75 research outputs found
A representation theorem for quantale valued sup-algebras
With this paper we hope to contribute to the theory of quantales and
quantale-like structures. It considers the notion of -sup-algebra and shows
a representation theorem for such structures generalizing the well-known
representation theorems for quantales and sup-algebras. In addition, we present
some important properties of the category of -sup-algebras.Comment: 6 page
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contents of former arXiv:0807.3859 (withdrawn), and the definition of Q-sheaf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
Enriched Stone-type dualities
A common feature of many duality results is that the involved equivalence
functors are liftings of hom-functors into the two-element space resp. lattice.
Due to this fact, we can only expect dualities for categories cogenerated by
the two-element set with an appropriate structure. A prime example of such a
situation is Stone's duality theorem for Boolean algebras and Boolean
spaces,the latter being precisely those compact Hausdorff spaces which are
cogenerated by the two-element discrete space. In this paper we aim for a
systematic way of extending this duality theorem to categories including all
compact Hausdorff spaces. To achieve this goal, we combine duality theory and
quantale-enriched category theory. Our main idea is that, when passing from the
two-element discrete space to a cogenerator of the category of compact
Hausdorff spaces, all other involved structures should be substituted by
corresponding enriched versions. Accordingly, we work with the unit interval
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
Convergence and quantale-enriched categories
Generalising Nachbin's theory of "topology and order", in this paper we
continue the study of quantale-enriched categories equipped with a compact
Hausdorff topology. We compare these -categorical compact
Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that
the presence of a compact Hausdorff topology guarantees Cauchy completeness and
(suitably defined) codirected completeness of the underlying quantale enriched
category
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