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 $𝑋$ . #m/def/cat In particular by the universal property of the equalizer it is a monomorphism.

Proof

Let $𝑓, 𝑔 : 𝑋 \to 𝑌$ and $𝑡 : 𝐸 \to 𝑋$ be their equalizer. Let $𝑎, 𝑏 : 𝑍 \to 𝑋$ so that $𝑡 𝑎 = 𝑡 𝑏 : = ℎ$ . 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 $𝖳 𝗈 𝗉$

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

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

Regular monomorphism

Examples

Footnotes