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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
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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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.
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
A convenient category of locally preordered spaces
 Applied Categorical Structures
, 2008
"... Abstract. As a practical foundation for a homotopy theory of abstract spacetime, ..."
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Abstract. As a practical foundation for a homotopy theory of abstract spacetime,
Worytkiewicz: A model category for local pospaces
 Homology, Homotopy and Applications
, 506
"... Abstract. Locally partialordered spaces (local pospaces) have been used to model concurrent systems. We provide equivalences for these spaces by ..."
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Cited by 7 (2 self)
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Abstract. Locally partialordered spaces (local pospaces) have been used to model concurrent systems. We provide equivalences for these spaces by
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.
Context for models of concurrency
 in Preliminary Proceedings of the Workshop on Geometry and Topology in Concurrency and Distributed Computing GETCO 2004, vol NS042 of BRICS Notes
, 2004
"... Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computerscientific information. That is, one wants to replace a given model with a simpler model with the ..."
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Cited by 5 (2 self)
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Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computerscientific information. That is, one wants to replace a given model with a simpler model with the same directed homotopytype. Unfortunately, the obvious definition of directed homotopy equivalence is too coarse. This paper introduces the notion of context to refine this definition. 1.
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