Finite closed coverings of compact quantum spaces

Piotr 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 -