Morphism

Regular monomorphism

A regular monomorphism1 is a morphism into some object which occurs as the equalizer of some parallel pair of morphisms out of . #m/def/cat In particular by the universal property of the equalizer it is a monomorphism.

Proof

Let and be their equalizer. Let so that . Since the universal property demands that the factorization of via be unique, it follows that .

Regular monomorphisms are a categorical generalization of an embedding, as demonstrated by the Examples. See Regular epimorphism for the dual notion.

Examples

#state/tidy | #lang/en | #SemBr

Footnotes

  1. In these notes, regular monomorphisms are implicitly denoted by , whereas denotes a monomorphism which may not be regular. ↩