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Philippe Gaucher

Publications and source records attributed to Philippe Gaucher.

At least 37 records · Page 2Linked to original sources
arXiv2022

Homotopy theory of Moore flows (I)

Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows.

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arXiv2022

Erratum to "Homotopy theory of Moore flows I"

The notion of reparametrization category is incorrectly axiomatized and it must be adjusted. It is proved that for a general reparametrization category $\mathcal{P}$, the tensor product of $\mathcal{P}$-spaces yields a biclosed semimonoidal structure. It is also described some kind of objectwise braiding for $\mathcal{G}$-spaces.

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arXiv2021

Homotopy theory of Moore flows (II)

This paper proves that the q-model structures of Moore flows and of multipointed $d$-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed $d$-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. It is even possible to write an inverse up to homotopy of this functor using Moore flows.

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arXiv2021

Left properness of flows

Using Reedy techniques, this paper gives a correct proof of the left properness of the q-model structure of flows. It fixes the preceding proof which relies on an incorrect argument. The last section is devoted to fix some arguments published in past papers coming from this incorrect argument. These Reedy techniques also enable us to study the interactions between the path space functor of flows with various notions of cofibrations. The proofs of this paper are written to work with many convenient categories of topological spaces like the ones of $k$-spaces and of weakly Hausdorff $k$-spaces and their locally presentable analogues, the $Δ$-generated spaces and the $Δ$-Hausdorff $Δ$-generated spaces.

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arXiv2020

Six model categories for directed homotopy

We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.

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arXiv2020

Flows revisited: the model category structure and its left determinedness

Flows are a topological model of concurrency which enables to encode the notion of refinement of observation and to understand the homological properties of branchings and mergings of execution paths. Roughly speaking, they are Grandis' $d$-spaces without an underlying topological space. They just have an underlying homotopy type. This note is twofold. First, we give a new construction of the model category structure of flows which is more conceptual thanks to Isaev's results. It avoids the use of difficult topological arguments. Secondly, we prove that this model category is left determined by adapting an argument due to Olschok. The introduction contains some speculations about what we expect to find out by localizing this minimal model category structure. Les flots sont un modèle topologique de la concurrence qui permet d'encoder la notion de raffinement de l'observation et de comprendre les propriétés homologiques des branchements et des confluences des chemins d'exécution. Intuitivement, ce sont des $d$-espaces au sens de Grandis sans espace topologique sous-jacent. Ils ont seulement un type d'homotopie sous-jacent. Cette note a deux objectifs. Premièrement de donner une nouvelle construction de la catégorie de modèles des flots plus conceptuelle grâce au travail d'Isaev. Cela permet d'éviter des arguments topologiques difficiles. Deuxièmement nous prouvons que cette catégorie de modèles est déterminée à gauche en adaptant un argument de Olschok. L'introduction contient quelques spéculations sur ce qu'on s'attend à trouver en localisant cette catégorie de modèles minimale.

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arXiv2019

Enriched diagrams of topological spaces over locally contractible enriched categories

It is proved that the projective model structure of the category of topologically enriched diagrams of topological spaces over a topologically enriched locally contractible small category is Quillen equivalent to the standard Quillen model structure of topological spaces. We give a geometric interpretation of this fact in directed homotopy.

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arXiv2017

Combinatorics of past-similarity in higher dimensional transition systems

The key notion to understand the left determined Olschok model category of star-shaped Cattani-Sassone transition systems is past-similarity. Two states are past-similar if they have homotopic pasts. An object is fibrant if and only if the set of transitions is closed under past-similarity. A map is a weak equivalence if and only if it induces an isomorphism after the identification of all past-similar states. The last part of this paper is a discussion about the link between causality and homotopy.

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arXiv2015

Left determined model categories

Several methods for constructing left determined model structures are expounded. The starting point is Olschok's work on locally presentable categories. We give sufficient conditions to obtain left determined model structures on a full reflective subcategory, on a full coreflective subcategory and on a comma category. An application is given by constructing a left determined model structure on star-shaped weak transition systems.

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arXiv2015

The choice of cofibrations of higher dimensional transition systems

It is proved that there exists a left determined model structure of weak transition systems with respect to the class of monomorphisms and that it restricts to left determined model structures on cubical and regular transition systems. Then it is proved that, in these three model structures, for any higher dimensional transition system containing at least one transition, the fibrant replacement contains a transition between each pair of states. This means that the fibrant replacement functor does not preserve the causal structure. As a conclusion, we explain why working with star-shaped transition systems is a solution to this problem.

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arXiv2015

The geometry of cubical and regular transition systems

There exist cubical transition systems containing cubes having an arbitrarily large number of faces. A regular transition system is a cubical transition system such that each cube has the good number of faces. The categorical and homotopical results of regular transition systems are very similar to the ones of cubical ones. A complete combinatorial description of fibrant cubical and regular transition systems is given. One of the two appendices contains a general lemma of independant interest about the restriction of an adjunction to a full reflective subcategory.

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arXiv2014

Towards a homotopy theory of higher dimensional transition systems

We proved in a previous work that Cattani-Sassone's higher dimensional transition systems can be interpreted as a small-orthogonality class of a topological locally finitely presentable category of weak higher dimensional transition systems. In this paper, we turn our attention to the full subcategory of weak higher dimensional transition systems which are unions of cubes. It is proved that there exists a left proper combinatorial model structure such that two objects are weakly equivalent if and only if they have the same cubes after simplification of the labelling. This model structure is obtained by Bousfield localizing a model structure which is left determined with respect to a class of maps which is not the class of monomorphisms. We prove that the higher dimensional transition systems corresponding to two process algebras are weakly equivalent if and only if they are isomorphic. We also construct a second Bousfield localization in which two bisimilar cubical transition systems are weakly equivalent. The appendix contains a technical lemma about smallness of weak factorization systems in coreflective subcategories which can be of independent interest. This paper is a first step towards a homotopical interpretation of bisimulation for higher dimensional transition systems.

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arXiv2014

Homotopy Theory of Labelled Symmetric Precubical Sets

This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determined model category by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems which was introduced in the first paper of this series is used to establish this Quillen equivalence. However, it is not a left Quillen functor. It is only a left adjoint. It is proved that the two model categories are related to each other by a zig-zag of Quillen equivalences of length two. The middle model category is still the model category of cubical transition systems, but with an additional family of generating cofibrations. The weak equivalences are closely related to bisimulation. Similar results are obtained by restricting the constructions to the labelled symmetric precubical sets satisfying the HDA paradigm.

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arXiv2010

Directed algebraic topology and higher dimensional transition system

Cattani-Sassone's notion of higher dimensional transition system is interpreted as a small-orthogonality class of a locally finitely presentable topological category of weak higher dimensional transition systems. In particular, the higher dimensional transition system associated with the labelled n-cube turns out to be the free higher dimensional transition system generated by one n-dimensional transition. As a first application of this construction, it is proved that a localization of the category of higher dimensional transition systems is equivalent to a locally finitely presentable reflective full subcategory of the category of labelled symmetric precubical sets. A second application is to Milner's calculus of communicating systems (CCS): the mapping taking process names in CCS to flows is factorized through the category of higher dimensional transition systems. The method also applies to other process algebras and to topological models of concurrency other than flows.

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arXiv2009

Homotopical interpretation of globular complex by multipointed d-space

Globular complexes were introduced by E. Goubault and the author in arXiv:math/0107060 to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CW-complex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows introduced in arXiv:math/0308054. The underlying category of this model category is a variant of M. Grandis' notion of d-space over a topological space colimit generated by simplices. This result enables us to understand the relationship between the framework of flows and other works in directed algebraic topology using d-spaces. It also enables us to prove that the underlying homotopy type functor of flows constructed in arXiv:math/0308063 can be interpreted up to equivalences of categories as the total left derived functor of a left Quillen adjoint.

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arXiv2009

Combinatorics of labelling in higher dimensional automata

The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled n-cube, in exactly one way. The main ingredient is the non-functorial construction called labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This non-functorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement. We prove in this paper that it is possible to force the labelled coskeleton functor to be well-behaved by working with labelled transverse symmetric precubical sets. Moreover, we prove that this solution is the only one. A transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy. Finally, we also prove that the two settings are equivalent from a directed algebraic topological viewpoint. To illustrate, a new semantics of CCS, equivalent to the old one, is given.

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arXiv2008

Towards a homotopy theory of process algebra

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 (HDA): one non-degenerate full $n$-dimensional 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 a 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.

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