Linear structures on locales

Abstract

We define a notion of morphism for quotient vector bundles that yields both a category $\text{QVBun}$ and a contravariant global sections functor $C:{\text{QVBun}}^{\text{op}}\to \text{Vect}$ whose restriction to trivial vector bundles with fiber $F$ coincides with the contravariant functor ${\text{Top}}^{\text{op}}\to \text{Vect}$ of $F$-valued continuous functions. Based on this we obtain a linear extension of the adjunction between the categories of topological spaces and locales: (i) a linearized topological space is a spectral vector bundle, by which is meant a mildly restricted type of quotient vector bundle; (ii) a linearized locale is a locale $△$ equipped with both a topological vector space $A$ and a $△$-valued support map for the elements of $A$ satisfying a continuity condition relative to the spectrum of $△$ and the lower Vietoris topology on $\mathrm{Sub}A$; (iii) we obtain an adjunction between the full subcategory of spectral vector bundles ${\text{QVBun}}_{\Sigma }$ and the category of linearized locales $\text{LinLoc}$, which restricts to an equivalence of categories between sober spectral vector bundles and spatial linearized locales. The spectral vector bundles are classified by a finer topology on $\mathrm{Sub}A$, called the open support topology, but there is no notion of universal spectral vector bundle for an arbitrary topological vector space $A$.

40 pages. Version 5 differs from 4 only in typesetting style and in the numbering of theorems, lemmas and examples (which now coincides with that of the published version). Version 6 corrects typos in pp. 2 and 20 ("span" was missing in the definition of gamma), and in the proof of Th. 6.16

References in corpus (5)

• Etale groupoids and their quantales
• On quantales that classify C*-algebras
• Inverse semigroup actions on groupoids
• Fell bundles over inverse semigroups and twisted etale groupoids
• Open quotients of trivial vector bundles

Cited by in corpus (3)

• Quantales and Fell bundles
• An abstract theory of physical measurements
• On the geometry of physical measurements: topological and algebraic aspects