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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.
A model category for the homotopy theory of concurrency
 Homology, Homotopy and Applications
"... Abstract. We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are Shomotopy equivalent. This result provides an interpretation of the notion ..."
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Cited by 38 (13 self)
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Abstract. We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are Shomotopy equivalent. This result provides an interpretation of the notion of Shomotopy equivalence in the framework of model
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.
A homotopy double groupoid of a Hausdorff space II: A van Kampen Theorem
 THEORY AND APPLICATIONS OF CATEGORIES
, 2005
"... This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal ..."
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Cited by 20 (11 self)
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This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.
Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Thin elements and commutative shells in cubical ωcategories
 Theory Appl. Categ
, 2005
"... The relationships between thin elements, commutative shells and connections in cubical ωcategories are explored by a method which does not involve the use of pasting theory or nerves of ωcategories (both of which were previously needed for this purpose; see [2], Section 9). It is shown that compos ..."
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Cited by 14 (0 self)
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The relationships between thin elements, commutative shells and connections in cubical ωcategories are explored by a method which does not involve the use of pasting theory or nerves of ωcategories (both of which were previously needed for this purpose; see [2], Section 9). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 in [7, 6].
On the Expressiveness of higher dimensional automata
 EXPRESS 2004, ENTCS
, 2005
"... Abstract In this paper I compare the expressive power of several models of concurrency based on their ability to represent causal dependence. To this end, I translate these models, in behaviour preserving ways, into the model of higher dimensional automata, which is the most expressive model under i ..."
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Cited by 11 (0 self)
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Abstract In this paper I compare the expressive power of several models of concurrency based on their ability to represent causal dependence. To this end, I translate these models, in behaviour preserving ways, into the model of higher dimensional automata, which is the most expressive model under investigation. In particular, I propose four different translations of Petri nets, corresponding to the four different computational interpretations of nets found in the literature. I also extend various equivalence relations for concurrent systems to higher dimensional automata. These include the history preserving bisimulation, which is the coarsest equivalence that fully respects branching time, causality and their interplay, as well as the STbisimulation, a branching time respecting equivalence that takes causality into account to the extent that it is expressible by actions overlapping in time. Through their embeddings in higher dimensional automata, it is now welldefined whether members of different models of concurrency are equivalent.
Investigating The Algebraic Structure of Dihomotopy Types
, 2001
"... This presentation is the sequel of a paper published in GETCO'00 proceedings where a research program to construct an appropriate algebraic setting for the study of deformations of higher dimensional automata was sketched. This paper focuses precisely on detailing some of its aspects. The main idea ..."
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Cited by 8 (1 self)
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This presentation is the sequel of a paper published in GETCO'00 proceedings where a research program to construct an appropriate algebraic setting for the study of deformations of higher dimensional automata was sketched. This paper focuses precisely on detailing some of its aspects. The main idea is that the category of homotopy types can be embedded in a new category of dihomotopy types, the embedding being realized by the Globe functor. In this latter category, isomorphism classes of objects are exactly higher dimensional automata up to deformations leaving invariant their computer scientific properties as presence or not of deadlocks (or everything similar or related). Some hints to study the algebraic structure of dihomotopy types are given, in particular a rule to decide whether a statement/notion concerning dihomotopy types is or not the lifting of another statement/notion concerning homotopy types. This rule does not enable to guess what is the lifting of a given notion/statement, it only enables to make the verification, once the lifting has been found.
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.