# Open maps of involutive quantales

## TL;DR

This paper introduces a new class of open maps between involutive quantales, characterized by a Frobenius reciprocity condition, and explores their properties and applications to Fell bundles on groupoids.

## Contribution

It defines and analyzes weakly open and open maps of involutive quantales, establishing conditions under which weakly open maps are open, with applications to groupoid theory.

## Key findings

- Weakly open surjections satisfying FR2 are open.
- Open maps are characterized by Frobenius reciprocity conditions.
- Applications to Fell bundles on groupoids.

## Abstract

By a map $p:Q\to X$ of involutive quantales is meant a homomorphism $p^*:X\to Q$. Calling a map $p$ weakly open if $p^*$ 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.