monomorphism

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

Relevant implications

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

Examples

There are 11 morphisms with 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
• identity map of a group
• identity map of a set
• inclusion of positive numbers
• 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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