Unsharp Values, Domains and Topoi

The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object, analogous to the state space of a classical system, and a quantity-value object, generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object -- hence the `values' of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as `unsharp values'. We will motivate and explain these aspects of the topos approach and show that the structure of the quantity-value object can be analysed using tools from domain theory, a branch of order theory that originated in theoretical computer science. Moreover, the base category of the topos associated with a quantum system turns out to be a domain if the underlying von Neumann algebra is a matrix algebra. For general algebras, the base category still is a highly structured poset. This gives a connection between the topos approach, noncommutative operator algebras and domain theory. In an outlook, we present some early ideas on how domains may become useful in the search for new models of (quantum) space and space-time.Comment: 32 pages, no figures; to appear in Proceedings of Quantum Field Theory and Gravity, Regensburg (2010

Topological Domain Theory

This thesis presents Topological Domain Theory as a powerful and flexible framework for denotational semantics. Topological Domain Theory models a wide range of type constructions and can interpret many computational features. Furthermore, it has close connections to established frameworks for denotational semantics, as well as to well-studied mathematical theories, such as topology and computable analysis.We begin by describing the categories of Topological Domain Theory, and their categorical structure. In particular, we recover the basic constructions of domain theory, such as products, function spaces, fixed points and recursive types, in the context of Topological Domain Theory.As a central contribution, we give a detailed account of how computational effects can be modelled in Topological Domain Theory. Following recent work of Plotkin and Power, who proposed to construct effect monads via free algebra functors, this is done by showing that free algebras for a large class of parametrised equational theories exist in Topological Domain Theory. These parametrised equational theories are expressive enough to generate most of the standard examples of effect monads. Moreover, the free algebras in Topological Domain Theory are obtained by an explicit inductive construction, using only basic topological and set-theoretical principles.We also give a comparison of Topological and Classical Domain Theory. The category of omega-continuous dcpos embeds into Topological Domain Theory, and we prove that this embedding preserves the basic domain-theoretic constructions in most cases. We show that the classical powerdomain constructions on omega-continuous dcpos, including the probabilistic powerdomain, can be recovered in Topological Domain Theory.Finally, we give a synthetic account of Topological Domain Theory. We show that Topological Domain Theory is a specific model of Synthetic Domain Theory in the realizability topos over Scott's graph model. We give internal characterisations of the categories of Topological Domain Theory in this realizability topos, and prove the corresponding categories to be internally complete and weakly small. This enables us to show that Topological Domain Theory can model the polymorphic lambda-calculus, and to obtain a richer collection of free algebras than those constructed earlier.In summary, this thesis shows that Topological Domain Theory supports a wide range of semantic constructions, including the standard domain-theoretic constructions, computational effects and polymorphism, all within a single setting

Problems in the Theory of Convergence Spaces

We investigate several problems in the theory of convergence spaces: generalization of Kolmogorov separation from topological spaces to convergence spaces, representation of reflexive digraphs as convergence spaces, construction of differential calculi on convergence spaces, mereology on convergence spaces, and construction of a universal homogeneous pretopological space. First, we generalize Kolmogorov separation from topological spaces to convergence spaces; we then study properties of Kolmogorov spaces. Second, we develop a theory of reflexive digraphs as convergence spaces, which we then specialize to Cayley graphs. Third, we conservatively extend the concept of differential from the spaces of classical analysis to arbitrary convergence spaces; we then use this extension to obtain differential calculi for finite convergence spaces, finite Kolmogorov spaces, finite groups, Boolean hypercubes, labeled graphs, the Cantor tree, and real and binary sequences. Fourth, we show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory; we then generalize these results to the cartesian closed category of convergence spaces. Finally, we show that every convergence space can be embedded into a homogeneous convergence space; we then use this result to construct a universal homogeneous pretopological space