Nuclear and Trace Ideals in Tensored *-Categories

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations''. The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. We introduce the notion of nuclear ideal to analyze these classes of morphisms. In compact closed categories, we see that all morphisms are nuclear, and in the category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored *-categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations which was the original motivating example. Finally, we extend the recent work of Joyal, Street and Verity on traced monoidal categories to this setting by introducing the notion of a trace ideal. For a given symmetric monoidal category, it is not generally the case that arbitrary endomorphisms can be assigned a trace. However, we can find ideals in the category on which a trace can be defined satisfying equations analogous to those of Joyal, Street and Verity. We establish a close correspondence between nuclear ideals and trace ideals in a tensored *-category, suggested by the correspondence between Hilbert-Schmidt operators and trace operators on a Hilbert space.Comment: 43 pages, Revised versio

A theory for game theories

International audienceGame semantics is a valuable source of fully abstract models of programming languages or proof theories based on categories of so-called games and strategies. However, there are many variants of this technique, whose interrelationships largely remain to be elucidated. This raises the question: what is a category of games and strategies? Our central idea, taken from the first author's PhD thesis, is that positions and moves in a game should be morphisms in a base category: playing move m in position f consists in factoring f through m, the new position being the other factor. Accordingly, we provide a general construction which, from a selection of "legal moves" in an almost arbitrary category, produces a category of games and strategies, together with subcategories of deterministic and winning strategies. As our running example, we instantiate our construction to obtain the standard category of Hyland-Ong games subject to the switching condition. The extension of our framework to games without the switching condition is handled in the first author's PhD thesis

A New Linear Logic for Deadlock-Free Session-Typed Processes

The π -calculus, viewed as a core concurrent programming language, has been used as the target of much research on type systems for concurrency. In this paper we propose a new type system for deadlock-free session-typed π -calculus processes, by integrating two separate lines of work. The first is the propositions-as-types approach by Caires and Pfenning, which provides a linear logic foundation for session types and guarantees deadlock-freedom by forbidding cyclic process connections. The second is Kobayashi’s approach in which types are annotated with priorities so that the type system can check whether or not processes contain genuine cyclic dependencies between communication operations. We combine these two techniques for the first time, and define a new and more expressive variant of classical linear logic with a proof assignment that gives a session type system with Kobayashi-style priorities. This can be seen in three ways: (i) as a new linear logic in which cyclic structures can be derived and a CYCLE -elimination theorem generalises CUT -elimination; (ii) as a logically-based session type system, which is more expressive than Caires and Pfenning’s; (iii) as a logical foundation for Kobayashi’s system, bringing it into the sphere of the propositions-as-types paradigm

A categorical framework for the quantum harmonic oscillator

This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role. Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Surprisingly, generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure

Involutive Categories and Monoids, with a GNS-correspondence

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive categories

Graphical Reasoning in Compact Closed Categories for Quantum Computation

Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning about such graphs and develop this into a generic proof system with a fixed logical kernel for equational reasoning about compact closed categories. Automating this reasoning process is motivated by the slow and error prone nature of manual graph manipulation. A salient feature of our system is that it provides a formal and declarative account of derived results that can include `ellipses'-style notation. We illustrate the framework by instantiating it for a graphical language of quantum computation and show how this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper published at AIS

A domain of spacetime intervals in general relativity

Beginning from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains.Comment: 25 page

Compactness of the space of causal curves

We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure.Comment: 15 page

A combined microfinance and training intervention can reduce HIV risk behaviour in young female participants.

OBJECTIVE: To assess effects of a combined microfinance and training intervention on HIV risk behavior among young female participants in rural South Africa. DESIGN: : Secondary analysis of quantitative and qualitative data from a cluster randomized trial, the Intervention with Microfinance for AIDS and Gender Equity study. METHODS: Eight villages were pair-matched and randomly allocated to receive the intervention. At baseline and after 2 years, HIV risk behavior was assessed among female participants aged 14-35 years. Their responses were compared with women of the same age and poverty group from control villages. Intervention effects were calculated using adjusted risk ratios employing village level summaries. Qualitative data collected during the study explored participants' responses to the intervention including HIV risk behavior. RESULTS: After 2 years of follow-up, when compared with controls, young participants had higher levels of HIV-related communication (adjusted risk ratio 1.46, 95% confidence interval 1.01-2.12), were more likely to have accessed voluntary counseling and testing (adjusted risk ratio 1.64, 95% confidence interval 1.06-2.56), and less likely to have had unprotected sex at last intercourse with a nonspousal partner (adjusted risk ratio 0.76, 95% confidence interval 0.60-0.96). Qualitative data suggest a greater acceptance of intrahousehold communication about HIV and sexuality. Although women noted challenges associated with acceptance of condoms by men, increased confidence and skills associated with participation in the intervention supported their introduction in sexual relationships. CONCLUSIONS: In addition to impacts on economic well being, women's empowerment and intimate partner violence, interventions addressing the economic and social vulnerability of women may contribute to reductions in HIV risk behavior