Membership values in arrow categories

In this paper we consider an internal representation of the lattice of membership values used by the relations in an arbitrary arrow category. This is achieved by defining a more general construction that pairs regular with membership values in one object. We will show basic properties of this construction including its relationship to the lattice of membership values. This characterization of the lattice of membership values allows to specify and reason about the lattice elements as elements of an object of the category. Furthermore, this representation also leads to an abstract approach to higher-order fuzziness based on arrow categories.