Quantale-valued dissimilarity

TL;DR

This paper develops a positive theory of dissimilarity valued in an involutive quantale, linking it to symmetric categories and exploring interactions with similarities, especially in the context of Girard quantales.

Contribution

It introduces a novel framework for $ ext{Q}$-valued dissimilarities without negation and connects them to symmetric categories and similarities using lax functors.

Findings

$ ext{Q}$-valued dissimilarities form symmetric categories.

Interactions between dissimilarities and similarities are characterized.

Dissimilarities and similarities are interdefinable in Girard quantales.

Abstract

Inspired by the theory of apartness relations of Scott, we establish a positive theory of dissimilarity valued in an involutive quantale $\mathsf{Q}$ without the aid of negation. It is demonstrated that a set equipped with a $\mathsf{Q}$-valued dissimilarity is precisely a symmetric category enriched in a subquantaloid of the quantaloid of back diagonals of $\mathsf{Q}$. Interactions between $\mathsf{Q}$-valued dissimilarities and $\mathsf{Q}$-valued similarities (which are equivalent to $\mathsf{Q}$-valued equalities in the sense of H{\"o}hle--Kubiak) are investigated with the help of lax functors. In particular, it is shown that similarities and dissimilarities are interdefinable if $\mathsf{Q}$ is a Girard quantale with a hermitian and cyclic dualizing element.