Regularity vs. constructive complete (co)distributivity

Abstract

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