6 research outputs found
Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory
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
Groupoid sheaves as quantale sheaves
Several notions of sheaf on various types of quantale have been proposed and
studied in the last twenty five years. It is fairly standard that for an
involutive quantale Q satisfying mild algebraic properties the sheaves on Q can
be defined to be the idempotent self-adjoint Q-valued matrices. These can be
thought of as Q-valued equivalence relations, and, accordingly, the morphisms
of sheaves are the Q-valued functional relations. Few concrete examples of such
sheaves are known, however, and in this paper we provide a new one by showing
that the category of equivariant sheaves on a localic etale groupoid G (the
classifying topos of G) is equivalent to the category of sheaves on its
involutive quantale O(G). As a means towards this end we begin by replacing the
category of matrix sheaves on Q by an equivalent category of complete Hilbert
Q-modules, and we approach the envisaged example where Q is an inverse quantal
frame O(G) by placing it in the wider context of stably supported quantales, on
one hand, and in the wider context of a module theoretic description of
arbitrary actions of \'etale groupoids, both of which may be interesting in
their own right.Comment: 62 pages. Structure of preprint has changed. It now contains the
contents of former arXiv:0807.3859 (withdrawn), and the definition of Q-sheaf
applies only to inverse quantal frames (Hilbert Q-modules with enough
sections are given no special name for more general quantales
Grothendieck quantaloids for allegories of enriched categories
For any small involutive quantaloid Q we define, in terms of symmetric
quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves
and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is
equivalent to the category of symmetric maps in the former. We prove that
Rel(Q) is the category of relations in a topos if and only if Q is a modular,
locally localic and weakly semi-simple quantaloid; in this case we call Q a
Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever
Q is a Grothendieck quantaloid. Any locale L is a Grothendieck quantale, and
Sh(L) is the topos of sheaves on L. Any small quantaloid of closed cribles is a
Grothendieck quantaloid, and if Q is the quantaloid of closed cribles in a
Grothendieck site (C,J) then Sh(Q) is equivalent to the topos Sh(C,J). Any
inverse quantal frame is a Grothendieck quantale, and if O(G) is the inverse
quantal frame naturally associated with an \'etale groupoid G then Sh(O(G)) is
the classifying topos of G.Comment: 28 pages, final versio