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.Comment: 21 pages, revised version accepted for publicatio