[Paper Review] Linear structures on locales

Pedro Resende, João Paulo Santos|arXiv (Cornell University)|Mar 21, 2016

Advanced Operator Algebra Research18 references3 citations

TL;DR

This paper introduces a linear extension of the classical adjunction between topological spaces and locales by defining spectral vector bundles and linearized locales, establishing an adjunction between spectral vector bundles and linearized locales that restricts to an equivalence for sober and spatial cases. It introduces the open support topology and shows that no universal spectral vector bundle exists for arbitrary topological vector spaces.

ABSTRACT

We define a notion of morphism for quotient vector bundles that yields both a category $ extit{QVBun}$ and a contravariant global sections functor $C: extit{QVBun}^{ extrm{op}} o extit{Vect}$ whose restriction to trivial vector bundles with fiber $F$ coincides with the contravariant functor $ extit{Top}^{ extrm{op}} o extit{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 $ riangle$ equipped with both a topological vector space $A$ and a $ riangle$-valued support map for the elements of $A$ satisfying a continuity condition relative to the spectrum of $ riangle$ and the lower Vietoris topology on $\operatorname{Sub} A$; (iii) we obtain an adjunction between the full subcategory of spectral vector bundles $ extit{QVBun}_Σ$ and the category of linearized locales $ extit{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 $A$.

Motivation & Objective

• To extend the classical adjunction between topological spaces and locales to a linear setting using vector bundles.
• To define a category of spectral vector bundles that generalizes locally trivial vector bundles and supports continuous sections.
• To introduce linearized locales as locales equipped with a topological vector space and a continuous support map satisfying adjoint conditions.
• To establish an adjunction between spectral vector bundles and linearized locales, with an equivalence for sober and spatial cases.
• To investigate the existence of universal spectral vector bundles and show their non-existence for general topological vector spaces.

Proposed method

• Define quotient vector bundles via open maps from trivial bundles, ensuring sufficient sections via evaluation maps.
• Introduce the spectrum of a topological vector space as Sub(A) equipped with the lower Vietoris topology.
• Construct support and restriction maps σ: Sub(A) → Ω(X) and γ: Ω(X) → Sub(A) that form an adjunction.
• Define a linearized locale as a triple (△, A, σ, γ) where σ is left adjoint to γ and γ restricted to the prime spectrum is continuous.
• Use the open support topology on Sub(A) to classify spectral vector bundles, refining the lower Vietoris topology.
• Prove that the spectral kernel map is continuous only under specific conditions, such as finite-dimensionality or sobriety.

Experimental results

Research questions

• RQ1 Can the adjunction between topological spaces and locales be linearized using vector bundles?
• RQ2 What conditions ensure that a quotient vector bundle is spectral, i.e., its kernel map is continuous?
• RQ3 Does a universal spectral vector bundle exist for an arbitrary topological vector space?
• RQ4 How does the open support topology refine the classification of quotient vector bundles?
• RQ5 Under what conditions is the spectrum of a linearized locale spatial or sober?

Key findings

• The category of spectral vector bundles QVBunΣ admits an adjunction with the category of linearized locales LinLoc, which restricts to an equivalence for sober spectral vector bundles and spatial linearized locales.
• The open support topology on Sub(A) provides a finer classification of spectral vector bundles than the lower Vietoris topology.
• For infinite-dimensional Hausdorff A, the universal bundle U∘A satisfies the open support property but its spectral kernel is not continuous, so it is not spectral.
• When dim A < ∞, the bundle U∘A is sober and thus spectral, as its spectral kernel is continuous.
• The zero section of U∘A is not closed for first-countable A with dim A > 2, implying that closed zero section does not imply Fell continuity.
• Prime open sets in Max∘A are of the form sob(V) only when dim A < ∞; otherwise, there exist non-sobrified prime opens, such as U_{0,A} = ∅, which are not of the form sob(V).