Sheaves as modules

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.

23 pages. Version 2 contains changes in structure in order to make the main results more explicit. Former section 4 has been eliminated. Added some material on matrix representations ($B$-sets), plus a proof that Hilbert modules with bases are locales (Lemma 3.14)

References in corpus (1)

• Etale groupoids and their quantales

Cited by in corpus (5)

• A perspective on non-commutative frame theory
• Groupoid sheaves as quantale sheaves
• Morita equivalence of pseudogroups
• Actions of étale-covered groupoids
• Functoriality of groupoid quantales. II