# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1904.05565