An equational approach to enriched distributivity

TL;DR

This paper extends the classical adjunction between ordered sets and completely distributive lattices to categories enriched over quantales, using a distributive law between down-set and up-set monads, under certain lattice conditions.

Contribution

It introduces a new equational framework for distributivity in enriched categories, generalizing known lattice distributive laws to quantale-enriched settings.

Findings

Distributive law formulated via operations, equations, and choice functions

Extension of adjunction to quantale-enriched categories

Conditions for concrete formulation in terms of lattice properties

Abstract

The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of "truth values"), via an appropriate distributive law between the "down-set" monad and the "up-set" monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices.