[Paper Review] Regularity vs. constructive complete (co)distributivity

Hongliang Lai, Lili Shen|arXiv (Cornell University)|Jun 29, 2016

Homotopy and Cohomology in Algebraic Topology34 references3 citations

TL;DR

This paper establishes a categorical equivalence between regular Q-distributors and constructively completely distributive (ccd) Q-categories for small quantaloids Q, proving that regularity of a Q-distributor ϕ is equivalent to Kϕ being (ccd) if and only if Q is a Girard quantaloid. The key contribution is a generalization of the classical relation between regular relations and completely distributive lattices to enriched category theory, with a new proof of Theorem 1.1 that avoids the Axiom of Choice.

ABSTRACT

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

Motivation & Objective

• To extend the classical theorem linking regular relations and completely distributive lattices to the enriched setting of Q-categories and Q-distributors.
• To investigate whether regularity of a Q-distributor ϕ is equivalent to the constructive complete distributivity (ccd) of the associated Q-category Kϕ.
• To determine the conditions under which the equivalence between regularity of ϕ and (ccd) of Kϕ holds, particularly in the case of commutative integral quantales.
• To clarify the role of the Axiom of Choice in the classical equivalence by providing a constructive proof in the case Q = 2.
• To identify necessary and sufficient conditions on Q for the equivalence between regularity and (ccd) to hold for all Q-distributors.

Proposed method

• Define Kϕ as the complete Q-category of fixed points of the Kan adjunction induced by a Q-distributor ϕ: A → B.
• Use the Q-enriched Yoneda embedding Y: A → PA and the sup-map sup: PA → A to define (ccd) Q-categories via a string of adjunctions T ⊣ sup ⊣ Y.
• Prove that if ϕ is regular, then Kϕ is (ccd), using properties of Kan adjunctions and fixed points in Q-categories.
• Show that if Kϕ is op(ccd), then ϕ is regular, establishing a dual implication that is technically challenging and central to the paper.
• Characterize Girard quantaloids as those for which (ccd) and op(ccd) are equivalent notions, using duality and the existence of a dualizing element.
• Apply the theory to commutative integral quantales, proving that the equivalence between regularity and (ccd) holds if and only if the quantale is Girard, using the bottom element as a dualizing element.

Experimental results

Research questions

• RQ1 Is the regularity of a Q-distributor ϕ equivalent to the constructive complete distributivity (ccd) of the Q-category Kϕ for any small quantaloid Q?
• RQ2 Under what conditions on Q does the equivalence between ϕ being regular and Kϕ being (ccd) hold?
• RQ3 Is the implication from Kϕ being op(ccd) to ϕ being regular true in general, and if not, what conditions make it hold?
• RQ4 For commutative integral quantales, what is the necessary and sufficient condition for the equivalence between regularity of ϕ and (ccd) of Kϕ to hold?
• RQ5 Can the classical result linking regular relations and completely distributive lattices be proven constructively, without the Axiom of Choice?

Key findings

• For any small quantaloid Q, if ϕ is a regular Q-distributor, then Kϕ is (ccd), establishing the implication ϕ regular ⇒ Kϕ (ccd).
• For any small quantaloid Q, if Kϕ is op(ccd), then ϕ is regular, establishing the dual implication Kϕ op(ccd) ⇒ ϕ regular.
• When Q is a Girard quantaloid, the three statements ϕ regular, Kϕ (ccd), and Kϕ op(ccd) are all equivalent for any Q-distributor ϕ.
• In the case Q = 2 (the two-element quantale), the classical result of Zarecki˘ı and Xu–Liu is recovered as a special case, with a new proof that does not rely on the Axiom of Choice.
• For a commutative integral quantale Q, the equivalence between regularity of ϕ and (ccd) of Kϕ holds for all ϕ if and only if Q is a Girard quantale.
• The bottom element ⊥ of a commutative integral quantale is a dualizing element if and only if Q is a Girard quantale, which characterizes the equivalence in the quantale case.