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Finite closed coverings of compact quantum spaces

Volume 98 / 2012

Piotr M. Hajac, Atabey Kaygun, Bartosz Zieliński Banach Center Publications 98 (2012), 215-237 MSC: Primary 18F20; Secondary 06D99. DOI: 10.4064/bc98-0-8

Abstract

We consider the poset of all non-empty finite subsets of the set of natural numbers, use the poset structure to topologise it with the Alexandrov topology, and call the thus obtained topological space the universal partition space. Then we show that it 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 partition 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 unital C*-algebras over \mathbb P^\infty.

Authors

  • Piotr M. HajacInstytut Matematyczny
    Polska Akademia Nauk
    Śniadeckich 8
    00-956 Warszawa, Poland
    and
    Katedra Metod Matematycznych Fizyki
    Uniwersytet Warszawski
    Hoża 74, 00-682 Warszawa, Poland
    e-mail
  • Atabey KaygunDepartment of Mathematics and Computer Science
    Bahçeşehir University
    Çırağan Cad.
    Beşiktaş 34353 Istanbul, Turkey
    e-mail
  • Bartosz ZielińskiDepartment of Theoretical Physics and Computer Science
    University of Łódź
    Pomorska 149/153
    90-236 Łódź, Poland
    and
    Mathematical Institute
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail

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