CatDat

strict monomorphism

A morphism $m : A \to B$ is a strict monomorphism if it is the joint equalizer of all pairs of morphisms $g,h : B \rightrightarrows C$ that it equalizes. That is, $m$ is a monomorphism, and a morphism $t : T \to B$ factors through $m$ if we have $g \circ t = h \circ t$ for all morphisms $g,h : B \rightrightarrows C$ that satisfy $g \circ m = h \circ m$. That is, the minimal requirement for a morphism to factor through $m$ is actually sufficient.
By the implications below, strict monomorphisms are closely related to effective 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: strict epimorphism
• Related properties: effective monomorphism, regular monomorphism
• nLab Link

Relevant implications

• regular monomorphism implies strict monomorphism
• strict monomorphism () implies effective monomorphism
• strict monomorphism implies monomorphism
• strict monomorphism implies strong 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 7 morphisms without this property.

• embedding of integer into rational 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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