CatDat

effective epimorphism

A morphism $p : A \to B$ is an effective epimorphism if the pullback $A \times_B A$ exists and $p$ is the coequalizer of the two projections $p_1,p_2 : A \times_B A \rightrightarrows A$.
By the implications below, effective epimorphisms are closely related to strict and regular epimorphisms. Every effective epimorphism is regular and hence strict, and in a category with pullbacks, every strict epimorphism is effective. Thus, in categories with pullbacks, all three mentioned classes of epimorphisms coincide.

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

Relevant implications

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

Examples

There are 4 morphisms with this property.

• identity map of a group
• identity map of a set
• map into the singleton set
• reduction modulo p

Counterexamples

There are 9 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
• multiplication with 2
• presentation of the walking idempotent
• 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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