Linear structures on locales

Pedro Resende ; Joao Paulo Santos

Theory and applications of categories, Tome 31 (2016), pp. 502-541

Résumé

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

Détail

Publié le : 2016-06-10

Classification : 06D22, 18B30, 18B99, 46A99, 46M20, 55R65
Keywords: Quotient vector bundles, locales, Banach bundles, lower Vietoris topology, Fell topology