Philippe Gaucher
Postal address
CNRS
IRIF
Université de Paris Cité
Bâtiment Sophie Germain
Case 7014
75205 Paris Cedex 13
France

Email
firstname.lastname@irif.fr

Location
Bâtiment Sophie Germain
8 place Aurélie Nemours
75013 PARIS
FRANCE

Introduction
Presentation: My work is about Directed Algebraic Topology, which is at the interface between the geometric models of concurrency (some of them topological, other ones combinatorial) and algebraic topology (
CV in English). The main geometric models I am interested in are several variants of Grandis' $d$spaces (multipointed or not), trace spaces giving rise to semicategorical variants called flows, Moore variants or these notions, and several improvements of the notion of precubical set (nonsymmetric and symmetric tranverse set, partial precubical set).
Keywords: homotopy, category, topology, combinatorics, concurrency.
Warnings: I do not work on higher category theory just because I don't need this theory. I did use strict globular and strict cubical higher dimensional categories in the past: that was a deadend for my work. The understanding of the link between the two geometric models of concurrency which are the multipointed dspaces and the flows does not require either to use some weak version of the notion of topologically enriched semicategory (unlike what I thought for several years). Instead, it is a Moore version of the notion of topologically enriched semicategory which is required (the socalled Moore flows introduced in
PDF, and explored further in
PDF,
PDF and
PDF). I do not work on homotopy type theory (HoTT) either.
Artificial Intelligence: Given all the nonsense one reads in the newspapers about AI, I feel obliged to state my point of view here. In French, I recommend the reading of the book
"Le mythe de la singularité. Fautil craindre l’intelligence artificielle ?" by JeanGabriel Ganascia. I don't know whether this book is translated in English. The book deconstructs the myth of the technological singularity, which is the day when a supposed general artificial intelligence will appear. Artificial Intelligence does not exist. It is simply a new programming tool. As far as generative AI is concerned, it simply produces probable sequences of words and symbols without really understanding what it's talking about. We talk about
stochastic parrot. Sometimes this leads to interesting things, sometimes to nonsense.
Publications
 Directed degeneracy maps for precubical sets Theory and Applications of
Categories, vol. 41, No. 7, 194237,
2024 (PDF).
Symmetric transverse sets were introduced to make the construction of the parallel product with synchronization for process algebras functorial. It is proved that one can do directed homotopy on symmetric transverse sets in the following sense. A qrealization functor from symmetric transverse sets to flows is introduced using a qcofibrant replacement functor of flows. By topologizing the cotransverse maps, the cotransverse topological cube is constructed. It can be regarded both as a cotransverse topological space and as a cotransverse Lawvere metric space. A natural realization functor from symmetric transverse sets to flows is introduced using Raussen's notion of natural $d$path extended to symmetric transverse sets thanks to their structure of Lawvere metric space. It is proved that these two realization functors are homotopy equivalent on cofibrant symmetric transverse sets by using the fact that the small category defining symmetric transverse sets is cReedy in Shulman's sense. This generalizes to symmetric transverse sets results previously obtained for precubical sets.
 Regular directed path and Moore flow, Rend. Mat. Appl. (7), Vol. 45 (12) (2024), 111151 (PDF).
Using the notion of tame regular $d$path of the topological $n$cube, we introduce the tame regular realization of a precubical set as a multipointed $d$space. Its execution paths correspond to the nonconstant tame regular $d$paths in the geometric realization of the precubical set. The associated Moore flow gives rise to a functor from precubical sets to Moore flows which is weakly equivalent in the hmodel structure to a colimitpreserving functor. The two functors coincide when the precubical set is spatial, and in particular proper. As a consequence, it is given a model category interpretation of the known fact that the space of tame regular $d$paths of a precubical set is homotopy equivalent to a CWcomplex. We conclude by introducing the regular realization of a precubical set as a multipointed $d$space and with some observations about the homotopical properties of tameness.
 Comparing cubical and globular directed paths, Fundamenta Mathematicae (2023) (PDF).
A flow is a directed space structure on a homotopy type. It is already known that the underlying homotopy type of the realization of a precubical set as a flow is homotopy equivalent to the realization of the precubical set as a topological space. This realization depends on the noncanonical choice of a qcofibrant replacement. We construct a new realization functor from precubical sets to flows which is homotopy equivalent to the previous one and which does not depend on the choice of any cofibrant replacement functor. The main tool is the notion of natural $d$path introduced by Raussen. The flow we obtain for a given precubical set is not anymore qcofibrant but is still mcofibrant. As an application, we prove that the space of execution paths of the realization of a precubical set as a flow is homotopy equivalent to the space of nonconstant $d$paths between vertices in the geometric realization of the precubical set.
 Comparing the nonunital and unital settings for directed homotopy, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LXIV2, Pages 176197 (2023) (PDF).
This note explores the link between the qmodel structure of flows and the Ilias model structure of topologically enriched small categories. Both have weak equivalences which induce equivalences of fundamental (semi)categories. The Ilias model structure cannot be leftlifted along the left adjoint adding identity maps. The minimal model structure on flows having as cofibrations the leftlifting of the cofibrations of the Ilias model structure has a homotopy category equal to the $3$element totally ordered set. The qmodel structure of flows can be rightlifted to a qmodel structure of topologically enriched small categories which is minimal and such that the weak equivalences induce equivalences of fundamental categories. The identity functor of topologically enriched small categories is neither a left Quillen adjoint nor a right Quillen adjoint between the qmodel structure and the Ilias model structure.
Note a mistake in the bibliography: The lastname of Ilias Amrani is Amrani, not Ilias as given in zbMath; therefore it is the Amrani model structure, not the Ilias model structure.
 Homotopy theory of Moore flows (II), Extracta Mathematicae, vol. 36 (2), 157239, 2021 (PDF)
This paper proves that the qmodel 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 qcofibrant objects (all objects are qfibrant). 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.
 Homotopy theory of Moore flows (I), Compositionality 3, 3 2021 (PDF)
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 qmodel category of Moore flows. It is proved that it is Quillen equivalent to the qmodel category of flows. This result is the first step to establish a zigzag of Quillen equivalences between the qmodel structure of multipointed dspaces and the qmodel structure of flows.
 Left properness of flows, Theory and Applications of
Categories, vol. 37, No. 19, 562612,
2021 (PDF)
Using Reedy techniques, this paper gives a correct proof of the left properness of the qmodel structure of flows. It fixes the preceding proof which relies on an incorrect argument. The last section is devoted to fixing 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 kspaces and of weakly Hausdorff kspaces and their locally presentable analogues, the $\Delta$generated spaces and the $\Delta$Hausdorff $\Delta$generated spaces.
 Six model categories for directed homotopy, Categories and General Algebraic Structures with Applications, vol 15(1), 145181, 2021 (PDF)
We construct a qmodel structure, a hmodel structure and a mmodel structure on multipointed $d$spaces and on flows. The two qmodel structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two mmodel structures and the two hmodel 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 mmodel structures, Quillen equivalent to the qmodel structure of the same category, are better behaved than the qmodel structures.
 Flows revisited: the model category structure and its left determinedness, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LXI2 (2020)
(PDF)
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 despaces au sens de Grandis sans espace topologique sousjacent. Ils ont seulement un type d'homotopie sousjacent. 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.
 Enriched diagrams of topological spaces over locally contractible enriched categories, NewYork
Journal of Mathematics 25 (2019), 1485–1510 (PDF)
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.
 Combinatorics of pastsimilarity in higher dimensional transition systems, Theory and Applications of
Categories, vol. 32, 11071164,
2017 (PDF) The key notion to understand the left determined Olschok model
category of starshaped CattaniSassone transition systems is
pastsimilarity. Two states are pastsimilar if they have homotopic
pasts. An object is fibrant if and only if the set of transitions is
closed under pastsimilarity. A map is a weak equivalence if and
only if it induces an isomorphism after the identification of all
pastsimilar states. The last part of this paper is a discussion
about the link between causality and homotopy.
 The choice of cofibrations of higher dimensional transition systems, NewYork
Journal of Mathematics 21 (2015), 11171151 (PDF)
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 starshaped transition systems is a solution to this problem.
 Left determined model categories, NewYork
Journal of Mathematics 21 (2015), 10931115 (PDF)
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 starshaped weak transition systems.
 The geometry of cubical and regular transition systems,
Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LVI4 (2015)
(PDF)
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.
 Erratum to ``Towards a homotopy
theory of higher dimensional transition systems'', Theory and Applications of
Categories, vol. 29, 1720,
2014 (PDF)
Counterexamples for Proposition 8.1 and
Proposition 8.2 are given. They are used in the paper only to prove
Corollary 8.3. A proof of this corollary is given without them. The
proof of the fibrancy of some cubical transition systems is
fixed.
 Homotopy Theory of Labelled
Symmetric Precubical Sets, NewYork
Journal of Mathematics 20 (2014), 93131 (PDF) 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 zigzag 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.
 Towards a homotopy theory of
higher dimensional transition systems, Theory and Applications of
Categories, vol. 25, 295341, 2011 (PDF) We
proved in a previous work that CattaniSassone's higher dimensional
transition systems can be interpreted as a smallorthogonality 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.
 Directed algebraic topology and
higher dimensional transition systems, NewYork
Journal of Mathematics 16 (2010), 409461 (PDF) CattaniSassone's
notion of higher dimensional transition system is interpreted as a
smallorthogonality 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 ncube turns out to be the free higher dimensional
transition system generated by one ndimensional 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.
 Combinatorics of labelling in
higher dimensional automata, Theoretical Computer Science (2010), 411(1113), pp 14521483 (PDF)
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 ncube, in exactly one way. The main
ingredient is the nonfunctorial 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 nonfunctorial 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 wellbehaved 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.
 Homotopical interpretation of
globular complex by multipointed dspace, Theory and Applications of
Categories, vol. 22, 588621, 2009 (PDF) Globular
complexes were introduced by E. Goubault and the author 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 CWcomplex. 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. The underlying
category of this model category is a variant of M. Grandis' notion of
dspace 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
dspaces. It also enables us to prove that the underlying homotopy
type functor of flows can be interpreted up to equivalences of
categories as the total left derived functor of a left Quillen
adjoint.
 Thomotopy and refinement
of observation (I) : Introduction,
Electronic Notes in Theoretical
Computer Sciences 230
(2009), 103110
(PDF) This paper is the extended
introduction of a series of papers about modelling Thomotopy by
refinement of observation. Thenotion of Thomotopy equivalence is
discussed. A new one is proposed and its behaviour with respect to
other construction in dihomotopy theory is explained. We also prove in
appendix that the tensor product of flows is a closed symmetric
monoidal structure.
Note: the version published in ENTCS is the wrong one !! Please
download this one which is a better abstract with an uptodate
bibliography.
 Towards a homotopy theory
of process algebra,
Homology,
Homotopy and Applications, vol. 10 (1):p.353388,
2008
(PDF) 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 nondegenerate 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.
 Globular realization and cubical
underlying homotopy type of time flow of process algebra, NewYork
Journal of Mathematics 14 (2008), 101137 (PDF) 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 Shomotopy. 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.
 Thomotopy and refinement
of observation (II) : Adding new Thomotopy equivalences,
Internat.
J. Math. Math. Sci., Article ID 87404, 20 pages
(2007) (PDF) This paper is the second part of a
series of papers about a new notion of Thomotopy of flows. It is
proved that the old definition of Thomotopy 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 Thomotopy equivalence is proposed,
following the intuition of refinement of observation. And it is proved
that up to weak Shomotopy, a old Thomotopy equivalence is a new
Thomotopy equivalence. The leftproperness of the weak Shomotopy
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.
 Thomotopy and refinement
of observation (III) : Invariance of the branching and merging
homologies, NewYork
Journal of Mathematics 12 (2006), 319348
(PDF) This series
explores a new notion of Thomotopy 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
Thomotopy equivalences preserve the branching and merging homology
theories of a flow. These homology theories are of interest in
computer science since they detect the nondeterministic branching and
merging areas of execution paths in the time flow of a
higherdimensional automaton. The proof is based on Reedy model
category techniques.
 Thomotopy and refinement
of observation (IV) : Invariance of the underlying homotopy
type, NewYork
Journal of Mathematics 12 (2006), 6395
(PDF) This series
explores a new notion of Thomotopy 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 Thomotopy
equivalences preserve the underlying homotopy type of a flow. The
proof is based on Reedy model category techniques.
 Inverting weak dihomotopy
equivalence using homotopy continuous flow,
Theory
and Applications of
Categories, vol. 16, 5983, 2006
(PDF) 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.
Erratum :
the class of morphisms $\mathcal{L}$ must be of course a subclass of
the class of monomorphisms for Proposition 3.18 to be
true.
 Flow does not model flows up
to weak dihomotopy, Applied Categorical Structures, vol. 13, p. 371388
(2005)
(PDF) 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.
 Homological properties of
nondeterministic branchings and mergings in higher dimensional
automata, Homology,
Homotopy and Applications, vol. 7 (1):p.5176, 2005 (PDF). 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.
 Comparing globular complex
and flow,
NewYork
Journal of Mathematics 11 (2005), 97150
(PDF) A functor is
constructed from the category of globular CWcomplexes to that of
flows. It allows the comparison of the Shomotopy equivalences
(resp. the Thomotopy equivalences) of globular complexes with the
Shomotopy equivalences (resp. the Thomotopy equivalences) of
flows. Moreover, it is proved that this functor induces an equivalence
of categories from the localization of the category of globular
CWcomplexes with respect to Shomotopy equivalences to the
localization of the category of flows with respect to weak Shomotopy
equivalences. As an application, we construct the underlying homotopy
type of a flow.
 The homotopy branching
space of a flow, Electronic Notes in Theoretical Computer
Science vol. 100 : pp 95109, 2004
(PDF)
In
this talk, I will explain the importance of the homotopy branching
space functor (and of the homotopy merging space functor) in
dihomotopy theory. Note : the definition of Thomotopy equivalence
given in this talk is now obsolete : it is conjecturally too
big.
 A model category for the homotopy
theory of concurrency, Homology,
Homotopy and Applications, vol. 5 (1):p.549599, 2003 (PDF). 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 categories.
 Concurrent Process up to
Homotopy (II),
C.
R.
Acad. Sci. Paris Ser. I Math., 336(8):647650, 2003
(French)
(PDF) On démontre que la catégorie des
CWcomplexes globulaires à dihomotopie près est équivalente à la
catégorie des flots à dihomotopie faible près. Ce théorème est une
généralisation du théorème classique disant que la catégorie des
CWcomplexes modulo homotopie est équivalente à la catégorie des
espaces topologiques modulo homotopie faible.
One proves that the category of globular CWcomplexes up to dihomotopy
is equivalent to the category of flows up to weak dihomotopy. This
theorem generalizes the classical theorem which states that the
category of CWcomplexes up to homotopy is equivalent to the category
of topological spaces up to weak homotopy.
 Concurrent Process up to
Homotopy (I),
C.
R.
Acad. Sci. Paris Ser. I Math., 336(7):593596, 2003
(French) (PDF) 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.
 (with Eric
Goubault) Topological Deformation of Higher
Dimensional Automata,
Homology,
Homotopy and Applications, vol. 5 (2):p.3982,
2003
(PDF) A local pospace is a gluing of
topological spaces which are equipped with a closed partial ordering
representing the time flow. They are used as a formalization of higher
dimensional automata which model concurrent systems in computer
science. It is known that there are two distinct notions of
deformation of higher dimensional automata, ``spatial'' and
``temporal'', leaving invariant computer scientific properties like
presence or absence of deadlocks. Unfortunately, the formalization of
these notions is still unknown in the general case of local pospaces.
We introduce here a particular kind of local pospace, the ``globular
CWcomplexes'', for which we formalize these notions of deformations
and which are sufficient to formalize higher dimensional automata. The
existence of the category of globular CWcomplexes was already
conjectured in "From Concurrency to Algebraic
Topology". After localizing the category of globular CWcomplexes
by spatial and temporal deformations, we get a category (the category
of dihomotopy types) whose objects up to isomorphism represent exactly
the higher dimensional automata up to deformation. Thus globular
CWcomplexes provide a rigorous mathematical foundation to study from
an algebraic topology point of view higher dimensional automata and
concurrent computations.
 The branching nerve of HDA and the
Kan condition, Theory
and Applications of Categories 11 n°3 (2003),
p.75106 (PDF) One can associate to any strict
globular $\omega$category three augmented simplicial nerves called
the globular nerve, the branching and the merging semicubical nerves.
If this strict globular $\omega$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
$\omega$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 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.
 Investigating The Algebraic
Structure of Dihomotopy Types, Electronic Notes in
Theoretical Computer Science 52 (2) 2002
(PDF)
This
presentation is the sequel of a paper published in the 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.
 About the globular homology
of higher dimensional automata,
Cahiers
de Topologie et Géométrie Différentielle Catégoriques, p.107156,
vol XLIII2 (2002)
(PDF) 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 "Homotopy invariants of higher dimensional
categories and concurrency in computer science" 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.
 Combinatorics of branchings
in higher dimensional automata,
Theory
and Applications of Categories 8 n°12 (2001),
p.324376
(PDF) We explore the combinatorial
properties of the branching areas of execution paths in higher
dimensional automata. Mathematically, this means that we investigate
the combinatorics of the negative corner (or branching) homology of a
globular $\omega$category and the combinatorics of a new homology
theory called the reduced branching homology. The latter is the
homology of the quotient of the branching complex by the subcomplex
generated by its thin elements. Conjecturally it coincides with the
non reduced theory for higher dimensional automata, that is
$\omega$categories freely generated by precubical sets. As
application, we calculate the branching homology of some
$\omega$categories and we give some invariance results for the
reduced branching homology. We only treat the branching side. The
merging side, that is the case of merging areas of execution paths is
similar and can be easily deduced from the branching
side.
 From
Concurrency to Algebraic Topology,
Electronic
Notes in Theoretical Computer Science 39 (2000), no. 2, 19p
(PDF) This paper is a survey of the new
notions and results scattered in other papers. However some
speculations are new. Starting from a formalization of higher
dimensional automata (HDA) by strict globular $\omega$categories, the
construction of a diagram of simplicial sets over the threeobject
small category $\leftarrow gl\rightarrow +$ 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.
 Homotopy
invariants of higher dimensional categories and concurrency in
computer science, Mathematical
Structure in Computer Science 10 (2000), no. 4, p.481524 (PDF) The strict globular $\omega$categories formalize the
execution paths of a parallel automaton and the homotopies between
them. One associates to such (and any) $\omega$category $\mathcal{C}$
three homology theories. The first one is called the globular
homology. It contains the oriented loops of $\mathcal{C}$. The two
other ones are called the negative (resp. positive ) corner homology.
They contain in a certain manner the branching areas of execution
paths or negative corners (resp. the merging areas of execution paths
or positive corners) of $\mathcal{C}$. Two natural linear maps called
the negative (resp. the positive ) Hurewicz morphism from the globular
homology to the negative (resp. positive) corner homology are
constructed. We explain the reason why these constructions allow the
reinterpretation of some geometric problems coming from computer
science.
 Lambdaopérations sur
l'homologie d'une algèbre de Lie de matrices, KTheory,
vol. 13(2), p.151167, 1998
(PDF)
 Produit tensoriel de matrices,
homologie cyclique, homologie des algèbres de Lie, Ann.
Inst. Fourier (Grenoble), vol. 44(2), p.413431, 1994 (PDF)
 Lambdaopérations et homologie des
matrices,
C. R. Acad. Sci. Paris Sér. I Math., 313(10):663666, 1991 (PDF) One extends LodayProcesi $\lambda$operations from
the cyclic homology of $A$ to the homology of the Lie algebra
$\bf{gl}_{\infty}( A)$ using exterior powers of matrices. In this way,
we obtain an interpretation of these $\lambda$operations, originally
defined in combinatorial terms, in terms of matrix operations. One
shows a formula giving their behavior with respect to the direct sum
of matrices. It uses the coproduct and the structure of ring objet
induced by the tensor product of matrices.
 Produit tensoriel de
matrices et homologie cyclique, C. R.
Acad. Sci. Paris
Sér. I Math., 312(1):1316, 1991 (PDF) If $A$
is
an associative and commutative $\mathbb{Q}$algebra with unit, the
tensor product of matrices enables us to define on the homology of the
Lie algebra $\bf{gl}_{\infty}( A)$ a product which give it with the
usual sum a graded ring structure which is commutative. One gives an
explicit formula for this product. After restriction to the primitive
part, this product coincides with the LodayQuillen's product on
cyclic homology.
Unpublished
 Natural homotopy of multipointed dspaces (PDF)
We identify Grandis' directed spaces as a full reflective subcategory of the category of multipointed $d$spaces. When the multipointed $d$space realizes a precubical set, its reflection coincides with the standard realization of the precubical set as a directed space. The reflection enables us to extend the construction of the natural system of topological spaces in BauesWirsching's sense from directed spaces to multipointed $d$spaces. In the case of a cellular multipointed $d$space, there is a discrete version of this natural system which is proved to be bisimilar up to homotopy. We also prove that these constructions are invariant up to homotopy under globular subdivision. These results are the globular analogue of Dubut's results. Finally, we point the incompatibility between the notion of bisimilar natural systems and the qmodel structure of multipointed $d$spaces. This means that either the notion of bisimilar natural systems is too rigid or new model structures should be considered on multipointed $d$spaces.
 Towards a theory of natural directed paths (PDF)
We introduce the abstract setting of presheaf category on a thick category of cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets and the new category of (nonsymmetric) transverse sets are examples of this structure. All these presheaf categories share the same metric and homotopical properties from a directed homotopy point of view. This enables us to extend Raussen's notion of natural $d$path for each of them. Finally, we adapt Ziemia\'{n}ski's notion of cube chain to this abstract setting and we prove that it has the expected behavior on precubical sets. As an application, we verify that the formalization of the parallel composition with synchronization of process algebra using the coskeleton functor of the category of symmetric transverse sets has a category of cube chains with the correct homotopy type.
 Homotopy theory of Moore flows (III) (PDF).
The previous paper of this series shows that the qmodel categories of $\mathcal{G}$multipointed $d$spaces and of $\mathcal{G}$flows are Quillen equivalent. In this paper, the same result is established by replacing the reparametrization category $\mathcal{G}$ by the reparametrization category $\mathcal{M}$. Unlike the case of $\mathcal{G}$, the execution paths of a cellular $\mathcal{M}$multipointed $d$space can have stop intervals. The technical tool to overcome this obstacle is the notion of globular naturalization. It is the globular analogue of Raussen's naturalization of a directed path in the geometric realization of a precubical set. The notion of globular naturalization working both for $\mathcal{G}$ and $\mathcal{M}$, the proof of the Quillen equivalence we obtain is valid for the two reparametrization categories. Together with the results of the first paper of this series, we then deduce that $\mathcal{G}$multipointed $d$spaces and $\mathcal{M}$multipointed $d$spaces have Quillen equivalent qmodel structures. Finally, we prove that the saturation hypothesis can be added without any modification in the main theorems of the paper.
 Erratum to "Homotopy theory of Moore flows I" (PDF).
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. The original paper being fixed, this erratum is no longer useful.
 About transfinite compositions of weak equivalences of higher dimensional transition systems (PDF)
This note will be never published. In two published papers "Towards a homotopy theory of
higher dimensional transition systems" and "Homotopy Theory of Labelled
Symmetric Precubical Sets", it is implicitely
assumed that the classes of weak equivalences of the model
structures constructed are closed under transfinite composition
because they are finitely accessible and accessibly embedded. It
turns out that the argument which is given can only prove that they
are accessible and accessibly embedded. In this note, this strong
argument is replaced by a weaker one which is easy to check.
 Closed symmetric monoidal structure and flow (PDF).
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.
 Homotopical equivalence of combinatorial and categorical semantics of process algebra (PDF)
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 noncanonical 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.
 Le Monopoly pour les nuls (French) (PDF,HTML)
Le but de cet exposé est de prouver que, contrairement à une idée reçue (cf par exemple l'article de Ian Stewart dans le ``Pour La Science'' de Juin 1996), les différentes cases du Monopoly ne sont pas équiprobables. Nous avons fait des tests sur le Monopoly français. Nous verrons même qu'il y a des disparités entre les cases, entre les lotissements, et à l'intérieur des lotissements.
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The Cost of Knowledge against Elsevier. In particular, this means that I do not attend any conference where people are forced to publish in such a journal. I do not write any review for a paper submitted to these journals. I do not submit any paper in these journals either. The scientific papers are financed by our salaries (the salaries of the authors, of the editors finding reviewers, and of the anonymous reviewers writing reports), and therefore by the taxpayers who therefore must have access to the papers at a reasonable cost, and if possible for free. The scientific knowledge does not belong to private investors making money with it and who do not even finance research in any way. It does not cost much to maintain a website. In fact, it is astonishing how slow and poorly designed the websites of some very expensive scientific journals are, which moreover display stupidities like the hit parade of mostread papers, as if one was listening to a radio station ranking hits. This reflects a false conception of science, which is unfortunately echoed by many politicians who believe that science is a matter of competition, when in fact it is a matter of collaboration.
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 quiver: a modern commutative diagram editor (HTML).
To draw diagrams easily.
 Detexify (HTML).
To find a symbol easily.