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60
Presheaf Models for Concurrency
, 1999
"... 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 wo ..."
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Cited by 49 (19 self)
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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 builtin notion of bisimulation. We show how
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... 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 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 28 (11 self)
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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 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Homotopy theory of simplicial sheaves in completely decomposable topologies
, 2000
"... decomposable topologies ..."
2011): Nominal terms and nominal logics: from foundations to metamathematics
 In: Handbook of Philosophical Logic
"... ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rew ..."
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Cited by 15 (9 self)
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ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rewriting, algebra, and firstorder logic. What characterises the languages of this chapter is that they are firstorder in character, and yet they can specify and reason on names. In the languages we develop, it will be fairly straightforward to give firstorder ‘nominal ’ axiomatisations of namerelated things like alphaequivalence, captureavoiding substitution, beta and etaequivalence, firstorder logic with its quantifiers—and as we shall see, also arithmetic. The formal axiomatisations we arrive at will closely resemble ‘natural behaviour’; the specifications we see typically written out in normal mathematical usage. This is possible because of a novel namecarrying semantics in nominal sets, through which our languages will have namepermutations and termformers that can bind as primitive builtin features.
FirstOrder Logical Duality
, 2008
"... Generalizing Stone duality for Boolean algebras, an adjunction between Boolean coherent categories—representing firstorder syntax—and certain topological groupoids—representing semantics—is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2valued models is re ..."
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Cited by 14 (4 self)
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Generalizing Stone duality for Boolean algebras, an adjunction between Boolean coherent categories—representing firstorder syntax—and certain topological groupoids—representing semantics—is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2valued models is replaced by an embedding of a Boolean coherent category, B, into a topos of equivariant sheaves on a topological groupoid of setvalued models and isomorphisms between them. The latter is a groupoid representation of the topos of coherent sheaves on B, analogously to how the Stone space of a Boolean algebra is a spatial representation of the ideal completion of the algebra, and the category B can then be recovered from its semantical groupoid, up to pretopos completion. By equipping the groupoid of sets and bijections with a particular topology, one obtains a particular topological groupoid which plays a role analogous to that of the discrete space 2, in being the dual of the object classifier and the object one ‘homs into ’ to recover a Boolean coherent category from its semantical groupoid. Both parts of the adjunction, then, consist of ‘homming into sets’, similarly to how both parts of the equivalence between Boolean algebras and Stone spaces consist of ‘homming into 2’. By slicing over this groupoid (modified to display an alternative setup), Chapter 3 shows how the adjunction specializes to the case of firstorder single sorted logic to yield an adjunction between such theories and an independently characterized slice category of topological groupoids such that the counit component at a theory is an isomorphism. Acknowledgements I would like, first and foremost, to thank my supervisor Steve Awodey. I would like to thank the members of the committee: Jeremy Avigad, Lars
Profunctors, open maps and bisimulation
 Mathematical Structures in Computer Science, To appear. Available from the Glynn Winskel’s web
, 2000
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A Comparison of Two ToposTheoretic Approaches to quantum theory,” arXiv:1010.2031 [mathph
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Logic Programming in Tau Categories
 in Computer Science Logic '94 , LNCS 933
, 1995
"... Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally nonground 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 domai ..."
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Cited by 9 (4 self)
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Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally nonground 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 (manysorted) 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 nary 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...