Open maps of involutive quantales

Abstract

By a map $p:Q\to X$ of involutive quantales is meant a homomorphism $p^{\ast }:X\to Q$. Calling a map $p$ weakly open if $p^{\ast }$ has a left adjoint $p_{!}$ which satisfies the Frobenius reciprocity condition (i.e., $p_{!}$ is a homomorphism of $X$-modules), we say that $p$ is open if it is stably weakly open. We also study a two-sided version, FR2, of the Frobenius reciprocity condition, and show that the weakly open surjections that satisfy FR2 are open. Maps of the latter kind arise in the study of Fell bundles on groupoids.

Version 2 contains a more detailed description of the coproduct of involutive quantales

References in corpus (4)

• Etale groupoids and their quantales
• On quantales that classify C*-algebras
• Functoriality of groupoid quantales. I
• Quantales and Fell bundles

Cited by in corpus (3)

• The many groupoids of a stably Gelfand quantale
• Quantales and Fell bundles
• On the geometry of physical measurements: topological and algebraic aspects