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Linear structures on locales

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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.

http://www.tac.mta.ca/tac/volumes/31/20/31-20.pdf

Revised 2016-12-07. Original version at:

http://www.tac.mta.ca/tac/volumes/31/20/31-20a.pdf

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