Regularity vs. constructive complete (co)distributivity

Hongliang Lai and Lili Shen

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

Keywords: Quantaloid, Girard quantaloid, Quantale, Girard quantale, Regular $\mathcal{Q}$-distributor, Complete distributivity, Kan adjunction

2010 MSC: 18D20, 18B35, 18A40, 06D10, 20M17

Theory and Applications of Categories, Vol. 33, 2018, No. 18, pp 492-522.

Published 2018-05-29.

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