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 for their associated categories L.
Keywords: Axiomatic Cohesion, Topos theory, Geometric morphisms, Intensive quality
2020 MSC: 18B25, 03G30, 18F99
Theory and Applications of Categories, Vol. 36, 2021, No. 9, pp 250-279.