Quantale-valued maps and partial maps

Abstract

Let $Q$ be a commutative and unital quantale. By a $Q$-map we mean a left adjoint in the quantaloid of sets and $Q$-relations, and by a partial $Q$-map we refer to a Kleisli morphism with respect to the maybe monad on the category $Q\text{-}\mathrm{Map}$ of sets and $Q$-maps. It is shown that every $Q$-map is symmetric if and only if $Q$ is weakly lean, and that every $Q$-map is exactly a map in Set if and only $Q$ is lean. Moreover, assuming the axiom of choice, it is shown that the category of sets and partial $Q$-maps is monadic over $Q\text{-}\mathrm{Map}$.