Quantale-valued dissimilarity

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.