teaching and research institutions in France or abroad, or from public or private research centers.

-Preuve de solidité logique de la non-relativité lorentzienne

Preuve de solidité logique de la non-relativité lorentzienne

Jean Stratonovitch Dans l'article Le paradoxe des jumeaux de Langevin et l'expérience d'aller et retour d'un cylindre en rotation, nous avons montré que cette expérience produit dans le cadre de la relativité restreinte deux résultats contradictoires.

Dans l'article La modification minimale à apporter à la relativité restreinte pour qu'elle supporte l'expérience d'aller et retour d'un cylindre en rotation, nous avons montré que le problème ne provient pas de la transformation de Lorentz, mais de la valeur postulée universelle et absolue du principe de relativité. Les lois régissant le comportement mécanique de la matière élastique ne peuvent pas être toutes relativistes, pas plus que ne peuvent l'être celles qui décrivent le Ciel lointain. L'éventail des espaces galiléens doit être muni d'un zéro permettant d'en individualiser les éléments, de la même façon qu'on installe sur une droite une origine des abscisses. Si le principe de relativité était d'une validité sans faille, ce serait impossible autrement que de façon purement arbitraire, mais tel n'est pas le cas. Dans le désert interstellaire où nous nous installons par la pensée notre laboratoire galiléen, un zéro naturel existe, c'est l'espace isotrope local (E is ), celui relativement auquel le Ciel lointain se montre statistiquement isotrope. Les autres espaces galiléens n'étant pas isotropes, la simultanéité-lumière relative à eux doit être tenue pour biaisée, et il n'existe dans ce modèle qu'une simultanéité véritable, celle relative à (E is ).

Une règle ou une horloge en mouvement relativement à cet espace, parce que l'univers est alors anisotrope relativement à elle, ne peut être considérée comme parfaitement identique, en elle-même, à ce qu'elle serait si elle était immobile relativement à (E is ) : les altérations lorentziennes des règles et des horloges, quand elles sont évaluées relativement à cet espace, doivent être considérées comme objectives.

Nous avons appelé nonrelativité lorentzienne le modèle ainsi obtenu. Il est voisin de celui qu'on appelle parfois la « relativité de Lorentz », à ceci près que l'espace isotrope n'est en rien identifiable à un espace absolu, et que la problématique de l'éther est hors de propos.

Un point crucial est à vérifier, auquel nous consacrerons cet article : la solidité logique de ce modèle engendré par la nécessité de remédier au caractère contradictoire de la mécanique relativiste lorentzienne du corps élastique.

-LA MÉTHODE

Il s'agit de montrer que quelle que soit l'expérience de pensée de mécanique des corps élastiques étendus qu'on considère, elle n'engendre aucune contradiction. Le problème peut sembler désespérément compliqué, mais en fait ce n'est pas le cas.

Car une méthode existe pour vérifier la solidité logique d'une théorie, c'est d'en construire un modèle à l'intérieur d'une autre. Par exemple, Beltrami puis Poincaré montrèrent la solidité de la géométrie de Lobatchevski en en construisant des modèles à l'intérieur de la géométrie euclidienne. Si la géométrie de Lobatchevski recelait une contradiction, ce serait du coup une contradiction de la géométrie euclidienne : la géométrie de Lobatchevski est donc au moins aussi solide que celle d'Euclide ; et, puisque cette dernière est constructible dans le cadre de la théorie des ensemblescomme toutes les mathématiques, au demeurantla géométrie de Lobatchevski est au moins aussi solide que la théorie des ensembles.

Bien sûr, on peut douter de tout, et même de la solidité de la théorie des ensembles. Mais pour démontrer quelque chose, il faut La méthode que nous emploierons donc pour vérifier la solidité logique de la mécanique lorentzienne non-relativiste des corps élastiques sera de montrer qu'on peut en construire un modèle à l'intérieur des mathématiques -c'est-à-dire, indirectement, à l'intérieur de la théorie des ensembles.

-CONSTRUCTIBILITÉ ENSEMBLISTE DE LA MÉCANIQUE LORENTZIENNE NON-RELATIVISTE DES CORPS ÉLASTIQUES CONSIDÉRÉS DANS LEUR ÉTENDUE

Partons d'un modèle de la cinématique lorentzienne tel que sa construction a été rapidement exposée dans l'article Le paradoxe des jumeaux de Langevin et l'expérience d'aller et retour d'un cylindre en rotation : un espace galiléen (E 0 ) identifié à , muni d'un temps identifié à , des points, qui sont des applications dérivables du temps dans cet espace, dont la dérivée est constamment inférieure à , des espaces galiléens formés des points allant à la même vitesse constante inférieure à , munis de coordonnées spatiotemporelles engendrées à partir de celles de (E 0 ) par la transformation de Lorentz.

À ce jour, un siècle et demi après le modèle de Beltrami, et plus d'un siècle après ceux de Poincaré, on ne trouve nulle part -s'il existe il est bien caché !de modèle ensembliste ou géométrique de la relativité restreinte incluant la mécanique des corps élastiques. Si un tel modèle existait, il n'aurait pas manqué d'être opposé à ceux qui voient des difficultés logiques dans cette théorie.

Mais son existence est impossible, parce que la théorie est contradictoire à ce niveau. Il sera intéressant de pointer l'endroit du raisonnement de constructibilité en lequel la relativité restreinte se dérobe.

On identifie (E is ) à (E 0 ) : c'est par rapport à lui que seront décrites les situations de la mécanique des corps élastiques et les lois qui les régissent, et que seront calculées les évolutions des systèmes.

Dans ce modèle de la cinématique lorentzienne, on définit ce qu'est un corps élastique considéré à un instant t. C'est en premier lieu une famille infinie (K) de points deux fois continûment dérivables, -dont l'ensemble (K)( ) des images dans (E 0 ) à l'instant est borné et homéomorphe à une boule ouverte, ou est une réunion finie connexe de tels ensembles, et telle que le champ des vitesses de ses points à cet instant soit continu.

Cette description n'est évidemment pas suffisante, et nous devrons la compléter. L'ensemble des corps élastiques considérés à un instant , et encore incomplètement définis, est ce stade constructible dans le cadre de la théorie des ensembles, et modélise les possibilités cinématiques raisonnables que peut avoir un corps élastique donné à un instant donné. De même qu'on resterait dans le raisonnable en attribuant au champ des vitesses une plus forte régularité mathématique, par exemple celle d'être continûment différentiable ; et que l'ensemble des corps élastiques considérés à l'instant resterait constructible dans le cadre de la théorie des ensembles.

À forme géométrique égale à un instant donné, un corps effectif peut avoir des contraintes internes différentesdes « déformations invisibles » différentes. Ces dernières ne doivent être installées qu'en ce seul instant , car aux autres instants elles seront calculables en même temps que l'évolution du système. Les décrire à cet instant revient à décrire la forme d'un voisinage infinitésimal de chaque point M du corps élastique relativement à la forme qu'aurait ce voisinage s'il était extrait du corps et au repos dans l'espace isotrope. Décrire tous les possibles raisonnables de ces déformations revient à installer sur le corps de toutes les façons possibles des champs de tenseurs de déformations, qu'on suppose continus, voire différentiables, là encore c'est sans incidence sur la constructibilité, puisque l'une et l'autre possibilité laissent place à des ensembles de solutions ni vides ni triviaux.

Nous complétons donc la définition du corps élastique considéré à l'instant en lui attribuant, de toutes les façons possibles, un tel système de déformations. L'ensemble des corps élastiques ainsi modifiédémultipliéreste constructible.

Ayant installé à l'instant t l'ensemble de toutes les éventualités cinématiques « externes » possibles de tous les corps élastiques possibles multiplié par l'ensemble de toutes leurs déformations « internes » possibles, il faut encore le multiplier par l'ensemble de toutes les caractéristiques mécaniques possibles, qui en un point donné sont par l'hypothèse de l'élasticité indépendantes du temps. Là encore, une description des caractéristiques mécaniques d'un corps est possible dans le cadre de la théorie des ensembles, au moyen d'applications de l'ensemble de ses points dans un espace de paramètres, fini ou infini, peu importe. Nous ne nous soucions pas de leur donner pour le moment une signification, ce ne sont que des paramètres en réserve.

Nous pouvons donc construire, dans le cadre de la théorie des ensembles, une entité que nous appelons encore corps élastique considéré à un instant . Cette entité est complexe, mais son support reste fait de points, qui sont des applications deux fois dérivables du temps dans l'espace. Elle n'est modélisée qu'à l'instant , aux autres instants les points qui la composent ont chacun leur trajectoire propre. Un même corps élastique effectif considéré à l'instant existe donc dans notre modèle d'une infinité de façons, qui toutes coïncident à cet instant tant pour les positions de leurs points que pour leurs vitesses, que pour leurs déformations internes que pour leurs caractéristiques mécaniques, et qui représentent tous les devenirs cinématiquement possibles de ce corps élastique effectif, sans se soucier des lois de la mécanique.

Une dernière fois, modifions notre définition, et appelons corps élastique considéré à l'instant l'ensemble infini de tous ces corps identiques à cet instant, chacun ayant son évolution particulière parmi toutes celles qui sont potentiellement possibles.

Une situation est la donnée d'une famille finie de corps élastiques considérés à un instant , disjoints deux à deux à cet instant, en ce sens qu'aucun point de l'un d'entre eux n'a les mêmes coordonnées spatiales qu'aucun point d'un autre. C'est un objet constructible dans le cadre de la théorie des ensembles, tout comme l'est l'ensemble des situations.

Les lois de la mécanique, de ce point de vue ensembliste, sont une fonction qui à toute situation (S) considérée à un certain instant associe le champ des accélérations de ses points à cet instant.

Ce champ est calculable en M par l'algorithme suivant : -On considère un voisinage infinitésimal de M, par exemple un pavé élémentaire centré en M ; on calcule les contraintes exercées sur ce voisinage, et leur résultante ;

on calcule la masse m de ce voisinage ; on en déduit, par la loi fondamentale de la mécanique l'accélération du voisinage infinitésimal de M relativement à (E 0 ) à l'instant t ; donc celle de M.

Du point de vue ensembliste qui est le nôtre en ce moment, les « contraintes », la « masse » sont des objets dont il importe peu qu'ils aient une quelconque signification physique. Ce sont des variables auxiliaires calculables quelle que soit la situation, et rien de plus. On pourrait remplacer les règles pertinentes que la physique a fini par élaborer par d'autres complètement farfelues que cela ne poserait aucun problème de consistance logique, pourvu que ces nouvelles règles déterminent pour toute situation une évolution instantanée sans équivoque, et soient suffisamment régulières mathématiquement.

LOIS DE LA MÉCANIQUE

Connaissant le champ des positions et des vitesses à l'instant , nous connaissions celui des positions à l'instant

; et maintenant que nous connaissons le champ des accélérations à l'instant , nous connaissons aussi celui des vitesses à l'instant : nous avons donc déterminé quelle est la situation à l'instant . Nous pouvons itérer ce processus : la connaissance des lois de la physique, jointe à celle de la situation, nous fournit une équation différentielle « téradimensionnelle » qui, à partir des conditions initiales, et sous couvert de régularité mathématique suffisante des lois de la mécanique, ce que nous postulons, a en vertu du théorème de Cauchy-Lipschitz une solution maximale et une seule définie sur un intervalle semi-ouvert maximal . Il n'est pas certain que : rien n'exclut que certaines situations sortent au bout d'un certain temps du domaine de la mécanique des corps élastiques, par exemple parce que les contraintes en un lieu deviennent infinies. Mais c'est sans importance. Le point essentiel est qu'il existe un critère mathématique permettant de sélectionner, parmi toutes les éventualités cinématiques des corps élastiques appartenant à une situation quelconque donnée, l'unique évolution obéissant aux lois de la mécanique des corps élastiques. Comme la sélection des éléments d'un ensemble selon un critère formulable en langage mathématique fournit légitimement un ensemble, on peut construire, dans le cadre de la cinématique lorentzienne nonrelativiste, l'ensemble de toutes les évolutions conformes aux lois de la mécanique de toutes les situations possibles impliquant des corps élastiques.

La mécanique lorentzienne non-relativiste des corps élastiques est constructible dans le cadre de la théorie des ensembles.

En particulier, elle ne peut engendrer de contradiction. Ce qui ne veut pas dire qu'elle soit automatiquement pertinente. Les lois farfelues que nous évoquions plus haut pourraient peut-être, sans contradiction logique, interne au modèle, transformer un porte-avions en un plat de spaghettis. Mais si nous supposons raisonnables ces lois, le cylindre (C) restera durant l'expérience d'aller et retour constamment semblabledifféomorphe -à ce qu'il est dans la première partie de l'expérience, lorsque son mouvement est uniforme dans toutes ses composantes ; il continuera de croiser le plan (P) dans le sens aller puis dans le sens retour en tournant sur lui-même, à vitesse a priori non uniforme ; la génératrice (G) gravée sur lui restera une courbe longitudinale rencontrant le plan (P) en N durant tout l'aller et retour ; le point M gravé sur elle traversera à l'aller et au retour (P) en deux instants et ; et l'expérience, qui restera globalement la même qu'en relativité restreinte, fournira les nombres et de tours fait par M et N dans l'intervalle .

Comme la mécanique lorentzienne relativiste est modélisable dans le cadre de la théorie des ensembles : La contradiction à laquelle aboutit l'expérience d'aller et retour dans le cadre de la relativité restreinte disparaît dans le cadre lorentzien non-relativiste.

L'entier

dépend continûment des conditions de l'expérience, qu'on peut faire varier continûment en déplaçant (P), jusqu'à une situation où ; par continuité, il garde la même valeur, qui est alors . Donc .

Dans le cadre non-relativiste lorentzien, les points N et M font exactement le même nombre de tours autour de l'axe du cylindre.

(C) (D) (P) N M (G)

-ÉCHEC DE LA DÉMONSTRATION DE CONSTRUCTIBILITÉ DE LA MÉCANIQUE DES CORPS ÉLASTIQUES DANS LE CADRE DE LA RELATIVITÉ RESTREINTE

On l'aura observé, le fait que la cinématique soit lorentzienne ou non n'intervient pas dans le raisonnement qui vient d'être fait. On peut en le décalquant quasiment à l'identique montrer la consistance de la mécanique galiléenne des corps élastiques.

Dans le contexte de la relativité restreinte, il peut être refait pas à pas relativement à un espace galiléen particulier (E), et il détermine l'évolution de la situation (S) à partir de l'instant de (E). Il n'y a jusqu'ici aucune différence.

Soit un évènement décrit relativement à (E), pris dans le domaine de (S) à cet instant. Si (E') est un autre espace galiléen, soit l'instant de (E') tel que . Puisque toutes les simultanéités ont la même valeur, il est tout aussi légitime de calculer l'évolution de la situation depuis (E') à l'instant , avec les mêmes règles de calcul. Cependant, puisque les simultanéités sont différentes, le panel des données initiales n'est plus le même.

Il ne coïncide avec le précédent que sur un ensemble de mesure nulle. On peut faire varier M d'une infinité de façons, on obtient une infinité de panels de données initiales aussi légitimes les unes que les autres. Et pour chaque M on peut faire varier (E') d'une infinité de façons, qui multiplient à l'infini les panels de données initiales légitimes pour calculer l'évolution de (S).

(E) (E') M Ainsi, l'évolution d'une situation à un instant donné, en nonrelativité lorentzienne, dépend d'un seul panel de données initiales, et se trouve par là à l'abri de tout conflit ; tandis qu'en relativité restreinte, elle dépend de n'importe quel panel parmi une infinité de possibles, qui tous doivent engendrer via la même fonction de calcul la même évolution effective. Cette « multi-contrainte » supplémentaire fait que le raisonnement montrant la constructibilité à l'intérieur de la théorie des ensembles de la mécanique des corps élastiques ne peut pas s'appliquer à la relativité restreinte.

-LA NON-RELATIVITÉ LORENTZIENNE OUVRE DES DEGRÉS DE LIBERTÉ DONT LA RELATIVITÉ RESTREINTE NE DISPOSE PAS

Lorsqu'une théorie est contradictoire, c'est que sa barque axiomatique est trop chargée. Elle porte en elle des contraintes impossibles à faire coexister. Pour lui éviter de couler, il faut jeter du lest, la remplaçant ainsi par une théorie moins contraignante dans laquelle la contradiction pourra s'évanouir.

Or telle est bien la position de la non-relativité lorentzienne par rapport à la relativité restreinte.

Dans cette dernière, en effet, chaque espace galiléen est isotrope, et constitue donc un « centre de symétrie » de l'univers ; tandis que dans la non-relativité lorentzienne, il n'y a plus qu'un seul espace isotrope, un seul « centre de symétrie ». C'est donc bien un allègement des contraintes.

Nous savons toutefois que la mécanique relativiste lorentzienne des points matériels isolés est consistante.

Dans le modèle proposé pour cette mécanique, les points matériels isolés ont hors interaction des vitesses uniformes, et les différents panels de conditions initiales sont alors immédiatement déductibles les uns des autres via l'uniformité des mouvements et la transformation de Lorentz, si bien que leur multiplicité n'est qu'apparente ; d'autre part l'exigence relativiste quant aux lois du choc conduit, dans ce cas simplifié où la forme des corps n'intervient pas, à des conditions mathématiques possibles à satisfaire.

De même, contrairement à ce qu'on pourrait hâtivement croire, postuler l'unicité de la simultanéité physique est moins contraignant que d'admettre qu'il en existe une infinité, tout aussi valables les unes que les autres. C'est en effet le corollaire de l'alternative entre l'existence d'un unique ou d'une infinité d'espaces isotropes.

La vitesse de la lumière n'est postulée égale à que relativement à l'espace isotrope : c'est donc aussi un allègement des contraintes -même si on démontre par la suite qu'avec les choix instrumentaux qui sont faits, elle vaut relativement à tout espace galiléen.

Seul point où il n'y a pas allègement, le postulat que la matière élastique avec laquelle nous fabriquons règles et horloges est un objet ondulatoire synchronisé entre ses différents lieux par la lumière ; et qu'elle conserve à l'identique son architecture vibratoire lorsque change sa vitesse de repos galiléen ; mais il adhère suffisamment bien à ce que nous savons aujourd'hui de la matière pour paraître hors de doute. Il opère en outre, sous une forme très générale, un rapprochement basique entre la mécanique quantique et les fondements de la cinématique qui comble en partie le fossé entre elles.

Reste que globalement, c'est bien un allègement. La preuve en est qu'on se retrouve devant une théorie non seulement solide logiquement, mais encore infiniment plus souple. Alors que la relativité restreinte, en raison de l'absolue rigidité du principe de relativité, ne supporte aucune loi possible pour la mécanique des corps élastiques, la non-relativité lorentzienne peut supporter n'importe lesquelles, sous couvert d'une régularité mathématique suffisante. Cela ne veut bien entendu pas dire qu'elles seront pertinentes, mais simplement qu'elles n'ouvriront la porte à aucune contradiction logique interne à la théorie.

Parmi toutes les lois possibles, on peut évidemment choisir celles basées sur le formulaire relativiste de la masse et de la quantité de mouvement, dont on sait qu'il est pertinent. Mais, au contraire de Ce postulat oblige la matière à modifier sa géométrie objective, mais la contraction de Lorentz n'est pas la seule réponse possible : toutes les déformations proportionnelles aux lorentziennes conviendraient aussi ; aussi ce postulat s'accompagne-t-il du choix particulier de la contraction lorentzienne, qui parmi elles est la seule à engendrer une cinématique relativiste. la relativité restreinte, où ces formules, étant des théorèmes, sont supposées d'une exactitude parfaite, la non-relativité lorentzienne peut sans devenir contradictoire supporter le cas peut-être pas impossible où leur validité n'irait pas indéfiniment au-delà de la milliardième décimale, et laisse ouvert un espace de liberté pour un éventuel affinage.

Table des matières

  1 -LA MÉTHODE ........................................................................................................... 2 2 -CONSTRUCTIBILITÉ ENSEMBLISTE DE LA MÉCANIQUE LORENTZIENNE NON-RELATIVISTE DES CORPS ÉLASTIQUES CONSIDÉRÉS DANS LEUR ÉTENDUE .................. 3 3 -ÉCHEC DE LA DÉMONSTRATION DE CONSTRUCTIBILITÉ DE LA MÉCANIQUE DES CORPS ÉLASTIQUES DANS LE CADRE DE LA RELATIVITÉ RESTREINTE ........................... 9 4 -LA NON-RELATIVITÉ LORENTZIENNE OUVRE DES DEGRÉS DE LIBERTÉ DONT LA RELATIVITÉ RESTREINTE NE DISPOSE PAS ................................................................... 10


Hadrien Hours

Ernst Biersack

Patrick Loiseau

Causal study of Network Performance

Keywords: Bayesian graph, Causality, Network Performance, TCP Protocol

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The TCP protocol supports more than 80% of the traffic going through the Internet and several studies of its behavior exist to better understand the different parameters and their roles in the performance one can expect when using this protocol. Theoretical models [Padhye 98] as well as statistical ones [Mirza 07] have been proposed. These models do not take into account the impact that parameters can have between each others and the approach is highly depending on the TCP version that is considered. For the second class of models, as relying on statistical correlation, there is the additional limitation of creating a model that is invalid as soon as it is used for predicting intervention that could change the statistics under which the model was inferred. The causal approach we propose in this paper answers these different issues.

The different works made in the domain of causal models inference and their representations as Bayesian graph, often Directed Acyclic Graphs (DAGs), as support for predicting interventions [Pearl 09,Spirtes 01] give access to new perspectives in the study and understanding of complex systems.

While causality has been used to study network performance [Tariq 08], we place ourselves in a situation where resource, both in terms of data and computational power, is a limiting factor. We also introduce what we believe to be more formal and robust methods for both graph inference and intervention predictions.

In this paper we present the Causal Model Inference approach in Section 2 and show one example of its application to the study of FTP traffic in Section 3. We will conclude this paper in Section 4.

Background

Correlation is not causality, two parameters can be found correlated but this correlation does not give us any information on whether one is the cause of the other, or the opposite, or if there exists a third parameter causing these two parameters. This basic notion illustrates the big difference between causal model and statistical model.

Causal model inference

In our work we use the PC algorithm [Spirtes 01] to infer our causal graph. This algorithm is based on leading independence tests to, gradually, build a graph where these independences are verified. The key, then, is the choice of the independence test that will assess the properties of our system. While the classical approach is to test partial correlation between the residuals of linear regression (as in Z-Fisher test), we are

Prediction

Assuming now that we have our causal graph representing our system, it is possible, using simple graphical criteria [Pearl 09, Spirtes 01], to predict the effects of an intervention on one of the parameters of our model. In the graph each vertex corresponds to one parameter of our system and an edge between two vertices represents a direct causal effect from the parent to the child (in X → Y, X is the parent and Y the child). A (set of) node(s), Z, is said to block a path between X and Y if (i) every collider (→ W ←), or its descendants, is not in Z and (ii) at least one non collider is in Z.

The notation do(X = x) represents the manual intervention of setting the parameter X to the value x.

Definition 1 (Back-door criterion) A set of variables, Z, is said to satisfy the Back-door criterion, relative to an ordered pair of variables (X i ,X j ), in a DAG G if: (i) No node in Z is a descendant of X i . (ii) Z blocks every path between X i and X j that contains an arrow into X i

Theorem 1 (Back-door adjustment) If a set of variables Z satisfies the Back-door criterion relative to (X,Y), then the causal effect of X on Y is identifiable and given by the formula:

P(Y = y | do(X = x)) = Z P(Y = y | X = x, Z = z)P(Z = z) (1)
Due to space constraints we will not present the theory of d-separation, at the source of this criterion, but we redirect the reader to [Pearl 09] for a formal explanation and justification of the Back-door criterion.

Study of FTP traffic

For illustrating our approach we decided to study FTP traffic. One reason for choosing this protocol is the absence of the application limiting the performance of TCP. The throughput is only limited by the Network or the client Receiver Window.

We first define the different parameters of our model and explain how do we build our dataset. Then, we present the causal model we obtain, with the PC algorithm, before predicting the effect of intervening on the RTT, on the Throughput.

Dataset

For our experiments we set up a FTP server where all the traffic is recorded. Using Intrabase [Siekkinen 05] and the Tstat tool † we obtain the different metrics presented in Table 1. This table also presents a summary of our dataset consisting in 1000 downloads from different clients, in Spain, Germany and France.

Causal model

The model we obtained is presented in Figure 1. Due to space constraints we cannot describe all the properties of this model and focus on prediction, presented in the following section.

Predictions

As we are working with variables where no distributions can be assumed, we use Copula Theory [Jaworski 10] to estimate the densities, and conditional densities, of the parameters present in the Back Door adjustment formula, Equation (1).

The presentation of Copulae, as well as the choice of the Copula to model the distributions of our parameters, are out of the scope of this paper.

For predicting the effect of intervening on RTT, we use the Back Door Criterion met by the set of variables Z RT T , Equation (2), with Z RT T = {T.O.D., N.L.A.C.}:

f T PUT |do(RT T ) (T put = ∆|do(RT T = θ)) = Z RT T f T put|RT T,Z RT T (T put = ∆|RT T = θ, Z RT T ) f Z RT T dZ RT T (2)
Figure 2 presents the estimated Throughput post-intervention (solid line), obtained with Equation (2). For comparison, we present the Throughput obtained from observations, in the initial dataset, for which the RTT value is the one corresponding to the intervention we want to predict (dotted line). We also added the information of the number of samples (bar plot) from which the distribution of the Throughput postintervention was estimated. For values of RTT in [250, 350] we can see that there are not enough samples for estimating the post-intervention distribution which leads to inconsistent estimates.

Removing the inconsistent estimates of the post-intervention distribution, we can see that the Throughput post intervention is smaller than the one we observe in the initial dataset for a given value of RTT. This observation can be explained by the causal graph presented Figure 1 as, by conditioning on a given RTT value, we also take into account the Back door effects of variables such as RWin, N.L.A.C., T.O.D. or T.O.R. which are spurious associations blocked by Z RT T in Equation (2). This result shows that adopting a naive approach of estimating the Throughput directly from the pre intervention samples will overestimate the effect that an intervention on the RTT will have on the Throughput.

Conclusion

This work is a first attempt to apply causal theory to the study of network performance. We present the inference of causal models, represented by Directed Acyclic Graphs, and their use to predict the effects of interventions on complex systems from passive observations. We show the different challenges that arise when applying theoretical theory to a real case study, the study of FTP traffic, and propose several solutions. As every model, causal models present their limits. We present the ones we judge as the most important for our work and propose solutions to overcome them. We introduce several methods that show very promising

RT T (ms)

Figure 2: Comparison between causal approach and naive approach results. The use of the Hilbert Schmidt Criterion, for testing independences in the PC algorithm, and copulae, for estimating the multidimensional probability density functions, are the two main ones.

As we can see from Equation (1), the variety of situations observed defines the range of predictions that we can make. It will be necessary to increase the number of samples we have to reach a greater accuracy in our predictions. We are now working on network simulated experiments where we will have access to more parameters and the possibility to verify the accuracy of our predictions.

One important limitation of our approach is the definition of static parameters, by averaging metrics such as RTT or RWin, to model a dynamic protocol. Using the web10G tool, we plan to have access to low level TCP parameters and to sample the parameters at a finer grained timescale.

Figure 1 :

 1 Figure 1: Causal graph model of FTP traffic performance

Table 1 :

 1 Summary of FTP traffic dataset observing parameters which are not normally distributed and with non linear dependences. Consequently, in our work, we use the Hilbert Schmidt Independence Criterion[Zhang 12], a criterion that does not rely on normality or linearity to test parameters independences.

	Parameter	Definition	Min	Max	Avg	Coeff. Var.
	Dist (km)	Distance between Server and Client	14	620	250	0.95
	T.O.D. (s)	Time of the day, when the connection was started	740	82000	46000	0.53
	Nbbytes (MB)	Number of bytes sent by the server	6	60	31.5	0.51
	N.L.A.C. (kbps)	Narrow Link available capacity	47.8	42.9e+3 5.9e+3	1.7
	Nbhops (units)	Number of hops between Client and Server	9	27	11	0.24
	RTT (ms)	Round Trip Time	60	710	270	0.76
	BufferingDelay (ms)	Part of the RTT due to queuing delay	4.2	470	84	1.2
	RetrScore (no unit)	Fraction of retransmitted packets	0.001	0.014	0.0018	0.92
	p (no unit)	Fraction of retransmitted windows of packets	3e-5	0.011	0.0007	1.3
	T.O.R. (no unit)	Fraction of retransmitted packets due to Time Outs	0	0.01	0.0006	1.7
	RWin (kbytes)	Receiver Window	10.7	253.9	136.7	0.68
	Tput (kbps)	Throughput	24	928	332	0.77

† http://tstat.tlc.polito.it/


Eric Hoffbeck
email: hoffbeck@math.univ-paris13.fr

Christine Vespa
email: vespa@math.unistra.fr

LEIBNIZ HOMOLOGY OF LIE ALGEBRAS AS FUNCTOR HOMOLOGY

Keywords: Mathematics Subject Classification: 17B55, 18D50, 17B56 Functor homology, operads, Lie algebras, Leibniz homology Contents

We prove that Leibniz homology of Lie algebras can be described as functor homology in the category of linear functors from a category associated to the Lie operad.

Introduction

Introduction

In the 90's, Robinson and Whitehouse defined and studied in [START_REF] Robinson | Operads and Γ-homology of commutative rings[END_REF] Γ-homology of commutative algebras, a homology theory suited in the Date: April 22, 2014.

context of differential graded modules over a field of positive characteristic. Pirashvili and Richter proved in [START_REF] Pirashvili | Robinson-Whitehouse complex and stable homotopy[END_REF] that this homology theory can be interpreted as functor homology, for functors from the category Γ of finite pointed sets. This category can be viewed as a category associated to the commutative set operad. For associative algebras, similar results are obtained by Pirashvili and Richter in [START_REF]Hochschild and cyclic homology via functor homology[END_REF]. In this paper, the authors interpret usual Hochschild and cyclic homology of associative algebras as functor homology. In this setting, the category Γ is replaced by a category associated to the associative set operad. In [START_REF] Livernet | An interpretation of E n -homology as functor homology[END_REF], Livernet and Richter give a description of E n -homology of nonunital commutative algebras as functor homology. In this setting the category Γ is replaced by a suitable category of epimorphisms related to planar trees with n-levels.

In all these results, homology theories are obtained as Tor functors in a category of functors, between a Loday functor and a functor t playing the role of the base ring. In [START_REF] Pirashvili | Hodge decomposition for higher order Hochschild homology[END_REF], Pirashvili uses the interpretation of homology theories as functor homology to give purely homological proofs of Hodge decompositions of higher order Hochschild homology of commutative algebras.

This paper is motivated by the following natural question: is it possible to describe homologies of Lie algebras as functor homology?

We obtain that the Leibniz homology of Lie algebras can be interpreted as functor homology. More precisely, we prove the following theorem:

Theorem. For A a Lie algebra and M a A-module, one has an isomorphism:

H Leib * (A, M) ≃ T or

Γ Lie sh *
(t, L Lie sh (A, M)) where Γ Lie sh is a suitable linear category associated to the Lie operad and L Lie sh (A, M) is a generalized Loday functor.

Leibniz homology was defined by Loday in [START_REF] Loday | Une version non commutative des algèbres de Lie: les algèbres de Leibniz[END_REF] in order to understand periodicity phenomena in algebraic K-theory (see [START_REF]Algebraic K-theory and the conjectural Leibniz K-theory[END_REF]). This homology can be defined via an explicit complex, obtained as the noncommutative analogue of the Chevalley-Eilenberg complex and it is defined for Leibniz algebras, which are non-commutative variants of Lie algebras. In Leibniz algebras, the bracket is not required to be anti-commutative anymore, and the Jacobi relation is replaced by the Leibniz relation, which can be seen as a lift of the Jacobi relation in the non-commutative context. As Lie algebras are a particular case of Leibniz algebras, the Leibniz homology can be used to compute homological invariants of Lie algebras.

The most striking result concerning Leibniz homology is the following Loday-Cuvier theorem [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF][START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]: for an associative algebra A, the Leibniz homology of the Lie algebra gl(A) is isomorphic to the free associative algebra over the Hochschild homology of A. This theorem is the non-commutative variant of the Loday-Quillen-Tsygan theorem [START_REF] Loday | Cyclic homology and the Lie algebra homology of matrices[END_REF][START_REF] Feȋgin | Additive K-theory, K-theory, arithmetic and geometry[END_REF] which states that the Chevalley-Eilenberg homology of gl(A) is isomorphic to the free exterior algebra over the cyclic homology of A. Recall that cyclic homology is naturally isomorphic to the additive K-theory which is the analogue of K-theory obtained by replacing the general linear group GL(A) by the Lie algebra gl(A).

In order to write Leibniz homology of Lie algebras as functor homology, two objects have to be defined: a category playing for Lie algebras the role of Γ and a variant of the Loday functor for this category. One main difference with the cases of commutative and associative algebras is that the operad encoding Lie algebras is not a set operad. This requires the use of linear categories (i.e. categories enriched over the category of k-modules), instead of usual categories.

The proof of the main theorem is based on a characterization of the Tor functors. The principal difficulty is to prove that the homology vanishes on projective generators. This result is the heart of this paper. To prove it, we need to consider a linear category associated to the operad Lie, called Γ Lie sh , with a shuffle condition, requiring some maps to preserve a part of the order. This condition is the main reason why we obtain Leibniz homology as functor homology and not Chevalley-Eilenberg homology.

The proof of the vanishing of the Leibniz homology on projective generators can be decomposed into three steps. We begin to describe a basis of the morphisms spaces in the linear category Γ Lie sh . This allows us to define a filtration on the k-module associated to the complex computing the homology of projective generators. The proof of the compatibility of this filtration with the differential requires to use a basis of the operad Lie and to understand its behaviour with respect to composition. Then we identify the associated graded complex with a sum of acyclic complexes. This step is heavily based on the thorough understanding of the combinatorial objects associated to the basis of the morphisms spaces in the linear category Γ Lie sh .

In a recent work [START_REF] Benoit Fresse | Functor homology and operadic homology[END_REF], Fresse proves that operadic homology can be obtained as functor homology, by a method different than ours. For the commutative operad and the associative operad, Fresse recovers results of Pirashvili and Richter [START_REF] Pirashvili | Robinson-Whitehouse complex and stable homotopy[END_REF][START_REF]Hochschild and cyclic homology via functor homology[END_REF]. For the operad Lie, Fresse obtains a description of Chevalley-Eilenberg homology of Lie algebras as functor homology over the category Γ Lie . Our theorem shows that functor homology over the category Γ Lie sh is naturally related to Leibniz homology of Lie algebras. The category Γ Lie sh has the advantage to be more manageable than the category Γ Lie for effective computations.

The paper is organized as follows: Section 1 consists in recollections on functor homology of enriched categories and on some operadic definitions. In Section 2 we define our category Γ Lie sh and the associated Loday functor. After some recollections of Leibniz homology, we state in Section 3 the main theorem of the paper. The last section is devoted to the proof of the vanishing of the Leibniz homology on projective generators.

The first author would like to thank Birgit Richter for explaining to him the usual strategy to interpret homology theories as functor homology. The authors gratefully acknowledge the hospitality of the Isaac Newton Institut in Cambridge. The first author is supported in part by the Sorbonne-Paris-Cité IDEX grant Focal and the ANR grant ANR-13-BS02-0005-02 CATHRE. The second author is supported by the ANR grant ANR-11-BS01-0002 HOGT.

Notations:

The following categories will be useful throughout the whole paper.

• Set is the category of sets with morphisms the set maps;

• k-Mod is the category of modules over a fixed commutative ground ring k; • k-grMod is the category of N-graded modules over k; • ∆ is the simplicial category, i.e. the category with objects ordered finite sets [n] and morphisms order preserving maps; • Γ is the skeleton of the category of finite pointed sets having as objects [n] = {0, 1, . . . , n} with 0 as basepoint and morphisms the set maps f : [n] → [m] such that f (0) = 0; • Γ surj is the category having as objects finite pointed sets and as morphisms the pointed surjective maps; • Γ sh is the category with objects ordered finite sets [n] and morphisms pointed shuffling maps f , that is maps such that min(f -1 (i)) < min(f -1 (j)) whenever i < j; • Γ surj sh is the category with objects finite ordered sets [n] and morphisms pointed shuffling surjections;

The Lie algebras we consider in this paper are algebras over the Lie operad. In particular, in characteristic 2, a Lie bracket is just antisymmetric and we do not have in general [x, x] = 0.

Recollections on enriched category, functor homology and operads

In this section we briefly recall some definitions and facts about enriched categories, functor homology and operads useful in the sequel. (representing the composition of morphisms in D and the identity morphism on i). These structure morphisms are required to be associative and unital in the obvious sense.

Example 1.2.

(1) A category enriched over (Set, ×, [0]) is an usual category.

(2) A category enriched over (k-Mod, ⊗, k) is a k-linear category and a functor enriched over (k-Mod, ⊗, k) is a k-linear functor.

F (B, B ′ ) : B(B, B ′ ) → D(F (B), F (B ′ ))
for every pair of objects B, B ′ ∈ B that are associative and unital.

Definition 1.4. A natural transformation enriched over C, σ : F → F ′ , between two functors enriched over C, F, F ′ : B → D consists in giving, for every object B ∈ B, a morphism:

σ B : 1 → D(F (B), F ′ (B))
in C, for every object B of B, that satisfy obvious commutativity conditions for a natural transformation.

We denote by C-Nat(F, F ′ ) the object of natural transformations enriched over C between F and F ′ . Theorem 1.5 (Enriched Yoneda lemma). Let B be a small category enriched over C. For every object B ∈ B and every functor enriched over C, F : B → C, the object of natural transformations enriched over C from B(B, -) to F exists and there is an isomorphism in C:

C-Nat(B(B, -), F ) ≃ F (B)
which is natural both in F and in B.

d,d ′ ∈D F (d ′ ) ⊗ D(d, d ′ ) ⊗ G(d) U , 2 B , 2 d∈D F (d) ⊗ G(d) , 2 F ⊗ D G in C
where the map U is induced by the composite

F (d ′ )⊗D(d, d ′ )⊗G(d) 1⊗F (d,d ′ ) -----→ F (d ′ )⊗C(F (d ′ ), F (d))⊗G(d) ev⊗1 ---→ F (d)⊗G(d)
where F (d, d ′ ) is the morphism in C given because F is a C-functor and ev is the evaluation map. The map B is induced by a similar composite with G in place of F .

Recall that the evaluation map is the counit of the adjunction on C of the monoidal product with the internal-hom.

(F ) ≃ G ⊗ D F , (3)
H i (F ) = 0 for all projective D-modules F and i > 0, then, for all F ∈ D-mod, we have a natural isomorphism

H i (F ) ≃ T or D i (G, F ).
In the rest of the paper, all our categories will be enriched over (k-Mod, ⊗ k , k).

Algebraic operads.

A good reference for operads is the book of Loday and Vallette [START_REF] Loday | Algebraic operads, Grundlehren der Mathematischen Wissenschaften[END_REF]. We denote by OrdSet the category of finite ordered sets (with order-preserving bijections as morphisms) and by F in the category of finite sets (with bijections as morphisms).

Definition 1.9. A collection (resp. symmetric collection) is a contravariant functor from the category OrdSet (resp. F in) to the category k-Mod.

For a collection P , we denote P ([n -1]) by P (n) for n ≥ 0. Let P and Q be two symmetric collections. One defines their symmetric composition P • Q by

(P •Q)(n) = m P (m)⊗ Σm   α∈Set([n-1],[m-1]) Q(α -1 (0)) ⊗ . . . ⊗ Q(α -1 (m -1))   .
Let P and Q be two collections. One defines their nonsymmetric composition P • Q by

(P •Q)(n) = m P (m)⊗   α∈∆([n-1],[m-1]) Q(α -1 (0)) ⊗ . . . ⊗ Q(α -1 (m -1))   . When P (0) = Q(0) = 0, one defines their shuffle composition P • sh Q by (P • sh Q)(n) = m P (m)⊗   α∈Γ sh ([n-1],[m-1]) Q(α -1 (0)) ⊗ . . . ⊗ Q(α -1 (m -1))   .
In the rest of the paper, unless otherwise stated, all our collections will be reduced, that is P (0) = 0. This hypothesis will be required to get a functor from symmetric operads to shuffle operads.

Remark 1.10. Note that

(P • sh Q)(n) ≃ m P (m)⊗   α∈Γ surj sh ([n-1],[m-1]) Q(α -1 (0)) ⊗ . . . ⊗ Q(α -1 (m -1))   since Q(0) = 0.
Remark 1.11. Note that the previous definitions and remark can be extended to a symmetric monoidal category (C, ⊗, 1) having an initial object 0 satisfying

C ⊗ 0 ≃ 0 ≃ 0 ⊗ C, ∀C ∈ C. This is the case for (Set, ×, [0]
), the initial object being ∅.

Definition 1.12. A linear nonsymmetric (resp. shuffle) operad is a monoid P in the category of collections equipped with the nonsymmetric (resp. shuffle) composition. A linear symmetric operad is a monoid P in the category of symmetric collections equipped with the symmetric composition.

An algebra A over an operad P is a k-module endowed with an action of P , that is there are maps θ : P (n) ⊗ A ⊗n → A compatible with the monoid structure of P (and with the symmetric group action in the symmetric case).

For P a symmetric operad, an A-module M is a k-module endowed with M ⊕ A has a P -algebra structure extending the structure on A and satisfying

θ(P (n) ⊗ (M ⊗k ⊗ A ⊗n-k )) = 0 if n ≥ k ≥ 2.
Example 1.13.

(1) Usual commutative algebras can be seen as algebras over a symmetric operad called Com, determined by Com(n) = k for all n ≥ 1 and Com(0) = 0, and equipped with the obvious composition. The notion of A-modules for A a commutative algebra is the usual one. The case of unital commutative algebras can be dealt using the symmetric operad called uCom, determined by uCom(n) = k for all n ≥ 0 and equipped with the obvious composition.

(2) Lie algebras can be seen as algebras over a symmetric operad called Lie. This operad is generated by a symmetric bracket which satisfies the Jacobi relation. The notion of A-modules for A a Lie algebra is the usual one.

There exists a forgetful functor (-) sh from symmetric collections to nonsymmetric collections. This functor allows us to make a direct link between the notion of symmetric operads and the notion of shuffle operads because of the following lemma, proved by Dotsenko and Khoroshkin: Lemma 1.14. [3, Proposition 3] The forgetful functor (-) sh is monoidal.

Therefore this functor induces a forgetful functor from symmetric operads to shuffle operads.

Example 1.15. The usual Lie symmetric operad induces a linear shuffle operad.

The fundamental property we will use later in the paper is the following: For P a symmetric operad, the k-modules P(n) and P sh (n) are the same. For instance they have the same bases. This observation also implies that for a P-algebra A, the maps θ : P (n) ⊗ A ⊗n → A defining the algebra structure can be seen as maps P sh (n) ⊗ A ⊗n → A. The same holds for maps defining a A-module structure on M.

Functorial constructions associated to an operad

2.1. Enriched category associated to an operad. The notion of enriched category associated to an operad is quite classical, already appearing in a paper of May and Thomason [START_REF] May | The uniqueness of infinite loop space machines[END_REF]. We recall the classical definition and then extend it to the shuffle context. Definition 2.1.

• The linear category of pointed operator Γ P associated to a reduced symmetric linear operad P is the linear category whose objects are pointed finite sets [n] and such that:

Γ P ([n], [m]) = α∈Γ([n],[m]) P(α -1 (0)) ⊗ . . . ⊗ P(α -1 (m)).
Composition of morphisms is prescribed by symmetric operad structure maps in P.

• The linear category of operator Γ P sh associated to a reduced shuffle linear operad P is the linear category whose objects are pointed finite sets [n] and such that:

Γ P sh ([n], [m]) = α∈Γ sh ([n],[m]) P(α -1 (0)) ⊗ . . . ⊗ P(α -1 (m)).
Composition of morphisms is prescribed by shuffle operad structure maps in P.

Remark 2.2. Note that for a symmetric operad P satisfying P(0) = 0, we have

Γ P ([n], [m]) ≃ α∈Γ surj ([n],[m]) P(α -1 (0)) ⊗ . . . ⊗ P(α -1 (m))
and

Γ P sh ([n], [m]) ≃ α∈Γ surj sh ([n],[m]) P(α -1 (0)) ⊗ . . . ⊗ P(α -1 (m)).
For a shuffle operad P sh coming from a symmetric operad P, we ease notation by abbreviating Γ P sh sh to Γ P sh . Note that Γ P sh ([n], [0]) = P sh ([n]) = P([n]) = P(n + 1). Example 2.3. We explain graphically how the composition works in Γ P sh for P a shuffle operad. Let (α, f ) be the generator of Γ P sh ([9], [START_REF] Benoit Fresse | Functor homology and operadic homology[END_REF]) where α ∈ Γ sh ([9], [START_REF] Benoit Fresse | Functor homology and operadic homology[END_REF]) is given by:

α(0) = α(4) = 0; α(1) = 1; α(2) = α(3) = α(7) = 2; α(5) = 3; α(6) = α(9) = 4; α(8) = 5 and f = f 0 ⊗ f 1 ⊗ f 2 ⊗ f 3 ⊗ f 4 ⊗ f 5 where ∀i ∈ {0, 1, 2, 3, 4, 5} f i ∈ P(α -1 (i))
. We can represent this morphism by the following picture:

0 ✺ ✺ ✺ 4 ✠ ✠ ✠ 1 2 ❍ ❍ ❍ ❍ ❍ 3 7 ✈ ✈ ✈ ✈ ✈ 5 6 ✺ ✺ ✺ 9 ✠ ✠ ✠ 8 f 0 f 1 f 2 f 3 f 4 f 5 0 1 2 3 4 5
Let g be the generator of Γ P sh ([5], [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]) represented by the following picture:

0 ✺ ✺ ✺ ✺ 3 ✠ ✠ ✠ ✠ 1 2 ❍ ❍ ❍ ❍ ❍ 4 5 ✈ ✈ ✈ ✈ ✈ g 0 g 1 g 2 0 1 2
To compose them, we use the composition of the shuffling maps α ∈ Γ sh and the shuffle composition of the operad P.

We obtain the following element in Γ P sh ([9], [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]), where the dotted lines shows where the shuffle composition of P has to be done.

0 ✺ ✺ ✺ 4 ✠ ✠ ✠ 5 1 2 ❍ ❍ ❍ ❍ ❍ 3 7 ✈ ✈ ✈ ✈ ✈ 6 ✺ ✺ ✺ 9 ✠ ✠ ✠ 8 f 0 f 3 f 1 f 2 f 4 f 5 g 0 g 1 g 2 0 1 2
This element can also be written as

0 ❍ ❍ ❍ ❍ ❍ 4 5 ✈ ✈ ✈ ✈ ✈ 1 2 ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ 3 6 ✺ ✺ ✺ 7 ✠ ✠ ✠ 8 ♥ ♥ ♥ ♥ ♥ ♥ ♥ 9 ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ h 0 h 1 h 2 0 1 2
where the element h 2 ∈ P({2, 3, 6, 7, 8, 9}) is given by the composition g 2 • (f 2 , f 4 , f 5 ) associated to the shuffling map α : {2, 3, 6, 7, 8, 9} → {0, 1, 2} defined by α -1 (0) = {2, 3, 7}, α -1 (1) = {6, 9} and α -1 (2) = {8}. The elements h 0 and h 1 are defined in a similar way.

For P = Lie, there are some particular maps in Γ Lie and Γ Lie sh which are central in our later constructions. Definition 2.4. For every n ≥ 0 and for

0 ≤ i < j ≤ n + 1, there is one element in Γ Lie sh ([n + 1], [n]) called d i,j associated to the single map f ∈ Γ sh ([n + 1], [n]
) sending i and j to i, where the element in Lie( 1) is the identity and the element µ in Lie(2) is the Lie bracket.

0 1 . . . i ❀ ❀ j ✄ ✄ i + 1 . . . ĵ . . . n + 1 µ 0 1 . . . i i + 1 . . . . . . n Remark
T or C (F, G) ≃ T or k[C] (k[F ], k[G]).
Observation 2.7. Let P be a symmetric operad. By the enriched Yoneda lemma, the representable functors Γ P ([n], -) for n ≥ 0 are projective generators of the category of k-linear functors from Γ P to k-Mod. Similarly, the representable functors P n = Γ P sh ([n], -) for n ≥ 0 are projective generators of the category of linear functors from Γ P sh to k-Mod.

Notice that for G a k-linear functor from (Γ P ) op to k-Mod and F a k-linear functor from Γ P to k-Mod, we have:

G ⊗ Γ P Γ P ([n], -) ≃ G([n]); Γ P (-, [n]) ⊗ Γ P F ≃ F ([n]).
2.2. Loday functor. We recall the definition of the Loday functor, appearing in [START_REF]Grundlehren der Mathematischen Wissenschaften[END_REF]. This functor appears in various contexts (associative or commutative algebras, pointed or unpointed sets) Let A be a unital commutative algebra and M an A-module. The Loday functor L(A, M)

: Γ → k-Mod is defined by L(A, M)([n]) = M ⊗ A ⊗n and for f an element of Γ([n], [m]) the induced map f * : L(A, M)([n]) → L(A, M)([m]) is given by f * (a 0 ⊗ a 1 ⊗ . . . ⊗ a n ) = b 0 ⊗ . . . ⊗ b m where b i = j∈f -1 (i) a j or b i = 1 if f -1 (i) = ∅.
We first extend this definition for algebras over a symmetric operad, and then see how it can be adapted in the shuffle context.

Let P be a symmetric operad, A be a P-algebra and M a A-module. Let us call θ the structure maps.

Lemma 2.8. There exists a linear functor:

L P (A, M) : Γ P → k-Mod
defined on an object [n] by:

L P (A, M)([n]) = M ⊗ A ⊗n and on a generator (α, f ) of Γ P ([n], [m]) where α : [n] → [m] and f = f 0 ⊗ . . . ⊗ f m (with f i ∈ P(α -1 (i)))
, by:

L P (A, M)(α, f )(a 0 ⊗ a 1 ⊗ . . . ⊗ a n ) = b 0 ⊗ . . . ⊗ b m where b i = θ(f i ⊗ j∈α -1 (i) a j ).
We extend the definition on a morphism in Γ P ([n], [m]) by linearity.

Proof. We need to prove first that L P (A, M)(α, f ) is well-defined, and then show the functoriality.

First, note that b 0 is in M because a 0 is in M, the other a j 's are in A and because α(0) = 0. Moreover, the linearity is obtained by construction.

Concerning the functoriality, recall that the composition in Γ P is defined by the operadic composition. Recall also that the structure maps θ (for the module M and the algebra A) are required to be compatible with the operadic composition. This requirement gives exactly L P (A, M)(id, 1 P ⊗ . . . ⊗ 1 P ) = id and L Remark 2.11. For A a commutative algebra and M a A-module, by [START_REF] Pirashvili | Hodge decomposition for higher order Hochschild homology[END_REF] and [START_REF] Pirashvili | Robinson-Whitehouse complex and stable homotopy[END_REF] we have

P (A, M)((α, f )(β, g)) = L P (A, M)(α, f )L P (A, M)(β, g).
H Γ * (A, M) ≃ T or Γ (t, L(A, M)) where H Γ * (A, M)
is the Γ-homology of A with coefficients in M and t : Γ op → Ab is the cokernel of Γ(-, [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]) → Γ(-, [START_REF] Borceux | Handbook of categorical algebra[END_REF]). By the previous remark and Remark 2.6 this result can be rephrased in linear categories as:

H Γ * (A, M) ≃ T or Γ uCom (k[t], L uCom (A, M)).
Let P be a symmetric operad, A be a P-algebra and M a A-module. Recall that P sh is the shuffle operad associated to P. Let us call θ the structure maps.

Lemma 2.12. There exists a linear functor: L P sh (A, M) : Γ P sh → k-Mod defined on an object [n] by:

L P sh (A, M)([n]) = M ⊗ A ⊗n and on a generator (α, f ) of Γ P sh ([n], [m]) where α : [n] → [m] and f = f 0 ⊗ . . . ⊗ f m (with f i ∈ P sh (α -1 (i)))
, by:

L P sh (A, M)(α, f )(a 0 ⊗ a 1 ⊗ . . . ⊗ a n ) = b 0 ⊗ . . . ⊗ b m where b i = θ(f i ⊗ j∈α -1 (i) a j ).
We extend the definition on a morphism in Γ P sh ([n], [m]) by linearity. Proof. The proof is mostly the same as in the symmetric case. The only difference comes from the shuffle context. Recall that the structures maps in the symmetric context induce structures maps in the shuffle context. These induced maps are compatible with the shuffle operadic composition.

Definition 2.13. The functor L P sh (A, M) : Γ P sh → k-Mod is called the shuffle Loday functor associated to the P-algebra A and the A-module M.

Leibniz homology of a functor

Leibniz homology of a Lie algebra was defined by Loday. In this section, we first recall the usual Leibniz complex of a Lie algebra, and then generalize it to define the Leibniz homology of a Γ Lie sh -module. In fact, Leibniz homology of L Lie sh (A, M) is Leibniz homology of the Lie algebra A with coefficients in M. Then we state the main result of this paper.

(A, M) → C Leib n-1 (A, M) is given by: d(x⊗a 1 ⊗. . .⊗a n ) = 1≤i<j≤n (-1) j x⊗a 1 ⊗. . . a i-1 ⊗[a i , a j ]⊗. . .⊗ a j ⊗. . .⊗a n + 1≤j≤n (-1) j [x, a j ] ⊗ a 1 ⊗ . . . ⊗ a j ⊗ . . . ⊗ a n . Remark 3.2.
• If we replace x by the symbol a 0 , we can notice that: 3.2. Leibniz homology as functor homology. Before stating our theorem, we first need to define a right Γ Lie sh -module, which will serve as basepoint.

d(a 0 ⊗a 1 ⊗. . .⊗a n ) = 0≤i<j≤n (-1) j a 0 ⊗a 1 ⊗. . . a i-1 ⊗[a i , a j ]⊗. . .⊗ a j ⊗. . .
Definition 3.6. The contravariant linear functor t from Γ Lie sh to k-Mod is defined by coker Γ Lie sh (-, [START_REF] Borceux | Handbook of categorical algebra[END_REF]) (t, T ).

(d 0,1 ) * ----→ Γ Lie sh (-, [ 0
To prove this theorem, we use the characterization of homology theories given in 1.8, which relies on three hypotheses.

The first hypothesis (H Leib * sending short exact sequences to long exact sequences) is satisfied, as H Leib * is defined via the homology of a complex.

The functor t has been defined in a way to satisfy the second hypothesis (concerning H 0 ). Proposition 3.8. For all left Γ Lie sh -modules T , we have

H Leib 0 (T ) = t ⊗ Γ Lie sh T. Proof.
We follow the ideas of the proof of Theorem 1.3 of [START_REF]Hochschild and cyclic homology via functor homology[END_REF]. By the definition of t, we have an exact sequence: Γ Lie sh (-, [START_REF] Borceux | Handbook of categorical algebra[END_REF])

(d 0,1 ) * ----→ Γ Lie sh (-, [ 0
]) → t → 0. For T a left Γ Lie sh -module, by Proposition 1.7 the functor -⊗ Γ Lie sh T is right exact, so we deduce the following exact sequence:

Γ Lie sh (-, [1]) ⊗ Γ Lie sh T → Γ Lie sh (-, [0]) ⊗ Γ Lie sh T → t ⊗ Γ Lie sh T → 0.
By Observation 2.7, Γ Lie sh (-, [START_REF] Borceux | Handbook of categorical algebra[END_REF])

⊗ Γ Lie sh T ≃ T ([1]) and Γ Lie sh (-, [0]) ⊗ Γ Lie sh T ≃ T ([0]) so: t ⊗ Γ Lie sh T ≃ T ([0])/Im T ([1]) → T ([0]) = H Leib 0 (T ).
The third hypothesis consists in the vanishing of the homology of the projective generators. The proof of this part is detailed in the remaining section of the paper.

Vanishing of the homology of the projective generators

The goal of this section is to prove the following proposition.

Proposition 4.1. For i > 0 and n ≥ 0,

H Leib i (P n ) = 0
where P n = Γ Lie sh ([n], -). We can note that for n = 0, we directly obtain

H Leib i (P 0 ) = 0, for i > 0, because P 0 ([m]) = 0 if m > 0.
From now on, we fix a postive integer n.

Let us first give a survey of the three steps of the proof of this proposition. In the first step, we give a description of bases of the Lie operad and of Γ Lie sh ([n], [m]). In Lemma 4.10, we identify these bases with easy combinatorial objects (tuples, or ordered partitions into tuples). One important point is to consider all possible values of m at the same time. This allows us to use the ordering between tuples to obtain, in Proposition 4.14, a filtration of the k-module ⊕ m C Leib m (P n ). Then we show, in Proposition 4.21 that this filtration is compatible with the differential of the complex C Leib * (P n ). This step in based on the crucial lemma 4.15 describing the bracket of two elements of the basis of Lie in terms of elements of the basis. In the last part of the proof we identify the associated graded complex with a sum of acyclic complexes. This step is heavily based on the thorough understanding of relationships between tuples and ordered partitions into tuples. We deduce Proposition 4.1 by a classical spectral sequence argument.

4.1. Filtration as a k-module.

4.1.1. Basis of the Lie operad. Recall that one can define the free symmetric operad generated by a symmetric collection, and that the notion of operadic ideal exists (cf [START_REF] Loday | Algebraic operads, Grundlehren der Mathematischen Wissenschaften[END_REF]). Therefore symmetric operads can be defined by generators and relations.

Our operad of interest in this paper, Lie, can then be defined by the free symmetric operad generated by the symmetric collection kµ concentrated in arity 2, where µ denotes an antisymmetric bracket, quotiented by the ideal generated by the Jacobi relation.

Recall that the Jacobi relation can be written operadically as (4.1.1)

1 ✽ ✽ ✽ ✽ ✽ ✽ 2 ❂ ❂ 3 ✂ ✂ µ ⑧ ⑧ µ = 1 ❂ ❂ 2 ✂ ✂ 3 ✝ ✝ ✝ ✝ ✝ ✝ µ ❄ ❄ µ - 1 ❂ ❂ 3 ✂ ✂ 2 ✝ ✝ ✝ ✝ ✝ ✝ µ ❄ ❄ µ .
In the rest of the paper, all internal vertices of the trees are labelled with the bracket µ, therefore we do not write it anymore.

Methods have been developed in [START_REF] Hoffbeck | A Poincaré-Birkhoff-Witt criterion for Koszul operads[END_REF] and [START_REF] Dotsenko | Gröbner bases for operads[END_REF] to find monomial bases with good properties with respect to an ordering.

A basis of the Lie operad is given by the following planar trees (cf [START_REF] Hoffbeck | A Poincaré-Birkhoff-Witt criterion for Koszul operads[END_REF]):

               1 ❄ ❄ ❄ σ(2) ⑧ ⑧ ❄ ❄ ❄ ❄ σ(3) ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ σ(n) ⑧ ⑧ n ≥ 1, σ permutation of {2, . . . , n}                .
Note that this operadic basis corresponds to a basis of the multilinear part of the free Lie algebra on x 1 , . . . , x n given by elements of the form

[. . . [[x 1 , x σ(2) ], x σ(3) ], . . . , x σ(n) ]. (See [17, Section 5.6.2]).
We denote the part of the basis in arity n by B Lie (n). We define similarly B Lie (I) for I a finite ordered set of cardinality n, using the order preserving bijection between I and {1, . . . , n}.

We denote the whole basis by B Lie = n B Lie (n).

Explicit description of the category associated to the Lie operad.

To describe the category Γ Lie sh , it is enough to describe the vector spaces of morphisms and to understand the composition of elements of the basis. We first give a k-basis for Γ Lie sh

([n], [m]). Recall that Γ Lie sh ([n], [m]) = α∈Γ sh ([n],[m]) Lie(α -1 (0)) ⊗ . . . ⊗ Lie(α -1 (m)).
Obviously, if m > n, the basis is empty. If m ≤ n, we can describe a basis B Γ Lie sh (n, m) of Γ Lie sh ([n], [m]) via shuffle maps and a tuple of elements of a Lie basis:

B Γ Lie sh (n, m) = α∈Γ sh ([n],[m]) B Lie (α -1 (0)) × . . . × B Lie (α -1 (m)).
We also define B Γ Lie sh (n, N) = m B Γ Lie sh (n, m). 4.1.3. Notations for elements of the bases. To avoid drawing huge forests of trees, we first identify the elements of B Γ Lie sh (n, N) as some particular tuples with splittings. We begin by defining these particular tuples and introduce some notations. Observation 4.2. All trees appearing in B Lie (n) are left combs, so they are characterized by their leaves. An element of B Lie (n) can be identified with a n-tuple (1, σ(2), . . . , σ(n)) where σ is a permutation of {2, . . . , n}, just by reading the inputs from left to right. So we have a bijection B Lie (n) ≃ S n-1 , where S n-1 is the permutation group. In the sequel, we identify these two sets.

Example 4.3. Let (α, f ) be the generator of Γ Lie sh ([6], [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]) where α ∈ Γ sh ([6], [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]) is given by:

α(0) = α(1) = α(5) = 0; α(2) = 1; α(3) = α(4) = α(6) = 2 and f = f 0 ⊗ f 1 ⊗ f 2 where: f 0 = Id ∈ B Lie (3) ≃ S 2 ; f 1 = Id ∈ B Lie (1) ≃ S 0 ; f 2 = τ 1,2 ∈ B Lie (3) ≃ S 2 .
We can represent this morphism by the following picture:

0 ❄ ❄ ❄ 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ 5 ⑧ ⑧ ⑧ 0 2 1 3 ❄ ❄ ❄ 6 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ 4 ⑧ ⑧ ⑧ 2 Definition 4.4. A m-split (n + 1)-tuple is an ordered partition of [n] into m + 1 tuples: B 0 ∪ B 1 ∪ . . . ∪ B m where m ≥ 0 and B i = (k i,0 , . . . , k i,ℓ i ) satisfying two conditions: (1) 0 ≤ i < j ≤ m ⇒ k i,0 < k j,0 , (2)
∀ 0 ≤ i ≤ m, ∀ 0 ≤ q ≤ ℓ i , k i,0 < k i,q .
To ease notations, we denote

B 0 ∪ B 1 ∪ . . . ∪ B m by (B 0 | B 1 | . . . | B m ).
Remark 4.5. The conditions impose that k 0,0 = 0. The first condition can be read as "the first term of a tuple is smaller than the first term of the following tuples". The second condition can be read as "the first term of a tuple is smaller than the other terms of this tuple". Notation 4.6. We denote by ST uple(n, m) the set of m-split (n + 1)tuples, by ST uple(n, N) the set ∪ m ST uple(n, m) and by T uple(n) the set ST uple(n, 0) corresponding to the set of (n + 1)-tuples starting with 0.

Example 4.7. For n = 2, the set ST uple(2, N) is composed of 6 elements:

(0 | 1 | 2), (0 | 1, 2), (0, 1 | 2), (0, 2 | 1), (0, 1, 2), (0, 2, 1).
The set T uple(2) is composed of only 2 elements: (0, 1, 2) and (0, 2, 1).

Lemma 4.8. The two conditions (1) and (2) in Definition 4.4 are jointly equivalent to the following single condition:

(3) ∀ 0 ≤ i ≤ j ≤ m, ∀ 0 ≤ q ≤ ℓ j , k i,0 < k j,q .
This condition can be read as "the first term of a tuple is smaller than the following terms".

Proof. Each direction is easy to check.

If B i and B i+1 are two consecutives blocks of a split tuple, we denote by B i B i+1 their concatenation. Then

(B 0 | B 1 | . . . | B i B i+1 | . . . | B m ) is also a split tuple.
Proof. The condition (3) in the above lemma for (

B 0 | B 1 | . . . | B m ) directly implies the condition for (B 0 | B 1 | . . . | B i B i+1 | . . . | B m ).
We now use these tuples and split tuples to describe the bases B Γ Lie sh (n, m) and B Γ Lie sh (n, N).

Lemma 4.10.

• The set B Γ Lie sh (n, m) is in bijection with ST uple(n, m). In particular, B Γ Lie sh (n, 0) is in bijection with T uple(n) • The set B Γ Lie sh (n, N) is in bijection with ST uple(n, N).

Proof. Let (α, σ 0 , . . . , σ m ) be an element in

B Γ Lie sh (n, m) where α ∈ Γ sh ([n], [m]) and ∀i ∈ {0, . . . , m}, σ i ∈ B Lie (α -1 (i)) ≃ S |α -1 (i)|-1 . We consider for all i ∈ {0, . . . , m} the ordered set α -1 (i) = {k i,0 < k i,1 < . . . < k i,|α -1 (i)| }. We associate to (α, σ 0 , . . . , σ m ) the ordered partition of [n] into m+ 1 tuples (B 0 | B 1 | . . . | B m ) where for all i ∈ {0, . . . , m}, B i = {k i,0 , k i,σ i (1) , . . . , k i,σ i (|α -1 (i)| )}. We have (B 0 | B 1 | . . . | B m ) ∈ ST uple(n,
Ψ : B Γ Lie sh (n, m) → ST uple(n, m) by: Ψ(α, σ 0 , . . . , σ m ) = (B 0 | B 1 | . . . | B m ).
This map Ψ is a bijection whose inverse is given by:

Ψ -1 (B 0 | B 1 | . . . | B m ) = Ψ -1 (k 0,0 , . . . , k 0,l 0 | . . . | k n,0 , . . . , k n,ln ) = (α, σ 0 , . . . , σ m )
where for all i ∈ {0, . . . , m}, α(k i,0 ) = . . . = α(k i,l i ) = i and σ i is the permutation such that k i,0 < k i,σ -1 i (1) < k i,σ -1 i (2) < . . . < k i,σ -1 i (l i ) . The map α is a shuffle map by condition (1) in Definition 4.4

Example 4.11. The following basis element of Γ Lie sh ([6], [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]): 0

❄ ❄ ❄ 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ 5 ⑧ ⑧ ⑧ 0 2 1 3 ❄ ❄ ❄ 6 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ 4 ⑧ ⑧ ⑧ 2 corresponds to the 2-split 7-tuple in ST uple(6, 2) (0, 1, 5 | 2 | 3, 6, 4)
by the bijection in Lemma 4.10.

In the rest of the paper, we identify the set B Γ Lie sh (n, m) with ST uple(n, m).

4.1.4. Ordering and filtration. We now define an ordering on the set B Γ Lie sh (n, N) and use it to endow m Γ Lie sh ([n], [m]) with a filtration (note that this direct sum is actually finite). Observation 4.12. Forgetting the vertical bars of an element of ST uple(n, N) gives an element of T uple(n). This defines a surjective map p : ST uple(n, N) → T uple(n).

Using the bijection between B Γ Lie sh (n, N) and ST uple(n, N), we define p : B Γ Lie sh (n, N) → T uple(n), which is again a surjection. We order the set T uple(n) with the usual total lexicographical ordering.

We now define an ascending filtration on m Γ Lie sh ([n], [m]).

Lemma 4.13. Let u be an element in T uple(n). The k-modules: We have to show that this filtration is compatible with the differential. This is the point of the next subsection. 4.2. Compatibility of the filtration with the differential. We first need to rewrite in B Lie a product of two basis elements. This allows us to understand the image by P n (d i,j ) = (d i,j ) * (where - * denotes the postcomposition) of an element in B Γ Lie sh (n, N). We actually do not need all the terms of the image, but only its leading term relatively to the ordering defined in the previous section. 4.2.1. Products of elements in the basis of Lie. We need to rewrite in B Lie such a composite of two basis elements:

F u :=
i 0 ❄ ❄ i 1 ⑧ ⑧ j 0 ❄ ❄ j 1 ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ i n ⑧ ⑧ ❄ ❄ ❄ ❄ j m ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧
where i 0 < j 0 , i 0 < i k for all 1 ≤ k ≤ n and j 0 < j k for all 1 ≤ k ≤ m.

(-1) k i 0 ❄ ❄ i 1 ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ i n ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ❄ ❄ ❄ ❄ • • • ⑧ ⑧ ⑧
where:

-the inputs between i 0 and i n are in the same order as before, -j 0 is in the (n + 2 + k)-th position, -the inputs between i n and j 0 are labelled by the j ℓ for ℓ ∈ S (with ℓ decreasing from top to bottom), -the inputs below j 0 are labelled by the j ℓ for ℓ / ∈ S (with ℓ increasing from top to bottom).

Let us denote by b S

k the element of the basis appearing for the indices k of the first sum and S of the second sum.

Sketch of proof. We proceed by induction on m.

For m = 0, there is nothing to prove. For m = 1, we apply the Jacobi relation.

To prove the formula for m + 1 if m ≥ 1, the idea is to apply the formula for m to the tree where j 0 ■ ■ j 1 ✉ ✉ has been replaced by j 0 . Then we graft j 0 ■ ■ j 1 ✉ ✉ back in every term where j 0 appears. Finally we use the Jacobi relation again, and obtain the desired formula for m + 1. We refer the reader to the proof of [START_REF] Reutenauer | Free Lie algebras[END_REF]Theorem 5.1] for details. Remark 4.17. This lemma moreover implies that the basis satisfies a property very similar to the Poincaré-Birkhoff-Witt property of associative algebras. It is then possible to show that the category Γ Lie sh is actually a Koszul category.

4.18. i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ j 0 ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ = i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ - i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ - i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ - i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ + i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ + i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ + i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ - i 0 ❄ ❄ ❄ i 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 3 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 2 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ j 0 ⑧ ⑧ ⑧ = b ∅ 0 -b {1} 1 -b {2} 1 -b {3} 1 + b {1,2} 2 + b {1,3} 2 + b {2,3} 2 -b {1,2,3} 3
.

Using our notations for the basis of Γ Lie sh ([n], [m]), we can notice that the previous example can be seen as applying (d 0,1 ) * to (i 0 , i 1 |j 0 , j 1 , j 2 , j 3 ) ∈ Γ Lie sh ([5], [START_REF] Borceux | Handbook of categorical algebra[END_REF]), which gives an element in Γ Lie sh ([5], [0]). And the sum of eight terms we obtain is the decomposition of (d 0,1 ) * (i 0 , i 1 |j 0 , j 1 , j 2 , j 3 ) in the basis of Γ Lie sh ([5], [0]). The sum begins with : (i 0 , i 1 , j 0 , j 1 , j 2 , j 3 ) -(i 0 , i 1 , j 1 , j 0 , j 2 , j 3 ) -. . . We can notice that the first term of the sum is in the same stage of the filtration as the element we were starting with, while the other 7 terms are in a higher stage of the filtration.

Let us make some additional observations about Lemma 4.15 in the general case:

(1) For k = 0, the sum on subsets S is composed of a single term for S = ∅. This term b ∅ 0 in the basis is identified with the tuple (i 0 , i 1 , . . . , i n , j 0 , j 1 , . . . , j n ), and has a +1 coefficient. This means that, if we denote B i = (i 0 , . . . , i n ) and B j = (j 0 , . . . , j m ), we obtain the equality b

∅ 0 = B i B j . Notice that p(b ∅ 0 ) = p(B i B j ) = p(B i | B j ). ( 2
) For k ≥ 1, notice that all the terms b S k appearing in the sum have a (n+2)th label equal to j l for l > 0. As j 0 is the (n+2)th label of p(B i | B j ) and j l > j 0 , using the lexicographical ordering in T uple(n), we obtain the inequality p(b S k ) > p(B i | B j ). (

i) If 0 ≤ i < m, then (d i,i+1 ) * (b) is in F u . Moreover, in gr u , we obtain (d i,i+1 ) * (b) = (B 0 | . . . | B i B i+1 | . . . | B m ). (ii) If 0 ≤ i < m and i+2 ≤ j ≤ m, then d i,j (b) is in F u . Moreover,
in gr u , we obtain d i,j (b) = 0.

That means that (d i,i+1 ) * (b) has one term (with a plus sign) in the same stage of the filtration as b, and other terms in higher stages. And (d i,j ) * (b) for j > i + 1 has only terms in stages of the filtration higher than the stage containing b, and thus vanishes in the graded object. Let us now consider an example of the second case: (d 0,2 ) * applied to the following element of the basis of Γ Lie sh ([6], [START_REF] Cuvier | Algèbres de Leibnitz: définitions, propriétés[END_REF]):

b = 0 ❄ ❄ ❄ 1 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ 5 ⑧ ⑧ ⑧ 0 2 1 3 ❄ ❄ ❄ 6 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ 4 ⑧ ⑧ ⑧ 2 .
The computation gives a sum of four terms of the form 0

❄ ❄ ❄ 1 ⑧ ⑧ ⑧ 2 ❄ ❄ ❄ ❄ 5 ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ σ(3) ⑧ ⑧ ❄ ❄ ❄ ❄ σ(6) ⑧ ⑧ ❄ ❄ ❄ ❄ σ(4)
⑧ ⑧ 0 1 where σ is a bijection of the set {3, 4, 6}.

As split tuples, the equality reads Clearly, when projecting on Tuples(6), the four terms of the right hand side are larger than the term of the left hand side. Thus in the associated graded object, the equality reduces to Proof. This is a direct consequence of the previous lemma, using that the differential in C Leib * (P n ) is d = 0≤i<j≤n (-1) j (d i,j ) * . 4.3. Acyclicity of the associated graded complex. We now identify the associated graded complex with a sum of acyclic ones.

(
First, let us describe the differential in the associated graded complex. From Lemma 4.19, we obtain For a fixed u = (0, k 1 , . . . , k n ) ∈ T uple(n), we want to describe the elements b ∈ ST uple(n) such that p(b) = u. We use the definition of a split tuple with the condition (3) from Lemma 4.8, which characterizes the first terms of a block. Definition 4.22. We call an admissible cut of the tuple u = (0, k 1 , . . . , k n ) an index i with 1 ≤ i ≤ n satisfying the condition ∀ 1 ≤ j ≤ m, j > i ⇒ k j > k i .

We denote by Cuts(u) the set of admissible cuts of u.

Notice that Cuts(u) is non-empty as n always belongs to Cuts(u) (the condition of the definition is empty for this index). Recall that for any non-empty finite set E = {e 0 , . . . , e n }, there exists an associated chain complex C * (E) whose basis is the power set P(E), a subset of cardinality k giving a generator in degree k. The differential

1. 1 .

 1 Enriched category. One of the standard references for symmetric monoidal categories and enriched categories is the book of Borceux [1, Chapter 6]. Let (C, ⊗, 1) be a closed symmetric monoidal category. Recall that a symmetric monoidal category C is closed when for each object C ∈ C, the functor -⊗C : C → C admits a right adjoint denoted by [C, -] : C → C. Definition 1.1. A category D enriched over C (or a C-category) consists of a class I (representing the objects of D) and for any objects i, j, k ∈ I an object of C: D(i, j) (representing the morphisms from i to j in D) and morphisms in C D(i, j) ⊗ D(j, k) → D(i, k) and 1 → D(i, i)

( 3 )Definition 1 . 3 .

 313 The category C can be provided with the structure of an enriched category over (C, ⊗, 1) using the bifunctor [-, -] : C op × C → C whose composition with the forgetful functor C(1, -) : C → Set is just C(-, -) : C op × C → Set. Let B and D categories enriched over C. A functor enriched over C (or a C-functor) from B to D, F : B → D consists of an object of D, F (B) for every object B ∈ B and of morphisms in C:

1. 2 .

 2 Functor homology. Definition 1.6. For D a C-category and F : D op → C and G : D → C a pair of C-functors, the enriched tensor product of F and G, F ⊗

Proposition 1 . 7 .

 17 For a category D enriched over C, we call left D-modules covariant C-functors from D to C and right D-modules contravariant C-functors from D to C. Let D-mod (resp. mod-D ) the category of left (resp. right) D-modules. If C is an abelian category, the categories D-mod and mod-D are abelian. In the sequel C is an abelian category. The bifunctor -⊗ D -: mod-D × D-mod → C is right exact with respect to each variable. We have the following characterization of homology theories in the enriched setting: Proposition 1.8. Let D be a C-category and G a right D-module. If H * is a functor from D-mod to C such that (1) H * sends short exact sequences to long exact sequences, (2) for all F ∈ D-mod we have a natural isomorphism H 0

2 . 5 .

 25 These elements d i,j are actually generators (via linear combination and composition) of n,m Γ Lie sh ([n], [m]). Remark 2.6. If S is a symmetric set operad, we consider the corresponding linear symmetric operad kS defined as kS(n) := k[S(n)] with k[-] denoting the free k-module. The k-linearization of a category C , which will be denoted by k[C], is the k-linear category which has the same objects as C and whose sets of morphisms k[C](c, c ′ ) are the k-modules k[C(c, c ′ )] generated by the sets of morphisms of C. It is easy to check that Γ kS = k[Γ(S)], where Γ(S) denotes the category associated to S. For example, Γ uCom = k[Γ] and Γ Com = k[Γ surj ], where Com and uCom are the symmetric linear operad defined in Example 1.13. Furthermore, for F : C → k-Mod and G : D op → k-Mod and k[F ] : k[C] → k-Mod and k[G] : k[D op ] → k-Mod their linearizations, we have:

Definition 2 . 9 .

 29 The functor L P (A, M) : Γ P → k-Mod is called the symmetric Loday functor associated to the P-algebra A and the Amodule M.Remark 2.10. For A a unital commutative algebra and M an A-module, we have L uCom (A, M) = k[L(A, M)] where uCom is the symmetric operad defined in Example 1.13.

3. 1 .

 1 Leibniz complex. Let A be a Lie algebra and M a A-module. Definition 3.1. [9] The Leibniz homology of A with coefficients in M, denoted by HL * (A, M), is the homology of the complex (C Leib * (A, M), d) where C Leib n (A, M) = M ⊗ A ⊗n and the differential d : C Leib n

.Remark 3 . 5 .

 35 ⊗a n Note that d 2 = 0 is proved in the same way as for the Chevalley-Eilenberg complex. • Leibniz homology and Chevalley-Eilenberg homology are related by a map H Leib * (A, M) → H CE * (A, M) induced by the anti-symmetrisation map C Leib * (A, M) → C CE * (A, M). Proposition 3.3. Let T be a linear functor from Γ Lie sh to k-Mod. The map d : T ([n]) → T ([n -1]) defined by d = T ( 0≤i<j≤n (-1) j d i,j ) satisfies d 2 = 0. Therefore the sequence: . . . d -→ T ([n]) d -→ T ([n -1]) d -→ . . . is a well-defined complex, denoted by (C Leib * (T ), d). Proof. The relation d 2 = 0 is obtained by exactly the same computation as d 2 = 0 for the Leibniz complex of an algebra. Definition 3.4. Let T be a linear functor from Γ Lie sh to k-Mod. The Leibniz homology of T , denoted by H Leib(T ) * , is the homology of the complex (C Leib * (T ), d). Leibniz homology produces a functor from left Γ Lie sh -modules to k-grMod called H Leib * If one chooses the Loday functor for T , the definition coincides with the previous one: (C Leib * (L Lie sh (A, M)), d) = (C Leib * (A, M), d).

Theorem 3 . 7 .

 37 ]) , where - * denotes the postcomposition and d 0,1 ∈ Γ Lie sh ([1], [0]) is defined in Definition 2.4. It is easy to check that (Im (d 0,1 ) * )[n] = Γ Lie sh ([n], [0]) for n ≥ 1 and (Im (d 0,1 ) * )[0] = 0. Therefore t([n]) = 0 for n ≥ 1 and t[0] = Γ Lie sh ([0], [0]) = k. We can now state the main theorem of the paper: Let T be a left Γ Lie sh -module. Then H Leib * (T ) = T or Γ Lie sh *

Lemma 4 . 9 .

 49 Let (B 0 | B 1 | . . . | B m ) be a split tuple, and 0 ≤ i < m.

  m) since condition (1) in Definition 4.4 corresponds to the shuffling property of α ∈ Γ sh ([n], [m]) and condition (2) is obviously satisfied. So we define the map

  ,N),p(b)≥u kb define an ascending filtration of the k-module m Γ Lie sh ([n], [m]). The associated graded object is gr m Γ Lie sh ([n], [m]) = u∈T uple(n) gr u where gr u is b∈B Γ Lie sh (n,N),p(b)=u kb. Recall that C Leib m (P n ) = Γ Lie sh ([n], [m]) as k-modules. Thus we obtain the following proposition: Proposition 4.14. We obtain a filtration on C Leib * (P n ) (seen as a kmodule) indexed by elements u in T uple(n).

Lemma 4 . 15 .

 415 The decomposition of this element in the basis B Lie is the following sum: m k=0 S⊂{1,...,m},|S|=k

Remark 4 .

 4 [START_REF] Pirashvili | Hodge decomposition for higher order Hochschild homology[END_REF]. In[START_REF] Reutenauer | Free Lie algebras[END_REF] Theorem 5.1] the description of the basis B Lie corresponds to the fact that Lyndon words form a Hall set and the previous lemma corresponds to the triangularity property of the corresponding Hall basis. Notice that a careful study of the proof of[START_REF] Reutenauer | Free Lie algebras[END_REF] Theorem 5.1] gives the more precise formula given in the statement of Lemma 4.15.

Example

4. 2 . 2 .

 22 Compatibility of the filtration with the differential. We now want to see how the observation from the previous example can be generalized to (d i,j ) * (b) where b is a basis element Γ Lie sh ([n], [m]).

Lemma 4 . 19 .

 419 Let b = (B 0 | . . . | B m ) be in B Γ Lie sh (n, m) where m > 0. Let u denote p(b).

Proof.

  By Lemma 4.15 we see that (d i,i+1 ) * (B 0 | . . . | B m ) is a sum of the term (B 0 | . . . | B i B i+1 | . . . | B m ) with the terms ±(B 0 | . . . | b S k | . . . | B m ) where k > 0.Using the projection to T uple(n) and the lexicographical ordering there, we noticeu = p(B 0 | . . . | B i B i+1 | . . . | B m ) and u < p(B 0 | . . . | b S k | . . . | B m ).Thus we obtain the part (i) of the lemma.For the second part, first notice that(d i,j ) * (B 0 | . . . | B m ) is a sum of terms whose smallest one is (B 0 | . . . | B i B j | . . . | B j | . . . | B m ). But p(B 0 | . . . | B i B j | . . . | B j | . . . | B m ) is larger than p(B 0 | . . . | B m ) because k i+1,0 < k j,0 as i + 1 < j by condition (1) in Definition 4.4.Thus all terms appearing in d i,j (B 0 | . . . | B m ) are in F u and vanish in gr u . This concludes the proof of (ii).

Example 4 .

 4 20. An example of the behaviour of (d i,i+1 ) * was already given in the observation after Example 4.18.

  d 0,2 ) * (0, 1, 5 | 2 | 3, 6, 4) = (0, 1, 5, 3, 6, 4 | 2)-(0, 1, 5, 6, 3, 4 | 2)-(0, 1, 5, 4, 3, 6 | 2)+(0, 1, 5, 6, 4, 3 | 2).

(d 0, 2 )

 2 * (0, 1, 5 | 2 | 3, 6, 4) ∼ = 0. Proposition 4.21. The differential of C Leib * (P n ) is compatible with the filtration.

d

  gr (B 0 | . . . | B m ) = 0≤i<m (-1) i+1 (B 0 | . . . | B i B i+1 | . . . | B m ).

Example 4 .

 4 23. Let u be (0, 2, 1, 5, 3, 4). The set Cuts(u) is {2, 4, 5}. These indices 2, 4 and 5 are exactly the ranks where it is possible to split the tuple. For instance, (0, 2 | 1, 5 | 3 | 4) is the split tuple obtained by cutting as most as possible.

Lemma 4 .

 4 24. The set of b ∈ ST uple(n) such that p(b) = u = (0, k 1 , . . . , k n ) is in bijection with the power set P(Cuts(u)), via (B 0 | . . . | B m ) → {|B 0 |, |B 0 | + |B 1 |, . . . , |B 0 | + . . . + |B m-1 |}The inverse bijection is given by{i 1 , . . . , i m } → (0, k 1 , . . . , k i 1 -1 | k i 1 , k i 1 +1 , . . . . . . | k i 2 . . . | . . . | k im , k im+1 , . . .).

Acknowledgements 1. Recollections on enriched category, functor homology and operads

Acknowledgements

d : C l+1 → C l is given by d({e i 0 , . . . e i ℓ }) = ℓ j=0 (-1) j {e i 0 , . . . , e i j , . . . , e i ℓ }.

This complex C * (E) is acyclic (for instance it can be seen as a complex associated to a simplex).

Lemma 4.25.

There is an isomorphism of complexes between (gr u , d gr ) and (C * (Cuts(u)), d).

Proof. Recall that in the graded object, the part d i,i+1 of the differential is the concatenation of the blocks B i and B i+1 , that is the removal of the (i + 1)th cut. The part d i,i+1 comes with the sign (-1) i+1 , and in the differential of the chain complex (C * (Cuts(u)), d), removing the (i + 1)th cut comes with the same sign. This isomorphim and the acyclicity of the complex C * (Cuts(u)), d imply the following lemma: Lemma 4.26. The complex (gr u , d gr ) is acyclic for u ∈ T uple(n).

We can now finish the proof of Proposition 4.1.

Proof of Proposition 4.1. The filtration of C Leib * (P n ) is bounded since T uple(n) is a finite set. So, by the classical convergence theorem of the spectral sequence associated to a filtration (see [START_REF] Weibel | An introduction to homological algebra[END_REF]Theorem 5.5.1]), this spectral sequence converges to H Leib * (P n ). The previous lemma implies that grC Leib *

gr u is a sum of acyclic complexes, and therefore is also acyclic. We deduce that H Leib * (P n ) = 0