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Philippe Gaucher

Publications and source records attributed to Philippe Gaucher.

At least 55 records · Page 3Linked to original sources
arXiv2007

Homotopical equivalence of combinatorial and categorical semantics of process algebra

It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomorphic if and only if the associated flows are weakly equivalent. The interest of the categorical semantics is that combinatorics totally disappears. Last but not least, a part of the categorical semantics of CCS goes down to a pure homotopical semantics of CCS using A. Heller's privileged weak limits and colimits. These results can be easily adapted to any other process algebra for any synchronization algebra.

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arXiv2007

Globular realization and cubical underlying homotopy type of time flow of process algebra

We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying homotopy type is naturally isomorphic to the homotopy type of the standard cubical complex associated with the precubical set.

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arXiv2007

Abstract homotopical methods for theoretical computer science

The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author's paper ``A model category for the homotopy theory of concurrency''. It is presented generalizations of the classical Whitehead theorem inverting weak homotopy equivalences between CW-complexes using weak factorization systems. It is also presented methods of calculation of homotopy limits and homotopy colimits using Quillen adjunctions and Reedy categories.

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arXiv2007

T-homotopy and refinement of observation (II) : Adding new T-homotopy equivalences

This paper is the second part of a series of papers about a new notion of T-homotopy of flows. It is proved that the old definition of T-homotopy equivalence does not allow the identification of the directed segment with the 3-dimensional cube. This contradicts a paradigm of dihomotopy theory. A new definition of T-homotopy equivalence is proposed, following the intuition of refinement of observation. And it is proved that up to weak S-homotopy, a old T-homotopy equivalence is a new T-homotopy equivalence. The left-properness of the weak S-homotopy model category of flows is also established in this second part. The latter fact is used several times in the next papers of this series.

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arXiv2006

T-homotopy and refinement of observation (III) : Invariance of the branching and merging homologies

This series explores a new notion of T-homotopy equivalence of flows. The new definition involves embeddings of finite bounded posets preserving the bottom and the top elements and the associated cofibrations of flows. In this third part, it is proved that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. These homology theories are of interest in computer science since they detect the non-deterministic branching and merging areas of execution paths in the time flow of a higher dimensional automaton. The proof is based on Reedy model category techniques.

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arXiv2006

T-homotopy and refinement of observation (IV) : Invariance of the underlying homotopy type

This series explores a new notion of T-homotopy equivalence of flows. The new definition involves embeddings of finite bounded posets preserving the bottom and the top elements and the associated cofibrations of flows. In this fourth part, it is proved that the generalized T-homotopy equivalences preserve the underlying homotopy type of a flow. The proof is based on Reedy model category techniques.

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arXiv2006

Inverting weak dihomotopy equivalence using homotopy continuous flow

A flow is homotopy continuous if it is indefinitely divisible up to S-homotopy. 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 S-homotopy 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.

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arXiv2006

Flow does not model flows up to weak dihomotopy

We prove that the category of flows cannot be the underlying category of a model category whose corresponding homotopy types are the flows up to weak dihomotopy. Some hints are given to overcome this problem. 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 version of the category of flows.

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arXiv2006

T-homotopy and refinement of observation (I) : Introduction

This paper is the extended introduction of a serie of papers about modelling T-homotopy by refinement of observation. The notion of T-homotopy equivalence is discussed. A new one is proposed and its behaviour with respect to other construction in dihomotopy theory is explained.

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arXiv2005

Comparing globular complex and flow

A functor is constructed from the category of globular CW-complexes to that of flows. It allows the comparison of the S-homotopy equivalences (resp. the T-homotopy equivalences) of globular complexes with the S-homotopy equivalences (resp. the T-homotopy equivalences) of flows. Moreover, it is proved that this functor induces an equivalence of categories from the localization of the category of globular CW-complexes with respect to S-homotopy equivalences to the localization of the category of flows with respect to weak S-homotopy equivalences. As an application, we construct the underlying homotopy type of a flow.

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arXiv2003

A model category for the homotopy theory of concurrency

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 S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories.

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arXiv2003

Closed symmetric monoidal structure and flow

The category of flows is not cartesian closed. We construct a closed symmetric monoidal structure which has moreover a satisfactory behavior from the computer scientific viewpoint.

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arXiv2003

The homotopy branching space of a flow

In this talk, I will explain the importance of the homotopy branching space functor (and of the homotopy merging space functor) in dihomotopy theory. The paper is a detailed abstract of math.AT/0304112 and math.AT/0305169.

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arXiv2003

The branching nerve of HDA and the Kan condition

One can associate to any strict globular $ω$-category three augmented simplicial nerves called the globular nerve, the branching and the merging semi-cubical nerves. If this strict globular $ω$-category 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 $ω$-category 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 semi-globular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semi-cubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers.

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arXiv2003

Concurrent Process up to Homotopy (I)

Globular CW-complexes 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 computer-scientific properties like the presence or not of deadlock. One constructs an embedding from globular CW-complexes to flows and one proves that two globular CW-complexes are dihomotopic if and only if the corresponding flows are dihomotopic. This note is the first one presenting some of the results of math.AT/0201252.

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