The motivating result is that the freeness of actions of compact quantum groups on unital C*-algebras (principality of comodule algebras) is preserved under one-surjective pullbacks. This talk is an elementary review of an unexpected by-product result on multi-morphism pullbacks: the opposite category of ordered N-coverings of algebras is equivalent to the category of flabby sheaves of algebras over the projective space P(N-1) (Z/2Z) with topology generated by the covering of affine spaces. Here an ordered N-covering of an algebra is an ordered family of N algebra surjections whose ideals intersect to zero and generate a distributive lattice. A key step in the proof is to show that all non-empty open subsets of this projective space form a free distributive lattice. (Determining the number of elements in this lattice is the celebrated Dedekind problem open since 1897.) As a corollary, we obtain an equivalence of the opposite category of flabby sheaves of commutative unital C*-algebras and the category of ordered closed coverings of compact Hausdorff spaces.
Based on joint work with U. Kraehmer, R. Matthes, E. Wagner and B. Zielinski.
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Last updated 6 May 2008. This page was created and is maintained by Stephen Wills (s.wills@ucc.ie).