CatDat

effective monomorphism

A morphism $m : A \to B$ is an effective monomorphism if the pushout $B \sqcup_A B$ exists and $m$ is the equalizer of the two coprojections $i_1,i_2 : B \rightrightarrows B \sqcup_A B$.
By the implications below, effective monomorphisms are closely related to strict and regular monomorphisms. Every effective monomorphism is regular and hence strict, and in a category with pushouts, every strict monomorphism is effective. Thus, in categories with pushouts, all three mentioned classes of monomorphisms coincide.

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

Relevant implications

• effective monomorphism implies regular monomorphism
• isomorphism implies effective monomorphism
• strict monomorphism () implies effective monomorphism

Examples

There are 4 morphisms with this property.

• embedding of A3 into S3
• identity map of a group
• identity map of a set
• map from the empty set

Counterexamples

There are 9 morphisms without this property.

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