strong monomorphism

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

Relevant implications

• extremal monomorphism () implies strong monomorphism
• monomorphism () implies strong monomorphism
• strict monomorphism implies strong monomorphism
• strong monomorphism implies extremal monomorphism

Examples

There are 6 morphisms with this property.

• Baer-Specker relations
• embedding of A3 into S3
• handle of the universal fork
• identity map of a group
• identity map of a set
• map from the empty set

Counterexamples

There are 9 morphisms without this property.

• embedding of integer into rational numbers
• example of a non-strong extremal monomorphism
• inclusion of positive numbers
• 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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