# Finite closed coverings of compact quantum spaces

Piotr M. Hajac; Atabey Kaygun; Bartosz Zieliński

Banach Center Publications (2012)

- Volume: 98, Issue: 1, page 215-237
- ISSN: 0137-6934

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topPiotr M. Hajac, Atabey Kaygun, and Bartosz Zieliński. "Finite closed coverings of compact quantum spaces." Banach Center Publications 98.1 (2012): 215-237. <http://eudml.org/doc/281837>.

@article{PiotrM2012,

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 $ℙ^∞$ 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 $ℙ^∞$.},

author = {Piotr M. Hajac, Atabey Kaygun, Bartosz Zieliński},

journal = {Banach Center Publications},

keywords = {partially ordered sets; Alexandrov topology; distributive lattices; flabby sheaves of algebras; equivalence of categories; universal partition space},

language = {eng},

number = {1},

pages = {215-237},

title = {Finite closed coverings of compact quantum spaces},

url = {http://eudml.org/doc/281837},

volume = {98},

year = {2012},

}

TY - JOUR

AU - Piotr M. Hajac

AU - Atabey Kaygun

AU - Bartosz Zieliński

TI - Finite closed coverings of compact quantum spaces

JO - Banach Center Publications

PY - 2012

VL - 98

IS - 1

SP - 215

EP - 237

AB - 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 $ℙ^∞$ 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 $ℙ^∞$.

LA - eng

KW - partially ordered sets; Alexandrov topology; distributive lattices; flabby sheaves of algebras; equivalence of categories; universal partition space

UR - http://eudml.org/doc/281837

ER -

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