Morita Equivalence for Quantales

TL;DR

The work extends Morita theory to quantales with identity by establishing a complete progenerator-based characterization: two quantales $Q$ and $R$ are Morita equivalent precisely when there exists a progenerator ${}_{Q}P$ such that $R \cong \operatorname{End}_{Q}(P)$. It proves a quantale analogue of the Eilenberg–Watts theorem, yielding an equivalence of module categories ${}_{Q}{\mathcal{M}}$ and ${}_{R}{\mathcal{M}}$ via the bimodules $P$ and $P^*$. The results unify representation-theoretic aspects of quantales and their modules with classical Morita theory for rings and semirings, and provide explicit isomorphisms between endomorphism quantales and the Morita data. The paper also develops the tensor-product framework and adjunctions necessary to transfer structure along equivalences, enabling a robust, categorical view of Morita phenomena in quantales.

Abstract

We study Morita equivalence in the context of quantales with identity, in the wake of Katsov and Nam's analogous work on semirings. Among a number of other results, we prove a characterization of Morita equivalence and an Eilenberg-Watts-type Theorem for quantales.

Theorems & Definitions (52)

• Definition 2.1
• Theorem 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• proof
• Proposition 3.6
• proof