Quantale-valued maps and partial maps
Lili Shen, Xiaoye Tang
TL;DR
This work develops a categorical framework for quantale-valued maps and partial maps by treating $\mathsf{Q}$-maps as left adjoints in the quantaloid $\mathsf{Q}\text{-}\mathbf{Rel}$ and defining partial $\mathsf{Q}$-maps via the maybe monad on $\mathsf{Q}\text{-}\mathbf{Map}$. It identifies precise conditions under which $\mathsf{Q}$-maps are symmetric (iff $\mathsf{Q}$ is weakly lean) and when they coincide with graphs of Set-maps (iff $\mathsf{Q}$ is lean), connecting fuzzy-valued maps to ordinary functions. Under the axiom of choice, it proves an equivalence between the Eilenberg-Moore and Kleisli categories for the maybe monad on $\mathsf{Q}$-Map, establishing monadicity of the partial-map construction over $\mathsf{Q}$-Map. Together, these results unify classical partial map theory with quantale-valued relations, offering criteria for when fuzzy maps reduce to classical functions and a monadic framework for partial $\mathsf{Q}$-maps.
Abstract
Let $\mathsf{Q}$ be a commutative and unital quantale. By a $\mathsf{Q}$-map we mean a left adjoint in the quantaloid of sets and $\mathsf{Q}$-relations, and by a partial $\mathsf{Q}$-map we refer to a Kleisli morphism with respect to the maybe monad on the category $\mathsf{Q}\text{-}\mathbf{Map}$ of sets and $\mathsf{Q}$-maps. It is shown that every $\mathsf{Q}$-map is symmetric if and only if $\mathsf{Q}$ is weakly lean, and that every $\mathsf{Q}$-map is exactly a map in $\mathbf{Set}$ if and only $\mathsf{Q}$ is lean. Moreover, assuming the axiom of choice, it is shown that the category of sets and partial $\mathsf{Q}$-maps is monadic over $\mathsf{Q}\text{-}\mathbf{Map}$.