strong epimorphism

A morphism $e : A \to B$ is a strong epimorphism if it is an epimorphism that is left orthogonal to any monomorphism. That is, for every commutative diagram $$\begin{CD} A @>e>> B \\ @VVV @VVV \\ C @>>m> D \end{CD}$$ in which $m : C \to D$ is a monomorphism, there is a unique morphism $B \to C$ such that both triangles commute. $$\begin{CD} A @>e>> B \\ @VVV \swarrow @VVV \\ C @>>m> D \end{CD}$$ Uniqueness is actually for free, and it suffices to demand commutativity of one triangle, as the other one follows.
If the category has equalizers, the orthogonality condition already implies that $e$ is an epimorphism, but in general, we need to demand this.
By the implications below, strong epimorphisms are closely related to extremal epimorphisms: every strong epimorphism is extremal, and the converse holds when pullbacks exist. See also this overview.

Relevant implications

• epimorphism () implies strong epimorphism
• extremal epimorphism () implies strong epimorphism
• strict epimorphism implies strong epimorphism
• strong epimorphism implies extremal epimorphism

Examples

There are 6 morphisms with this property.

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

Counterexamples

There are 9 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
• universal morphism

Unknown

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

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