Introducing sheaves over commutative semicartesian quantales

Abstract

We extend the classic definition of sheaves on locales introducing an original notion of sheaves on semicartesian quantales. We show that the resulting category and the category of sheaves on locales share similar categorical properties, and discuss the difficulties in concluding whether our sheaves on quantales form a Grothendieck topos. We also prove a base change theorem, which may be useful not only to study the relation between sheaves on locales and sheaves on quantales, but also may be applied in the presence of an isomorphism of commutative and unital rings.