Regularity vs. constructive complete (co)distributivity

Abstract

It is well known that a relation $\phi$ between sets is regular if, and only if, $\mathcal{K}\phi$ is completely distributive (cd), where $\mathcal{K}\phi$ is the complete lattice consisting of fixed points of the Kan adjunction induced by $\phi$. 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 $\mathcal{K}\phi$ is (ccd) $⟹$ $\phi$ is regular $⟹$ $\mathcal{K}\phi$ is (ccd)" for any $\mathcal{Q}$-distributor $\phi$. 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 $\phi$ if, and only if, $\mathcal{Q}$ is a Girard quantale.

31 pages, final version

References in corpus (3)

• Categorical structures enriched in a quantaloid: categories, distributors and functors
• Categorical structures enriched in a quantaloid: tensored and cotensored categories
• Chu connections and back diagonals between $\mathcal{Q}$-distributors

Cited by in corpus (2)

• Completely distributive enriched categories are not always continuous
• Diagonals between $\mathcal{Q}$-distributors