Categories of relations for variable-basis fuzziness

Arrow categories establish a categorical and algebraic description of LL-fuzzy relations, i.e., relations that use membership values from an arbitrary but fixed complete Heyting algebra LL. With other words arrow categories describe the fixed-basis case. In this paper we are interested in the variable-basis case, i.e., the case where relations between different objects may use different membership values. We will investigate the structure of the collection of lattices of membership values within a given Dedekind category. This will lead to a complete characterization of the variable-basis case in this context.