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Homotopy Invariants of Higher Dimensional Categories and Concurrency in Computer Science
, 1999
"... The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other one ..."
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Cited by 49 (10 self)
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The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of C. Two natural linear maps called the negative (resp. the positive) Hurewicz morphism from the globular homology to the negative (resp. positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.
SOME GEOMETRIC PERSPECTIVES IN CONCURRENCY THEORY
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.95–136
, 2003
"... Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on ..."
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Cited by 43 (3 self)
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Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the “direction ” of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: “directed ” or dihomotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ωcategorical and topological techniques.
Combinatorics Of Branchings In Higher Dimensional Automata
 Theory Appl. Categ
, 2001
"... We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory ca ..."
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Cited by 35 (9 self)
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We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the subcomplex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is #categories freely generated by precubical sets. As application, we calculate the branching homology of some #categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.
From Concurrency to Algebraic Topology
, 2000
"... This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !categories, the construction of a diagram of simplicial sets over the threeobject small category gl ! + is exposed. Some of ..."
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Cited by 24 (8 self)
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This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !categories, the construction of a diagram of simplicial sets over the threeobject small category gl ! + is exposed. Some of the properties discovered so far on the corresponding simplicial homology theories are explained, in particular their links with geometric problems coming from concurrency theory in computer science.
About the Globular Homology of Higher Dimensional Automata
, 2000
"... We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau99] disappear. Moreover the important morphisms which associate to every globe its co ..."
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Cited by 23 (8 self)
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We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau99] disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets.
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 : : : : : : : : : : : : : ...
The branching nerve of HDA and the Kan condition
 Theory and Applications of Categories
, 2003
"... One can associate to any strict globular omegacategory three augmented simplicial nerves called the globular nerve, the branching and the merging semicubical nerves. If this strict globular omegacategory is freely generated by a precubical set, then the corresponding homology theories contain dif ..."
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Cited by 7 (6 self)
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One can associate to any strict globular omegacategory three augmented simplicial nerves called the globular nerve, the branching and the merging semicubical nerves. If this strict globular omegacategory is freely generated by a precubical set, then the corresponding homology theories contain different informations about the geometry of the higher dimensional automaton modeled by the precubical set. Adding inverses in this omegacategory to any morphism of dimension greater than 2 and with respect to any composition laws of dimension greater than 1 does not change these homology theories. In such a framework, the globular nerve always satisfies the Kan condition. On the other hand, both branching and merging nerves never satisfy it, except in some very particular and uninteresting situations. In this paper, we introduce two new nerves (the branching and merging semiglobular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semicubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers.
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 : : ...
and
, 2000
"... We show the equivalence of two kinds of strict multiple category, namely the wellknown globular ocategories, and the cubical ocategories with connections. # 2002 ..."
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We show the equivalence of two kinds of strict multiple category, namely the wellknown globular ocategories, and the cubical ocategories with connections. # 2002