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Relations in Concurrency
"... The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the seman ..."
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Cited by 304 (36 self)
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The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higherorder processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higherorder processes. For this it is useful to generalise event structures to allow events which “persist.”
Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
"... ..."
Categories and groupoids
, 1971
"... In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, ..."
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Cited by 67 (2 self)
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In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, 37, 58, 65] 1). By contrast, the use of groupoids was confined to a small number of pioneering articles, notably by Ehresmann [12] and Mackey [57], which were largely ignored by the mathematical community. Indeed groupoids were generally considered at that time not to be a subject for serious study. It was argued by several wellknown mathematicians that group theory sufficed for all situations where groupoids might be used, since a connected groupoid could be reduced to a group and a set. Curiously, this argument, which makes no appeal to elegance, was not applied to vector spaces: it was well known that the analogous reduction in this case is not canonical, and so is not available, when there is extra structure, even such simple structure as an endomorphism. Recently, Corfield in [41] has discussed methodological issues in mathematics with this topic, the resistance to the notion of groupoids, as a prime example. My book was intended chiefly as an attempt to reverse this general assessment of the time by presenting applications of groupoids to group theory
Initial Algebra and Final Coalgebra Semantics for Concurrency
, 1994
"... The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial ..."
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Cited by 56 (8 self)
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The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
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Cited by 51 (3 self)
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... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
Convenient Categories of Smooth Spaces
, 2008
"... A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual ..."
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Cited by 38 (0 self)
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A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual
The SheafTheoretic Structure Of NonLocality and Contextuality
, 2011
"... Locality and noncontextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic nogo theorems by Bell, KochenSpecker, et al. is to show that nonlocality and contextuality are necessary features of any theory whose predic ..."
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Cited by 36 (11 self)
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Locality and noncontextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic nogo theorems by Bell, KochenSpecker, et al. is to show that nonlocality and contextuality are necessary features of any theory whose predictions agree with those of quantum mechanics. We use the mathematics of sheaf theory to analyze the structure of nonlocality and contextuality in a very general setting. Starting from a simple experimental scenario, and the kind of probabilistic models familiar from discussions of Bell’s theorem, we show that there is a very direct, compelling formalization of these notions in sheaftheoretic terms. Moreover, on the basis of this formulation, we show that the phenomena of nonlocality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. We give linear algebraic methods for computing these obstructions, and use these methods to obtain a number of new insights into nonlocality and contextuality. For example, we distinguish a proper hierarchy of strengths of nogo theorems, and show that three leading examples — due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively — occupy successively higher levels of this hierarchy. We show how our abstract setting can be represented in quantum mechanics. In doing so, we uncover a strengthening of the usual nosignalling theorem, which shows that quantum mechanics obeys nosignalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.
Étale groupoids and their quantales
, 2004
"... We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantal ..."
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Cited by 34 (9 self)
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We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal
Arithmetic compactifications of PELtype Shimura varieties
, 2010
"... In this thesis, we constructed minimal (SatakeBailyBorel) compactifications and smooth toroidal compactifications of integral models of general PELtype Shimura varieties (defined as in Kottwitz [72]), with descriptions of stratifications and local structures on them extending the wellknown ones ..."
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Cited by 33 (3 self)
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In this thesis, we constructed minimal (SatakeBailyBorel) compactifications and smooth toroidal compactifications of integral models of general PELtype Shimura varieties (defined as in Kottwitz [72]), with descriptions of stratifications and local structures on them extending the wellknown ones in the complex analytic theory. This carries out a program initiated by Chai, Faltings, and some other people more than twenty years ago. The approach we have taken is to redo the FaltingsChai theory [39] in full generality, with as many details as possible, but without any substantial casebycase study. The essential new ingredient in our approach is the emphasis on level structures, leading to a crucial Weil pairing calculation that enables us to avoid unwanted boundary components in naive constructions.