# Philippe Gaucher

Laboratoire PPS
Univ Paris Diderot
Case 7014
75205 Paris Cedex 13
France

## Email,Téléphone,Fax

prenom.nom@pps.univ-paris-diderot.fr
firstname.lastname@pps.univ-paris-diderot.fr
Tél: +33 01 57 27 92 16
Fax: +33 01 57 27 92 97

Batiment Sophie Germain
8 Place FM/13
75013 PARIS
Métro: Bibliothèque
Bureau: 3048
I do not work in any way for Elsevier, Springer, John Wiley & Sons and Informa. Of course, I do work for free for non-profit journals. Read for example about the obscene profits of commercial scholarly publishers, more than 35% of profits as a percentage of annual revenue, at the level of Apple and Microsoft, two other "specialists" of captive market.

# Publications

• Erratum to Towards a homotopy theory of higher dimensional transition systems'' (PDF,BIB), Theory and Applications of Categories, vol. 29, 17-20, 2014. 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 (PDF,BIB). New-York Journal of Mathematics 20 (2014), 93-131. 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 (PS,PDF,BIB), Theory and Applications of Categories, vol. 25, 295-341, 2011. 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 (PDF,BIB). New-York Journal of Mathematics 16 (2010), 409-461. 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 (PS,PDF,BIB), ArXiv, Theoretical Computer Science (2010), 411(11-13), pp 1452-1483, doi:10.1016/j.tcs.2009.11.013. Final PDF version available on request. 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.
• Towards a homotopy theory of process algebra (PS,PDF,BIB), Homology, Homotopy and Applications, vol. 10 (1):p.353-388, 2008. 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 (PS,PDF,BIB). New-York Journal of Mathematics 14 (2008), 101-137. 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.
• Homotopical interpretation of globular complex by multipointed d-space (PS,PDF,BIB), Theory and Applications of Categories, vol. 22, 588-621, 2009. 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 (PS,PDF,BIB) Electronic Notes in Theoretical Computer Sciences 230 (2009), 103-110. (GETCO 2005). 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.
• T-homotopy and refinement of observation (II) : Adding new T-homotopy equivalences (PDF,BIB), pusblished in the Internat. J. Math. Math. Sci., Article ID 87404, 20 pages (2007). 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 (PS,PDF,BIB). New-York Journal of Mathematics 12 (2006), 319-348. 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 (PS,PDF,BIB). New-York Journal of Mathematics 12 (2006), 63-95. 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 (PS,PDF,BIB), Theory and Applications of Categories, vol. 16, 59-83, 2006. 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 (PS,PDF,BIB). Applied Categorical Structures, vol. 13, p. 371-388 (2005). 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 (PS,PDF,BIB), Homology, Homotopy and Applications, vol. 7 (1):p.51-76, 2005. 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 (PS,PDF,BIB). New-York Journal of Mathematics 11 (2005), 97-150. 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 (PS,PDF,BIB). 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 (PS,PDF,BIB), Homology, Homotopy and Applications, vol. 5 (1):p.549-599, 2003. 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) (PS,PDF,BIB). C. R. Acad. Sci. Paris Ser. I Math., 336(8):647-650, 2003 (French). 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) (PS,PDF,BIB). C. R. Acad. Sci. Paris Ser. I Math., 336(7):593-596, 2003 (French). 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 (PS,PDF,BIB), Homology, Homotopy and Applications, vol. 5 (2):p.39-82, 2003. 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.
• Investigating The Algebraic Structure of Dihomotopy Types, Electronic Notes in Theoretical Computer Science 52 (2) 2002 (PS,PDF,BIB). 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.
• The branching nerve of HDA and the Kan condition (PS,PDF,BIB), Theory and Applications of Categories 11 n°3 (2003), p.75-106. 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.
• About the globular homology of higher dimensional automata (DJVU,PDF,BIB), Cahiers de Topologie et Géométrie Différentielle Catégoriques, p.107-156, vol XLIII-2 (2002). 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 (PS,PDF,BIB), Theory and Applications of Categories 8 n°12 (2001), p.324-376. 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 (PS,PDF,BIB), Electronic Notes in Theoretical Computer Science 39 (2000), no. 2, 19p (GETCO 2000) 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 (PS,PDF,BIB). Mathematical Structure in Computer Science 10 (2000), no. 4, p.481-524. 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 (PS,BIB). K-Theory, vol. 13(2), p.151-167, 1998.
• Lambda-opérations et homologie des matrices (PDF,BIB). C. R. Acad. Sci. Paris Sér. I Math., 313(10):663-666, 1991. 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, homologie cyclique, homologie des algèbres de Lie. Ann. Inst. Fourier (Grenoble), vol. 44(2), p.413-431, 1994 (PDF,BIB).
• Produit tensoriel de matrices et homologie cyclique (PDF,BIB). C. R. Acad. Sci. Paris Sér. I Math., 312(1):13-16, 1991. 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.

# Transparents/slides

• Towards a Homotopy Theory of Higher Dimensional Transition Systems, SIC Mars 2011, Paris (26 pages, PDF) Un mot pour l'audience: il n'y avait pas d'erreur dans le transparent page 24. Je sais précisément caractériser les équivalences faibles de la double localisation de Bousfield avec le foncteur $\underline{\mathbf{L}}_\mathcal{O}$.
• About Higher Dimensional Transition Systems, GETCO 2010, Aalborg (Danemark), 14 janvier 2010 (26 pages : PDF) I will introduce the category of weak higher dimensional transition systems which contains the Cattani-Sassone ones as a small-orthogonality class. I will explain its categorical and homotopical properties, and the links with process algebras, bisimulation, and the topological models of concurrency.
• Higher Dimensional Transition System and Labelled Symmetric Precubical Set, Séminaire Itinérant de Catégorie, Paris 24 octobre 2009 (20 pages : PS,PDF) Résumé à télécharger (PS,PDF)
• Combinatorics of labelling in higher dimensional automata, Rencontre du projet ANR "Invariants algébriques des systèmes informatiques", Lyon 25-26 septembre 2008 (27 pages : PS,PDF)
• Cubes, homotopy and process algebra, ATMCSIII, Paris le 7-11 juillet 2008 (57 pages : PS,PDF) In directed algebraic topology, the concurrent execution of n actions is abstracted by a full $n$-cube. Each coordinate corresponds to one of the n actions. This $n$-cube may be viewed as a representable presheaf of the category of precubical sets, as a topological $n$-cube equipped with some continuous paths modelling the possible execution paths up to homotopy, and as a commuting $n$-cube, i.e usually the small category associated with the poset of vertices of the $n$-cube. In fact, we have to remove the identity maps for various reasons, e.g., because the full $n$-cube does not contain any loop. In this talk, all these points of view are related to one another by considering Milner's calculus of communicating systems (CCS). All operators of this process algebra are given a higher dimensional interpretation. The restriction to dimension 1 corresponds to the usual structural operational semantics.
• A topological model for studying branching and merging homologies of flows, CT08, Calais, le 23-28 juin 2008 (14 pages : PS,PDF) There exist various methods for modeling time flows associated with concurrent systems. The study of the branching and merging homologies of a time flow requires a very specific feature: the mapping taking an object to its set of non-constant execution paths must be functorial. The first model I studied is the one of non-contracting Kan strict globular $\omega$-categories consisting of the category of strict globular $\omega$-categories \mathcal{C} such that $\mathbb{P}\mathcal{C} = \mathcal{C}_1 \cup\mathcal{C}_2 \cup \dots$ is a strict $\omega$-groupoid (hence the word Kan'', $\mathcal{C}_n$ being the set of $n$-morphisms) and of $\omega$-functors $f:\mathcal{C} \longrightarrow \mathcal{D}$ inducing a map of strict $\omega$-groupoids $\mathbb{P}f: \mathbb{P}\mathcal{C} \longrightarrow \mathbb{P}\mathcal{D}$ (hence the word non-contracting''). The second model I studied is a version of the preceding model in which the path space $\mathbb{P}\mathcal{C}$ is topologized, hence the notion of flows (small category without identities enriched over topological spaces). The third and last model I am studying now is a variant of M. Grandis' notion of d-space, the multipointed d-spaces, in which not only the path space is topologized, but also the direction of time. The three models are locally presentable, by working with Delta-generated spaces. The last one of multipointed d-spaces is also topological. In this talk, I will present the last model and its relation with the model of flows.
• Homotopical semantics of parallel composition in CCS, PSSL 86, Nancy le 8-9 septembre 2007 (20 pages : PS,PDF)
• Globular realization and cubical underlying homotopy type, Rencontre du projet ANR "Invariants algébriques des systèmes informatiques", Nancy le 7 septembre 2007 (18 pages : PS,PDF)
• Sémantique purement homotopique des algèbres de processus (Paris; 6 avril 2007), INVAL avril 2007 (32 pages : PS,PDF) Je vais présenter une construction purement homotopique de l'espace des chemins et des homotopies de dimension supérieure des algèbres de processus (je me concentrerai sur CCS) : la restriction en dimension 1 redonnant la construction habituelle en terme de systèmes de transition étiquetés. Pour cela, je partirai d'une construction à valeur dans les ensembles précubiques (étiquetés), élaborée en partant d'une idée de K. Worytkiewitcz, mais un peu modifiée pour tenir compte du paradigme des automates de haute dimension : un et un seul $n$-cube plein pour l'exécution concurrente de n transitions. Puis en utilisant un foncteur réalisation des ensembles précubiques dans les flots (étiquetés), on verra comment les particularités algébriques et homotopiques de cette dernière catégorie permettent d'obtenir notamment une formalisation du produit parallèle avec synchonisation complètement débarassée de toute combinatoire, c'est-à-dire sans construction cosquelette, et en fait n'utilisant que des colimites homotopiques.
• Flots temporels à homotopie près : entre processus concurrents et catégories de modèles (Nice, 3 novembre 2006, GDR Topologie Algébrique et Applications, 45 slides in English) (PS,PDF) La dihomotopie (pour homotopie dirigée) est une équivalence entre flots temporels préservant la structure causale du flot et le type d'homotopie de l'espace d'états sous-jacent. La principale difficulté pour modéliser la dihomotopie est la contractibilité des chemins non bouclés et les définitions naïves qui préservent uniquement l'antériorité temporelle détruisent donc la structure causale. On introduit dans cet exposé les flots non-étiquetés et étiquetés, obtenant ainsi un modèle de parallélisme (contenant les automates de haute dimension, les algèbres de processus avec n'importe quelle algèbre de synchronisation, les systèmes de transitions asynchrones, les structures d'évènements etc...). Puis on introduit une catégorie de modèles dont les équivalences faibles sont les S-homotopies faibles. En Bousfield localisant par rapport à un ensemble de cofibrations modélisant le rafinement de l'observation, on obtient une catégorie de modèles dont les équivalences faibles sont appelées quasidihomotopie. La dihomotopie est entre la S-homotopie et la quasidihomotopie. Cette dernière est comme la dihomotopie sauf dans des parties non-observables du flot temporel. Les objets locaux sont notamment les flots avec les parties non-observables remplies. Je présenterai des problèmes ouverts et quelques résultats, notamment concernant des invariants détectant les zones de branchements ou de confluences non-déterministes, et les relations entre la dihomotopie et la quasidihomotopie. Si le temps le permet, je parlerai d'une autre approche de la dihomotopie des flots utilisant des flots de Segal (i.e. des flots faibles) et des catégories de modèles de Rezk. Cette approche encore très expérimentale est nécessaire pour de futurs développements homologiques.
Dihomotopy (for directed homotopy) is an equivalence between time flows preserving the causal structure and the homotopy type of the underlying state space. The main difficulty to model dihomotopy is that non-looped continuous paths are contractible. Therefore the naive definitions preserving only the order of time fail to preserving the causal structure. In this talk, I introduce the unlabeled and labeled flows, obtaining this way models for concurrency (containing higher dimensional automata, process algebras with any synchronization algebra, asynchronous transition systems, event structures etc...). I then introduce a model category structure whose weak equivalences are called S-homotopy. By Bousfield localizing with respect to a set of cofibrations modeling refinement of observation, one obtains a new model category whose weak equivalences are called quasidihomotopy. Dihomotopy is between S-homotopy and quasidihomotopy. Quasidihomotopy is like dihomotopy except in non-observable areas of the time flow. In particular, local objects are flows with all non-observable areas filled. Open problems and a few results will be presented, concerning some invariants detecting the non-deterministic branching and merging areas and the relation between dihomotopy and quasidihomotopy. If the time permits, another approach of dihomotopy using Segal flows (i.e. weak flows) and Rezk model category structures will be presented. The latter approach is very experimental but it is needed for future homological developments.
• T-homotopy and Refinement of Observation (32 slides with overlays in English), Talk given the 21th of August 2005, San Francisco (PS,PDF), GETCO 2005. Globular complexes and flows are designed to modelling spatial and temporal deformations of higher dimensional automata, modelling this way the invariance of the time flow of a HDA by subdivision and refinement of observation. A HDA is modelled in the framework of flows by a set of states (the 0-skeleton) and between each pair (a,b) of states by a topological space whose elements play the role of the execution paths from a to b. These data are equipped with an associative composition law which plays the role of composition of execution paths. Spatial deformations (S-homotopy) are very well interpreted by a Quillen model category structure on the category of flows. Temporal deformations (T-homotopy) are considerably more difficult to model and to understand. A convincing formalization will be proposed which will make use of the preceding model category structure. It will be explained why the underlying space of the time flow (defined only up to homotopy) is preserved and why the topological configurations of branching and merging areas of execution paths of the time flow are also preserved.
An application of the preceding constructions will be proposed : an analogue of Whitehead's theorem for the full dihomotopy relation, and not only for S-homotopy as in previous works of the author, will be presented. This theorem says that a morphism between cofibrant homotopy continuous flows is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy and that every flow is weakly dihomotopy equivalent to a cofibrant homotopy continuous flow. This cofibrant homotopy continuous representative is of course unique up to dihomotopy. This first structural theorem for the full dihomotopy relation of Flow proves at least one thing : dihomotopy is really a new kind of homotopy.
• Towards a homotopy theory of higher dimensional automata (29 slides) (PDF), Talk given the 17th of June 2004, Minneapolis, USA. We give an overlook of our work about globular complexes and flows. Globular complexes, flows, S-homotopy and T-homotopy are explained by examples. Several model categories are presented. And some open questions are discussed.
• T-homotopy and Quillen model category (33 slides in English), Talk given the 18th of March 2005, Montpellier (PS,PDF). Les flots sont un modèle de parallélisme permettant de définir une notion de dihomotopie sans utiliser un choix non-canonique de pointage des états. La dihomotopie des flots contient la S-homotopie qui commence à être bien comprise et qui donne déjà quelques résultats (par exemple une longue suite exacte pour les branchements et les confluences en haute dimension), et la T-homotopie . Après avoir défini cette dernière, j'expliquerai pourquoi toute structure modèle sur les flots ou bien n'a pas assez d'équivalences faibles, ou bien a trop d'équivalences faibles, les équivalences faibles en trop étant des morphismes qui modifient ou bien l'homologie des branchements, ou bien l'homologie des confluences. Et comment cela m'a donné l'idée d'introduire une catégorie de préfaisceaux sur une petite sous-catégorie des flots. Puis en admettant le principe de Vopenka, j'expliquerai pourquoi aucune des structures modèles connues (ou du moins que je connais) ni aucune de leur localisation homotopique ne convient pour l'étude de la T-homotopie sur cette catégorie de préfaisceaux car toutes ont, encore, des équivalences faibles parasites qui modifient l'homologie des branchements ou des confluences. En conclusion, je ne répondrai probablement pas à la question posée mais je donnerai au cours de l'exposé les arguments mi-géométriques mi-catégories de modèles qui me font penser qu'une telle catégorie de modèles existe.
• Flow does not model flows up to weak dihomotopy (PDF) (9 slides in English), talk given the 4th of May 2004, Paris. 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.
• Structures modèles pour la dihomotopie (PDF) (French, 15 transparents). Exposé du 27 novembre 2003 donné à Paris 7, Jussieu. On explique dans ces transparents les trois structures de catégorie modèle trouvées jusqu'ici sur la catégorie des flots.
• Investigating The Algebraic Structure of Dihomotopy Types (PS) (slides in English), talk given at GETCO 2001. 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.

# Non-publié/Unpublished

• Homotopical equivalence of combinatorial and categorical semantics of process algebra (PS,PDF), ArXiv (UNDER REVISION with a future improved redaction: this version is correct as far as I know). 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.
• T-homotopy and refinement of observation (V) : Strom model structure for branching and merging homologies (PS,PDF), ArXiv. (We check that there exists a model structure on the category of flows whose weak equivalences are the S-homotopy equivalences. As an application, we prove that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. The method of proof is completely different from the one of the third part of this series of papers). UNDER REVISION. The main result (the Cole-Strom model structure) is correct, not the link with the application: Theorem 8.17 is false. The revised paper will provide a new application which will be used in a future paper. The title will be also changed.
• Déformation des Flots de Chemins Continus : Théorie et Applications (PS,PDF). Mémoire d'habilitation, 2001.
My habilitation thesis, in French.
• Closed symmetric monoidal structure and flow (PS,PDF), ArXiv. 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.

# Monopoly (French)

• Le Monopoly pour les nuls (HTML)