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- Homotopie, catégorie, topologie algébrique dirigée
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- Homotopy theory of Moore flows (II), Extracta Mathematicae, vol. 36 (2), 157-239, 2021 (PDF) 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.
- Homotopy theory of Moore flows (I), Compositionality 3, 3 2021 (PDF) A reparametrization category is a small topologically enriched symmetric semimonoidal category such that the semimonoidal structure induces a structure of a commutative 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 closed 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. Erratum available here: PDF.
- Left properness of flows, Theory and Applications of Categories, vol. 37, No. 19, 562-612, 2021 (PDF) 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 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 k-spaces and of weakly Hausdorff k-spaces 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), 145-181, 2021 (PDF) 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.
- Flows revisited: the model category structure and its left determinedness, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LXI-2 (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 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. - Enriched diagrams of topological spaces over locally contractible enriched categories,
*New-York 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 past-similarity in higher dimensional transition systems, Theory and Applications of Categories, vol. 32, 1107-1164, 2017 (PDF) 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.
- The choice of cofibrations of higher dimensional transition systems,
*New-York Journal of Mathematics*21 (2015), 1117-1151 (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 star-shaped transition systems is a solution to this problem. - Left determined model categories,
*New-York Journal of Mathematics*21 (2015), 1093-1115 (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 star-shaped weak transition systems. - The geometry of cubical and regular transition systems, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LVI-4 (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, 17-20, 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,
*New-York Journal of Mathematics*20 (2014), 93-131 (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 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. - Towards a homotopy theory of higher dimensional transition systems, Theory and Applications of Categories, vol. 25, 295-341, 2011 (PDF) 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.
- Directed algebraic topology and
higher dimensional transition systems,
*New-York Journal of Mathematics*16 (2010), 409-461 (PDF) 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. - Combinatorics of labelling in
higher dimensional automata,
*Theoretical Computer Science*(2010), 411(11-13), pp 1452-1483 (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 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. - Homotopical interpretation of
globular complex by multipointed d-space,
*Theory and Applications of Categories*, vol. 22, 588-621, 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 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. 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 can be interpreted up to equivalences of categories as the total left derived functor of a left Quillen adjoint. - T-homotopy and refinement
of observation (I) : Introduction,
Electronic Notes in Theoretical
Computer Sciences 230
(2009), 103-110
(PDF) This paper is the extended
introduction of a series of papers about modelling T-homotopy by
refinement of observation. Thenotion of T-homotopy 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 up-to-date bibliography. - Towards a homotopy theory
of process algebra,
*Homology, Homotopy and Applications*, vol. 10 (1):p.353-388, 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 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. - Globular realization and cubical
underlying homotopy type of time flow of process algebra,
*New-York Journal of Mathematics*14 (2008), 101-137 (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 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. - T-homotopy and refinement
of observation (II) : Adding new T-homotopy 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 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. - T-homotopy and refinement
of observation (III) : Invariance of the branching and merging
homologies,
*New-York Journal of Mathematics*12 (2006), 319-348 (PDF) 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 nondeterministic 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. - T-homotopy and refinement
of observation (IV) : Invariance of the underlying homotopy
type,
*New-York Journal of Mathematics*12 (2006), 63-95 (PDF) 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. - Inverting weak dihomotopy
equivalence using homotopy continuous flow,
*Theory and Applications of Categories*, vol. 16, 59-83, 2006 (PDF) 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.

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. 371-388 (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
non-deterministic branchings and mergings in higher dimensional
automata,
*Homology, Homotopy and Applications*, vol. 7 (1):p.51-76, 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,
*New-York Journal of Mathematics*11 (2005), 97-150 (PDF) 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. - The homotopy branching
space of a flow,
*Electronic Notes in Theoretical Computer Science*vol. 100 : pp 95-109, 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 T-homotopy 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.549-599, 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 S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories. - Concurrent Process up to
Homotopy (II),
*C. R. Acad. Sci. Paris Ser. I Math.*, 336(8):647-650, 2003 (French) (PDF) On démontre que la catégorie des CW-complexes 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 CW-complexes modulo homotopie est équivalente à la catégorie des espaces topologiques modulo homotopie faible.

One proves that the category of globular CW-complexes 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 CW-complexes 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):593-596, 2003 (French) (PDF) Les CW-complexes 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 CW-complexes globulaires dans les flots et on démontre que deux CW-complexes globulaires sont dihomotopes si et seulement si les flots associés sont dihomotopes.

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. - (with Eric
Goubault) Topological Deformation of Higher
Dimensional Automata,
*Homology, Homotopy and Applications*, vol. 5 (2):p.39-82, 2003 (PDF) A local po-space 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 po-spaces. We introduce here a particular kind of local po-space, the ``globular CW-complexes'', for which we formalize these notions of deformations and which are sufficient to formalize higher dimensional automata. The existence of the category of globular CW-complexes was already conjectured in "From Concurrency to Algebraic Topology". After localizing the category of globular CW-complexes 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 CW-complexes 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.75-106 (PDF) One can associate to any strict globular $\omega$-category three augmented simplicial nerves called the globular nerve, the branching and the merging semi-cubical 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 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. - 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.107-156, vol XLIII-2 (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.324-376 (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 sub-complex 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 three-object 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.481-524 (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. - Lambda-opérations sur
l'homologie d'une algèbre de Lie de matrices,
*K-Theory*, vol. 13(2), p.151-167, 1998 (PDF) - Produit tensoriel de matrices,
homologie cyclique, homologie des algèbres de Lie,
*Ann. Inst. Fourier (Grenoble)*, vol. 44(2), p.413-431, 1994 (PDF) - Lambda-opérations et homologie des
matrices,
*C. R. Acad. Sci. Paris Sér. I Math.*, 313(10):663-666, 1991 (PDF) One extends Loday-Procesi $\lambda$-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):13-16, 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 Loday-Quillen's product on cyclic homology.

Passer le pointeur sur le titre pour afficher le résumé.

- Directed degeneracy maps for precubical sets (PDF). Transverse (symmetric precubical) 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 transverse sets in the following sense. A q-realization functor from transverse sets to flows is introduced using a q-cofibrant 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 transverse sets to flows is introduced using Raussen's notion of natural $d$-path extended to transverse sets thanks to their structure of Lawvere metric space. It is proved that these two realization functors are homotopy equivalent on cofibrant transverse sets by using the fact that the small category defining transverse sets is c-Reedy in Shulman's sense. This generalizes to transverse sets results previously obtained for precubical sets.
- Regular directed path and Moore flow (PDF). Using the notion of regular $d$-path of the topological $n$-cube, we construct the regular realization of a precubical set as a multipointed $d$-space. Its execution paths correspond to the tame regular $d$-paths of the precubical set between two vertices. By considering the associated Moore flow, we obtain a colimit-preserving functor from precubical sets to Moore flows. As a consequence, we obtain that the associated Moore flow is always m-cofibrant. This provides a model category interpretation of the known fact that the space of tame regular $d$-paths of a precubical set between two vertices is homotopy equivalent to a CW-complex. Finally, we prove that the flow associated with the regular realization of a precubical set coincides with the natural realization of a precubical set as a flow.
- Comparing cubical and globular directed paths (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 non-canonical choice of a q-cofibrant 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 q-cofibrant but is still m-cofibrant. 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 $d$-paths of the geometric realization of the precubical set as a Grandis $d$-space.
- Comparing the non-unital and unital settings for directed homotopy (PDF). This note explores the link between the q-model structure of flows and the Ilias model structure of topologically enriched small categories. Both have weak equivalences which preserve fundamental (semi)categories. The Ilias model structure cannot be left-lifted along the left adjoint adding identity maps. The minimal model structure on flows having as cofibrations the left-lifting of the cofibrations of the Ilias model structure has a homotopy category equal to the $3$-element totally ordered set. The q-model structure of flows can be right-lifted to a q-model structure of topologically enriched small categories which is minimal and such that the weak equivalences preserve fundamental categories. The identity functor of topologically enriched small categories is neither a left Quillen adjoint nor a right Quillen adjoint between the q-model structure and the Ilias model structure.
- 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.
- About locally finite cellular multipointed d-spaces (PDF) It is proved that the set of execution paths of a locally finite cellular multipointed $d$-space equipped with the compact-open topology is $\Delta$-generated. Thus the space of execution paths of a locally finite cellular multipointed $d$-space is metrizable with the distance of the uniform convergence. Note: this note will be never published as it is; the part about locally finite cellular spaces must be considered as well known and the part about locally finite multipointed $d$-spaces does not have yet any application.
- 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 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.
- 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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- ArXiv profile (HTML) Unlike this homepage, my ArXiv profile only contains preprints. It also contains abandoned preprints which were splitted or merged in other ones.
- Mathoverflow profile (HTML) Mathoverflow is a very interesting web site. I encourage mathematicians to read and participate. It is time consuming as soon as we start reading the sometimes quite long answers that some people write. Its main issue OMHO is the scoring system which gives too much power to people asking too general questions.
- Linkedin profile (HTML) Linkedin is a well-known social network. It contains a short CV in English, with hopefully a correct translation of the degrees...

- Cahier de topologie et géométrie différentielle catégoriques (HTML)
- Theory and Applications of Categories (HTML)
- Categories and General Algebraic Structures with Applications (HTML)
- New York Journal of Mathematics (HTML)
- Compositionality (HTML)
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- quiver: a modern commutative diagram editor (HTML). To draw diagrams easily.
- Detexify (HTML). To find a symbol easily.