The canonical intensive quality of a cohesive topos

Abstract

We strengthen a result of Lawvere by proving that every pre-cohesive geometric morphism p: E --> S has a canonical intensive quality s: E --> L. We also discuss examples among bounded pre-cohesive p: E --> S and, in particular, we show that if E is a presheaf topos then so is L.This result lifts to Grothendieck toposes but the sites obtained need not be subcanonical.To illustrate this phenomenon, and also the subtle passage from E to L,we consider a particular family of bounded cohesive toposes over Set and build subcanonical sites fortheir associated categories L.