
• handle of the universal fork
• identity map of a group
• identity map of a set
• map from the empty set
• multiplication with 2
• universal morphism

Counterexamples

There are 4 morphisms without this property.

• map into the singleton set
• presentation of the walking idempotent
• reduction modulo p
• 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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