Sheaves as modules

TL;DR

This paper reinterprets sheaves on locales within the framework of quantale modules, establishing a categorical equivalence through Hilbert modules and operator adjoints, thus providing a new algebraic perspective.

Contribution

It introduces a novel module-theoretic approach to sheaves on locales, connecting local homeomorphisms with Hilbert modules and adjointable homomorphisms.

Findings

Equivalence between local homeomorphisms and sheaves via Hilbert modules

Homomorphisms are necessarily adjointable, forming a self-dual category

Morphisms of sheaves correspond to operator adjoints of local homeomorphism maps

Abstract

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms $p:X\to B$ are identified with the Hilbert $B$-modules that are equipped with a natural notion of basis. The homomorphisms of these modules are necessarily adjointable, and the resulting self-dual category yields a description of the equivalence between local homeomorphisms and sheaves whereby morphisms of sheaves arise as the ``operator adjoints'' of the maps of local homeomorphisms.