Morphism

Regular monomorphism

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

Proof

Let 𝑓,𝑔 :𝑋 →𝑌 and 𝑡 :𝐸 →𝑋 be their equalizer. Let 𝑎,𝑏 :𝑍 →𝑋 so that 𝑡𝑎 =𝑡𝑏 :=ℎ. Since the universal property demands that the factorization of ℎ via 𝑡 be unique, it follows that 𝑎 =𝑏.

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

Examples

• Regular monomorphisms in the category of topological spaces

---

tidy | en | SemBr

Footnotes

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