In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a built-in notion of bisimulation. We show how

The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2-category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this

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decomposable topologies

Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally non-ground character, and the uniform way in which such languages have been extended to typed domains, subject to constraints, suggest that a categorical treatment of constraint domains, of programming syntax and of semantics may be closer in spirit to what declarative programming is really about, than conventional settheoretic semantics. We generalize the notion of a (many-sorted) logic program and of a resolution proof, by defining them both over a (not necessarily free) -category C , a category with products enriched with a mechanism for canonically manipulating n-ary relations [8]. Computing over this domain includes computing over the Herbrand Universe, and over equationally presented constraint domains as special cases. We give a categorical treatment of the fixpoint semantics of Kowalski and van Emden, which establishes completeness in a very general setting. 1 In...

. The ultimate aim of this work is to find a semantics for reasoning about actions and change in cooperating agent scenarios based on the concept of so called logical fiberings. As a first step in this direction we describe a small but non-trivial 3-agent-robotics scenario using two different methods, viz. resource-oriented deductive planning and the mathematical framework of logical fiberings. By means of this scenario both methods are illustrated, compared and the correspondences between the basic notions for both methods are clarified. The fiberings method is found to be very useful in modeling communication and interaction between cooperating agents, due to the possibility to switch between a local/global point of view which is inherent to this framework. We formulate a generic modeling principle using this notion. Furthermore, possible extensions of the framework, like formulas depending on space and/or time, are discussed. 1 Introduction The objective of this work is to provide ...

This thesis is concerned with formal semantics and models for concurrent computational systems, that is, systems consisting of a number of parallel computing sequential systems, interacting with each other and the environment. A formal semantics gives meaning to computational systems by describing their behaviour in a mathematical model. For concurrent systems the interesting aspect of their computation is often how they interact with the environment during a computation and not in which state they terminate, indeed they may not be intended to terminate at all. For this reason they are often referred to as reactive systems, to distinguish them from traditional calculational systems, as e.g. a program calculating your income tax, for which the interesting behaviour is the answer it gives when (or if) it terminates, in other words the (possibly partial) function it computes between input and output. Church's thesis tells us that regardless of whether we choose the lambda calculus, Turing machines, or almost any modern programming language such as C or Java to describe calculational systems, we are able to describe exactly the same class of functions. However, there is no agreement on observable behaviour for concurrent reactive systems, and consequently there is no correspondent to Church's thesis. A result of this fact is that an overwhelming number of di#erent and often competing notions of observable behaviours, primitive operations, languages and mathematical models for describing their semantics, have been proposed in the litterature on concurrency.

Abstract. Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by

. A new description of the exact completion C ex/reg of a regular category C is given, using a certain topos Shv(C) of sheaves on C; the exact completion is then constructed as the closure of C in Shv(C) under finite limits and coequalizers of equivalence relations. An infinitary generalization is proved, and the classical description of the exact completion is derived. 1. Introduction A category C with finite limits is said to be regular if every morphism factorizes as a strong epimorphism followed by a monomorphism, and moreover the strong epimorphisms are stable under pullback; it follows that the strong epimorphisms are precisely the regular epimorphisms, namely those arrows which are the coequalizer of their kernel pair. Every kernel pair is an equivalence relation; a regular category is said to be exact if every equivalence relation is a kernel pair. Thus a regular category is a category with finite limits and coequalizers of kernel pairs, satisfying certain exactness condition...

September, 2005 This paper studies categorical models of predicative formal systems, like Aczel’s CZF or Martin-Löf Type Theory. Following the suggestion of Moerdijk and Palmgren that the collection of such categorical models should have the closure properties of toposes, I study the stability under sheaves of five possible axiomatisations. The conclusion will be that all the notions of a “predicative topos ” that I consider, are stable under presheaves, while most are stable under sheaves. This opens up the possibility of using sheaf models for studying predicative theories. 1