Regularity vs. constructive complete (co)distributivity

Abstract

It is well known that a relation $\varphi$ between sets is regular if, and only if, $\mathcal{K}\varphi$ is completely distributive (cd), where $\mathcal{K}\varphi$ is the complete lattice consisting of fixed points of the Kan adjunction induced by $\varphi$. For a small quantaloid $\mathcal{Q}$, we investigate the $\mathcal{Q}$-enriched version of this classical result, i.e., the regularity of $\mathcal{Q}$-distributors versus the constructive complete distributivity (ccd) of $\mathcal{Q}$-categories, and prove that "the dual of $\mathcal{K}\varphi$ is (ccd) $\implies$ $\varphi$ is regular $\implies$ $\mathcal{K}\varphi$ is (ccd)" for any $\mathcal{Q}$-distributor $\varphi$. Although the converse implications do not hold in general, in the case that $\mathcal{Q}$ is a commutative integral quantale, we show that these three statements are equivalent for any $\varphi$ if, and only if, $\mathcal{Q}$ is a Girard quantale.