Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory

TL;DR

This paper explores the relationship between two module-theoretic approaches to sheaves on involutive quantales, establishing conditions under which modules have canonical Hilbert structures and applying these results to various sheaf theories.

Contribution

It clarifies the connection between Hilbert structures and principally generated modules on involutive quantales, identifying when a canonical Hilbert structure exists.

Findings

Modules with symmetry conditions have canonical Hilbert structures.

Over modular quantal frames, Hilbert structures correspond exactly to principally generated symmetric modules.

Applications to sheaves on locales, quantal frames, and sites are discussed.

Abstract

We explain the precise relationship between two module-theoretic descriptions of sheaves on an involutive quantale, namely the description via so-called Hilbert structures on modules and that via so-called principally generated modules. For a principally generated module satisfying a suitable symmetry condition we observe the existence of a canonical Hilbert structure. We prove that, when working over a modular quantal frame, a module bears a Hilbert structure if and only if it is principally generated and symmetric, in which case its Hilbert structure is necessarily the canonical one. We indicate applications to sheaves on locales, on quantal frames and even on sites.