Dedekind categories with cutoff operators

Dedekind categories provide a suitable categorical framework for lattice-valued binary relations. It is known that the notion of crispness cannot be described by the basic tools of this theory only. In this paper we will study Dedekind categories with a cutoff operator in order to circumvent this shortage. We will introduce and investigate the properties of the operator and its relationship with other tools previously used for the same purpose. The main result of this paper is a representation theorem for Dedekind categories with a cutoff operator satisfying the point axiom.