2008/09/25 by Hans Heymans, Heymans, Hans, Isar Stubbe +1
Mathematics · #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA) #math.CT #math.QA
paper · pdf · doi:10.48550/arxiv.0809.4336
21 pages, revised version accepted for publication
arxiv created 2009/06/11 · arxiv updated 2009/12/01
We explain the precise relationship between two module-theoretic descriptions of sheaves on an involutive quantale, namely the description via so-called Hilbert structures on modules and that via so-called principally generated modules. For a principally generated module satisfying a suitable symmetry condition we observe the existence of a canonical Hilbert structure. We prove that, when working over a modular quantal frame, a module bears a Hilbert structure if and only if it is principally generated and symmetric, in which case its Hilbert structure is necessarily the canonical one. We indicate applications to sheaves on locales, on quantal frames and even on sites.