A Topos Foundation for Theories of Physics: IV. Categories of Systems
arXiv:quant-ph/0703066 · doi:10.1063/1.2883826
Abstract
This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. 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 arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this programme for a single system. In the present paper, we turn to considering a collection of systems: in particular, we are interested in the relation between the topos representation for a composite system, and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category.
arXiv comment38 pages, no figures
References in corpus (5)
- A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
- A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
- A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
- Kochen-Specker theorem for von Neumann algebras
- Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity
Cited by in corpus (43)
- A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
- `What is a Thing?': Topos Theory in the Foundations of Physics
- A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
- A topos for algebraic quantum theory
- A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
- Intuitionistic quantum logic of an n-level system
- Abelian subalgebras and the Jordan structure of a von Neumann algebra
- Topos theory and `neo-realist' quantum theory
- Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme
- Classical and Quantum Probabilities as Truth Values
- Quantum Measure Theory: A New Interpretation
- Quantum Mechanics from Focusing and Symmetry
- Weak Values as Context Dependent Values of Observables and Born's Rule
- Exotic smooth R^4, noncommutative algebras and quantization
- Contextuality and noncommutative geometry in quantum mechanics
- Self-adjoint Operators as Functions II: Quantum Probability
- A Comparison of Two Topos-Theoretic Approaches to Quantum Theory
- Categorical Semantics for Schrödinger's Equation
- Topos Quantum Logic and Mixed States
- Topos Analogues of the KMS State
- Topos Models for Physics and Topos Theory
- Independence Conditions for Nets of Local Algebras as Sheaf Conditions
- Complex Numbers, One-Parameter of Unitary Transformations and Stone's Theorem in Topos Quantum Theory
- The coevent formulation of quantum theory
- Sheaf-Theoretic Methods in Quantum Mechanics and Quantum Information Theory
- Bohrification of local nets
- Lectures on Topos Quantum Theory
- Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics
- How Logic Interacts with Geometry: Infinitesimal Curvature of Categorical Spaces
- Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices
- Coevents as Beables
- Topos Quantum Theory on Quantization-Induced Sheaves
- Synthetic Approach to the Singularity Problem
- Negations and Meets in Topos Quantum Theory
- A Topos Model for Loop Quantum Gravity
- An Introduction to Topos Physics
- Topos Quantum Theory Reduced by Context-Selection Functors
- A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory
- Gravity in the smallest
- Can category-theoretic semantics resolve the problem of the interpretation of the quantum state vector?
- Superlogic Manifolds and Geometric approach to Quantum Logic
- A Topos Theory Foundation for Quantum Mechanics
- Consistent Histories as Valuations