Morita theory for quantales

TL;DR

The paper develops Morita theory for quantales by leveraging the comparison theorem for adjunctions and monads to characterize when a quantaloid is equivalent to a module category over a quantale; it then provides necessary and sufficient criteria for Morita equivalence of quantales via projective generators and idempotents, including a matrix-based description. It shows that every quantale A is Morita equivalent to a matrix-based endomorphism quantale aMat_X(A)a for suitable X and full idempotent a, and it applies the theory to internal sup-lattices in Grothendieck toposes, yielding equivalences with external modules over Sub(B×B) and recovering established results for locales. The framework thus transfers classical Morita concepts to quantale- and quantaloid-enriched contexts, with concrete consequences for topos theory and locale theory. The work provides a unified categorical account of module categories over quantales and their Morita invariants, enabling structural translations across quantales, quantaloids, and toposes.

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.

Theorems & Definitions (33)

• Theorem 2.2
• Remark 3.4
• Theorem 4.3
• Remark 4.4
• Theorem 4.5
• proof
• Theorem 4.6
• Theorem 4.8
• Theorem 5.1
• Theorem 5.2