Morita theory for quantales

TL;DR

This paper develops Morita theory for quantales, providing a characterization of quantaloids equivalent to modular categories over quantales and establishing conditions for Morita equivalence between quantales.

Contribution

It introduces a Morita theory framework for quantales, characterizing when quantaloids are equivalent to modular categories and when two quantales are Morita-equivalent.

Findings

Characterization of quantaloids equivalent to modular categories over quantales

Necessary and sufficient conditions for Morita equivalence of quantales

Equivalence of internal sup-lattices in a Grothendieck topos to module categories over quantales

Abstract

Morita theory for quantales is developed. The main result of the paper is a characterization of those quantaloids (categories enriched in the symmetric monoidal closed category of sup-lattices) that are equivalent to modular categories over quantales. Based on this characterization, necessary and sufficient conditions are derived for two quantales to be Morita-equivalent, i. e. have equivalent module categories. As an application, it is shown that the category of internal sup-lattices in a Grothendieck topos is equivalent to the module category over a suitable chosen ordinary quantale.