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Modeling Concurrency with Geometry
"... The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer ..."
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Cited by 124 (13 self)
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The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one home over the other? We identify dimension as the culprit: 1dimensional automata are skeletons permitting only interleaving concurrency, whereas true nfold concurrency resides in transitions of dimension n. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one. We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude with a formal definition of higher dimensional automaton as an ncomplex or ncategory, whose two essential axioms are associativity of concatenation within dimension and an interchange principle between dimensions.
Pasting Schemes for the Monoidal Biclosed Structure on
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a gen ..."
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Cited by 18 (0 self)
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this. Contents 1 Introduction 3 2 Cubes and cubical sets 5 2.1 Cubes combinatorially : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 A model category for cubes : : : : : : : : : : : : : : : : : : : : : 6 2.3 Generating the model category for cubes : : : : : : : : : : : : : : 7 2.4 Cubical sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Duality : : : : : : : : : : : : : ...
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Cited by 6 (0 self)
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Higherdimensional Mac Lane's pentagon and Zamolodchikov equations
, 1999
"... An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that \all diagrams commute". This paper gives a higherdimensional generalization of Mac Lane's pentagon: a 6dimensional diagram whose commutativity is ..."
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An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that \all diagrams commute". This paper gives a higherdimensional generalization of Mac Lane's pentagon: a 6dimensional diagram whose commutativity is needed in order for all diagrams in somewhat weak teisi to commute. Looping twice gives a 4dimensional diagram in somewhat weak braided teisi, of which ve 3dimensional edges can be interpreted as proofs of ve dierent Zamolodchikov equations in braided monoidal 2categories. Hence higherdimensional Mac Lane's pentagon expresses the relations between these proofs concisely. 1 Introduction The coherence theorem for tricategories states that every tricategory is triequivalent to a Graycategory [6]. But there is also another coherence theorem for tricategories, stating that tricategories are (algebras for a) contractible (operad) [1], which roughly says that \all diagrams in a tricategory...
Pasting Presentations for OmegaCategories
, 1995
"... The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !ca ..."
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The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...
Linear Term rewriting systems are higher dimensional string rewriting systems
, 1991
"... This paper illustrates a categorical approach to the theory of rewriting systems along the lines adumbrated by Buchberger [3]. It shows how 2categories provide a simple framework for string rewriting systems and briefly reviews the theory of string rewriting to indicate its simplicity. In contrast ..."
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This paper illustrates a categorical approach to the theory of rewriting systems along the lines adumbrated by Buchberger [3]. It shows how 2categories provide a simple framework for string rewriting systems and briefly reviews the theory of string rewriting to indicate its simplicity. In contrast the theory of term rewriting systems is complicated by the interaction of substitution and rewriting. We outline a 3categorical framework for linear term rewriting systems and show by a detailed example how this separates substitution and rewriting and reduces linear term rewriting to the simplicity of string rewriting, but with one extra dimension. 1 Introduction Rewriting systems have broad applications in Computer Science in areas as diverse as automated theorem proving [12], polynomial ideal theory [2], computational algebra [10], Petri nets, and general models of computation, especially using constructions of free categories with structure [16],[14],[4]. A rewriting rule is a "basic d...
ON DEFORMATIONS OF PASTING DIAGRAMS
"... Abstract. We adapt the work of Power [14] to describe general, notnecessarily composable, notnecessarily commutative 2categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in the 2category of klinear categories, para ..."
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Abstract. We adapt the work of Power [14] to describe general, notnecessarily composable, notnecessarily commutative 2categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in the 2category of klinear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack [9], proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy Galgebras of Gerstenhaber and Voronov [10]. 1.
Section Headings ¤1. Outline of the program
, 1995
"... subject which might be called postmodern algebra (or even Òpostmodern mathematicsÓ ..."
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subject which might be called postmodern algebra (or even Òpostmodern mathematicsÓ
On Deformations of Pasting Diagrams 1
, 709
"... Abstract: We adapt the work of Power [13] to describe general, notnecessarily composable, notnecessarily commutative 2categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in klinear categories, paralleling that provid ..."
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Abstract: We adapt the work of Power [13] to describe general, notnecessarily composable, notnecessarily commutative 2categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in klinear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack [9], proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy Galgebras of Gerstenhaber and Voronov [10].