3,873 research outputs found
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding
modalities to the language of propositional geometric logic. Models for this
logic are based on coalgebras for an endofunctor on some full subcategory of
the category of topological spaces and continuous functions. We investigate
derivation systems, soundness and completeness for such geometric modal logics,
and we we specify a method of lifting an endofunctor on Set, accompanied by a
collection of predicate liftings, to an endofunctor on the category of
topological spaces, again accompanied by a collection of (open) predicate
liftings. Furthermore, we compare the notions of modal equivalence, behavioural
equivalence and bisimulation on the resulting class of models, and we provide a
final object for the corresponding category
The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach
We compare two influential ways of defining a generalized notion of space.
The first, inspired by Gelfand duality, states that the category of
'noncommutative spaces' is the opposite of the category of C*-algebras. The
second, loosely generalizing Stone duality, maintains that the category of
'pointfree spaces' is the opposite of the category of frames (i.e., complete
lattices in which the meet distributes over arbitrary joins). One possible
relationship between these two notions of space was unearthed by Banaschewski
and Mulvey, who proved a constructive version of Gelfand duality in which the
Gelfand spectrum of a commutative C*-algebra comes out as a pointfree space.
Being constructive, this result applies in arbitrary toposes (with natural
numbers objects, so that internal C*-algebras can be defined). Earlier work by
the first three authors, shows how a noncommutative C*-algebra gives rise to a
commutative one internal to a certain sheaf topos. The latter, then, has a
constructive Gelfand spectrum, also internal to the topos in question. After a
brief review of this work, we compute the so-called external description of
this internal spectrum, which in principle is a fibered pointfree space in the
familiar topos Sets of sets and functions. However, we obtain the external
spectrum as a fibered topological space in the usual sense. This leads to an
explicit Gelfand transform, as well as to a topological reinterpretation of the
Kochen-Specker Theorem of quantum mechanics, which supplements the remarkable
topos-theoretic version of this theorem due to Butterfield and Isham.Comment: 12 page
Generalized Vietoris Bisimulations
We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our
notion of bisimulation is sound and complete for behavioural equivalence, and
generalizes Vietoris bisimulations [4]. The main result of our paper is that
bisimulation for a coalgebra is the topological closure of
bisimulation for the underlying coalgebra
A Hilbert C*-module admitting no frames
We show that every infinite-dimensional commutative unital C*-algebra has a
Hilbert C*-module admitting no frames. In particular, this shows that
Kasparov's stabilization theorem for countably generated Hilbert C*-modules can
not be extended to arbitrary Hilbert C*-modules.Comment: Minor change. To appear in Bull. Lond. Math. So
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