91 research outputs found
Inverse semigroup actions as groupoid actions
To an inverse semigroup, we associate an \'etale groupoid such that its
actions on topological spaces are equivalent to actions of the inverse
semigroup. Both the object and the arrow space of this groupoid are
non-Hausdorff. We show that this construction provides an adjoint functor to
the functor that maps a groupoid to its inverse semigroup of bisections, where
we turn \'etale groupoids into a category using algebraic morphisms. We also
discuss how to recover a groupoid from this inverse semigroup.Comment: Corrected a typo in Lemma 2.14 in the published versio
Intuitionistic quantum logic of an n-level system
A decade ago, Isham and Butterfield proposed a topos-theoretic approach to
quantum mechanics, which meanwhile has been extended by Doering and Isham so as
to provide a new mathematical foundation for all of physics. Last year, three
of the present authors redeveloped and refined these ideas by combining the
C*-algebraic approach to quantum theory with the so-called internal language of
topos theory (see arXiv:0709.4364). The goal of the present paper is to
illustrate our abstract setup through the concrete example of the C*-algebra of
complex n by n matrices. This leads to an explicit expression for the pointfree
quantum phase space and the associated logical structure and Gelfand transform
of an n-level system. We also determine the pertinent non-probabilisitic
state-proposition pairing (or valuation) and give a very natural
topos-theoretic reformulation of the Kochen--Specker Theorem. The essential
point is that the logical structure of a quantum n-level system turns out to be
intuitionistic, which means that it is distributive but fails to satisfy the
law of the excluded middle (both in opposition to the usual quantum logic).Comment: 26 page
A topos for algebraic quantum theory
The aim of this paper is to relate algebraic quantum mechanics to topos
theory, so as to construct new foundations for quantum logic and quantum
spaces. Motivated by Bohr's idea that the empirical content of quantum physics
is accessible only through classical physics, we show how a C*-algebra of
observables A induces a topos T(A) in which the amalgamation of all of its
commutative subalgebras comprises a single commutative C*-algebra. According to
the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter
has an internal spectrum S(A) in T(A), which in our approach plays the role of
a quantum phase space of the system. Thus we associate a locale (which is the
topos-theoretical notion of a space and which intrinsically carries the
intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which
is the noncommutative notion of a space). In this setting, states on A become
probability measures (more precisely, valuations) on S(A), and self-adjoint
elements of A define continuous functions (more precisely, locale maps) from
S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to
propositions about the system, the pairing map that assigns a (generalized)
truth value to a state and a proposition assumes an extremely simple
categorical form. Formulated in this way, the quantum theory defined by A is
essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical
Physic
Pre-torsors and Galois comodules over mixed distributive laws
We study comodule functors for comonads arising from mixed distributive laws.
Their Galois property is reformulated in terms of a (so-called) regular arrow
in Street's bicategory of comonads. Between categories possessing equalizers,
we introduce the notion of a regular adjunction. An equivalence is proven
between the category of pre-torsors over two regular adjunctions
and on one hand, and the category of regular comonad arrows
from some equalizer preserving comonad to on
the other. This generalizes a known relationship between pre-torsors over equal
commutative rings and Galois objects of coalgebras.Developing a bi-Galois
theory of comonads, we show that a pre-torsor over regular adjunctions
determines also a second (equalizer preserving) comonad and a
co-regular comonad arrow from to , such that the
comodule categories of and are equivalent.Comment: 34 pages LaTeX file. v2: a few typos correcte
Continuous Truth II: Reflections
Abstract. In the late 1960s, Dana Scott first showed how the Stone-Tarski topological interpretation of Heytingâs calculus could be extended to model intuitionistic analysis; in particular Brouwerâs continuity prin-ciple. In the early â80s we and others outlined a general treatment of non-constructive objects, using sheaf modelsâconstructions from topos theoryâto model not only Brouwerâs non-classical conclusions, but also his creation of ânew mathematical entitiesâ. These categorical models are intimately related to, but more general than Scottâs topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In Continuous Truth I, presented at Logic Colloquium â82 in Florence, we showed that general principles of continuity, local choice and local com-pactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more gen-eral analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman & Grayson 1982), and correct two long-standing errors therein
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
Causal categories: relativistically interacting processes
A symmetric monoidal category naturally arises as the mathematical structure
that organizes physical systems, processes, and composition thereof, both
sequentially and in parallel. This structure admits a purely graphical
calculus. This paper is concerned with the encoding of a fixed causal structure
within a symmetric monoidal category: causal dependencies will correspond to
topological connectedness in the graphical language. We show that correlations,
either classical or quantum, force terminality of the tensor unit. We also show
that well-definedness of the concept of a global state forces the monoidal
product to be only partially defined, which in turn results in a relativistic
covariance theorem. Except for these assumptions, at no stage do we assume
anything more than purely compositional symmetric-monoidal categorical
structure. We cast these two structural results in terms of a mathematical
entity, which we call a `causal category'. We provide methods of constructing
causal categories, and we study the consequences of these methods for the
general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure
`What is a Thing?': Topos Theory in the Foundations of Physics
The goal of this paper is to summarise the first steps in developing a
fundamentally new way of constructing theories of physics. The motivation comes
from a desire to address certain deep issues that arise when contemplating
quantum theories of space and time. In doing so we provide a new answer to
Heidegger's timeless question ``What is a thing?''.
Our basic contention is that constructing a theory of physics is equivalent
to finding a representation in a topos of a certain formal language that is
attached to the system. Classical physics uses the topos of sets. Other
theories involve a different topos. For the types of theory discussed in this
paper, a key goal is to represent any physical quantity with an arrow
\breve{A}_\phi:\Si_\phi\map\R_\phi where \Si_\phi and are two
special objects (the `state-object' and `quantity-value object') in the
appropriate topos, .
We discuss two different types of language that can be attached to a system,
. The first, \PL{S}, is a propositional language; the second, \L{S}, is
a higher-order, typed language. Both languages provide deductive systems with
an intuitionistic logic. With the aid of \PL{S} we expand and develop some of
the earlier work (By CJI and collaborators.) on topos theory and quantum
physics. A key step is a process we term `daseinisation' by which a projection
operator is mapped to a sub-object of the spectral presheaf \Sig--the topos
quantum analogue of a classical state space. The topos concerned is \SetH{}:
the category of contravariant set-valued functors on the category (partially
ordered set) \V{} of commutative sub-algebras of the algebra of bounded
operators on the quantum Hilbert space \Hi.Comment: To appear in ``New Structures in Physics'' ed R. Coeck
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