strict epimorphism

A morphism $e : A \to B$ is a strict epimorphism if it is the joint coequalizer of all pairs of morphisms $f,g : C \rightrightarrows A$ that it coequalizes. That is, $e$ is an epimorphism, and a morphism $t : A \to T$ factors through $e$ if we have $t \circ f = t \circ g$ for all morphisms $f,g : C \rightrightarrows A$ that satisfy $e \circ f = e \circ g$. That is, the minimal requirement for a morphism to factor through $e$ is actually sufficient.
By the implications below, strict epimorphisms are closely related to effective 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. See also this overview.

Relevant implications

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

Examples

There are 5 morphisms with this property.

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

Counterexamples

There are 10 morphisms without 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
• inclusion of positive numbers
• map from the empty set
• multiplication with 2
• presentation of the walking idempotent
• universal morphism

Unknown

There are 0 morphisms for which the database has no information on whether they satisfy this property.

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