Quantale-valued maps and partial maps

TL;DR

This paper explores the structure of quantale-valued maps and partial maps, characterizing their symmetry and exactness conditions, and establishing a monadic relationship under the axiom of choice.

Contribution

It introduces a categorical framework for quantale-valued maps and partial maps, providing new characterizations and a monadicity result.

Findings

Every $ extsf{Q}$-map is symmetric iff $ extsf{Q}$ is weakly lean.

Every $ extsf{Q}$-map is a set map iff $ extsf{Q}$ is lean.

Category of sets and partial $ extsf{Q}$-maps is monadic over $ extsf{Q}$-maps under the axiom of choice.

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 $\mathsf{Q}\text{-}\mathbf{Map}$.