# Linear structures on locales

## Pedro Resende and Joao Paulo Santos

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.

Keywords: Quotient vector bundles, locales, Banach bundles, lower Vietoris topology, Fell topology

2010 MSC: 06D22, 18B30, 18B99, 46A99, 46M20, 55R65

Theory and Applications of Categories, Vol. 31, 2016, No. 20, pp 502-541.

Published 2016-06-10.

Revised 2016-12-07. Original version at:
http://www.tac.mta.ca/tac/volumes/31/20/31-20a.pdf

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