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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contents of former arXiv:0807.3859 (withdrawn), and the definition of Q-sheaf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
Quantum channels as a categorical completion
We propose a categorical foundation for the connection between pure and mixed
states in quantum information and quantum computation. The foundation is based
on distributive monoidal categories.
First, we prove that the category of all quantum channels is a canonical
completion of the category of pure quantum operations (with ancilla
preparations). More precisely, we prove that the category of completely
positive trace-preserving maps between finite-dimensional C*-algebras is a
canonical completion of the category of finite-dimensional vector spaces and
isometries.
Second, we extend our result to give a foundation to the topological
relationships between quantum channels. We do this by generalizing our
categorical foundation to the topologically-enriched setting. In particular, we
show that the operator norm topology on quantum channels is the canonical
topology induced by the norm topology on isometries.Comment: 12 pages + ref, accepted at LICS 201
Categories, norms and weights
The well-known Lawvere category R of extended real positive numbers comes
with a monoidal closed structure where the tensor product is the sum. But R has
another such structure, given by multiplication, which is *-autonomous.
Normed sets, with a norm in R, inherit thus two symmetric monoidal closed
structures, and categories enriched on one of them have a 'subadditive' or
'submultiplicative' norm, respectively. Typically, the first case occurs when
the norm expresses a cost, the second with Lipschitz norms.
This paper is a preparation for a sequel, devoted to 'weighted algebraic
topology', an enrichment of directed algebraic topology. The structure of R,
and its extension to the complex projective line, might be a first step in
abstracting a notion of algebra of weights, linked with physical measures.Comment: Revised version, 16 pages. Some minor correction
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