45 research outputs found
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contents of former arXiv:0807.3859 (withdrawn), and the definition of Q-sheaf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
Bohrification
New foundations for quantum logic and quantum spaces are constructed by
merging algebraic quantum theory and topos theory. Interpreting Bohr's
"doctrine of classical concepts" mathematically, given a quantum theory
described by a noncommutative C*-algebra A, we construct a topos T(A), which
contains the "Bohrification" B of A as an internal commutative C*-algebra. Then
B has a spectrum, a locale internal to T(A), the external description S(A) of
which we interpret as the "Bohrified" phase space of the physical system. As in
classical physics, the open subsets of S(A) correspond to (atomic)
propositions, so that the "Bohrified" quantum logic of A is given by the
Heyting algebra structure of S(A). The key difference between this logic and
its classical counterpart is that the former does not satisfy the law of the
excluded middle, and hence is intuitionistic. When A contains sufficiently many
projections (e.g. when A is a von Neumann algebra, or, more generally, a
Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be
compared with the traditional quantum logic, i.e. the orthomodular lattice of
projections in A. This time, the main difference is that the former is
distributive (even when A is noncommutative), while the latter is not.
This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in
"Deep Beauty" (ed. H. Halvorson
Extending obstructions to noncommutative functorial spectra
Any functor from the category of C*-algebras to the category of locales that
assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on
algebras of nxn-matrices for n at least 3. This obstruction also applies to
other spectra such as those named after Zariski, Stone, and Pierce. We extend
these no-go results to functors with values in (ringed) topological spaces,
(ringed) toposes, schemes, and quantales. The possibility of spectra in other
categories is discussed
Exponentiable Grothendieck categories in flat Algebraic Geometry
We introduce and describe the -category of
Grothendieck categories and flat morphisms between them. First, we show that
the tensor product of locally presentable linear categories
restricts nicely to . Then, we characterize exponentiable
objects with respect to : these are continuous Grothendieck
categories. In particular, locally finitely presentable Grothendieck categories
are exponentiable. Consequently, we have that, for a quasi-compact
quasi-separated scheme , the category of quasi-coherent sheaves
is exponentiable. Finally, we provide a family of examples
and concrete computations of exponentials.Comment: Minor revision. The proofs of Sec 5 have been expanded to make the
paper self containe