Quantale-valued dissimilarity

In the same spirit of the theory of apartness relations of Scott, a positive theory of dissimilarity valued in an involutive quantale $\mathsf{Q}$ is established without the aid of negation, which dualizes the theory of $\mathsf{Q}$-valued sets in the sense of Hohle--Kubiak that can be understood as sets equipped with a similarity valued in $\mathsf{Q}$. It is demonstrated that sets equipped with a $\mathsf{Q}$-valued dissimilarity are precisely symmetric categories enriched in a quantaloid constructed out of $\mathsf{Q}$, whose morphisms are certain back diagonals of $\mathsf{Q}$. Interactions between similarities and dissimilarities valued in $\mathsf{Q}$ are investigated with the help of certain lax functors. In particular, it is shown that similarities and dissimilarities are interdefinable if $\mathsf{Q}$ is a Girard quantale.