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Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
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Cited by 20 (6 self)
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We investigate the development of theories of types and computability via realizability.
Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)
"... This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where ..."
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Cited by 12 (7 self)
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This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologicallytrivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes ' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C*algebras.
Inequilogical spaces, directed homology and noncommutative geometry
 Homology Homotopy Appl
"... Abstract. We introduce a preordered version of D. Scott's equilogical spaces [Sc], called inequilogical spaces, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients ' of spaces, which are not topological spaces and are of interest in noncommutat ..."
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Cited by 7 (5 self)
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Abstract. We introduce a preordered version of D. Scott's equilogical spaces [Sc], called inequilogical spaces, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients ' of spaces, which are not topological spaces and are of interest in noncommutative geometry, with finer results than the ones obtained with equilogical spaces, in a previous paper. This setting is compared with other structures which have been recently used for Directed Algebraic Topology: spaces equipped with an order, or a local order, or distinguished paths or distinguished cubes.
The Extensive Completion Of A Distributive Category
 Theory Appl. Categ
, 2001
"... A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for wh ..."
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Cited by 7 (1 self)
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A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for which D ex is extensive; that is, for all objects A and B the functor D ex /A D ex /B # D ex /(A + B) is an equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive preorder. 1.
Elementary Axioms for Categories of Classes (Extended Abstract)
 In Proceedings of 14th Annual Symposium on Logic in Computer Science
, 1999
"... ) Dedicated to Ana Alex K. Simpson LFCS, Division of Informatics, University of Edinburgh, JCMB, King's Buildings, Edinburgh, EH9 3JZ Alex.Simpson@dcs.ed.ac.uk Abstract We axiomatize a notion of "classic structure" on a regular category, isolating the essential properties of the category of cla ..."
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Cited by 7 (3 self)
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) Dedicated to Ana Alex K. Simpson LFCS, Division of Informatics, University of Edinburgh, JCMB, King's Buildings, Edinburgh, EH9 3JZ Alex.Simpson@dcs.ed.ac.uk Abstract We axiomatize a notion of "classic structure" on a regular category, isolating the essential properties of the category of classes together with its full subcategory of sets. Like the axioms for a topos, our axiomatization is very simple, but has powerful consequences. In particular, we show that our axiomatized categories provide a sound and complete class of models for Intuitionistic ZermeloFraenkel set theory. 1. Introduction Almost thirty years on, Lawvere and Tierney's axiomatization of (elementary) toposes [10] remains a milestone in the development of contemporary mathematics. Nowadays, toposes are defined by axioms of exceptional simplicity, which nonetheless entail an astonishing richness of categorical structure [12]. Moreover, toposes are ubiquitous in mathematics. In particular, they have arisen in ge...
The unfolding of general Petri nets ∗
"... ABSTRACT. The unfolding of (1)safe Petri nets to occurrence nets is well understood. There is a universal characterization of the unfolding of a safe net which is part and parcel of a coreflection ..."
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Cited by 5 (3 self)
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ABSTRACT. The unfolding of (1)safe Petri nets to occurrence nets is well understood. There is a universal characterization of the unfolding of a safe net which is part and parcel of a coreflection
A Note On The Exact Completion Of A Regular Category, And Its Infinitary Generalizations
, 1999
"... . 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 ..."
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Cited by 4 (1 self)
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. 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...