Linear Structures on Locales

We define a notion of morphism for quotient vector bundles that yields both a category QVBun and a contravariant global sections functor C : QVBun → Vect whose restriction to trivial vector bundles with fiber F coincides with the contravariant functor Top → 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 4 equipped with both a topological vector space A and a 4-valued support map for the elements of A satisfying a continuity condition relative to the spectrum of 4 and the lower Vietoris topology on SubA; (iii) we obtain an adjunction between the full subcategory of spectral vector bundles QVBunΣ and the category of linearized locales 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 SubA, called the open support topology, but there is no notion of universal spectral vector bundle for an arbitrary topological vector space A.