Finite closed coverings of compact quantum spaces

TL;DR

This paper establishes a classification of finite closed coverings of compact quantum spaces using sheaves over a projective space with the Alexandrov topology, linking algebraic and topological structures.

Contribution

It introduces a functorial equivalence between finite closed coverings of compact quantum spaces and sheaves over a specific projective space, providing a new categorical framework.

Findings

Equivalence between categories of sheaves and algebraic structures

Representation of finite coverings via sheaves over P^ (Z/2)

Application of Gelfand transform to relate coverings and sheaves

Abstract

We show that a projective space P^\infty(Z/2) endowed with the Alexandrov topology is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over this projective space. In technical terms, we prove that the category of finitely supported flabby sheaves of algebras is equivalent to the category of algebras with a finite set of ideals that intersect to zero and generate a distributive lattice. In particular, the Gelfand transform allows us to view finite closed coverings of compact Hausdorff spaces as flabby sheaves of commutative C*-algebras over P^\infty(Z/2).