Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory

Details Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory

2008-09-25

arxiv.org/abs/0809.4336v2

Mathematics

Category Theory

21 pages, revised version accepted for publication

Scientific paper

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.

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