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-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-Map.
NETL
NSTL
Elsevier
