Linear structures on locales

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