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24
Stabilization of model categories
, 1998
"... monoidal structure which is compatible with the model structure. Given a monoidal model category, we consider the homotopy theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker ..."
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Cited by 203 (8 self)
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monoidal structure which is compatible with the model structure. Given a monoidal model category, we consider the homotopy theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker, than the results of [10]. In particular, our results apply to the monoidal model category of topological symmetric spectra [7].
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
Homological properties of nondeterministic branchings and mergings in higher dimensional automata
 Homology, Homotopy and Applications
"... Abstract. The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (r ..."
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Cited by 11 (8 self)
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Abstract. The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (resp. mergings) satisfy a long exact sequence. Contents
Flow does not model flows up to weak dihomotopy
 Applied Categorical Structures
, 2005
"... In particular, a new approach of dihomotopy involving simplicial presheaves over an appropriate small category is proposed. This small category is obtained by taking a full subcategory of a locally presentable ..."
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Cited by 10 (4 self)
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In particular, a new approach of dihomotopy involving simplicial presheaves over an appropriate small category is proposed. This small category is obtained by taking a full subcategory of a locally presentable
Concurrent process up to homotopy (I
 C. R. Acad. Sci. Paris Ser. I Math
, 2003
"... Abstract. Les CWcomplexes globulaires et les flots sont deux modélisations géométriques des automates parallèles qui permettent de formaliser la notion de dihomotopie. La dihomotopie est une relation d’équivalence sur les automates parallèles qui préserve des propriétés informatiques comme la prése ..."
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Cited by 7 (4 self)
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Abstract. Les CWcomplexes globulaires et les flots sont deux modélisations géométriques des automates parallèles qui permettent de formaliser la notion de dihomotopie. La dihomotopie est une relation d’équivalence sur les automates parallèles qui préserve des propriétés informatiques comme la présence ou non de deadlock. On construit un plongement des CWcomplexes globulaires dans les flots et on démontre que deux CWcomplexes globulaires sont dihomotopes si et seulement si les flots associés sont dihomotopes. Globular CWcomplexes and flows are both geometric models of concurrent processes which allow to model in a precise way the notion of dihomotopy. Dihomotopy is an equivalence relation which preserves computerscientific properties like the presence or not of deadlock. One constructs an embedding from globular CWcomplexes to flows and one proves that two globular CWcomplexes are dihomotopic if and only if the corresponding flows are dihomotopic. 1. Rappels sur les CWcomplexes globulaires Cette note est la première de deux notes présentant quelques résultats de [3].
Towards a homotopy theory of process algebra
 2008) 353–388 (electronic). MR2426108 (2009d:68115), Zbl 1151.68037
"... Abstract. This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between them for every process name of every process algebr ..."
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Cited by 6 (4 self)
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Abstract. This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between them for every process name of every process algebra with any synchronization algebra using a notion of labelled flow. This interpretation of process algebra satisfies the paradigm of higher dimensional automata: one nondegenerate full ndimensional cube (no more no less) in the underlying space of the time flow corresponding to the concurrent execution of n actions. This result will enable us in future papers to develop an homotopical approach of process algebras. Indeed, several homological constructions related to the causal structure of time flow are possible only in the framework of flows. Contents
Homotopy Branching Space and Weak Dihomotopy
"... The branching space of a flow is the topological space of germs of its nonconstant execution paths beginning in the same way. However, there exist weakly Shomotopy equivalent flows having non weakly homotopy equivalent branching spaces. This topological space is then badly behaved from a computersc ..."
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Cited by 5 (5 self)
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The branching space of a flow is the topological space of germs of its nonconstant execution paths beginning in the same way. However, there exist weakly Shomotopy equivalent flows having non weakly homotopy equivalent branching spaces. This topological space is then badly behaved from a computerscientific viewpoint since weakly Shomotopy equivalent flows must correspond to higher dimensional automata having the same computerscientific properties. To overcome this problem, the homotopy branching space of a flow is introduced as the left derived functor of the branching space functor from the model category of flows to the model category of topological spaces. As an application, we use this new functor to correct the notion of weak dihomotopy equivalence, which did not identify enough flows in its previous version.
Inverting weak dihomotopy equivalence using homotopy continuous flow
 Theory Appl. Categ
"... Abstract. A flow is homotopy continuous if it is indefinitely divisible up to Shomotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and ..."
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Cited by 5 (3 self)
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Abstract. A flow is homotopy continuous if it is indefinitely divisible up to Shomotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provides an implementation of Whitehead’s theorem for the full dihomotopy relation, and not only for Shomotopy as in previous works of the author. This fact is not the consequence of the existence of a model structure on the category of flows because it is known that there does not exist any model structure on it whose weak equivalences are exactly the weak dihomotopy equivalences. This fact is an application of a general result for the localization of a model category with respect to a weak factorization system. Contents