Regularity vs. constructive complete (co)distributivity
Hongliang Lai, Lili Shen · Theory and applications of categories · 2018
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.