Remark 4.21.
The previous lemma implies that any quotient map of logoi is a monomorphism when regarded as a morphism in \(\Topos\). This is analogous to the fact that an epimorphism \(R \to S\) of commutative rings induces a monomorphism \(\Spec(S) \hookrightarrow \Spec(R)\) of affine schemes. Recall that there are two common constructions for commutative rings that give rise to epimorphisms:
Given a commutative ring \(R\) and an ideal \(I\), we may form the quotient ring \(R \to R/I\) by killing off all elements in \(I\).
Given a commutative ring \(R\) and a set \(S\) of elements of \(R\), we may form the localization \(R \to R[S^{-1}]\) by formally inverting all elements in \(S\).
The quotient map \(R \to R/I\) corresponds to a closed immersion of affine schemes. A localization map \(R \to R[S^{-1}]\) induces a flat monomorphism \(\Spec(R[S^{-1}])\to\Spec(R)\); it is an open immersion when \(S\) is generated by finitely many elements, equivalently when the localization may be written as \(R\to R[f^{-1}]\) for a single \(f\in R\). We will discuss the analogous notions of open and closed immersions of topoi in Section 5.4 below.