Higher-Order Arrow Categories

Abstract

Arrow and Goguen categories were introduced as a suitable categorical and algebraic description of \({\mathcal L}\)-fuzzy relations, i.e., of relations using membership values from an arbitrary complete Heyting algebra \({\mathcal L}\) instead of truth values or elements from the unit interval [0,1]. Higher-order fuzziness focuses on sets or relations that use membership values that are fuzzy themselves. Fuzzy membership values are functions that assign to a each membership value a degree up to which the value is considered to be the membership degree of the element in question. In this paper we want to extend the theory of arrow categories to higher-order fuzziness. We will show that the arrow category of type (n + 1)-fuzziness is in fact the Kleisli category over the category of type n-fuzziness for a suitable monad.