J'ai eu l'immense honneur d'eectuer ma thèse sous la direction de Yann Palu et d'Alexander Zimmermann. Je ne les remercierai jamais assez de leur patience, de leur gentillesse. Alexander, merci pour tes explications toujours très claires, pour ce cours de M2 qui m'a initiée à la théorie des catégories, et merci pour le livre de chevet. Yann, merci pour les après midi entières passées dans ton bureau, les explications que tu donnais et redonnais inlassablement, merci de m'avoir écoutée, de ta bienveillance éternelle, merci pour les cafés, friandises et j'en passe. J'aimerais exprimer ma reconnaissance à Karin Baur et Peter Jorgensen qui m'ont fait l'honneur de rapporter cette thèse. Je souhaite aussi remercier vivement Claire Amiot et Bernhard Keller qui ont accepté de faire partie du jury.

Je tiens profondément à remercier les membres du LAMFA pour leur ambiance conviviale, pour les discussions en salle café, les encouragements sans faille. Merci aux "grands" d'ici. Ce sont eux qui ont conrmé mon goût pour l'algèbre lors de ma scolarité à l'Université de Picardie Jules Verne. Je ne pourrais citer tous les noms sans en oublier. Néanmoins, parmi eux, je remercie particulièrement Isabelle et Christelle, et leur organisation hors pair, et je m'étonne toujours qu'elles répondent à mes mails à peine envoyés. Je tiens à remercier les doctorants qui ont su créer un climat de conance entre nous. En particulier, merci à Aktham, entre autres pour les nombreuses machines à café que j'ai cassées, et merci à Maxime, qui a soutenu sa thèse trop tôt.

Toujours au sein de l'Université de Picardie Jules Verne, je tiens à remercier chaleureusement l'équipe de l'ESPE, dont je fais partie maintenant, pour avoir pris en compte mon activité de recherche, et pour m'avoir intégrée dès le début dans leurs rangs avec chaleur et humanité. Merci à Sabine, Ramla, Virginie, Michèle. J'espère sincèrement ne pas froisser celles et ceux que j'aurais oubliés. 1 Merci à tous mes amis non "mathémiciens" comme ils m'appellent, qui ont su me garder les pieds sur terre dans un monde scientique trop cérébral. Je remercie en particulier Angie, Chacha, Grand, Tchiot, pour le nombre incalculable de soirées, Alex pour les après-midi de jeux terminés à refaire le monde et Cathy et Stéphane, ainsi que tous les amis du café-atelier Couleurs d'antan pour les après-midi en terrasse. Encore une fois je ne peux citer tout le monde.

Enn, je veux remercier ma femme qui m'a toujours soutenue dans les bons, comme dans les mauvais moments. Me supporter est un exploit, et tu le réalises quotidiennement : merci inniment mon Ed'.

I Geometric realizations of m-cluster categories 49

1 A geometric realization of the m-cluster categories of type Dn 51

1.1 How to associate a colored quiver with an (m + 2)-angulation . . . 51 1.1.1 Geometric realization and ips . . . . . . . . . . . . .

Pour le cas Dn , on trouve la gure qui correspond à l'étude, on démontre la compa- tibilité entre le ip d'une (m + 2)-angulation, et la mutation de carquois coloré. On trouve une bijection entre les objets m-rigides et chaque arc dit admissible, puis entre les objets amas-basculants et les (m + 2)-angulations. De plus, on démontre la compatibilité entre la réduction d'Iyama-Yoshino, et le fait de couper le long d'un arc, qu'on dénira formellement.

Nous démontrons aussi qu'une catégorie exacte est une catégorie de prébration au sens de Anderson-Brown-Cisinski, qui vérie le théorème de Quillen, et une catégorie de Frobenius est munie d'une structure de modèle, compatible avec le passage à la catégorie stable, qui est triangulée.

We show that a subcategory of the m-cluster category of type Dn is isomorphic to a category consisting of arcs in an (n -2)m-gon with two central (m -1)-gons inside of it. We show that the mutation of colored quivers and m-cluster-tilting objects is compatible with the ip of an (m + 2)-angulation.

In this thesis, we study the geometric realizations of m-cluster categories of Dynkin types A, D, Ã and D. We show, in those four cases, that there is a bijection between (m+2)-angulations and isoclasses of basic m-cluster tilting objects. Under these bijections, ips of (m + 2)-angulations correspond to mutations of m-cluster tilting objects. Our strategy consists in showing that certain Iyama-Yoshino reductions of the m-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. This allows to infer results from small cases to the general ones.

Let E be a weakly idempotent complete exact category with enough injective and projective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective and projective objects, and stable under taking direct sums and summands. In this paper, E is equipped with the structure of a prebration category with cobrant replacements. As a corollary, we show, using the results of Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF], that the category of nite presentation modules on the costable category M is a localization of E. We also deduce that E → modM admits a calculus of fractions up to homotopy. These two corollaries are analogues for exact categories of results of Buan and Marsh in [START_REF] Aslak | From triangulated categories to module categories via localisation[END_REF], [START_REF] Aslak | From triangulated categories to module categories via localization II: calculus of fractions[END_REF] (see also [START_REF] Beligiannis | Rigid objects, triangulated subfactors and abelian localizations[END_REF]) that hold for triangulated categories.

If E is a Frobenius exact category, we enhance its structure of prebration category to the structure of a model category (see the article of Palu in [?] for the case of triangulated categories). This last result applies in particular when E is any of the Hom-nite Frobenius categories appearing in relation to cluster algebras.

Introduction 1.1 Vers les catégories amassées supérieures

Le concept d'algèbre amassée est né au début des années 2000, introduit par Fomin et Zelevinsky dans [START_REF] Fomin | Cluster algebras. I. Foundations[END_REF], an entre autres, de donner un cadre combinatoire pour étudier les bases canoniques et la positivité totale, deux sujets de théorie de Lie algébrique.

Il s'est avéré plus tard que les algèbres amassées répondaient à beaucoup d'autres questions, notamment en théorie des représentations des carquois et algèbres de dimensions nie. Une belle introduction des algèbres amassées est donnée par Keller dans [START_REF] Keller | Cluster algebras and cluster categories[END_REF].

Une algèbre amassée est une Q-algèbre commutative, munie de générateurs appelés variables d'amas, regroupés en amas, possédant toujours le même nombre de variables. Ces amas sont construits à partir de mutations successives d'un amas particulier appelé graine initiale, contenant un carquois à n sommets et n variables. Ainsi, l'ensemble des variables d'amas peut être ni ou inni. Fomin et Zelevinsky ont démontré dans [START_REF] Fomin | Cluster algebras. II. Finite type classication[END_REF] que l'algèbre amassée contient un nombre ni de variables d'amas si et seulement si le carquois de la graine initiale est équivalent par mutation à un carquois de type Dynkin (où il est à noter que l'algèbre amassée est un invariant de la classe de mutation du carquois initial).

An de démontrer certains résultats liés aux algèbres amassées, il a été nécessaire de catégoriser cette notion. Dans le cas où Q est un carquois acyclique (et k un corps algébriquement clos), la catégorie amassée associée à Q a été créée en 2006 par Buan, Marsh, Reineke, Reiten et Todorov (voir [BMR + 06] pour plus de 7 Chapitre 1. Introduction détails). Il s'agit de la catégorie d'orbite

D b (kQ)/τ -1 [1]
où D b (kQ) désigne la catégorie dérivée associée à l'algèbre kQ, τ est la translation d'Auslander-Reiten et [1] le shift. Les détails seront fournis dans les préliminaires. Le cas où Q est de type A n fut traité en détail par Caldero, Chapoton et Schifer dans [START_REF] Caldero | Quivers with relations arising from clusters (A n case)[END_REF]. Le lien entre catégories amassées et algèbres amassées est le suivant : via l'application de Caldero-Chapoton (voir [START_REF] Caldero | Cluster algebras as Hall algebras of quiver representations[END_REF]), il y a une bijection explicite entre les amas et les objets amas-basculants (basiques), et entre les variables d'amas et les objets rigides indécomposables. Pour une étude détaillée, voir l'article de Keller, [START_REF] Keller | Cluster algebras and cluster categories[END_REF]. Peu après la dénition de catégories amassées au sens de [BMR + 06], Keller a prouvé que ces catégories étaient triangulées dans l'article [START_REF] Keller | On triangulated orbit categories[END_REF]. Il a en réalité démontré un résultat plus général. Sous certaines conditions, la catégorie d'orbite d'une catégorie dérivée associée à une algèbre héréditaire par une gentille auto-équivalence est une catégorie triangulée. Cet article a notamment attiré l'attention d'autres catégories, les

D b (kQ)/τ -1 [m]
où m est un entier naturel non nul. Elle sont alors baptisées catégories amassées supérieures.

L'étude des catégories amassées supérieures peut commencer. Dans son article [Tho07], Thomas en donne une dénition formelle, où les objets m-amas basculants correspondent aux m-amas (dénis par Fomin et Reading dans l'article [START_REF] Fomin | Generalized cluster complexes and Coxeter combinatorics[END_REF]), de la même manière que, dans les catégories amassées, les objets amas basculants correspondent aux amas de l'algèbre amassée. On dénit les catégories amassées supérieures à la section 3.2 En 2009, Amiot dans [START_REF] Amiot | On generalized cluster categories[END_REF] a déni les catégories amassées généralisées, pour un carquois quelconque en tant que localisation de Verdier, puis Guo, en 2010, dénit les catégories m-amassées généralisées associées à une algèbre satisfaisant certaines conditions.

Réalisation géométrique des catégories amassées (supérieures)

La dénition de catégorie amassée de Caldero-Chapoton-Schier dans l'article [START_REF] Caldero | Quivers with relations arising from clusters (A n case)[END_REF] a suscité chez certains l'envie de trouver une description géométrique de 1.2. Réalisation géométrique ces catégories dans les cas les plus simples de carquois, d'abord, les carquois de Dynkin, le type A ayant déjà été traité, puis dans le cas Ã. Pour plus de détails, voir la section 3.5.

Dans tous les cas, il s'agit de trouver une surface simple, d'eectuer sur celle-ci des triangulations, puis de démontrer que chaque triangulation est isomorphe à un objet amas basculant. Pour le cas A n , il s'agit d'un polygone à n + 3 côtés. Voici un exemple de triangulation :

Le cas D n a été traité par Schier dans [START_REF] Schier | A geometric model for cluster categories of type D n[END_REF]. Cette fois-ci, un polygone ne susait pas à illustrer le carquois, donc il y a rajouté une ponction au centre. Voici un exemple de triangulation : Fomin, Schapiro et Thurston dans l'article [START_REF] Fomin | Cluster algebras and triangulated surfaces. I. Cluster complexes[END_REF], généralisèrent des résultats au cas d'une surface marquée épointée (sans ponction). Brüstle et Zhang (voir [START_REF] Brüstle | On the cluster category of a marked surface without punctures[END_REF] pour plus de détails) catégorisent ces résultats. Marsh et Palu dans [START_REF] Marsh | Coloured quivers for rigid objects and partial triangulations: the unpunctured case[END_REF] démontrent la compatibilité de cette réalisation géométrique avec les mutations de carquois et la réduction d'Iyama-Yoshino. Ils utilisent la dénition de catégorie amassée généralisée d'Amiot, et un résultat de Keller-Yang montrant l'invariance de la catégorie amassée par mutation de carquois.

Baur et Marsh, quant à eux, s'intéressèrent au cas des catégories amassées supérieures, et plutôt que d'utiliser des triangulations, tracèrent des (m + 2)angulations, pour une catégorie m-amassée (de la même manière que l'on trace des triangles pour une catégorie amassée). Le cas A n a été traité dans [START_REF] Baur | A geometric description of m-cluster categories[END_REF], et le cas D n dans [START_REF] Baur | A geometric description of the mcluster categories of type D n[END_REF].

Torkildsen traita le cas Ãn dans le cas général, en illustrant par un "anneau" formé de deux polygones l'un dans l'autre, le premier ayant p sommets, où p est le nombre de èches dans le carquois allant dans un sens, et le second q, où q = n + 1 -p (qui correspond au nombre de èches dans l'autre sens). On pourra consulter [START_REF] André | A geometric realization of the m cluster category of type[END_REF] pour une étude en détail.

Dans cette thèse, nous traitons le cas Dn en profondeur et démontrons quelques résultats additionnels aux travaux de Baur, Marsh et Torkildsen sur les réalisations géométriques des catégories amassées supérieures. Pour le cas Dn , on trouve la gure qui correspond à l'étude, on démontre la compatibilité entre le ip d'une (m + 2)-angulation, et la mutation de carquois coloré au théorème 2.1.4. On trouve une bijection entre les objets m-rigides et chaque arc dit admissible, puis entre les objets amas-basculants et les (m + 2)-angulations au théorème 2.1.8. De plus, on démontre la compatibilité entre la réduction d'Iyama-Yoshino, et le fait de couper le long d'un arc, qu'on dénira formellement. Ceci est fait au théorème 2.1.6.

Catégories triangulées, algèbre homotopique et généralisation aux catégories exactes

Buan, Marsh et Reiten dans [START_REF] Bakke Buan | Cluster-tilted algebras[END_REF], souhaitent retrouver les diérentes catégories de modules sur les algèbres d'endomorphismes des objets amas basculants à partir de la catégorie amassée. Ils ont démontré que le quotient d'une catégorie amassée par le shift d'un objet amas basculant était une catégorie de module. Buan et Marsh ont généralisé ce résultat aux catégories triangulées dans l'article [START_REF] Aslak | From triangulated categories to module categories via localisation[END_REF], et la localisation d'une catégorie triangulée est une catégorie de module. Puis, Palu étudia le problème d'un point de vue algèbre homotopique, et démontra en 2014 dans [START_REF] Palu | From triangulated categories to module categories via homotopical algebra[END_REF] qu'on pouvait munir une catégorie triangulée d'une structure de mo-1.3. Catégories triangulées, algèbre homotopique et généralisation aux catégories exactes11 dèle faible, ce qui permettait de respecter les hypothèses du théorème de Quillen sur l'algèbre homotopique (voir ici le théorème 3.12.3 ou [START_REF] Daniel | Homotopical algebra[END_REF]). On rappelle que ce théorème donne le résultat suivant : pour une catégorie de modèle faible C, il existe une équivalence de catégorie entre la sous-catégorie des objets brants et cobrants modulo une relation d'homotopie, et la catégorie homotopique de C.

La réexion part de ce constat : La catégorie stable d'une catégorie de Frobenius est munie d'une structure triangulée (voir pour une démonstration l'excellent [START_REF] Happel | Triangulated categories in the representation theory of nite-dimensional algebras[END_REF] de Happel). Et celles-ci sont (faiblement) des catégories de modèle, et respectent le théorème de Buan et Marsh. Est-il alors possible de "remonter" aux catégories de Frobenius, et, plus encore, aux catégories exactes contenant assez d'injectifs (ou de projectifs) ?

C'est le but de la seconde partie de cette thèse. Nous démontrons qu'une catégorie exacte est une catégorie de prébration au sens de Anderson-Brown-Cisinski (voir 2.2.2), qui vérie le théorème de Quillen, et une catégorie de Frobenius est munie d'une structure de modèle (voir 2.2.8), compatible avec le passage à la catégorie stable, qui est triangulée.

Chapter 2

Summary of results

All throughout this thesis, if C is a category, we denote by C(X, Y ) the set of morphisms from X to Y . All the notions will be introduced properly in chapter 3.

2.1 Geometric realization of a higher cluster category of type A n , D n , Ãn and Dn

In the rst part of this thesis, we are interested in geometric realizations of higher cluster categories in dierent cases. Indeed, though the cases A n , D n , and Ãn have been treated respectively by Baur and Marsh in [BM08] and [BM07], and by Torkildsen in [START_REF] André | A geometric realization of the m cluster category of type[END_REF], we show on these cases that the Iyama-Yoshino reduction corresponds to cutting along an arc. Moreover, we treat the case Dn , by giving an explicit bijection between rigid objects and admissible arcs. We also show that cutting along an arc corresponds to Iyama-Yohsino reduction.

Notation

The letter K denotes a eld (we prefer to use the small letter k for an integer). For a triangulated category C endowed with a shift functor Σ, we consider that the sets Ext 1 (X, Y ) and C(X, ΣY ) are the same.

We denote by Q a quiver. In this part, n and m are integers, where n is the number of vertices of Q, n ≥ 4.

If A is an object in a category C, A ⊥ is the class of all objects X such that Ext i (X, A) = 0 for all i ∈ {1, • • • , m}. For a quiver Q (respectively for an (m + 2)angulation ∆), we note µ k (Q) (respectively µ k (∆) the mutation (respectively the ip) at vertex k (respectively at arc k) of Q (respectively ∆).

13

We denote by mod(KQ) the category of nitely generated right modules over the path algebra KQ. The letter τ stands for the Auslander-Reiten translation. The application [1] is the shift functor of the (triangulated) higher cluster category.

Results

We start with a denition, common to all cases, of an (m + 2)-angulation.

Denition 2.1.1. Let n be a positive integer, and let C m be the higher cluster category associated with a quiver Q of type A n , D n , Ãn or Dn . Let P be:

A polygon with nm + 2 sides for case A n .

A polygon with nm -m + 1 sides for case D n .

An annulus with p+q marked points, where p and q are the number of arrows in one direction and the other for type Ãn . An (n -2)m-gon with two (m + 1)-gon inside of it for case Dn . An (m + 2)-angulation of P is a maximal set of noncrossing m-diagonals. Remark 2.1.2. The notion of m-diagonal is dened dierently in each case.

As Buan and Thomas dened a colored quiver from an m-cluster-tilting object, with each (m + 2)-angulation, we dene a colored quiver.

Denition 2.1.3. Let ∆ be an (m + 2)-angulation of the gure P in one of the above cases. We dene the colored quiver Q ∆ associated with ∆ in the following way:

1. The vertices of Q ∆ are in bijection with the m-diagonals of ∆.

2. If i and j form two sides of the same polygon and only one, then we draw an arrow from i to j. The color of the corresponding arrow is the number of edges between both m-diagonals, counted clockwise from i.

As Buan and Thomas dened the mutation of colored quiver, we can dene the ip of an (m + 2)-angulation. An important result in this thesis, is that both notions coincide. Theorem 2.1.4. Let n be a positive integer, and let C m be the higher cluster category associated with a quiver Q of type A n , D n , Ãn or Dn . Let P be as in denition 2.1.1.

Let ∆ be any (m + 2)-angulation of P . Let Q ∆ be the colored quiver associated with ∆. Then, for any k ∈ {1, • • • , n}, we have the equality

µ k (Q ∆ ) = Q µ k (∆) .

Higher cluster categories of type A n , D n , Ãn and Dn

From the m-diagonals we dene in this thesis, we can build a new category as the mesh category of m-diagonals, with well-chosen morphisms between the arcs. This gives rise to an equivalence of categories between the category of arcs, and a subcategory of the higher cluster category. Moreover, with each m-diagonal α, we can associate an m-rigid object X α . The following theorem helps to build a bijection between m-cluster-tilting objects and (m + 2)-angulations.

Theorem 2.1.5. Let α and β be two arcs in the gure P . Let X α and X β be the m-rigid objects associated with. Then, ∀i ∈ {1, • • • , m}, Ext i C (X α , X β ) = 0 if and only if α and β do not cross each other.

The following theorem shows that cutting along an arc corresponds to Iyama-Yoshino reduction, as in the paper of Marsh and Palu, [MP14, Section 4].

Theorem 2.1.6. Let C be the m-cluster category of type A n , D n , Ãn or Dn . Let α be an m-ear. Let X α be the m-rigid object associated with α.

Let U = {Y ∈ C, Ext i C (X α , Y ) = 0}.
Let C be the Iyama-Yoshino reduction of C, i.e. C = U/(X α ). Then, we have the following equivalence of categories :

C C (m) Q/α
where Q/α is the quiver obtained from Q by removing α and all its incident arrows.

Finally, as Buan and Thomas showed that there was a compatibility between mutation of m-cluster-tilting objects and colored quivers, we show that the ip of an (m + 2)-angulation and mutation of m-cluster-tilting object is compatible.

This was not obvious. Indeed, a colored quiver is not associated with a unique m-cluster-tilting object (or (m + 2)-angulation, as we will see at theorem 2.1.8). There can be several m-cluster-tilting objects associated with the same colored quiver. Then two compatibilities did not imply directly the third.

Theorem 2.1.7. Let X be an m-cluster-tilting object in C. Let ∆ X be its associated (m + 2)-angulation. Let µ i be the ip at the arc α i in ∆ X as well as the mutation of the m-cluster-tilting object X at summand i. Then we have:

µ i (∆ X ) = ∆ µ i (X)
Theorem 2.1.8. There is a bijection between (m + 2)-angulations and m-clustertilting objects.

Th corollary of the theorem is that the quivers of an (m + 2)-angulation and of an m-cluster-tilting object coincide.

Theorem 2.1.9. Let ∆ be an (m+2)-angulation. Let Q ∆ be the associated coloured quiver. Let X ∆ be the m-cluster-tilting object associated with ∆, and let Q X ∆ be the quiver associated with X ∆ in the sense of Buan and Thomas in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF]. Then

Q ∆ = Q X ∆
We also work on a deeper study of case Dn which had not been treated yet: we give the admissible arcs corresponding to projective modules, injective modules and regular modules. We give an isomorphism between regular modules and some particular m-diagonals in the geometric realization.

On the homotopy category of an exact category

The end of the thesis is devoted to the construction of a model structure on exact categories. Indeed, Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF] showed that the quotient of an exact category E by a homotopy relation is a module category. The goal of this second part is to show that this quotient is actually the homotopy category of E. This permits to satisfy the hypotheses of the theorem of Quillen, and to provide an explicit homotopy category.

In the same way Palu in [START_REF] Palu | From triangulated categories to module categories via homotopical algebra[END_REF] showed that a triangulated category was equipped with a weak model structure, we show that a Frobenius category is equipped with a model structure, thus checks the hypotheses of the theorem of Quillen, and then we show that an exact category E (under some quite nice assumptions) is a prebration category, which still permits to apply the theorem of Quillen.

Notations

We denote by E an exact category. Let f : X → Y and g : A → B be two morphisms in E. We say that f g when, for any commutative square

A G G X B G G b b
Y there exists a morphism B → X such that both triangles commute. For a set A of morphisms of E, we call by A = {g, ∀f ∈ A, f g} and A = {f, ∀g ∈ A, f g}.

Results

This part of the thesis is devoted to the study of exact categories in a homotopical point of view. Indeed, we show the theorem of Quillen on homotopy theory for exact categories. Moreover, we show that there is a prebration structure on such categories, and that Frobenius categories are equipped with a model structure.

Lemma 2.2.1. Let E be a weakly idempotent complete exact category with enough injective and projective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective and projective objects, and stable under taking direct sums and summands. Let

prM = {X ∈ E, ∃M 1 , M 0 ∈ M, 0 → M 1 → M 0 → X → 0} and M = {X ∈ E, ∃M ∈ M, I ∈ Inj, 0 → M → I → X → 0}.
Then, there exist some prM-approximation. Theorem 2.2.2. Let E be a weakly idempotent complete exact category with enough injective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective objects, and stable under taking direct sums and summands.

Then there exist two classes of morphisms Fib and W (the same weak equivalences as in the Frobenius case) such that (E, Fib, W) has a prebration sutrcture in the sense of Anderson-Brown-Cisinski (see [START_REF] Radulescu-Banu | Ams-latex, 158 pages[END_REF] for a deeper study). All the objects are brant, and an object is cobrant if and only if it belongs to prM. Moreover, under the assumption that M contains the projective objects, there exist some cobrant replacements for any object of E. Corollary 2.2.3. Let E be a weakly idempotent complete exact category with enough injective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective objects, and stable under taking direct sums and summands. Suppose moreover that M contains the projective objects, and that there exist some M-approximation. Let

prM = {X ∈ E, ∃M 1 , M 0 ∈ M, 0 → M 1 → M 0 → X → 0} and M = {X ∈ E, ∃M ∈ M, I ∈ Inj, 0 → M → I → X → 0}. Let G : E → ModM X → E(-, X)/M
which induces the following equivalence of categories prM/ M mod M.

Then the homotopy category is equivalent to mod M. Corollary 2.2.4. We deduce from this that there is a homotopy calculus of fractions in the sense of Radulescu-Banu in [START_REF] Radulescu-Banu | Ams-latex, 158 pages[END_REF].

The theorem 2.2.2 not only shows the previous result. It shows that the objects of prM are exactly the brant and cobrant objects (as all objects are brant, we say that the subcatebory prM is the subcategory of cobrant objects).

Theorem 2.2.5. Each object of an exact category E satisfying the hypotheses of corollary 2.2.3 has a cobrant replacement. This means that prM-approximations are weak equivalences (they even are trivial brations).

We also characterize trivial cobrations (morphisms which satisfy a lift property on brations):

Lemma 2.2.6. Suppose that M → Y is a right M-approximation. A mor- phism f : X → Y is in (J ) if and only if it is a retract of the canonical injection X → X ⊕ M .
We nd a homotopy relation on morphisms of the category E, compatible with the usual notion of left and right homotopy.

Lemma 2.2.7. For two morphisms f and g from an object X to Y ,f and g are homotopic if and only if f -g factorizes through M ⊥ .

Next, from now we suppose that E is a Frobenius category. As it has much more properties, we can equip it with a stronger structure : a model structure.

Theorem 2.2.8. Let E be a Frobenius category. Let M be a full subcategory of E containing the injective objects, and assume that M is contravariantly nite. Then, there exist three classes of morphisms, Fib, Cof and W (the last one being the weak equivalences, which means the morphisms such that their image under the functor G is an isomorphism), forming a model structure on E. More precisely, all the objects are brant, and an object is cobrant if and only if it belongs to prM.

We characterize precisely weak equivalences, brations and trivial brations: Lemma 2.2.9. Let X, Y ∈ E. Let f : X → Y . Then f is a weak equivalence if and only if, for

X f G G ι X Y g I X u G G π X Z X X
we have

I X ⊕ Y (u g) G G Z ∈ M ⊥ and, for X f G G Y Z G G g y y P Y y y Y y y Y y y we have Z G G X ⊕ P Y ∈ M ⊥ . Lemma 2.2.10.
Suppose that E is a weakly idempotent complete category (see the article of Bühler, [START_REF] Bühler | Exact categories[END_REF]). Then f is a bration if and only if f is a deation, and g ∈ M ⊥ , where g is a cone of f , dened by

X f G G ι X Y g I X u G G π M Z X X .
Note that all these results are compatible by studying the stable category, which is triangulated (it means that, stabilizing the Frobenius category E, we nd back the results of Palu in [START_REF] Palu | From triangulated categories to module categories via homotopical algebra[END_REF]).

Chapter 3 Preliminaries

Triangulated categories

Triangulated categories behave like exact categories. Triangles are the equivalent for short exact sequences in exact categories. For a gentle introduction to triangulated categories, see the book of Holm and Jorgensen [START_REF] Holm | Triangulated categories: denitions, properties, and examples[END_REF]. For a deeper study of triangulated categories, see the book of Zimmermann, [START_REF] Zimmermann | Representation theory[END_REF].

Axioms of a triangulated category

Denition 3.1.1. Let C be an additive category. Let Σ be an automorphism of C, called the suspension functor. We call by triangles the diagrams of the form X → Y → Z → ΣX, stables under isomorphisms. Then C, endowed with such triangles is triangulated if it satises the following axioms : TR 1 For any morphism X → Y , there exists a triangle X → Y → Z → ΣX.

For any XinC, the diagram

X 1 G G X G G 0 G G ΣX is a triangle. TR 2 The diagram X f G G Y g G G Z h G G ΣX is a triangle is and only if the diagram Y g G G Z h G G ΣX -Σf G G ΣY . TR 3 For any triangles X f G G Y g G G Z h G G ΣX and X f G G Y g G G Z h G G ΣX , for any commutative square X f G G α Y β X f G G Y , 21 there exists γ : Z → Z such that X f G G α Y g G G β Z h G G γ ΣX Σα X f G G Y g G G Z h G G ΣX
is a morphism of triangle. TR 4 Suppose that there exist three triangles

X a G G Y b G G Z c G G ΣX , Y α G G Z β G G X γ G G ΣY and X µ G G Z ν G G Y ξ G G ΣX such that µ = α • a.
Then there exists a triangle

Z d G G Y e G G X f G G ΣZ such that                d • b = ν • α c = ξ • d b[1] • γ = f a[1] • ξ = γ • e e • ν = β (3.1)
We can illustrate this axiom by the following diagram: if we have the full arrows, then we have the dotted arrows.

Σ -1 X Σ -1 γ Σ -1 X Σ -1 f X d G G Y α b G G Z d c G G ΣX X µ G G Z β ν G G Y e ξ G G ΣX X X
with all the squares commutative.

Remark 3.1.2. The last axiom is called the octahedral axiom. It can be formulated dierently. The following one illustrates well the name of the axiom:

Z d Õ Õ β % % Y e G G ξ ~X γ } } f Ò Ò X µ g g a G G Y b Õ Õ α y y Z c d y y
but here the triangles are quite dicult to see. Remark 3.1.3. The morphism in TR 3 is unique, up to non-unique isomorphism. This causes many problems we do not have in abelian categories.

Some properties of triangulated categories

Throughout this section, C is a triangulated category.

Lemma 3.1.4 (May's lemma). The axiom TR 3 is a consequence of the axioms TR 1, TR 2 and TR 4.

It is thought that axiom TR 4 is a consequence of TR1, TR 2 and TR 3, but the question is still without any answer.

Proposition 3.1.5. Let X f G G Y g G G Z h G G ΣX be a triangle. Then we have g • f = 0 and h • g = 0. Lemma 3.1.6 (Triangulated ve lemma). Let X f G G α Y g G G β Z h G G γ ΣX Σα X f G G Y g G G Z h G G ΣX
be a morphism of triangle. If two of the vertical arrows are isomorphisms, then the third one is one too.

We now tell a useful lemma about long exact sequences.

Lemma 3.1.7. Let X f G G Y g G G Z h G G ΣX be a triangle. Then, we have a long exact sequence:

• • • → C(U, Σ -1 Z) → C(U, X) → C(U, Y ) → C(U, Z) → C(U, ΣX) → • • •
for any U ∈ C. We also have

• • • → C(ΣX, U ) → C(Z, U ) → C(Y, U ) → C(X, U ) → C(Σ -1 X, U ) → • • •
for any U ∈ C.

Examples of triangulated categories

Now, the category C is not necessarily triangulated anymore.

Derived categories

Historically, derived categories are the rst example of triangulated categories, due to Verdier in his posthumous printed thesis [START_REF] Verdier | Des catégories dérivées des catégories abéliennes[END_REF]. Here, we dene derived categories. For a nive lecture note, see [Kel96]. For a deep study of derived categories, see [START_REF] Zimmermann | Representation theory[END_REF]. Denition 3.1.8. Let C be an abelian category (we often study mod A, which is the category of nitely generated right modules over a nite dimensional k-algebra, where k is an algebraically closed eld). We introduce C(C) the new category of complexes, given by: Objects: Complexes of objects of C. Morphisms: morphisms of degree 0, which commute with the dierential. Natural composition.

From this category, we dene another category, the homotopy category K(C).

Denition 3.1.9. Let f : M → N be a morphism of complexes. The morphisms f is said to be null-homotopic if there exists a morphism h : where f ∈ K(X , Y ), and w is a quasi-isomorphism. Two such diagrams are called equivalent when there exists a commutative diagram:

M → N , of degree -1, such that f = d • h + h • d,
X } } 3 3 X Z y y o o G G Y X a a
The composition of two diagrams is given by the following diagram:

X 4 4 } } X ~4 4 X | | 2 2 X Y Z
Theorem 3.1.11 (Verdier,[START_REF] Verdier | Des catégories dérivées des catégories abéliennes[END_REF]). Let C be an additive category. Then the derived category D(C) of C is triangulated.

For more details, see the excellent book of Alexander Zimmermann, [START_REF] Zimmermann | Representation theory[END_REF].

The stable category of a Frobenius category

We dene at section 3.6 what a Frobenius category is. In his book [START_REF] Happel | Triangulated categories in the representation theory of nite-dimensional algebras[END_REF], Happel discovered a new example of triangulated categories. Denition 3.1.12. Let E be a Frobenius category. For any two objects A and B, we denote by I(A, B) the subgroup of morphisms from A to B which factors through an injective object.

The stable category E, associated with E, is the category whose objects are the same as the ones of E, and the set of morphisms, is

E(A, B) = E(A, B)/I(A, B).
The following theorem is due to Happel.

Theorem 3.1.13 (Happel,[START_REF] Happel | Triangulated categories in the representation theory of nite-dimensional algebras[END_REF]). The stable category of a Frobenius category is triangulated, with suspension functor the object ΣA appearing in the triple

A I A ΣA.
Remark 3.1.14. The homotopy category, as above, is the stable category of a Frobenius category. Then it is automatically a triangulated category.

Higher cluster categories

In this thesis, we are interested in higher cluster categories. Cluster categories have been dened in [BMR + 06], in order to categorify the concept of cluster algebras, Keller in [START_REF] Keller | On triangulated orbit categories[END_REF] studied these and also higher cluster categories. For a survey of the two next section, see the article of Buan, [START_REF] Buan | An introduction to higher cluster categories[END_REF].

Denition of higher cluster categories

We recall the denition of a cluster category: Consider H a basic hereditary nite dimensional algebra over an algebraically closed eld k. Let D b (H) be the bounded derived category of H. For convenience, we denote by [1] the shift functor of D b (H), and τ the Auslander-Reiten translation on the derived category. We refer to [START_REF] Zimmermann | Representation theory[END_REF] for a deeper study of derived categories. The cluster category associated with H is the orbit category

D b (H)/τ -1 [1].
Denition 3.2.1. Let H be a basic hereditary nite dimensional algebra over an algebraically closed eld k. Let D b (H) be the bounded derived category of H. Then the m-cluster category associated with H of degree m is dened by

C (m) H = D b (H)/τ -1 [m]
where [m] is the functor [1] repeated m times. Theorem 3.2.2 (Keller,[START_REF] Keller | On triangulated orbit categories[END_REF]). The higher cluster category is triangulated. Theorem 3.2.3 (Buan, Marsh, Reineke, Reiten, Todorov, [BMR + 06]). The higher cluster category is Krull-Schmidt.

Example of a 2-cluster category

We take this example from Buan, in [START_REF] Buan | An introduction to higher cluster categories[END_REF]. Let Q be the quiver

1 G G 2 3 o o G G 4 .
Let A be the path algebra kQ. We draw the innite Auslander-Reiten quiver of Q:

I 1 [-1] 4 4 P 1 1 1 I 4 0 0 P 4 [1] 5 5 I 2 [-1] X X 6 6 P 2 1 1 c c M 1 1 c c I 3 3 3 a a P 3 [1] • • • I 3 [-1] 4 4 P 3 c c 1 1 I 2 d d 0 0 P 2 [1] Y Y 5 5 • • • I 4 [-1] X X P 4 c c N c c I 1 a a P 1 [1]
Then, the 2-cluster category of Q consists in taking each object and the rst copy (rst shift) of each object. Then, it is composed of 24 indecomposable objects (10 plus 10 shifts plus one additional copy of the projective objects shifted twice).

The Auslander-Reiten quiver of the higher cluster category is thus cyclic:

P 1 1 1 I 4 0 0 P 4 [1] 5 5 Y [1] 4 4 I 1 [1] 4 4 P 1 [2]
3 3

P 1 P 2 1 1 c c M 1 1 c c I 3 3 3 a a P 3 [1] 5 5 Y Y I 2 [1] 4 4 `P2 [2] 5 5 Y Y P 2 1 1 c c P 3 c c 1 1 I 2 d d 0 0 P 2 [1] Y Y 5 5 X[1] 4 4 `I3 [1] 4 4 `P3 [2]
3 3

a a P 3 P 4 c c N c c I 1 a a P 1 [1] Y Y I 4 [1] `P4 [2] Y Y P 4 c c
The left and right most objects are identied.

Cluster-tilting subcategories

In triangulated categories, there are some interesting objects or subcategories, which reveal the category, and helps to understand it: the cluster-tilting objects (or subcategories). In this section, we present this.

Denition of cluster-tilting categories

Denition 3.3.1. Let k be a eld, and let C be a triangulated k-linear category with split idempotents and a suspension functor [1]. We suppose that C is Homnite, and admits a Serre functor S. Then C is said to be (m + 1)-Calabi-Yau if there exists an isomorphism

[m + 1] S.
We now dene m-cluster-tilting subcategory. Denition 3.3.2. Let T be a subcategory of C. We say that T is an m-clustertilting subcategory if it satises the following:

The category T is a k-linear subcategory.

The functors C(-, X)|T and C(X, -)|T are nitely generated for any X ∈ C. For all i ∈ {1, • • • , m}, and any T, T ∈ T , we have Ext i (T, T ) = 0. For X ∈ C, if Ext i (T, X) = 0 for all i ∈ {1, • • • , m} and any T ∈ T , then X ∈ T . Denition 3.3.3. Let C be an m-cluster category. An object T of C is called rigid if

Ext i C (T, T ) = 0
for all i ∈ {1, • • • , m}. An m-cluster-tilting object is a rigid object T satisfying moreover the following property:

∀X ∈ C, Ext i C (T, X) = 0 ∀i ∈ {1, • • • , m} =⇒ X ∈ addT.
Remark 3.3.4. We note that, if T ∈ C, then T is rigid if and only if the above implication is an equivalence. Zhu in [START_REF] Zhu | Generalized cluster complexes via quiver representations[END_REF] and Wraalsen in [Wr a09] showed independently that maximal rigid objects and m-cluster-tilting objects coincide. If A is a nite dimensional hereditary algebra containing exactly n simple modules up to isomorphism, they also showed that a rigid object T is m-cluster-tilting if and only if it has n isomorphism classes of indecomposable direct summands.

3.3.2 Some properties of m-cluster-tilting objects Theorem 3.3.5 (Wraalsen, Zhou-Zhu, [Wr a09], [START_REF] Zhou | Cluster combinatorics of d-cluster categories[END_REF]). Let T = ⊕T k be an mcluster-tilting object. We denote by B k = T /T k an almost m-cluster-tilting object.

The objects X such that B k ⊕X is an m-cluster-tilting object are called complement to B k . Then B k has m + 1 complements. Denition 3.3.6. Let T = ⊕T k be an m-cluster-tilting object in a triangulated category C. Then the map f :

T k → B k is a minimal left add B k -approximation of T k if: B k ∈ add B k Any map T k → B, B ∈ add B k factors through f . If, for any g : B k → B k , g • f = f ,
then the map g is an isomorphism. By TR 1 in the category C, we have the following triangle

T k → B k → T * k → T k [1].
This triangle is called an exchange triangle. Theorem 3.3.7 (Iyama-Yoshino, [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF]). The m + 1 complements found in the previous theorem appear in the m + 1 exchange triangles. They are exactly the T (c) k for c ∈ {0, • • • , m} occurring in the exchange triangles

T (c) k → B (c) k → T (c+1) k → T (c) k [1].
Here we only cite a very useful theorem of Keller and Reiten in [START_REF] Keller | Acyclic Calabi-Yau categories[END_REF], we will use all throughout the rst part of the thesis. Theorem 3.3.8. [KR08, Theorem 4.2] Let C be a Hom-nite algebraic (m + 1)-Calabi-Yau category. Let T be an m-cluster-tilting object in C, such that ∀i ∈ {1, • • • , m} Ext -i C (T, T ) = 0 and End C (T ) KQ T , for some quiver Q T . Then C C m Q T . Remark 3.3.9. These categories from theorem 3.3.8 are Iyama-Yoshino reductions. They are algebraic (see the article of Buan, Iyama, Reiten and Scott [BIRS09, Theorem I.1.8] for example). Remark 3.3.10. In his paper [START_REF] Keller | On triangulated orbit categories[END_REF], Keller has shown that orbit categories are also algebraic.

Example

We still take the previous example due to Buan

Quivers and colored quivers

Before dening colored quivers, let us introduce the mutation of a classic Gabriel quiver. Denition 3.4.1. Let Q be a quiver, it means a set of vertices Q 0 , a set of arrows Q 1 and two maps s and t from Q 1 to Q 0 , which associate respectively with an arrow its source and its target. Then the mutation at vertex i of the quiver Q, denoted as µ i (Q) is given by: Replace each j → i → k by an arrow j → k. Reverse all arrow coming from or going to i. Remove all 2-cycles.

Denition of a colored quiver

A deeper study on colored quivers can be found in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF] written by Buan and Thomas.

Denition 3.4.2. Let m, n be positive integers. An m-colored quiver consists of:

A set Q 0 of n vertices.

A set Q 1 of arrows. Two maps s, t : Q 1 → Q 0 which associate with each arrow respectively the source and the target.

A map Q 1 → {1, • • • , m} which associates with each arrow a color c ∈ {1, • • • , m}.
We write the color above the arrow. The term q (c) ij corresponds to the number of arrows from i to j of color c.

For the study, we ask our colored quivers to satisfy the following properties:

No loops which means that q

(c) ii = 0 for all c ∈ {1, • • • , m} and i ∈ {1, • • • , n}. Monochromaticity which means that if q (c) ij = 0, then q (c ) ij = 0 for all c = c. Skew-symmetry which means that q (c) ij = q (m-c) ji .
We now dene the mutation of colored quivers. Denition 3.4.3. [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF] Let Q be a colored quiver, and let k be a vertex of Q. We dene the new quiver µ k (Q) with the same vertices, and the new number of arrows q(c) ij given by:

q(c) ij =      q (c+1) ij if j = k q (c-1) ij if i = k max{0, q (c) ij -t =c q (t) ij + (q (c) ik -q (c-1) ik )q (0) kj + q (m) ik (q (c) kj -q (c+1) kj
)} else.

In section 3.4.3, an example is shown.

Main theorem about colored quivers

In this section, if T is an m-cluster-tilting object, we dene T , the m-clustertilting object which is obtained by mutation of T . Unfortunately, if Q T is the Gabriel quiver associated with T , for m > 1, there does not exist a quiver mutation µ j such that Q T = µ j (Q T ). Then, to remedy this lack, Buan and Thomas in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF] built a new quiver from T , which is denoted as the colored quiver associated with T . Proposition 3.4.4. Mutating a colored quiver as in denition 3.4.3 is equivalent to the following procedure:

1. For any i (c) G G k (0) G G j , if i = j and c is an integer in {0, • • • , m}, then draw an arrow i (c) G G j and an arrow j (m-c) G G i. 2. If condition 2 of monochromaticity in the restriction of colored quivers is not satised anymore from one vertex i to one vertex j, then remove the same number of arrows of each colour, in order to restore the condition. 3. For any arrow i (c) G G k , add 1 to the color c, and for any arrow k (c) G G j , subtract 1 to the color c.

We recall from Buan and Thomas in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF] that there are exchange triangles

T (c) k f (c) k G G B (c) k g (c+1) k G G T (c+1) k h (c+1) k G G T (c) k [1] (3.2)
With any m-cluster-tilting object T in the m-cluster category, we associate a corresponding colored quiver Q T such that:

1. The vertices of Q T are the integers from 1 to n where n is the number of indecomposable summands of T .

The number q

(c) ij is the multiplicity of T j in B (c) i in the exchange triangle (2.13).

We now state the main theorem about colored quivers and m-cluster-tilting objects a proof of which can be found in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF]:

Theorem 3.4.5. [BT09, Theorem 2.1] Let T = n i=1 T i and T = T /T k T (1)
k be m-cluster-tilting objects, where there is an exchange triangle

T k → B (0) k → T (1) k → T k [1]. Then Q T = µ k (Q T ).
Remark 3.4.6. In particular, the coloured quiver Q T only depends on the colored quiver Q T .

Example of the mutation of colored quivers

We give here a simple example of the mutation of a colored quiver:

4 (3) 0 0 2 (3) Ð Ð 1 (0) (0) d d (0) 0 0 (0) Ð Ð 5 (3) d d 3 (3) mutation at 3 -------→ 4 (3) 0 0 2 (3) Ð Ð 1 (0) (0) d d (1) 0 0 (0) Ð Ð 5 (3) d d

3

(2)

3.5 Geometric realization of cluster categories of type A, D, Ã and marked surfaces

This section is more an explanation of what have been done until now than a real preliminary background, necessary to understand the results of this thesis.

Researchers have tried to give a geometric realization of cluster categories, which could explain some curious results such as how the morphisms in the Auslander-Reiten quiver are set, and give a visual understanding of these categories. We now consider the cases A, D, and the very general case of marked surfaces.

Case A

This is the rst case studied, by Caldero, Chapoton and Schier in [START_REF] Caldero | Quivers with relations arising from clusters (A n case)[END_REF]. Let Q be a quiver of type A n . Let P be the regular (n + 3)-gon. Then with each diagonal (we keep the vocabulary of [START_REF] Caldero | Quivers with relations arising from clusters (A n case)[END_REF]) of a triangulation of P , we can associate a vertex of Q. We can draw arrows between diagonals (indeed, we draw an arrow from one diagonal to another if they are consecutive clockwise). This makes the triangulation correspond to a quiver.

We introduce the ip of a triangulation in the following way: let T be a triangulation. Let i be a diagonal of T . Then, the new triangulation T = µ i (T ), ipped at vertex i, is dened in the following way: T contains all the diagonals of T except i. The diagonal i obtained by removing i and taking the other diagonal of the quadrilateral created by removing i completes the triangulation T . In this situation, Caldero, Chapoton and Schier in [START_REF] Caldero | Quivers with relations arising from clusters (A n case)[END_REF] show the following lemma: Lemma 3.5.1. Let Q be a quiver of type A n . Let P be the regular (n + 3)gon. Let T be the triangulation corresponding to Q in P . Let i be a vertex of Q, corresponding to a diagonal of T . Then,

µ i (Q) = µ i (T ).

Geometric realization of cluster categories

Continuing from this, Caldero Chapoton and Schier built a new category where the objects arise from diagonals in P and where the irreducible morphisms arise from the arrows between diagonals, with morphisms arising from compositions of arrows between the objects, modulo mesh relations. They showed the following theorem: Theorem 3.5.2 (Caldero-Chapoton-Schier, Theorem 2.14, [START_REF] Caldero | Quivers with relations arising from clusters (A n case)[END_REF]). There is an equivalence of categories between the cluster category of type A and the category geometrically built from diagonals.

They even showed more than this: particular objects such as injective, projective, corresponded to special diagonals in the polygon P .

Case D

After this, Schier in [START_REF] Schier | A geometric model for cluster categories of type D n[END_REF] provided a model for type D n , for n ≥ 4. Now, P is a polygon with n vertices, and one puncture in its center. Schier denes what is an admissible diagonal, and elementary moves between two diagonals. In the same way as in case A n , he shows the following lemma: Lemma 3.5.3. Let Q be a quiver of type D n . Let P be the regular n-gon with one puncture. Let T be the triangulation corresponding to Q in P . Let i be a vertex of Q, corresponding to a diagonal of T . Then,

µ i (Q) = µ i (T ).
Finally, by building a category whose objects arise from the diagonals, and the morphisms the composition of elementary moves, modulo mesh relations, he shows the following theorem: Theorem 3.5.4 (Schier, Theorem 3.3, [START_REF] Schier | A geometric model for cluster categories of type D n[END_REF]). There is an equivalence of category between the category of diagonals, and the cluster category of type D.

Case

Ã

This case has been treated by Torkildsen in [START_REF] André | A geometric realization of the m cluster category of type[END_REF]. He shows similar results, taking into account the dierent form of the Auslander-Reiten quiver of Q (presence of tubes, which must correspond to regular modules). For a deeper study of these Euclidean quivers, see the book [START_REF] Simson | Elements of the representation theory of associative algebras[END_REF].

The rst part of his thesis consists in nding these results for case Ãn , and to show that cutting along an arc corresponds to apply Iyama-Yoshino reduction.

General case of the marked surface

Brüstle and Zhang in [START_REF] Brüstle | On the cluster category of a marked surface without punctures[END_REF], introduce the cluster category associated with a marked surface. Marsh and Palu in [START_REF] Marsh | Coloured quivers for rigid objects and partial triangulations: the unpunctured case[END_REF] generalized the results given above for these generalized cluster categories. Indeed, they use a theorem of Keller and Yang (see [START_REF] Keller | Derived equivalences from mutations of quivers with potential[END_REF]). Before giving it, we need to see the denition of the generalized cluster category. Denition 3.5.5. Let Q be a quiver, associated with a potential W (a linear combination of cycles in Q, up to cyclic permutation). We dene the Ginzburg dg algebra Γ(Q, W ) as follows: First dene a graded quiver Q which arrows are dened as follows:

The arrows of Q, of degree 0 For each arrow α : i → j, add an arrow α * : j → i of degree -1.

For each vertex i of Q, add a loop e * i at i of degree -2. The Ginzburg dg algebra is the path algebra of the graded quiver.

We now dene the cluster category for an arbitrary quiver Q (without assuming acyclicity). Denition 3.5.6 (Amiot). The generalized cluster category

C Q associated with Q is the Verdier localization perΓ(Q, W )/D b (Γ(Q, W )).
Amiot has shown that this category corresponds in the acyclic case to the classical cluster category. One can dene a quiver with potential from a triangulation of a marked surface. Then, Keller and Yang have shown this powerful theorem: Theorem 3.5.7 (Keller-Yang, [START_REF] Keller | Derived equivalences from mutations of quivers with potential[END_REF]). Let T be a triangulation, and (Q T , W T ) be the associated quiver with potential. Then, for any arc i of T , if µ i (T ) is the new triangulation obtained from ipping the arc i, there is a triangle equivalence: Then we have the equivalence

C Q T C Q µ i (T ) .
This theorem permits to associate with a surface, a unique cluster category. Indeed, Marsh and Palu show the following result: Theorem 3.5.8. There is a bijection between the arcs of the triangulation of a marked surface, and the cluster-tilting objects of the associated cluster category. Moreover, cutting along an arc corresponds to Iyama-Yoshino reduction. Finally, the quiver mutation is compatible with ip, as in the A n and D n cases.

Exact and Frobenius categories

In this section, we introduce exact and Frobenius categories as in Bühler (see [START_REF] Bühler | Exact categories[END_REF]). Exact categories have been invented by Quillen in [START_REF] Quillen | Higher algebraic K-theory: I [mr0338129[END_REF] in order to generalize the notion of short exact sequences in abelian categories.

Axioms of exact categories

In this section, we choose as a reference the article of Bühler [START_REF] Bühler | Exact categories[END_REF].

Denition 3.6.1. Let E be an additive category. A short exact sequence (i, p) in E is a pair of composable morphisms A i G G A p G G A such that i is a kernel of p and p is a cokernel of i. If a class C of short exact sequences on E is xed, an ination is a morphism i for which there exists a morphism p such that (i, p) ∈ C. Deations are dened dually.

An exact structure on E is a class C of short exact sequences, which is closed under isomorphisms and satises the following axioms: E0 The identity morphism is an ination. E0' The identity morphism is a deation. E1 The class of inations is closed under composition. E1' The class of deations is closed under composition. E2 The push-out of an ination along an arbitrary morphism exists and yields an ination. E2' The pull-back of a deation along an arbitrary morphism exists and yields a deation. Remark 3.6.2 (Vocabulary). Short exact sequences are also called kernel-cokernel pairs. Moreover, there exists a further pair of notions for ination and deation, which is admissible monic and admissible epic. Here we prefer to use the notations of Keller. Remark 3.6.3. In order to be clear, we note ination with an arrow and a deation with an arrow .

Some properties of exact categories

Let E be an exact category. To have proofs of these properties, we refer to Bühler ([Büh10]) Proposition 3.6.4. For any objects A and B, the diagram

A G G ( 1 0 ) G G A ⊕ B (0 1) G G G G B
is a short exact sequence. Proposition 3.6.5. The direct sum of two short exact sequences is short exact. Proposition 3.6.6. Consider the following commutative square:

A G G i G G f B f A G G i G G B .
Suppose that the horizontal arrows are inations. Then the following assertions are equivalent: i The square is a push-out ii The sequence

A G G ( i -f ) G G B ⊕ A (f i ) G G G G B
is short exact. iii The square is both a push-out and a pull-back. iv The square makes part of a commutative diagram

A G G i G G f B f p G G G G C A G G i G G B p G G G G C
where the rows are short exact. Proposition 3.6.7. The pull-back of an ination along a deation yields an ination.

Examples of exact categories

1. We remark that any abelian category is in particular exact.

2. Any full, closed under extension subcategory of an abelian category is exact.

3. If C is a category which has nite limits, is cartesian closed, and has a subobject classier (it is a morphism such that any monomorphism in the topos is the pullback of this morphism along a unique morphism), the category C is called a topos, then C is an exact category.

The particular case of Frobenius categories

Now we recall the denition of a Frobenius category. For further information, see the book of Happel, [START_REF] Happel | Triangulated categories in the representation theory of nite-dimensional algebras[END_REF]. Denition 3.6.8. Let E be an additive category.

An object P ∈ E is said to be projective if, for any deation π : A B, and any morphism f : P → B, there exists a lift g : P → A such that π • g = f .

An object I ∈ E is said to be injective if, for any ination ι : A B, and any morphism f : A → I, there exists a lift g : B → I such that ι • g = f . Denition 3.6.9. An additive category E has enough projective objects if, for any B ∈ E, there exists a deation π : P B with P projective.

An additive category E has enough injective objects if, for any A ∈ E, there exists an ination ι : A I with I injective. Denition 3.6.10. An exact category E is said to be Frobenius if is has enough projective and injective objects, and if the projective coincide with the injective.

Weakly idempotent complete categories

We now see a particularity of some categories, which we will use for exact categories (See [Büh10, Section 7]). This type of categories permits to build easily short exact sequences. Denition 3.7.1. In a category E (not necessarily exact), a morphism r : A → B is called a retraction if there exists a section s : B → A such that r • s = 1 B . Dually, a morphism c : A → B is a coretraction if there exists a section s :→ A such that s • c = 1. Proposition 3.7.2. If E is an additive category (not necessarily exact), the following are equivalent:

(i) Any coretration has a cokernel (ii) Any retraction has a kernel Denition 3.7.3. If one of the conditions of the previous lemma holds, then the category E is said to be weakly idempotent complete.

We now have the following useful proposition: Proposition 3.7.4. Let E be an exact category. Then the following are equivalent: (i) The category E is weakly idempotent complete.

(ii) For any two morphisms f : A → B and g : B → C, if g • f : A C is a deation then g is a deation.

Quotient of exact categories by cluster-tilting subcategories as module categories

In 2007, Buan, Marsh and Reiten in [START_REF] Bakke Buan | Cluster-tilted algebras[END_REF] showed that we can pass from triangulated categories to abelian categories by factoring out any cluster-tilting subcategory. The aim of this section is to present the analogous result shown by Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF] for exact categories. In fact, the main theom is more interesting than that, and this thesis uses it a lot in order to prove the theorem of Quillen in some cases (see the dedicated section on preliminaries).

Theorem 3.8.1 ( [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF]). Let E be an exact category. Let M be a full rigid subcategory which contains the injective objects. Let

prM = {X ∈ E, ∃M 1 , M 0 ∈ M, 0 → M 1 → M 0 → X → 0} and M = {X ∈ E, ∃M ∈ M, I ∈ Inj, 0 → M → I → X → 0}.
These are short exact sequences. Let G be the following functor:

G : E → ModM X → E(-, X)/M
Then, the functor G induces the following equivalence: prM/ M modM.

They show analogously:

Theorem 3.8.2 ( [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF]). Let E be an exact category. Let M be a full rigid subcategory which contains the projective objects. Let

coprM = {X ∈ E, ∃M 1 , M 0 ∈ M, 0 → X → M 0 → M 1 → 0} and ΩM = {X ∈ E, ∃M ∈ M, P ∈ Proj, 0 → X → P → X → M }.
These are short exact sequences. Let H be the following functor:

H : E → ModM X → Ext 1 E (-, X)/M
Then, the functor H induces the following equivalence:

coprM/ΩM modM.

This has important consequences:

Corollary 3.8.3. In the rst case, if we denote by M ⊥ the set {X ∈ E, E(M, X) = 0}, then we have that

M = M ⊥ ∩ prM.
In the dual case, if M is rigid and contravariantly nite (it means that there exist some M-approximations), then the quotient category coprM/ΩM is abelian.

Localization of triangulated categories

In their article [START_REF] Aslak | From triangulated categories to module categories via localisation[END_REF], Buan and Marsh built an equivalence of categories between a localization of a triangulated category, and a module category. This is what we generalize to Frobenius categories, and then exact categories. Theorem 3.9.1 (Buan-Marsh, [START_REF] Aslak | From triangulated categories to module categories via localisation[END_REF]). Let C be a skeletally small, Hom-nite, Krull-Schmidt triangulated category with a rigid object T . Let χ T be the full subcategory of C whose objects are the objects of C having no non-zero maps from T . Let S be the class of maps f that can be completed to a triangle f, g, h in which g, h ∈ (χ T ). Then, the induced functor G : C S → mod End C (T ) op is an equivalence.

Model categories

Model categories have been invented in order to give a reasonable context to do homotopy theory. It has been developed by Quillen in [START_REF] Daniel | Homotopical algebra[END_REF]. For a gentle introduction, see [START_REF] Dwyer | ski. Homotopy theories and model categories[END_REF], or for a deeper study, see the book of Hovey, [START_REF] Hovey | Model categories[END_REF].

Denition of a model category

We start with a preliminary denition. Denition 3.10.1. Given a commutative diagram

A a G G h X f B b G G Y a lift of the diagram is a morphism g : B → X such that g • h = a and f • g = b
(we say that both triangles commute). Denition 3.10.2. Let C be a category. Let W be the class of weak equivalences, Fib be the class of brations, and Cof be the class of cobrations. We call an element of Fib ∩ W (respectively an element of Cof ∩ W) an acyclic bration (respectively an acyclic cobration). The category C is said to have a model structure if the triple (W, Fib, Cof) and C satisfy the following properties: MC 1 Finite limits and colimits exist in C. MC 2 W has the "two-out-of-three" property, it means that, for two composable maps g and f , if two of the three maps f , g or gf are in W, so is the third. MC 3 The three sets Fib, Cof and W are stable under retracts. MC 4 For a commutative diagram

A a G G h X f B b G G b b
Y a lift exists if: either h is a cobration and f is an acyclic bration (meaning that f ∈ Fib ∩ W) or h is an acyclic cobration and f is a bration. MC 5 Any morphism can be factored in two ways: a cobration followed by an acyclic bration, and an acyclic cobration followed by a bration. Remark 3.10.3. Here, we observe that this is in fact the denition of a closed model category. Indeed, all throughout this thesis, we only work with closed model categories. Proposition 3.10.4. If C is a model category, then C op is also a model category, with brations and cobrations exchanged. Denition 3.10.5. An object X of a model category C is said to be brant if the morphism X → * is a bration (where * is the terminal object of C, existing from MC 1). It is said to be cobrant if the morphism ∅ → X is a cobration (where ∅ is the initial object of C).

Some properties of model categories

The following lemma is actually a corollary of the retract argument, which we do not need here. Lemma 3.10.6. If C is a model category, then cobrations (and acyclic cobrations) are closed under pushouts. That is to say, if we have a diagram

A a G G h X f B b G G Y ,
if h is a cobration (respectively an acyclic cobration), then f is one too. Dually, brations (and acyclic brations) are closed under pullbacks.

In his book [START_REF] Hovey | Model categories[END_REF], Hovey showed a very useful theorem about model categories. He has shown an equivalent denition for categories with all small limits and colimits.

Theorem 3.10.7 [START_REF] Hovey | Model categories[END_REF]). Let C be a category. Let W, Fib, Cof be three classes of morphisms, where Fib (respectively Cof) is obtained from an orthogonal relation on a set of morphisms J (respectively I). Then there is a model structure on C if and only if:

(i) The set of morphisms W has the "two-out-of-three" property, and is stable under retracts.

(ii) We have J ∩ W = I . (iii) We have (J ) ⊆ W ∩ (I ).
(iv) Any morphism can be factored through a weak cobration followed by a bration. Any morphism can be factored through a cobration followed by a weak bration. Remark 3.10.8. In his article, Hovey does not write the theorem in this way. As he uses the small object argument, he does not need the factorizations, but he needs that the domains of I (respectively J) are small relative to the class of colimits of elements of I (respectively J). It is this way he nds the factorizations.

We did not use it like this, because in some examples (for instance, see [START_REF] Burban | Cluster tilting for one-dimensional hypersurface singularities[END_REF]), we do not have all the colimits. Then, we nd directly the factorization without using the small object argument.

Examples of model categories

The category of topological spaces Top is equipped with a structure of model category with:

The morphism f is a weak equivalence if it is a weak homotopy equivalence.

The morphism f : X → Y is a cobration if it is a retract of a map X → Y ,
where Y is obtained from X by attaching cells.

The brations are Serre brations.

If C and D are model categories, we can construct C × D, the product category, which is, in the natural way, a model category.

Prebration and precobration categories

Some categories are not nice enough in order to build a model structure on them. For example, one can have cobrations, but no brations. Then, to deal with such situations, Anderson, Brown and Cisinski have given axioms for precobration categories, and prebration categories. This section is dedicated to this. A quick denition is given by Brown in [START_REF] Brown | Abstract homotopy theory and generalized sheaf cohomology[END_REF]. For a further study, see the book of Radulescu [START_REF] Radulescu-Banu | Ams-latex, 158 pages[END_REF].

Denition of prebration and precobration categories

Denition 3.11.1. An ABC prebration category (ABC stands for Anderson-Brown-Cisinski) consists of a category E, with two classes of maps, the brations Fib and the weak equivalences W satisfying the following axioms: F 1 E has a nal object which is brant. Fibrations are stable under composition.

All isomorphisms are weak equivalences, and all isomorphisms with brant codomain are trivial functions (brations which are also weak equivalences). F 2 W has the "two-out-of-three" property, meaning that, for two composable maps g and f , if two of the three maps f , g or gf are in W, so is the third. F 3 For any diagram

A a G G h X f B b G G Y
the pullback exists in E. Moreover, if f is a bration (respectively a trivial bration), then h is a bration (respectively a trivial bration). F 4 Any morphism f : A → B, with B brant, can be factored through a weak equivalence followed by a bration.

Dually, one can dene precobration categories.

Denition 3.11.2. An ABC precobration category consists of a category E, with two classes of maps, the cobrations Cof and the weak equivalences W satisfying the following axioms:

COF 1 E has an initial object which is cobrant. Cobrations are stable under composition. All isomorphisms are weak equivalences, and all isomorphisms with cobrant domain are trivial functions (cobrations which are also weak equivalences).

COF 2 W has the "two-out-of-three" property, meaning that, for two composable maps g and f , if two of the three maps f , g or gf are in W, so is the third. COF 3 For any diagram

A a G G h X f B b G G Y
the pushout exists in E. Moreover, if f is a cobration (respectively a trivial cobration), then h is a cobration (respectively a trivial cobration). COF 4 Any morphism f : A → B, with A cobrant, can be factored through a cobration followed by a weak equivalence.

Properties of such categories

In the following theorem, C does not need to be a prebration (or precobration) category.

Theorem 3.11.3 (Radulescu-Banu,[RB06], Theorem 6.4.2). If E, W is a category pair satisfying the following:

1. The two-of-the-three axiom.

For a diagram

A a G G h X f B b G G Y
the pullback exists in E. Moreover, if f is a weak equivalence, then h is a weak equivalence.

For any maps

A f G G g G G B t G G B with t ∈ W, and t • f = t • g, there exists t : A → A ∈ W such that f • t = g • t .
Then, we have the following results: 1. Each map h : A → B in the homotopy category can be written as a right fraction f • s -1 , with s ∈ W. 2. Two fractions f • s -1 and g • s -1 are equal in the homotopy category if and only if there exist weak equivalences s , t , as in the diagram below, and such that s

• s = t • t and f • s = g • t : A s } } t 3 3 A s ~f B B A g 2 2 t t t A B

The homotopy category of a model category

In this section, we explain the main theorem of Quillen on homotopy categories. He actually shows that the homotopy category of a nice category C is equivalent to the category of brant and cobrant objects quotiented by a homotopy relation. This result needs a litte bit of background on homotopy categories. For the reader who wants to know more, see [START_REF] Dwyer | ski. Homotopy theories and model categories[END_REF] of [START_REF] Hovey | Model categories[END_REF].

Denition of the homotopy category

This denition comes from Hovey in [START_REF] Hovey | Model categories[END_REF]. Denition 3.12.1. Let C be a category which contains a subcategory of weak equivalences, called W. Let us form the free category F C dened in the following way: the objects of F C are the same as the ones of C. A morphism of F C is a nite string of composable elementary morphism where the elementary morphisms are either morphisms of C or weak inverses of morphisms of W. The empty string is the identity, and the composition is dened naturally by the composition of strings.

Then the homotopy category Ho C of C is the quotient of F C by the relations (the notation dom f stands for the domain of f , and codom f stands for the codomain of f ):

1 A = (1 A ) ∀A ∈ C (f, g) = (g • f ) for any composable f, g 1 dom f = (f, f -1 ) for all f ∈ W 1 codom f = (f -1 , f ) for all f ∈ W.
At this moment, we do not know that Ho C is a category. Indeed, the class of morphisms is not a set. The theorem of Quillen will tell us that , if C is a model category, then Ho C is a category. If C is not a model category, it is necessary to nd a higher universe to consider Ho C as a category. We do not consider universe problems, and then we assume Ho C is a category. Denition 3.12.2. Let C be a model category. We call by C cf the full subcategory of C containing the objects which are both brant and cobrant. Theorem 3.12.3. There is an equivalence of categories

Ho C cf → Ho C.
Now we dene cylinder and path objects. Denition 3.12.4. Let X ∈ E. A cylinder object for X is a factorization of the morphism ∇ : X ⊕ X → X (which is the identity on each copy of X).

Let f, g : X → Y be two morphisms. A left homotopy from f to g is a morphism h : X → Y , where X is a cylinder object for X, such that h • ∇ = (f g).

Dually, we can dene path objects and right homotopies. Denition 3.12.5. Let

Y ∈ E. A path object for Y is a factorization of the morphism ∆ : Y → Y ⊕ Y (which is the identity on each copy of Y ). Let f, g : X → Y be two morphisms. A right homotopy from f to g is a morphism k : X → Y , where Y is a path object for Y , such that ∆ • k = f
g . Denition 3.12.6. Two morphisms f and g are said to be homotopic if they are both right and left homotopic. We note f ∼ g.

We say that a morphism f : A → B is a homotopy equivalence if there exists g : B → A such that g • f ∼ 1 and f • g ∼ 1.

Theorem of Quillen

Before claiming the theorem, we start with a useful lemma. Lemma 3.12.7. The homotopy relation on the morphisms of C cf is an equivalence relation compatible with composition. Thus, the category C cf / ∼ exists.

Moreover, weak equivalences are exactly homotopy equivalences. Theorem 3.12.8 (Quillen,[START_REF] Daniel | Homotopical algebra[END_REF]). Let C be a model category. Then there is an equivalence of categories

C cf / ∼ ∼ G G Ho C .
Isomorphisms in the homotopy category are exactly weak equivalences. Remark 3.12.9. In his proof in [START_REF] Hovey | Model categories[END_REF], Hovey only needs the existence of the second factorization on cobrant objects.

A weak model structure on triangulated categories

In his paper [START_REF] Palu | From triangulated categories to module categories via homotopical algebra[END_REF], Palu has shown that there was a weak model structure on triangulated categories, which satisfy the theorem of Quillen.

Theorem 3.13.1 (Palu,[START_REF] Palu | From triangulated categories to module categories via homotopical algebra[END_REF]). Let C be a triangulated category with a covariantly nite, rigid subcategory T . There is a left-weak model category structure on C. It means that C contains three classes of morphisms W, Fib, Cof such that:

1. Pullbacks of trivial brations along deations exist and are trivial brations.

If A is cobrant, then for any cobration

A → B, B is cobrant. If B is cobrant, then the inclusions A → A ⊕ B and A → B ⊕ A are cobrations. 3.
The class W has the "two-out-of-three" property. 4. The three classes W, Fib, Cof contain identity, are stable under retracts and composition. 5. We have W ∩ Cof ⊆ Fib and Cof ⊆ (W ∩ Fib) 6. Any morphism can be factored through a weak cobration followed by abration. Any morphism with cobrant domain can be factored through a cobration followed by a weak bration.

Some examples of such categories

As it has been said in the introduction, there are several examples of applications of this article. Denition 3.14.1. Let A be a nite-dimensional algebra over an algebraically closed eld K. Let Q = (Q 0 , Q 1 , s, t) be a Dynkin quiver. Let Q be the quiver obtained from Q by adding, for each arrow α of Q from i to j, an arrow α * from j to i. Let KQ be the path algebra over Q. Then the preprojective algebra associated with Q is dened as

Λ Q = KQ/I
where I is the ideal generated by the element

α∈Q 1 (α * • α -α • α * ).
Lemma 3.14.2. Let Q and Q be two quivers of the same Dynkin type. Then Λ Q is isomorphic to Λ Q .

Theorem 3.14.3. Let T be a maximal rigid Λ-module. Then the functor

Hom Λ (-, T ) : mod(Λ) → mod(End Λ (T ))
is fully faithful, and its image is the category of End Λ (T )-modules of projective dimension at most one.

Moreover, if T 1 and T 2 are two maximal rigid modules, then End Λ (T 1 ) and End Λ (T 2 ) are derived equivalent.

They also have shown similar results about more general quivers. Denition 3.14.4. Let Q be a nite connected quiver without oriented cycles and n vertices, and let Λ be its associated preprojective algebra. As KQ is a subalgebra of Λ, we can introduce π Q : mod(Λ) → mod(KQ) as the restriction functor.

Let M be a KQ-module. Then M is called terminal if the following hold:

M is preinjective If X is an indecomposable KQ-module, with Hom KQ (M, X) = 0, then X ∈ add M .
Any indecomposable injective KQ-module belongs to add M . Theorem 3.14.5. Let C M be the subcategory of all Λ-modules which image under π Q belongs to add M . Then the following holds:

The category C M is Frobenius with n indecomposable projectives. The stable category of C M is 2-Calabi-Yau.

If we consider the non-Frobenius case, Adachi in [START_REF] Adachi | τ -rigid-nite algebras with radical square zero[END_REF] has shown the following results: Theorem 3.14.6. Let Λ be a nite-dimensional algebra with radical square zero. Then the following are equivalent:

1. The algebra Λ is τ -rigid-nite 2. Any single subquiver of the separated quiver for Λ is a disjoint union of Dynkin quivers. Corollary 3.14.7. Under the same assumptions, if any indecomposable Λ-module is τ -rigid, then Λ is representation-nite.

2)-angulation

In this section we introduce the geometric realizations. We dene the ip of an (m + 2)-angulation, and build a colored quiver from an (m + 2)-angulation. The main theorem of the section, namely theorem 2.1.12, states the compatibility between the ip of an (m+2)-angulation and the mutation of the associated colored quiver.

Geometric realization and ips

Let n > 4 and m be a positive integer. Let P be an (n -2)m-gon with two central (m -1)-gons R and S inside of it (cf gure 1.1). We replace each vertex of R and S by a disk, which we henceforth call a thick vertex. If m = 1, then we consider an (n -2)-gon with two disks inside of it.

We now dene the notion of a tangent arc.

Denition 1.1.1. Consider one moment an arc starting at an arbitrary vertex of P , and ending at a thick vertex of P is called to be left tangent at R if this arc is C ∞ , tangent to a thick vertex, and if there exists a neighborhood of this arc such that the thick vertex of R is situated at the right of the arc. We similarly dene right tangency. Denition 1.1.2. Let us number the vertices of the polygon P from P 1 to P (n-2)m clockwise. Then, an admissible arc between P i and P j is dened in the following way: 1. If i = j, then an admissible arc is an oriented path from P i to P j , lying inside of P , which does not cross any of the two inner polygons, satisfying one of the following conditions: Either the arc crosses the space between both central polygons and cuts the gure into a km+1-gon and a k m+1-gon, for some k (k is entirely determined by k). This arc is of type 1.

Or, the arc is homotopic to the boundary path, and cuts the gure into a km-gon with both central polygons inside of it and a k m + 2-gon (where k is still entirely determined by k). This arc is of type 2. 2. If P i = P j , then an admissible arc is a path ending in P i , and the other end of the path is tangent to one of the thick vertices placed around R and S.

There are two more admissible arcs starting and ending at i, going around one of the inner polygons: the left loop and the right loop. They look the same on the picture, but they are labeled dierently, we will see later why. 3. Any arc being tangent to two disks, one arising from R, and one from S is admissible. Remark 1.1.3. We will often write i for P i . Denition 1.1.4. We denote by B ij the boundary path from i to j going clockwise around the boundary (where B ii denotes the trivial path at i). We call a "t-angle", a gure delimited by arcs and/or B ij for some distinct i and j and/or sides of the inner polygons R and S, where the total number of sides and arcs is t.

Note that we only consider unoriented arcs, the order of i and j does not matter. For convenience, we will nevertheless use the terminology "from i to j". Notation 1.1.5. We call a Dehn twist around R the action of rotating R. It means that if we consider an arc α hung to R, applying a Dehn twist of R makes α roll around R only.

Denition 1.1.6. An m-diagonal is an equivalence class of admissible arcs where the dierent classes are:

1. If i = j, type 1: If the arc cuts the gure into a km + 1-gon and a k m + 1gon, then a class up to homotopy of m-diagonals of type 1 is built for each value of k. 2. If i = j, type 2: If the arc cuts the gure into a km-gon with both central polygons inside of it and a k m + 2-gon, then a class up to homotopy of m-diagonals of type 2 is built for each value of k. 3. If i = j: a class contains all the images of Dehn twists of admissible arcs tangent to the left of a thick vertex of R plus the left loop described in the remark below. The other classes are given by all the images of Dehn twists of admissible arcs tangent to the right of a thick vertex. Similar classes are built in the same way for S. 4. Any admissible arc up to homotopy linking both central polygons form a class of m-diagonal. Remark 1.1.7. Let i be a vertex of P , and α be an arc tangent to the left of a thick vertex of R. Then the loop belonging to the class of α is the arc up to homotopy going around R without crossing α (and not homotopic to the boundary path). The same description can be done for right-tangency and S.

Note that there is only one loop for each class containing a tangent m-diagonal. Remark 1.1.8. If i = j, then, there exists an innity of classes of arcs tangents to the left or R. Indeed, the m-diagonals can roll around R and S, and this leads to two dierent classes of arcs. See gure 1.2 for an illustration. In gure 1.4, we can see an example of two arcs corresponding to the same m-diagonal, where we have applied a Dehn twist to one of the polygons. Denition 1.1.9. 1. If two arcs are not tangent to the same inner polygon, they are said to be noncrossing if their class under homotopy contains representatives which do not cross. If they both are tangent to the same inner polygon: If they have the same tangency (left for example), they are said to be noncrossing if their class under homotopy contain representatives which do not cross.

If not, replace one of them by a loop, and they are said to be noncrossing if their class under homotopy contain representatives which do not cross. Two m-diagonals linking both inner polygons are said to be noncrossing if their classes under homotopy contain representatives which do not cross.

2. An (m + 2)-angulation is a set of noncrossing m-diagonals, such that there exists representative admissible arcs which cut the gure into (m + 2)-angles. Such a choice of representatives is called a good set of representatives.

See gure 1.5 for examples of an (m + 2)-angulation.

Denition 1.1.10. For any m and n ≥ 4, we dene the initial (m + 2)-angulation in the following way: From the vertex 1 of P , we draw:

1. An arc tangent to the right of the disk composing a thick vertex R i of R.

Figure 1.5 On the left, a collection of arcs that is not a good set of representatives since it does not cut the gure into (m + 2)-angles. An example of a good set of representatives is shown on the right.

2. A left loop around R.

3. An arc tangent to the right of the disk composing a thick vertex S i of S. 4. A left loop around S. 5. The remaining arcs of the initial (m + 2)-angulations are all of type 1. The rst is from 1 to m + 1, the second from 1 to 2m + 1... and the last from 1 to (n -3)m + 1. There is an exception if n = 4. Then P has 2m sides, and there is only one arc of type 1, which links vertex 1 to vertex m. The arcs hung to R are dened similarly, and the arcs hung to S start at m (and not 1 as previously). See gure 1.6 for the example of m = 3. It is obvious that the arcs of the previous denition form a good set of representatives for the initial (m + 2)-angulation. See gure 1.7 for an example of initial 4-angulation.

Note that we do not have to label the loops, since their labels are automatically deduced from the other arc ending in the corresponding inner polygon. Lemma 1.1.11. Let ∆ be an (m+2)-angulation.

Let {α 1 , • • • , α n } and {β 1 , • • • , β n }
be two good sets of representatives for ∆. Then there exists θ = (θ R , θ S ) a Dehn twist on R and S such that, after exchanging some choices of left or right loops if necessary, for all i, θ(β i ) = α i up to homotopy. Proof. We may assume that all the loops which appear in the good sets of representatives are left loops. We are only interested in the arcs attached to R. The case is similar for S.

Let α i be an arc an end point of which is a thick vertex of R. Let θ i be a Dehn twist such that θ i (β i ) = α i . Let α j be the neighbor of α i according to R. Let θ j such that θ j (β j ) = α j . Then the number of sides of R between α i and α j is the same as the number of sides between β i and β j . Indeed, the sets {α 1 , • • • , α n } and {β 1 , • • • , β n } are (m + 2)-angulations, and the other end point of α i and β i (respectively α j and β j ) is a vertex a i of P (respectively a j ). Then to respect the notion of (m + 2)-angles in an (m + 2)-angulation, it means that if k is the number of edges of R between α i and α j , k is also the number of edges of R between β i and β j . Then θ i = θ j .

We can reproduce this process until we have treated all the arcs ending on R.

We now introduce the twist which will be used in order to dene the ip of an (m + 2)-angulation at some m-diagonal.

Denition 1.1.12. Let ∆ be an (m + 2)-angulation. Let α be an m-diagonal of ∆, linking the vertices a and b. If, in ∆, there are two m-diagonals which are left and right tangent to the same inner polygon, replace one of them by a loop. When α is tangent to an inner polygon (say R) and there are only two arcs tangent to R, always replace α by a loop. If α is tangent to R and the other arc is a loop, then replace this loop by the corresponding arc tangent to R, and α by a loop. The twist of α in ∆ is dened as follows:

Pick any good set of representatives for ∆ containing α. Let α a (respectively α b ) be the side of the (m + 2)-angle ending at a (respectively at b) consecutive to α. Then the twist of α, namely κ ∆ (α) is the path α a αα b . See gure 1.8 for an illustration of the twist. Remark 1.1.13. If necessary, we can always choose a set of representative where α is a loop. Indeed, if α is not a loop (hung to vertex i of P , left tangent for instance), the chosen good set of representatives for the (m + 2)-angulation contains an arc which is tangent to R to the right, and has a common endpoint with α (called i). Exchange α for the left loop.

After applying the loop, note that the set of representatives is still a good one. Lemma 1.1.14. The new equivalence class obtained by twisting the m-diagonal does not depend on the choice of the set of good representatives to which α belongs. Remark 1.1.15. Here, as in lemma 1.1.11, we note that we choose a set of good representatives in which the arc corresponding to the m-diagonal α is a loop. As explained in remark 1.1.13, this is always possible when necessary. In fact, applying the twist to α consists in "slipping" α clockwise. The previous lemma says that "slipping along" a side of R does not depend on the choice of a class under Dehn twist. Denition 1.1.16. Consider ∆ an (m+2)-angulation. Choose a good set {γ 1 , • • • , γ n } of representatives for ∆. Let α be an arc in it. From remark 1.1.13, we may assume that α is not a loop. The ip of the (m + 2)-angulation at α is dened by µ α ∆ = ∆ \ {α} ∪ {α * } where α * is given by κ ∆ (α), the twist of α.

Proof. Let {α 1 , • • • , α n } and {β 1 , • • • , β n }
Remark 1.1.17. Here again, the ip does not depends on the choice of the set of good representatives before ipping. Remark 1.1.18. If we want to ip an m-diagonal α which is tangent to the left to one side of R (when at the same time there is an m-diagonal tangent to the right on the same thick vertices as α), then we have to choose the left loop, in order to make sure we have chosen the good representative. The issue is the same for any arcs tangent to the right. Remark 1.1.19. 1. Note that the twist has an inverse, which consists in moving the arc counterclockwise. Then the ip is also invertible. 2. We note that the twist is not the same as the Dehn twist. 3. A ip does not change the number of m-diagonals in the (m + 2)-angulation.

→ Figure 1.10 First example of a ip

The twist is dened on the m-diagonals, which is a class of equivalence under Dehn twist, and the ip is dened on an (m + 2)-angulation, which is a set of mdiagonal. Nonetheless, we will sometimes apply them on arcs, implicitly assuming that we have chosen a good set of representatives. Proof. We show that any (m + 2)-angulation can be obtained from the initial (m + 2)-angulation by applying a sequence of ips.

We know that there are at least ve m-diagonals (as n ≥ 4). If needed, we ip the (m + 2)-angulation at some m-diagonals ending at R, in order to reduce to the case where only two arcs are attached to R i , a thick vertex of R. We do the same for S. We x a set of good representatives of the (m + 2)-angulation.

Indeed, there cannot be less than two arcs ending at R, and if there are more than two, then at most two end at the same disk R i , and others end at R j where j = i. It suces to ip several times the arcs which do not end at R i in a way that they link two dierent thick vertices of P . Then we are reduced to the case of gure 1.13 (which is not completed by the other arcs of the (m + 2)-angulation).

Then, we notice that this gure is similar to the A n case. The A n case has been treated by Baur and Marsh in [START_REF] Baur | A geometric description of m-cluster categories[END_REF]. From here, the remaining (m + 2)angulation is related by a sequence of ip to any (m + 2)-angulation, including the initial (m + 2)-angulation.

Corollary 1.1.21. With this lemma and the fact that the ip does not change the Figure 1.13 number of m-diagonals in an (m + 2)-angulation, we notice that all the (m + 2)angulations contain exactly n + 1 m-diagonals.

The following lemma comes from a result of Freedman, Hass, and Scott in [START_REF] Freedman | Closed geodesics on surfaces[END_REF] on general geodesics, which allows us to consider that the m-diagonals do not cross each other.

Lemma 1.1.22. If ∆ = {γ 1 , • • • , γ n } is an (m + 2)-angulation, then there exist {α 1 , • • • , α n } representatives of {γ 1 , • • • , γ n },
such that for any i and any j = i, α i and α j do not cross. Proof. We know from [START_REF] Freedman | Closed geodesics on surfaces[END_REF] that under homotopy, there exist {α 1 , • • • , α n } representatives which do not cross. It remains the case where some m-diagonals are hung to R or S, which can be treated in the same way. Now we treat the case where there exist γ i from a vertex of P to a thick vertex of R, and γ j a loop around R. If an arc crosses γ i , then it obviously crosses γ j . Else, if it crosses the loop γ j , it cannot be hung to a thick vertex of R, because that would not cut into (m + 2)-angles. So it crosses also γ i , and this shows the lemma.

Remark 1.1.23. We can also apply the "ip" on a maximal set ∆ of non-crossing m-diagonals. Indeed, the previous lemma allows us to pick representatives which are pairwise noncrossing. For α an m-diagonal, we can set κ ∆(α) in exactly the same way as in (m + 2)-angulations.

Before starting our next result on (m + 2)-angulations, we recall that a t-angle is a gure with t sides, made of:

Sides of P Sides of R or S m-diagonals.
Proposition 1.1.24. The denition of an (m + 2)-angulation is equivalent to the following one: an (m + 2)-angulation is a maximal set of noncrossing m-diagonals.

Proof. On the one hand, a maximal set of noncrossing m-diagonals cuts the polygon into (m + 2)-angles. Indeed, let us x ∆ a maximal set of noncrossing mdiagonals.

If ∆ contains two loops. Then, these two loops cut the gure into a (n -2)m + 2-gon (where +2 stands for the loops which become a new side of P ) without punctures. This reduces to case A n-2 (since the polygon in the description of Baur and Marsh in [START_REF] Baur | A geometric description of m-cluster categories[END_REF] in type A p contains pm + 2 sides). As the set is maximal, it cuts the gure into (m + 2)-angles.

If ∆ only contains one loop (at S, for instance). Then, this m-diagonal cuts the polygon into a new gure with one inner polygon (which behaves like a puncture) and (n -2)m + 1 sides. This reduces to case D n-1 . As the set is maximal, there exist two arcs i and j from two vertices of P to R. If they hang the same vertex, then replace one by a loop and this is the previous point. If not, we can apply the twist to one of them (for instance i) several times. Any times we apply the twist, it is impossible for i to hang both ends of it to vertices of P . Indeed, this contradicts the maximality of the m-diagonals (and the denition of m-diagonals). But one end of i nishes to "slip along" j. The only way is to become a loop. This leads us to the rst case.

If ∆ has no loops, we use the same argument as the previous case to create loops.

If ∆ contains an arc i from R to S, the maximality of ∆ implies that there exist one arc from a vertex of P to R and one arc from P to S (indeed, if this was not the case, we have only m-diagonals linking dierent vertices of P , plus an arc from R to S. We then can add an m-diagonal from a vertex of P to R, which does not cross th others and is admissible, and this contradicts the maximality). Then, we apply the twist to i as many times as necessary to hang one of its ends to the polygon P . Then, we come to the previous case. Note that two arcs linking R to S are not compatible (as in [START_REF] Fomin | Cluster algebras and triangulated surfaces. I. Cluster complexes[END_REF], for the case where m = 1).

This nally shows that ∆ cuts the gure into (m + 2)-angles. On the other hand, let ∆ be a set of noncrossing m-diagonals, cutting the picture into (m + 2)-angles. We rst note that the m-diagonals do not cross. In addition, the set is maximal. Indeed, if it was not the case: Let us add a noncrossing arc α in an (m + 2)-gon, then it cannot be an m-diagonal:

If m = 1, then we cannot cut any triangle without being homotopic to one of the edges. Else, the only way to cut an (m + 2)-angle by forming a (km + 1)-gon is to form a triangle. This is impossible given the denition of an m-diagonal.

Remark 1.1.25. We notice that the number of (m + 2)-angles in the gure is xed whatever the (m + 2)-angulation, because of the previous proposition and the fact that the ip does not change the number of (m + 2)-angles. Lemma 1.1.26. Any set of noncrossing m-diagonals, can be completed to an

(m + 2)-angulation. Proof. Suppose Λ = {α 1 , • • • , α k } is a set of noncrossing m-diagonals.
If the set is maximal, then it is already an (m + 2)-angulation. Else, we can add an new mdiagonal α k+1 . We repeat the operation with Λ ∪ {α k+1 } until the set is maximal. The process ends since P has a nite number of sides, and once we have cut into (m + 2)-angles, we cannot go further.

With these denitions on the polygon, we are now able to associate a quiver with an (m + 2)-angulation. We will see that the ip of an (m + 2)-angulation is compatible with quiver mutation at the same vertex.

Colored quivers and (m + 2)-angulations

First, we indicate how to associate a quiver with an (m + 2)-angulation and then we will see how to draw a colored quiver from an (m + 2)-angulation.

Denition 1.1.27. Let ∆ be an (m + 2)-angulation (where we have chosen good representatives, without any loops). We dene the quiver Q ∆ associated with ∆ in the following way:

1. The vertices of Q ∆ are in bijective correspondence with the m-diagonals of ∆.

2. Between two vertices i and j (corresponding to two m-diagonals in ∆), we draw an arrow from i to j when both diagonals share an oriented angle. We require j to be consecutive to i clockwise. If i and j are hung to the same disk of an internal polygon, the arrows incident to i are found by forgetting j, and the arrows incident to j are found by forgetting i. Remark 1.1.28. From there, we call m-diagonals and vertices by the same name. It should be clear from the context whether we talk about polygons or quivers.

This implies that Q ∆ from lemma 1.1.11 is well-dened, because it is independent from the choice of a system of representatives. Indeed, the Dehn twist does not aect the order of the m-diagonals, and this matter to nd the arrows of the quiver. More precisely, if we apply θ R and θ S two Dehn twists to the polygons R and S, let us recall i and j the images if i and j by θ S • θ R . The end of i (respectively j ) hung to P si the same as the one of i (respectively j). Then the order is the same, and there is an arrow if i is consecutive to j (which is the same as i being consecutive to j).

For example in case m = 1, if we take the following triangulation ∆ (gure 1.14): Now, following the rule, we associate the following quiver Q ∆ with this triangulation:

3 6 2 Ð Ð 1 o o G G 5 d d 0 0
4 7 Let us now mutate this quiver at vertex 2. We obtain this new quiver Q :

3 0 0 6 2 G G 1 G G g g w w 5 d d 0 0 4 d d 7
We remark that if we ip the previous triangulation at the m-diagonal 2, we obtain the new triangulation ∆ of gure 1.15.

It is noticeable that the quiver Q ∆ arising from the new triangulation corresponds to the mutation at 2 of the quiver Q ∆ . This is a result showed by Fomin, Shapiro and Thurston for m = 1.

Proposition 1.1.29 ([FST08], proposition 4.8). Let ∆ be any triangulation. Let Q ∆ be the quiver associated with ∆ as before. If ∆ i is the new triangulation ipped at i from ∆, then the quiver Q ∆ i associated with ∆ i corresponds to the mutation at vertex i of Q ∆ .

We will later exhibit a proof for the case of colored quivers of mutation-type D. We now describe how to associate a colored quiver with an (m + 2)-angulation.

Lemma 1.1.30. If m = 1, then the m-diagonals of P are in bijection with the tagged arcs of Fomin, Schapiro and Thurston in [START_REF] Fomin | Cluster algebras and triangulated surfaces. I. Cluster complexes[END_REF] for the twice punctured (n -2)m-gon and the obvious bijection respects the noncrossing conditions. Proof. It suces to read the article of [START_REF] Fomin | Cluster algebras and triangulated surfaces. I. Cluster complexes[END_REF] to see that the arcs are in bijection.

Arcs hung to a thick vertex of R (or S) correspond to arcs linking a puncture in the realization of [START_REF] Fomin | Cluster algebras and triangulated surfaces. I. Cluster complexes[END_REF]. In our point of view, there are two types of arcs, one left tangent, and one right tangent. In Fomin Schapiro and Thurston case, there are plain, or notched arcs, so two types of arcs again.

Denition 1.1.31. Let ∆ be an (m + 2)-angulation. We dene the colored quiver Q ∆ associated with ∆ in the following way:

1. The vertices of Q ∆ are in bijection with the m-diagonals of ∆.

2. If i and j form two sides of the same polygon, then we draw an arrow from i to j. The color of the corresponding arrow is the number of edges between both m-diagonals, counted clockwise from i. Remark 1.1.32. We note that if we only draw the arrows of color 0, then we nd the classical quiver associated with the (m + 2)-angulation as in denition 1.1.27. Proposition 1.1.33. There is an equivalent denition: the vertices are similarly dened, and for i and j two vertices, and c an integer,

q (c) ij = 1 if κ c
∆ (i) and j share an oriented angle 0 otherwise.

Proof. We only have to show that the arrows are the same. If i and j form two sides of the polygon, with a color c, it means that if we apply the twist to i, then there will be c -1 edges between κ ∆ (i) and j. Then if we apply the twist c times, there will be no edge between κ c ∆ (i) and j, and they will share an oriented angle. On the other hand, if κ c ∆ (i) and j share an oriented angle, it suces to apply the inverse of the twist c times to make sure that i and j form two sides of a polygon, and that there are c edges between i and j.

Theorem 1.1.34. Let ∆ be any (m + 2)-angulation. Let Q ∆ be the colored quiver associated with the (m + 2)-angulation ∆. If ∆ k is the new (m + 2)-angulation ipped at k from ∆, then the colored quiver Q ∆ k associated with ∆ k is the mutation at vertex k of the colored quiver Q ∆ (see gures 1.16 and 1.17 for an illustration).

1 4 5 2 3 ip at 3 ----→ 1 4 5 2 3 4 (3) 0 0 2 (3) Ð Ð 1 (0) (0) d d (0) 0 0 (0) Ð Ð 5 (3) d d 3 (3) mutation at 3 -------→ 4 (3) 0 0 2 (3) Ð Ð 1 (0) (0) d d (1) 0 0 (0) Ð Ð

5

(3)

d d 3 (2)
Figure 1.16 Quiver mutation is compatible with the ip of an (m+2)-angulation, for m = 3. Note that before ipping the arc 3, we have to choose a good set of representatives. Nonetheless, in order to keep a clear gure, we have not replaced the arcs 8 and 5 by loops.

7

(2)

0 0 4 (0) Ð Ð 6 (0) (0) Ð Ð (2) G G 1 (0) o o (2) G G 2 (0) o o (2) G G 3 (0) o o (2) d d (2) 0 0 8 (2) d d 5 (0) mutation at 3 -------→ 7 (2) 0 0 4 (1) Ð Ð (0) w w 6 (0) (0) Ð Ð (2) G G 1 (0) o o (2) G G 2 (0) o o (0) G G (2)
U U

(2)

9 9 3 (2) o o d d (1) 0 0 8 (2) d d 5 (0) g g Figure 1
.17 Quiver mutation is compatible with the ip of an (m+2)-angulation, for m = 2. In order to keep a clear gure, we have not replaced the arcs 8 and 5 by loops.

Lemma 1.1.35. The quiver fullls the conditions asked for colored quivers in section ??. In particular it is symmetric.

Proof of the lemma. By denition, the quiver contains no loops (i.e. no arrows from i to i).

If there is an arrow from i to j of color c, it means that i and j share two sides of an (m + 2)-angle. If we count from i to j, there are c edges between them. But if we count from j to i, as we deal with (m + 2)-angles, it means that from j to i there are m -c edges. So there is an arrow from j to i of color m -c. Then the symmetry is respected.

In addition, for condition 2, we divide the proof into three cases.

First case: m = 1. If we have i

(1) G G j and i (0) G G j (note that c can be either 1 or 0). It suces to draw a triangle to ensure this situation is impossible.

Second case: m > 2. If we have i (c) G G j and i (c ) G G j , it means that i and j share two (m + 2)-gons, one with c edges between i and j, and the other one with c edges. So we are in the type of situation of gure 1.18).

Let us denote by P 1 (respectively P 2 ) the number of vertices delimited from i to j (respectively from j to i). Let T i (respectively U i ) be the vertices along the dotted edges cutting on the side of P 1 (respectively P 2 ) the gure into two parts, one of type A and one of type D. On the gure, we can visualize T i in red and U i in blue. Let R 1 and R 2 (respectively S 1 and S 2 ) be the number of vertices as shown in gure 1.18 in R (respectively in S).

Then, by making a computation on the total number of vertices, we are led to a contradiction. Indeed, we have the following equations:

P1 P2 R1 S1 R2 S2 Ti Ui i j Figure 1.18 Pathological case                T i + R 1 + S 1 = m -1 U i + R 2 + S 2 = m -1 R 1 + R 2 = m -1 S 1 + S 2 = m -1 T i + U i = n -2 P 1 + P 2 = (n -2)m (1.1)
Then we have that

T i + R 1 + S 1 + U i + R 2 + S 2 = 2(m -1). So T i + U i = 0
This leads to T i = U i = 0 and this is impossible.

Third case: m = 2. We are going to number the dierent possible sub-cases.

First sub-case: if we have i (0) G G j and i

(1) G G j it means that i and j belong to a triangle, which is impossible since m = 2.

Second sub-case: if we have i (0) G G j and i

(2) G G j , it means that we have i (0) G G j and j (0) G G i which is also impossible.

Third sub-case: if we have i

(1) G G j and i

(2) G G j , this reduces to the rst case, since we have j (1) G G i and j (0) G G i .

We thus have shown that there is no way to have two arrows from i to j of dierent colors This show the lemma.

Proof of the theorem. Let ∆ be an (m + 2)-angulation, let Q ∆ be the associated colored quiver, and let k be a vertex of Q.

We want to show that

Q µ k (∆) = µ k (Q ∆ ).
Let us call by T the new (m + 2)angulation obtained from ∆ by ipping the arc k. There is an evident bijection between the vertices of Q ∆ and Q T . Let i and j be two vertices of

Q µ k (∆) (and of µ k (Q ∆ )). Let q(c) ij (respectively q (c) ij ) be the number of arrows of color c from i to j in Q µ k (∆) (respectively µ k (Q ∆ )). Let us show that q(c) ij = q (c) ij . First case: If k = i, then q (c) ij = q (c-1) ij and κ c T (i) = κ c+1 ∆ (i). Consequently q(c) ij = q (c-1) ij . Second case: If k = j, then q (c) ij = q (c+1) ij and κ c T (i) = κ c-1 ∆ (i). Consequently q(c) ij = q (c+1) ij . Third case: If, in Q ∆ , we have i (c) G G k (0)
G G j and i and j are not two sides of the same (m + 2)-angle. Then q

(c) ij = q (c) ij + 1.
We notice that we cannot have two arrows from k to j of color 0. We are in a situation of the following type:

i a k j
Let then a be the common vertex of k and j. Let b be the other vertex of j (where a or b can be central polygons). Then κ ∆ (k) and j have b as a common vertex. As, in ∆, κ c ∆ (i) and k have one common vertex a, then κ c T (i) and j have one common vertex. As a consequence,

q(c) ij = q (c) ij + 1.
There is no loss because there are not any arrows from i to j of color dierent from c. We note that there can be two arrows of dierent color appearing from j to k, but as this is forbidden, we remove them. This case is symmetric to the case where i and k share a common vertex.

To be clear, we draw an example at the remark just below the proof.

Fourth case: If we are in a situation of the following type:

i k a j
It means that we have a quiver:

i (c) G G (c+1) ) ) k (0) Ð Ð j Then q (c) ij = q (c)
ij -1 = 0, because the arrows get erased. Moreover, under the notations of the third case, κ ∆ (k) and j have b as a common vertex, and i and j do not share a common polygon anymore. Then q(c) ij = 0.

Remark 1.1.36. Let us draw an example in gure 1.19. We have i (0) G G k (0) G G j . When we mutate k, we obtain the right gure. The quiver becomes

i (0) G G (1) ) ) j (0)
k (where we do not note the inverse arrow for sake of clearance). Now let us mutate at vertex i. The gure becomes:

i k j

If we mutate the colored quivers, since two arrows of dierent color have been built from j to k (one of color (0) and one of color (1)), they get erased and the quiver becomes j → i → k of color (0). This exactly corresponds to the quiver of the gure.

Construction of the category of m-diagonals

We are now able to construct a new category, C, equivalent to a subcategory of the higher cluster category of tyê D by using the m-diagonals. We remark that the indecomposables of C correspond to the rigid indecomposables of the higher cluster category.

Before starting with the construction of the category we dene what will be the generators the morphisms of C: the elementary moves.

Elementary moves

Elementary moves are applications that send an m-diagonal to another one. They should not be confused with ips.

Let Q be a quiver of type Dn . We want the elementary moves to correspond to the arrows in the Auslander-Reiten quiver of C m Q .

Figure 1.20 An example of an m-diagonal of type 1.

We now dene the length of an m-diagonal of type 1.

Denition 1.2.1. Let a < b be two integers between 1 and (n -2)m. The length of the boundary path from a to b, denoted by l(a, b) is the number of edges between a and b counted clockwise. Denition 1.2.2. Let P be the polygon considered in section 1.1. Let us number the vertices of P clockwise. Let α be an m-diagonal from i to j. The translation τ α of α is dened as follows:

1. If i = j and α is of type 1, then τ α is the unique new arc obtained by composing α with the boundary paths B i i+m and B j j+m , linking i + m and j + m, and of the same type.

2. If i = j and α is homotopic to a boundary path, then τ α is the unique new arc obtained by by composing α with the boundary paths B i i+m and B j j+m , linking i + m and j + m, and of the same type. 3. If i = j, then τ α is the new arc tangent to the same vertex of the inner polygon, ending at i + m. If m is odd, then change the side of the tangency. 4. If α links both inner polygons, from R k to S k for instance, the new arc τ α links R k-m and S k +m modulo m -1. The side of the tangency changes only if m is odd.

For the last point, the new arc is uniquely dened, because it does not depends on the eventual Dehn twist of R or S, but only of the relative position before applying τ .

Remark 1.2.3. Applying τ several times to an arc of type 1 makes the arc "roll around" both central polygons.

Here in gure 1.22 we can see an example of a translation: There is an elementary move from α to β when: 1. α and β are of type 1, share a vertex a, and if l(b, c) = m (where α ends at b and β ends at c). 2. α is of type 1 and β links a vertex of P to a thick vertex of R (or equivalently S), α and β share a vertex a, and β is consecutive to α clockwise. 3. β is of type 1 and α links a vertex of P to a thick vertex of R (or equivalently S), β and α share a vertex a, and β is consecutive to α clockwise. 4. α and β are homotopic to boundary paths, share a vertex a, and if l(b, c) = m (where α ends at b and β ends at c). 5. α and β are from a unique thick vertex of R (respectively S) to a thick vertex of S (respectively R) and β is consecutive to α clockwise. Remark 1.2.5. The cases 2 and 3 are included in case 1 by using the loops.

Here is the tabular of elementary moves in the three rst cases for a polygon with (n -2)m sides (where e-m stands for "elementary move" and the m-diagonal α is black whereas β is red). For sake of clarity, we have not drawn the disks on each thick vertex of the central polygons. The dotted lines hide m edges.

The remaining cases in the denition will be treated in section 4.

type 1 type 2 type 1 type 2 no e-m
in the m-cluster category

The category of m-diagonals

In this section, we will make use of the elementary moves dened above in order to generate the morphism of a category of m-diagonals. Let us just give a lemma before starting.

Lemma 1.2.6. If α and β are two m-diagonals, then there exists an elementary move f : α → β if and only if there exists one f : τ β → α. Proof. There are several cases to examine. We can always reduce to the case where α or β are not loops. Note that if α is a transjective arc, there exists an elementary move from α to β if and only if β is also a transjective arc.

1. Let us rst study the case of transjective arcs. (a) If β is of type dierent from 1.

Then let a be its only vertex. To be compatible with the denition of an elementary move, it is necessary for α to be of type 1. Let b be the other vertex of α (the rst one being a, since both arcs have to share an oriented angle). The converse is similar.

Remark 1.2.7. This lemma seems natural since applying τ -1 corresponds to apply an elementary move to each end of the arc. Denition 1.2.8. The quiver Q is dened as follows:

1. The vertices are the m-diagonals.

2. There is an arrow between two m-diagonals α and β when there is an elementary move from α to β. The category C is dened to be the additive mesh category of Q.

It means that the category C is the additive Krull-Schmidt category where:

1. The indecomposable objects of C are the m-diagonals of denition 2.1.21. So the objects of C are the nite direct sums of m-diagonals.

2. The set of morphisms between two indecomposable objects X and Y is given by the k-vector space generated by the paths from X to Y in Q by the subvector-space generated (elements in the ideal generated by) mesh relations.

3. The composition is induced by the concatenation of paths in Q.

Mesh relations: By the previous lemma, we know that if we have f : β → α, then we also have f : τ α → β. Then the mesh relations are the relations:

R α = β,f :β→α f f .
We will see that C is equivalent to a sub-category of the higher cluster category C (m) Dn . Proposition 1.2.9. The quiver ( Q, τ ) is a stable translation quiver. Proof. Let x be a vertex in Q. Given a vertex y, we have to show that if there is an arrow from y to x, then there is an arrow from τ x to y. This is exactly the lemma 1.2.6.

In addition, no vertex is projective, and as τ is dened on all vertices, so is τ m . Figure 1.23 illustrates a small part of the Auslander-Reiten quiver of Chapter 1. A geometric realization of the m-cluster categories of type Dn Remark 1.2.10. We can nd on a slice of the Auslander-Reiten quiver of C m Q the following initial (m + 2)-angulation. In fact, each arc of the initial (m + 2)angulation corresponds to the image in the m-cluster category of a preprojective indecomposable representation. Remark 1.2.11. We note that, on the previous pictures, we have only drawn the rst preprojective component of the Auslander-Reiten quiver of Q, but there are m copies of it, where we can nd the m-diagonals shifted 1, • • • , m -1 times. Indeed, between the tubes, there are m copies of the preinjective and preprojective components. On gure 1.24 we can see how the preprojective, regular and preinjective components of a Auslander-Reiten quiver of type Dn are set.

C (m) Q . • • • ! ! " " " " • • • • • • ' ' 0 0 0 0 • • • r r d d 0 0 " " r r d d 0 0 " " s s g g ' ' ! ! • • • g g d d d d • • • • • • s s r r r r • • •

Mesh relations

Is is possible to make use of the m-diagonals so as to easily picture mesh relations. There are several types of mesh relations that can thus be described: mesh relations containing two or three monomials in the transjective component and mesh relations in the three tubes of rank greater than one.

The rst type of relation is the one of the following:

1 1 d d
It is easy to see that the end points of both arcs of type dierent from 1 correspond to the end points of the arc of type 1 linked with the other arcs, and this is a general case.

The second one is of this shape:

1 1 d d 1 1 d d
As in the rst mesh relation, we can see that the end points coincide. The third one is of this shape:

1 1 1 1 c c G G G G c c
The pictures on the end points on the left and the right are m-diagonals α and the end points of β = τ α are always of length m. The upper two pictures in the middle column (or the lower two if we are situated at the bottom of the Auslander-Reiten quiver) on the gures 1.23 and 1.29 are arcs tangent to one side of a central polygon, and the other end point is at the vertex shared by both α and β (as they are of length m and β = τ α, they share one vertex). Finally the lowest (or uppermost) picture is an m-diagonal of type 1, of length 2m, where the vertices are the ones of α and β which are not shared by each other. The bases of the tube will be described in the next section.

Equivalence of categories

In this section, assuming that Q is still a quiver of type Dn , we build an additive functor F from the category C to the m-cluster category.

Denition 1.2.12. Let P be the polygon considered in section 1.1. Let us number the vertices of P clockwise. Let α be an m-diagonal from i to j. The shift α[1] of α is dened as follows:

1. If i = j and α is of type 1, then α[1] is the unique new arc obtained by composing α with the boundary paths B i i+1 and B j j+1 , linking i + 1 and j + 1, and of the same type. 2. If i = j and α is homotopic to a boundary path, then α[1] is the unique new arc obtained by by composing α with the boundary paths B i i+1 and B j j+1 , linking i + 1 and j + 1, and of the same type. 3. If i = j, then α[1] is the new arc tangent to the same vertex of the inner polygon, ending at i + 1. If m is odd, then change the side of the tangency. 4. If α links both inner polygons, from R k to S k for instance, the new arc α [1] links R k-1 and S k +1 . The side of the tangency changes only if m is odd. Denition 1.2.13. If d is an integer in {1, • • • , m}, then we dene S d as the connected component in the quiver of denition 1.2.8 containing the d-th shift of the initial (m + 2)-angulation, the rst component being the one corresponding to the initial (m + 2)-angulation, and the following one corresponding to the initial (m + 2)-angulation where [1], [2], • • • has been applied.

Let S d be the component in the Auslander-Reiten quiver of C (m) Dn , containing objects of type τ s P [d] where P is a projective indecomposable object and s is an integer. Theorem 1.2.14. Let d ∈ {1, • • • , m}. Then we have an isomorphism between both components:

S d S d
Proof. For n ≥ 6, we x an orientation for our quivers. The orientation we choose is the one of the quiver corresponding to the initial m-angulation:

7 0 0 4 6 G G 1 G G 2 G G 3 d d 0 0 8 d d 5
Let ∆ 0 1 be the initial (m + 2)-angulation in the rst copy of the quiver Q. It corresponds to the m-cluster-tilting object T = P i at the initial slice of the Auslander-Reiten quiver of Q. With the m-diagonal α i (at vertex i), we associate the projective module P i = F (α i ). Then we associate with the elementary moves of m-diagonals, the irreducible morphisms in the cluster category. Indeed, according to the tabular of section 1.2.1, there is an elementary move from α to β when both share an oriented angle. As it has been told in denition 1.1.27, in the quiver it means that there is an arrow from α to β. So there is an irreducible morphism from F (α) to F (β).

Now that we have treated the case of the initial slice in the rst copy, still calling α i the m-diagonal at vertex i in this slice, we note that every m-diagonal

in the component S d in the quiver Q is of the form τ t α i [d], for some t ∈ Z and i ∈ {1, • • • , n + 1}. So it is natural to dene F (τ t α i [d]) = τ t (P i [d])∀t.
Therefore, there is a bijection between the arcs of S 1 and the indecomposables of the transjective components in the Auslander-Reiten quiver of C (m) Dn containing the image of the indecomposable projective objects.

Indeed, there are two things to show: First, for any arc α, there is a unique way to write it τ t α i [d]. If we had τ t α i [d] = τ s α j [d] in a component d, then τ t α i = τ s α j , so i = j because of the type of the diagonal (if it is tangent one or another side of the central polygon, or if it is of type 1, with length l, etc...), and if t = s it would mean that there is the same m-diagonal at two dierent vertices in the quiver Q, which is impossible. Second, if X is an indecomposable in the d-th component of Q, it can be written τ t P i [d], and this has a unique antecedent by F , which is

τ t α i [d].
As a matter of conclusion, F is bijective on the objects. Let us show that it is also a bijection on the elementary moves. Suppose that f : α → β is an elementary move. Then there exist s

∈ N, i ∈ {1, • • • , n + 1}, d ∈ {1, • • • , m} such that τ s α = α i [d] and τ s β = β * [d] (i.e α = τ -s α i [d] and β = τ -s β * [d],
where β * = α j when there is an elementary move α i → α j or β * = τ -1 α j where there is an elementary move α j → α i . In the rst case, there is an irreducible morphism from P i to P j thus from F (α i ) to F (β * ) and from

F (α) = τ -s F (α i )[d] to F (β) = τ -s F (β * )[d].
In the second case, there is an irreducible morphism from P j to P i thus from P i to τ -1 P j and from

F (τ s α) = F (α i [d]) → F (β i [d]) = F (τ s β), thus an arrow F (α) → F (β).
This proves the statement.

The case of regular modules

Now that we have seen an equivalence for the transjective components, we are interested in the case of tubes, and their correspondences in terms of arcs. When we introduced the dierent types of admissible m-diagonals, we voluntarily forgot some other types of diagonals. These remaining diagonals have the particularity of being cyclic. Indeed, if we apply τ several times to them, they come to their rst position at a rank r. This cannot be the case of the types of m-diagonals dened in 2.1.21 (because they wrap around the central polygons).

In gure 1.25, we can see the other types of diagonals. Note that we only have drawn the rst ones, but there are also their successive images under τ (of which there are a nite number because they are cyclic). These diagonals have to gure in the Auslander-Reiten quiver of Q, and it appears that we can see them in the tubes (it means in the regular part of the quiver).

In case Dn , as we can see in the course of Crawley-Boevey [CB], there are three types of tubes, two of period 2, and one of period n -2.

As we deal with the higher cluster categories, we have to notice, that we have m copies of each tube, it means we have 3m tubes. The rst tube contains the rst picture of gure 1.25 and all its images under τ . There are m copies of this 1.3. The case of regular modules 81 tube, which correspond to the successive shifts of the arcs. This is the same proces for the second and third tube. . . .

• • • 5 5 • • • 5 5 d t 2 b b objects identied f f τ -1 d t 2 Y Y τ -2 d t 2
It is known in [CB] that in a tube of size r, only the rst r -1 layers contain rigid objects. Graphically this corresponds to noncrossing arcs. It means that in Q, for the example of the only tube of size n -2, the (n -1) lowest lines are made of noncrossing arcs. Then the arc crosses itself, and this does not correspond anymore to an m-rigid object in the higher cluster category, and this is the same for the shifted arcs in the successive copies of this rst tube. Then with an arc situated on the r -1 rst lines of the tube, we associate the m-rigid object which actually takes place in the Auslander-Reiten quiver of C (m) Dn . In order to make sure that an arc corresponds to a unique m-rigid object in the tube, we have to choose a convention. Let ∆ be the initial (m + 2)-angulation. Let β be the red m-diagonal in gure 1.26. Then the vertex of Q associated with β is n -1 from the isomorphism of theorem 1.2.14. Let α = µ ∆ (β). We only need to associate an m-rigid object in order to nd all the objects of the tube of size n -2. We choose to associate with α the simple object τ -1 S n-1 at vertex n -1.

To be precise, we set at the base of the rst tube of size n -2 the ip of the arc corresponding to the preprojective at the rst slice of the Auslander-Reiten quiver. This arc is thus associated with the simple regular module which corresponds to the mutation of the preprojective. This means that we have mutated the object P n-1 in the (m + 2)-angulation containing the arcs corresponding to the sum of the indecomposable projective objects.

For the tubes of size 2, we set the only arcs linking both central polygons without self-crossing (see gure 1.25).

In this way we have a 1-1-correspondence between the arcs linking two dierent vertices which are homotopic to the border and the objects of the tube of size n-2 in the Aulslander-Reiten quiver of C We know by theorem 1.2.14 that an arc in the geometric realization can be interpreted as an object in the m-cluster-category. We use the following notation: if α is an arc in the category C let X α be the object associated in the m-cluster category.

Lemma 1.4.1. The application {m-diagonals} → {m-rigid indecomposable objects} α → X α up to isomorphism, is bijective.

Proof. With each arc α which does not cross itself we associate by theorem 1.2.14 an m-rigid object X α . Then ∀i ∈ {1, • • • , m} Ext i (X α , X α ) = 0. Indeed, rst, if α belongs to the preprojective part of the Auslander-Reiten quiver, it belongs in particular to a slice of it which forms a quiver of type Dn . If we add all the arcs forming this quiver, we obtain an (m + 2)-angulation. In the Auslander-Reiten quiver of C m Q , by theorem 1.2.14, this slice corresponds to an m-cluster-tilting object, made of the sum of all m-rigid objects at the vertices of the slice. Then, with α is associated an m-rigid object X α , and as α is in this part of the Auslander-Reiten quiver of Q, it does not cross itself.

Second, if α belongs to a tube, then if it does not cross itself, it is at the base of it, and by the previous section, this corresponds to an m-rigid object (because in the case of a tube of size r, X α is situated at one of the r-1 rst lines). Conversely, if X α is m-rigid, then it is at the base (in the rst r -1 lines) of the tube of size r, it means that it corresponds to an arc of the rst lines, which does not cross itself.

We have thus shown that α is an arc which does not cross itself if and only if

∀i ∈ {1, • • • , m}, Ext i (X α , X α ) = 0.
Then, there is a one-to-one bijection between arcs without any self-crossing and m-rigid objects.

Remark 1.4.2. We can extend this bijection to the arcs in the tubes including arcs with an auto-intersection. This can be easily visualized in gure 1.32. Remark 1.4.3. In a paper which is being drawn up, we show that there is a bijection between (m + 2)-angulations and m-cluster-tilting objects.

An example

Let us resume this paper with a complete study of an Euclidean quiver of type Dn , and its m-cluster category, taking n = 7 and m = 2. So we study the following category

C 2 D7 = D b (K D7 )/τ -1 [2],
where D7 is the following quiver:

7 0 0 4 6 G G 1 G G 2 G G 3 d d 0 0 8 d d 5 1.5

.1 Quadrangulations of the decagon and colored quiver

The geometric realization shown in gure 1.27 for this quiver is a decagon with two squares inside of it.

Figure 1.27 The decagon with two monogons

We are now drawing eight 2-diagonals in order to cut the decagon in quadrangles. This is in gure 1.28 the initial (m + 2)-angulation from which we will build later the translation quiver isomorphic to a sub-quiver of the Auslander-Reiten quiver of C If we follow the rules of denition 1.1.27, we associate with this quadrangulation the quiver at the beginning of this section.

Moreover, if we associate the colored quiver with this quadrangulation according to denition 2.1.9, we obtain this one.

7

(2)

0 0 4 (0) Ð Ð 6 (0) (0) Ð Ð (2) G G 1 (0) o o (2) G G 2 (0) o o (2) G G 3 (0) o o (2) d d (2) 0 0 8 (2) d d 5 (0)
Now we know that mutating a colored quiver corresponds to ipping an arc in an (m + 2)-angulation, we can ip in any way the quadrangulation, draw the corresponding colored quiver, and note that the colored quivers are related by mutation at the corresponding vertex.

2-diagonals and preprojective component of the Auslander-Reiten quiver

From the initial quadrangulation, we extract the arcs and put them in the rst component of the Auslander-Reiten quiver of D7 .

As the preprojective component of the Auslander-Reiten quiver is of the following form (where we have associated the vertices with the types of the arcs we

d d G G d d d d G G d d d d
are working with), we just have to apply τ and the shift functor to the arcs to be able to complete the preprojective component and obtain the result in gure 1.29. Chapter 1. A geometric realization of the m-cluster categories of type Dn

d 7 6 6 τ -1 d 7 6 6 τ -2 d 7 d 6 6 6 X X G G d 8 G G τ -1 d 6 6 6 X X G G τ -1 d 8 G G τ -2 d 6 X X G G τ -2 d 8 d 1 6 6 X X τ -1 d 1 6 6 X X τ -2 d 1 X X d 2 6 6 X X τ -1 d 2 6 6 X X τ -2 d 2 X X d 4 G G d 3 44 G G τ -1 d 4 G G τ -1 d 3 6 6 X X G G τ -2 d 4 G G τ -2 d 3 X X G G τ -3 d 4 d 5 c c τ -1 d 5 X X τ -2 d 5 X X 1 1 ' ' 1 1 c c G G G G 1 1 c c G G G G • • • • • • 1 1 c c 1 1 d d • • • g g 0 0 d d 1 1 c c • • • d d G G d d 1 1 G G G G 1 1 d d G G G G • • • d d G G • • • c c c c • • • c c
Figure 1.29 Arcs on the Auslander-Reiten quiver of D7

Note that there are two copies of this component, and the other one is about the same but all the arcs have been shifted once clockwise.

As the preprojective component is innite, we can along it, watch the arcs wrap around the squares. 1.5.3 2-diagonals in the tubes of the Auslander-Reiten quiver of D7

Concerning the regular modules, there are three types of tubes: The rst type, containing m tubes of size 5 and the two other types of tubes (each one containing m copies of it) of size 2. The one of size 5 corresponds to the following cyclic arc (in gure 1.30) and its images under the shift. Indeed, if we apply τ ve times, the arc returns to the origin, so it is cyclic of order 5. The two remaining arcs corresponding to the tubes of order 2 are the ones of gure 1.31, plus their successive images under the shift.

Figure 1.31 Arcs of order 2

Here we draw the tube of size 2 corresponding to the rst picture of gure 1.31. . . .

• • • 2 2 • • • 2 2 b b objects identied b b b b
Note that in this tube, only the rst line corresponds to rigid objects. The upper arcs cross themselves and thus no not correspond to rigid objects anymore.

The following picture gives the tube of size 5, with the corresponding arcs. We can notice that in the rst four ranks, the arcs do not cross since they match to a rigid object, and this is not the case anymore from rank 5. In order to have clear pictures, we have replaced the monogon inside of P with a simple disk. A bijection between m-cluster-tilting objects and (m + 2)-angulations in m-cluster categories 2.1 Geometric realizations

6 6 6 6 6 6 6 6 X X 1 1 X X 1 1 X X 1 1 X X 1 1 1 1 X X d d 1 1 d d 1 1 d d 1 1 c c 1 1 d d 1 1 d d 1 1 d d 1 1 c c 1 1 c c 1 1 c c 1 1 c c 1 1 c c 1 1 c c 1 1 c c c c c c c c c c

Case A [BM08]

In this section, we recall the geometric realization of the m-cluster category of a quiver of type A n , for an integer n, after Baur and Marsh [START_REF] Baur | A geometric description of m-cluster categories[END_REF].

Let Q be a quiver of type A n , with n vertices, and let C m Q be the m-cluster category associated with Q. Let P be a polygon with nm + 2 sides, numbered clockwise.

Denition 2.1.1. An m-diagonal α from the vertex i to j = i in P is a diagonal of P linking i and j such that α cuts the gure into two polygons, one with km + 2 sides, for some k ∈ N and one with lm + 2 sides, for some l ∈ N.

In gure 2.1 we draw an example of an m-diagonal.

Denition 2.1.2. An (m + 2)-angulation of P is a maximal set of noncrossing m-diagonals.

Remark 2.1.3. We note that this denition is equivalent to the following one : an (m + 2)-angulation is a set of m-diagonals cutting the polygon into (m + 2)-angles.

We can introduce the initial (m + 2)-angulation, that we will use later. All its m-diagonals have one end at the same vertex 1 (see gure 2.3). We can dene the twist and the ip of an (m + 2)-angulation, as Buan and Thomas did in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF]. Denition 2.1.4. Let ∆ be an (m + 2)-angulation. Let α be an m-diagonal of ∆, linking the vertices a and b. The twist of α in ∆ is dened as follows:

Let (a , a) (respectively (b, b )) be the side of the (m + 2)-angle ending at a (respectively at b ) consecutive to a clockwise (respectively preceding b). Then the twist of α, namely κ ∆ (α) is the m-diagonal (a , b ). Denition 2.1.5. Consider ∆ an (m + 2)-angulation. Let α be an arc in ∆. The ip of the (m + 2)-angulation at α is dened by µ α ∆ = ∆ \ {α} ∪ {α * } where α * is given by κ ∆ (α), the twist of α. Remark 2.1.6. 1. Note that the twist has an inverse, which consists in moving the arc counterclockwise. Then the ip is also invertible. 2. A ip does not change the number of m-diagonals in the (m + 2)-angulation.

In gure 2.4, we can see an example of a ip.

→ Figure 2.4 Example of a ip

Lemma 2.1.7. Any two (m + 2)-angulations are related by a sequence of ips. Proof. Let ∆ be an (m + 2)-angulation. We show that we can reach the initial (m + 2)-angulation by applying a sequence of ips.

If one of the arc has one end at vertex 1, then is suces to ip several times the neighbour arc in order to hang one end of it to vertex 1. We repeat the operation until all arcs are hung to vertex 1. This leads exactly to the initial (m + 2)angulation. If no arc has one end hung to vertex 1, then consider the (m + 2)-gon containing the vertex 1. Flip one arc of this polygon as many times as necessary in order to hang it at vertex 1.

Corollary 2.1.8. [START_REF] Tzanaki | Polygon dissections and some generalizations of cluster complexes[END_REF] With this lemma and the fact that the ip does not change the number of m-diagonals in an (m + 2)-angulation, we notice that all the (m + 2)-angulations contain exactly n m-diagonals.

Note that if Λ is a set of noncrossing m-diagonals, it can be completed in order to form an (m + 2)-angulation.

We now associate a colored quiver with an (m + 2)-angulation.

Denition 2.1.9. Let ∆ be an (m + 2)-angulation. We dene the colored quiver Q ∆ associated with ∆ in the following way:

1. The vertices of Q ∆ are in bijection with the m-diagonals of ∆.

2. If i and j form two sides of some (m + 2)-gon in ∆, then we draw an arrow from i to j and an arrow from j to i. The color of the corresponding arrow is the number of edges between both m-diagonals, counted clockwise from i (respectively from j). Proposition 2.1.10. There is an equivalent denition: the vertices are similarly dened, and for i and j two vertices, and c an integer,

q (c) ij = 1 if κ c
∆ (i) and j share a counterclockwise oriented angle 0 otherwise. Proof. We only have to show that the arrows are the same. If i and j form two sides of the polygon, with a color c, it means that if we apply the twist to i, then there will be c -1 edges between κ ∆ (i) and j. Then if we apply the twist c times, there will be no edge between κ c ∆ (i) and j, and they will share an oriented angle. On the other hand, if κ c ∆ (i) and j share an oriented angle, it suces to apply the inverse of the twist c times to make sure that i and j form two sides of a polygon, and that there are c edges between i and j. This only works if c ≤ m. If c > m, then apply this method to the other end of i.

Lemma 2.1.11. The quiver fulls the conditions asked for colored quivers in the article of Buan and Thomas [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF]. In particular it is symmetric. Proof. By denition, the quiver contains no loops (it means, no arrows from i to i).

The condition of monochromaticity is respected since two arcs can only share one polygon.

If there is an arrow from i to j of color c, it means that i and j share two sides of a triangle. If we count from i to j, there are c edges between them. But if we count from j to i, as we deal with (m + 2)-angles, it means that from j to i there are m -c edges. So there is an arrow from j to i of color m -c. Then the symmetry is respected.

We remark that we have the compatibility between the mutation of a colored quiver in the sense of Buan and Thomas, and the ip of an (m + 2)-angulation.

Theorem 2.1.12 (Buan-Thomas, [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF]). Let ∆ be any (m + 2)-angulation. Let Q ∆ be the colored quiver associated with the (m + 2)-angulation ∆. If ∆ k is the new (m + 2)-angulation ipped at k from ∆, then the colored quiver Q ∆ k associated with ∆ k is the mutation at vertex k of the colored quiver Q ∆ . Proof. This is immediate given the previous proposition.

Baur and Marsh in [BM08, Theorem 5.6] have shown that a category geometrically built from these m-diagonals is equivalent to the m-cluster category. In the next section, we will go further and show that cutting along an arc corresponds to the Iyama-Yoshino reduction, as Marsh and Palu showed it for the general case of Riemann surfaces for m = 1.

We also have the following theorem:

Theorem 2.1.13 ([BM08], Proposition 5.4). There is an explicit bijection between the m-diagonals without any self-crossing and the m-rigid objects of the m-cluster category.

This bijection is found in the following way: Buan and Marsh build a quiver from m-diagonals, which is aimed to be isomorphic to the Auslander-Reiten quiver of Q.

Remark 2.1.14. We notice that if we only draw the arrows of color 0, then we nd the Gabriel quiver of the endomorphism algebra of the m-cluster-tilting object associated with the (m + 2)-angulation.

Case D [BM07]

In this case, treated by Baur and Marsh in [START_REF] Baur | A geometric description of the mcluster categories of type D n[END_REF], we use a slightly dierent geometric realization, in order to simplify the notion of ip of an (m+2)-angulation. Baur and Marsh use a polygon with nm -m + 1 sides with a punction inside of it. We replace the puntion by an (m -1)-gon with, on each vertex of it, a disc. The realization is the same as the one of D (see [JM]), but with only one polygon inside ot it. The arcs are dened in the same way, and this respects the article of Baur and Marsh, since we have an evident bijection between the arcs of Baur and Marsh, and the ones we dened as in case D in [JM].

We also have the following theorem:

Theorem 2.1.15 ([BM07], Theorem 3.6). There is an explicit bijection between the m-diagonals without any self-crossing and the m-rigid objects of the m-cluster category.

Here again, Buan and Marsh build a quiver from m-diagonals, which is aimed to be isomorphic to the Auslander-Reiten quiver of Q.

Case

à [Tor12b]
The geometric description of case à has been completely treated by Torkildsen in [START_REF] André | A geometric realization of the m-cluster category of type[END_REF]. We recal part of his description. 96Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations Let m be an integer. Let Q be a quiver of type Ãn , with p arrows going one direction, and q arrows going the other. Let P be a regular mp-gon, with a regular mq-gon inside of it. In the following, we give the example with p = 4 and q = 3, for m = 2. We number the vertices of the outer polygon O 1 , • • • , O mp-1 and the vertices of the inner polygon I 1 , • • • , I mq-1 .

There are three types of m-diagonals:

A path from a vertex of the outer polygon to a vertex of the inner polygon A path from O i to O i+km+2 , where i + km + 2 ≥ 1, is counted modulo pm + 1 and i ∈ {1, • • • , pm + 1} homotopic to the boundary path of the outer polygon.

A path from I i to I i+m-1 for some i and some k ≥ 3 homotopic to the boundary path of the inner polygon. Denition 2.1.16. We call "(m + 2)-angulation" a set of p + q noncrossing mdiagonals cutting the gure into (m + 2)-angles.

With an (m + 2)-angulation, we can associate a colored quiver dened in this way: Denition 2.1.17. Let ∆ be an (m + 2)-angulation. Let Q ∆ be the quiver dened as follows:

The vertices are the m-diagonals There is an arrow from i to j if i and j bound a common (m + 2)-angle.

The color of the arrow is the number of edges between i and j counted from i counterclockwise.

Then, Torkildsen in [START_REF] André | A geometric realization of the m-cluster category of type[END_REF] dened the mutation of an (m + 2)-angulation:

Denition 2.1.18. Let ∆ be an (m + 2)-angulation. Let α be an m-diagonal of ∆. Then, remove α. There exist m + 1 m-diagonals which can complete the almost (m + 2)-angulation. Let α be the neighbor of α clockwise. The mutation of ∆ at α is dened to be the (m + 2)-angulation ∆ \ {α} ∪ {α } Theorem 2.1.19. [START_REF] André | A geometric realization of the m-cluster category of type[END_REF]Proposition 5.1] Let ∆ be any (m + 2)-angulation. Let Q ∆ be the colored quiver associated with the (m + 2)-angulation ∆. If ∆ k is the new (m + 2)-angulation ipped at k from ∆, then the colored quiver Q ∆ k associated with ∆ k is the mutation at vertex k of the colored quiver Q ∆ .

In the following article, Torkildsen builds an equivalence of categories and shows the following result:

Theorem 2.1.20 ([Tor12b], Theorem 7.3). There is an explicit bijection between the m-diagonals (called "diagonals" in his article) without any self-crossing and the m-rigid indecomposable objects of the m-cluster category.

Case

D

This case has been treated in [JM]. We give some quick explanations here: Let P be an (n -2)m-gon with two central (m -1)-gons R and S inside of it (cf gure 2.6). We replace each vertex of R and S by a disk, which we henceforth call a thick vertex. If m = 1, then we consider an (n -2)-gon with two disks inside of it. 1. If i = j: the homotopy classes of arcs crossing the space between both central polygons and cutting the gure into a km + 1-gon and a k m + 1-gon, for some k and k . In this case, it is said that the class is of type 1. 2. If i = j: the homotopy classes of arcs cutting the gure into a km-gon with both central polygons inside of it and a k m + 2-gon. 3. If i = j: all the admissible arcs tangent to the left of the disks of the inner polygon R, plus the left loop, form a rst class. The second class is given by all the arcs tangent to the right of the disks, plus the right loop. There are two more classes built in the same way for S. For more details, see [JM].

Any admissible arc linking both central polygons form a class of m-diagonal.

To be clear, if α is an arc whose rst end is tangent to a side of R, and second end is tangent to a side of S, then it is an m-diagonal. Denition 2.1.22. An (m + 2)-angulation is a set of noncrossing m-diagonals cutting P into (m + 2)-angles. In a similar way to that of Torkildsen, we dene the ip of an (m+2)-angulation, and build a colored quiver associated with an (m + 2)-angulation. We show the following result: Theorem 2.1.23. Let ∆ be any (m + 2)-angulation. Let Q ∆ be the colored quiver associated with the (m + 2)-angulation ∆. If ∆ k is the new (m + 2)-angulation ipped at k from ∆, then the colored quiver Q ∆ k associated with ∆ k is the mutation at vertex k of the colored quiver Q ∆ .

Then we show that there are some common points between the higher cluster category and a geometrically built category. More precisely, we build a category from these m-diagonals which is equivalent to a subcategory of the higher cluster category. We explicit all m-diagonals in the Auslander-Reiten quiver of C m Q , where Q is a quiver of type Dn .

We also show the following theorem:

Theorem 2.1.

([JM]

). There is an explicit bijection between the m-diagonals without any self-crossing and the m-rigid indecomposable objects of the m-cluster category.

From now and all throughout the paper, we x such a bijection.

Noncrossing arcs and extensions

In this section, we are going to show in types A, D, Ã and D the following theorem :

Theorem 2.2.1. Let α and β be two arcs in the polygon P . Let X α and X β be the associated m-rigid objects. If ∀i ∈ {1, • • • , m}, Ext i C (X α , X β ) = 0, then α and β do not cross each other. Remark 2.2.2. The result in cases A and D has already been shown by Thomas in [Tho07] and by Baur and Marsh in [START_REF] Baur | A geometric description of m-cluster categories[END_REF] for case A and [BM07] for case D.

We nontheless include a proof as it illustrates the method that will be applied in types à and D.

Our strategy to prove this consists in showing that cutting along an arc corresponds to applying the Iyama-Yoshino reduction. But rst, let us show a useful lemma:

Lemma 2.2.3. Let C be a Hom-nite triangulated category with a Serre functor. Let X ∈ C an m-rigid object. Let Y be an object of C which belongs to X ⊥ . Suppose that C(Y, X) = 0 and for all i ∈ {1, • • • , k}, where k ≤ m, we have

Ext -i C (Y, X) = 0, then ∀i ∈ {1, • • • , k}Y -i Y [-i]
where 1 denotes the shift in C and in the Iyama-Yoshino reduction X ⊥ /(X).

100Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

Proof. We show by induction on

i that Y -i = Y [-i],
where [1] denotes the shift in C and 1 is the shift in the Iyama-Yoshino reduction. It is dened on objects as follows: Let

T (c) k f (c) k G G B (c) k g (c+1) k G G T (c+1) k h (c+1) k G G T (c)
k [1] be the exchange triangles as seen in the preliminaries. If we use the notations of the theorem, we have that

Y = T (c) k . Then Y 1 is in fact the object T (c+1) k in the exchange triangle. First, Y -1 Y [-1]. Indeed, C(X, Y ) = 0.
Let us take an add X -approximation of Y . Then we have the following triangle:

Y [-1] G G Y -1 G G 0 0 G G Y .
As the right morphism is zero, we have that Y -1

Y [-1]. Suppose Y -i + 1 Y [-i + 1]. Then C(X, Y -i + 1 ) = C(X, Y [-i + 1]) = 0
Let us once again take an add X -approximation of Y -i + 1 . Then we have

Y [-i] G G Y -i G G 0 G G Y [-i + 1]. Then Y -i Y [-i]

Cases A and D

We start by a useful lemma.

Denition 2.2.4. We call an m-diagonal α an m-ear, when α divides P into an m+2-gon and an (n-1)m+2-gon for the A case (respectively (n-1)m-m+1-gon containing the interior polygon for case D).

Let Q be the quiver

1 → 2 → • • • → n or n -1 1 G G 2 G G • • • G G n -2 X X 7 7 n .
Lemma 2.2.5. Let C be the m-cluster category of type A n (respectively D n ). Let α be an m-ear. Let X α be the m-rigid object associated with α.

Let U = {Y ∈ C, Ext i C (X α , Y ) = 0}.
Let C be the Iyama-Yoshino reduction of C: C = U/(X α ). Then, we have an equivalence of categories :

C C m Q/α
where Q/α is the quiver obtained from Q by removing α and all incident arrows. Proof. This lemma is a consequence of theorem 3.3.8 of Keller and Reiten.

Let us nd an m-cluster-tilting object T in C satisfying the assumptions of theorem 3.3.8, such that End(T ) KA n-1 (respectively End(T ) KD n-1 ). We recall that we choose the clockwise convention, it means that we draw the arrows of the quiver of an (m + 2)-angulation clockwise. Moreover, we name by 1, the vertex of P which corresponds to the common vertex of the arcs of the rst slice of the Auslander-Reiten quiver of Q (this ensures that 1 is a source in Q).

We know from the bijection between m-rigid objects and m-diagonals that the m-ear α corresponds to a translation of the rst projective module P 1 up to some shift [j], for some j ≤ m. We may thus assume that X α = P 1 . Let T be the sum of all i in Q 0 for i = 1, viewed both as an object in C m Q and of C . We have that End C (T ) KA n-1 . Indeed, rst, we have that End mod (T ) = KA n-1 (respectively KD n-1 ) because X α = P 1 and 1 is a source in Q. Then we show that this remains the same in the higher cluster category by drawing the Auslander-Reiten quiver of Q. Applying the Iyama-Yoshino reduction does not change anything since no morphism is incident to α. Moreover, from Keller and Reiten in [KR08, Section 4], we have that ∀i ∈ {1, • • • , m} Ext -i C (T, T ) = 0. From lemma 2.2.3 and the fact that 1 is a source in Q, we then have ∀i ∈ {1, • • • , m} Ext -i C (T, T ) = 0 Then we have shown the lemma.

Before showing the next lemma, we are going to explicit the bijection {β does not cross α ↔ {β ∈ S/α} in cases A and D. For case A (respectively for case D), if β is an arc which does not cross α, then we can "cut along α" in order to have two new gures of type A (respectively one of type D and one of type A) and one of these contains the same arc as β, which we still call as β.

Lemma 2.2.6. Let P be an nm+2-gon (respectively an mn-m+1-gon) associated with a quiver Q of type A n (respectively D n ). Let α be an m-ear from i to j. Then cutting along α corresponds to applying the Iyama-Yoshino reduction of C m An (respectively C m Dn ) applied on X α . More precisely, let C be the m-cluster category associated with a quiver of type A n (respectively D n ), and let C = U/X α , where 102Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations X α is the m-rigid object associated with α, and

U = {Y, Ext l C ( X α , Y ) = 0 ∀l ∈ {1, • • • , m}}.
Let Q/α be the quiver Q where the vertex corresponding to α and all the incident arrows have been removed. The previous lemma tells us that we have the following result:

C C m Q/α .
Moreover, the following diagram is commutative.

{β does not cross α, β = α} G G y y {X ∈ U; X X α }/ y y {β ∈ S/α} G G {X ∈ C }/
where the second is given by the Iyama-Yoshino reduction. The horizontal arrows are maps sending β to X β . The symbol S/α means the surface obtained from S by cutting along α. Up to homeomorphism, this does not depends on the choice of a representative of α.

Proof. It suces to show that the arcs that cross α exactly correspond to the m-rigid which do not lie in U. Let us then take an m-diagonal β which cuts α (see gure 2.9). Let X β be the associated m-rigid object. Let us show that there exists If we take β, an arc which does not cross α. As there is a bijection between the Auslander-Reiten quiver of Q and the translation quiver built in [START_REF] Baur | A geometric description of m-cluster categories[END_REF] for case A and [START_REF] Baur | A geometric description of the mcluster categories of type D n[END_REF] for case D, the arc β corresponds to a unique object X β situated on the Auslander-Reiten quiver of Q. Cutting the Auslander-Reiten quiver rst, or the translation quiver built by Baur and Marsh rst (and then associating X β with β on the new cut quiver) is the same. Thus, the diagram is commutative.

k ∈ {1, • • • , m} such that Ext k C (X α , X β ) = 0.
Remark 2.2.7. We need to note that the cases are symmetric. Indeed, to shift β k times in order to nd a morphism from α to β[k] is the same as to shift α k times, which gives a morphism from β[k -(m + 1)] to α. Thanks to the (m + 1)-Calabi-Yau property, we know that C(β[k -(m + 1)], α) D(α, β[k]). This means no matter which vertex we shift.

1. In case A: we assume that α is the arc from 1 to m + 2 with no loss of generality.

In the Auslander-Reiten quiver of C, the m-rigid X α is situated at the bottom as we can see in the next picture where we identify an arc with the associated object in the higher cluster category. We give the name of the arcs by D 1j , where the arcs links 1 to j. Moreover, we draw the Hom-hammock in red.

9 9 9 9 D 1 5m+2 @ @ @ @ 9 9 U U D 1 4m+2 T T @ @ D m+1 5m+2 @ @ T T @ @ T T • • • 9 9 U U D 1 3m+2 U U 9 9 D m+1 4m+2 T T @ @ D 2m+1 5m+2 @ @ T T • • • D 1 2m+2 U U 9 9 D m+1 3m+2 T T @ @ D 2m+1 4m+2 T T @ @ D 3m+1 4m+2 @ @ T T X α = D 1 m+2 U U D m+1 2m+2 U U D 2m+1 3m+2 T T D 3m+1 4m+2
T T

D 4m+1 5m+2

If we draw the corresponding arcs on the Auslander-Reiten quiver, we realize that the ones on the slice arising from X α (on the gure, P 2 , P 3 , P 4 , P 5 ) have an end point equal to 1. We note moreover that those are all arcs having 1 as an end. Then β[k] belongs to one of them.

It is also known that these modules exactly correspond to the ones which have a nonzero morphism from X α . Then Ext k C (X α , X β ) = 0.

In case D:

In the Auslander-Reiten quiver of C, the m-rigid X α is situated at the bottom as we can see in the next picture. We name the diagonals by D ij in the same way as in A n case. Both particular diagonals are called B l 1 or B r 1 . We draw the gure for the case i = 1 for the sake of simplicity. 104Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

9 9 9 9 B l 1 9 9 B r 3 8 8 G G 8 8 V V G G G G D 1 4m+2 U U 9 9 G G B r 1 G G D m+1 1 @ @ U U G G B l 3 G G 8 8 W W G G • • • 9 9 U U D 1 3m+2 U U 9 9 D m+1 4m+2 T T @ @ D 2m+1 1 9 9 U U • • • D 1 2m+2 U U 9 9 D m+1 3m+2 T T @ @ D 2m+1 4m+2 T T @ @ D 3m+1 1 8 8 V V X α = D 1 m+2 U U D m+1 2m+2 U U D 2m+1 3m+2 T T D 3m+1 4m+2 U U D 4m+1 1
The Hom-hammock starting at X α contains precisely those X γ 's for which γ contains vertex 1. Then β[k] belongs to one of them. It is also known that these modules exactly correspond to the ones which have a nonzero morphism from X α . Then Ext k C (X α , X β ) = 0.

We have shown that cutting along an arc corresponds to the Iyama Yoshino reduction. Let us now prove theorem 2.2.1.

Proof of theorem 2.2.1. Let us suppose that α and β cross each other. If α is an m-ear, then the result is already shown.

Else, if we can shift β k < m times so that they have one common end point, we prove in a similar way to that of previously, that β[k] is situated on the Homhammock of α, and then there is a nonzero extension from X α to X β . If we cannot shift β k < m times as needed, it means that we can draw an m-ear γ, which does not cross α neither β, and from the previous theorem, we cut along it. By induction, there is some k ∈ {1, • • • , m} such that Ext k C (X α , X β ) = 0. From Iyama-Yoshino, we have that C(X, Y [i]) and C (X, Y i ) are isomorphic and this nishes the proof of the theorem.

Case

Ã

In this subsection, we will use the same sketch of proof. Let us now dene the notion of an m-ear: Denition 2.2.8. Let α be an m-diagonal. Then α is an m-ear if it lies in the outer or inner polygon, and links a vertex i to i + m + 1, and is homotopic to the boundary path (see gure 2.10 for an example of m-ear). Lemma 2.2.9. Let C be the m-cluster category of type Ãn . Let α be an m-diagonal which is either an m-ear or in the transjective component of the Auslander-Reiten quiver of Ãn . Let X α be the m-rigid associated with α.

Let U = {Y ∈ C, Ext i C (X α , Y ) = 0}.
Let C be the Iyama-Yoshino reduction of C: C = U/(X α ). Then, we have an equivalence of categories :

C C m Q/α
where Q/α is the quiver obtained from Q by removing α and all incident arrows.

Remark 2.2.10. We can show exactly the same results if α links two sides of the internal polygon, and is homotopic the the boundary of it.

Proof. There are two dierent cases.

First, if α is an m-ear: Let us begin by illustrating this fact with the Gabriel quivers. The mutation at vertex 1 leads to the following quiver:

1 Ð Ð 3 G G o o 5 0 0 0 U U 0 0 7 2 G G 4 G G 6 d d
Using the Iyama-Yoshino reduction at vertex 1 corresponds to forgetting this vertex and all incident arrows. By doing this, we are ensured to be reduced to a quiver of type Ãn-1 : 106Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

3 G G 5 0 0 0 U U 0 0 7 2 G G 4 G G 6 d d
Here again, we use the theorem of Keller and Reiten in [START_REF] Keller | Acyclic Calabi-Yau categories[END_REF], as in type A and D. We have to nd an m-cluster-tilting object T such that End C (T ) K Ãn-1 , and ∀i ∈ {1, • • • , m}Ext -i C (T, T ) = 0. Let T = ⊕P i . We know from Torkildsen (see gure 8 in [START_REF] André | A geometric realization of the m-cluster category of type[END_REF]), that T corresponds summand by summand, to the (m + 2)-angulation made of all mdiagonals linking the external polygon to the internal one (see gure 2.11) Figure 2.11 The "initial" (m + 2)-angulation of type Ã, for m = 4, p = 12, q = 6.

Let T be the mutation of T at P 1 the rst preprojective module. Then T = ⊕ i =1 P i ⊕ X is also an m-cluster-tilting object. Let us show that τ X corresponds to the simple module at the base of the rst tube (see gure 2.12 to visualize the mutation in terms of arcs). However, we do not know yet that the mutation of m-cluster tilting objects corresponds to the ip of (m + 2)-angulations.

-→ Figure 2.12 Geometric visualization of T

We have to show that τ X = S k , the simple module in k, which is situated at the bottom of the tube of size n -2.

Let us nd C(X[-1], T ). For all i = k, C(X[-1], P i ) C(X, P i [1]) = 0 since T is an m-cluster-tilting object. Now we focus on C(X[-1], P k ). We have that C(X[-1], P k ) C(X, P k [1]) K from Iyama and Yoshino in [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF].

From the (m + 1)-Calabi-Yau property, we have

C(X[-1], T ) C(X, T [1]) DC(T, τ X)
Then τ X S k and X = X α corresponds to the red arc, named α. Moreover, from the paper of Baur and Torkildsen, we can easily visualize the morphisms in the module category of type Ã.

We have End C (T ) K Ãn-1 . Indeed, in the module category, End mod (T ) K Ãn-1 , because, the objects of T (apart from X α ) are on the projective slice of the Auslander-Reiten quiver of Q. Thus there is no relation. If any morphism f : T → T factorizes through P k , then f = uv where T u G G P k v G G T and this is impossible given that there is no morphism from a regular module to a preprojective one. Now we show this for the higher cluster category. We have the following decomposition of morphisms (G is the functor τ -1 [m]):

C(M, N ) i∈Z D b (G i M, N ).
If m = 1, the result is already known, because if X is a preprojective object and Y a regular one, then Ext C (X, Y ) = 0.

If not, we use the decomposition just above. For i ≥ 1, we have that

D b (τ -1 [m]T, T ) = DD b (T, T [m + 1])
thanks to the duality. From the book [START_REF] Assem | Elements of the representation theory of associative algebras[END_REF], the algebra of a quiver of type à is hereditary and then the extension Ext 2 D b (T, T ) = 0. Then, for i ≥ 1, all the terms of the sum are zero. Then

End C (T ) End D b (T ).
Applying the Iyama-Yoshino reduction does not change anything since in the higher cluster category, there is no morphism incident to α.

It nally remains to prove that Ext -i C (T , T ) = 0 for all i ∈ {1, m -1}. Let us rst show that Ext -i C (T , T ) = 0, using the shift in C. Then we will use lemma 2.2.3 in order to conclude. 108Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

We claim that C(T , T [-i]) = 0. Indeed, from Keller and Reiten in [START_REF] Keller | Acyclic Calabi-Yau categories[END_REF], we know that C(⊕ j =k P j , T [-i]) = 0. Moreover, as X α and X α [-i] are not in the same tube, then C(X α , X α [-i]) = 0 in the module category (which from Wraalsen [Wr a09] or Zhou-Zhu [START_REF] Zhou | Cluster combinatorics of d-cluster categories[END_REF], immediately translates to the higher cluster category). In addition, there cannot be any morphism from a tubular component to a preprojective one. Then C(X α [-i], P j ) = 0 in the module category (which from Wraalsen [Wr a09] or Zhou-Zhu [START_REF] Zhou | Cluster combinatorics of d-cluster categories[END_REF], immediately translates to the higher cluster category). It remains to show that C(P j , X α [-i]) = 0 for any j = k and i ∈ {1, • • • , m -1}. By denition of the mutation, we have the exchange triangle

P k [-i] G G U [-i], U ∈ add ⊕ l =k P l G G X α [-i] G G P k [-i + 1] P j ∃ g i i f y y 0 V V . For i = 1, there is no morphism from P j to P k [-i + 1].
Then, there exists g :

P j → U [-i].
As U is only composed with projectives which are not P k , this shows that g = 0. Then f = 0.

For i = 1, the composition P j → X α [-1] → P k is zero because there is no morphism from tubular objects to preprojective objects. Then there exists g such as previously, but the composition with

U [-i] → X α [-i] is zero for the same reason. Then f = 0. This shows that Ext -i C (T , T ) = 0. From lemma 2.2.3, we have that T l < -i > T l [-i]. Finally, Ext -i C (T , T ) = 0.
We now have gathered all the information in order to apply the theorem of Keller and Reiten, and we have that

C C m Q/α .
Else, if α corresponds to a transjective module, we proceed in the same way, we have that Ext -i C (T , T ) = 0 from Keller and Reiten [KR08, Lemma 4.1], and we can apply Keller-Reiten theorem. In details, let T be the m-cluster-tilting object corresponding to a slice of the Auslander-Reiten quiver of Q (see the article of Baur and Torkildsen [START_REF] Baur | A geometric realization of tame categories[END_REF] for details). Then as there is an isomorphism between the Auslander-Reiten quiver of Q (except the homogeneous tubes) and the translation quiver Γ built by Baur and Torkildsen in [BT15, Proposition 3.7], the morphisms in the module category from T to T (where T is a slice of the Auslander-Reiten quiver) correspond to a quiver of type A n-1 . Then we have End mod (T ) KA n-1 . Lemma 2.2.11. Let α be an m-ear. Then cutting along α corresponds to applying the Iyama-Yoshino reduction. More precisely, the application which, with an indecomposable rigid object of the higher cluster category, associates an m-diagonal, induces a map {rigid indecomposable modules of U} → {m-diagonals which do not cross α}. Remark 2.2.12. We need to note that the cases are symmetric. Indeed, to shift β k times in order to nd a morphism from α to β[k] is the same as to shift α k times, which gives a morphism from α to β[m + 1 -k] thanks to the (m + 1)-Calabi-Yau property. This means, we do not care about which vertex we shift. Proof. Here, as in cases A and D, it suces to show that, if β is an m-diagonal crossing α, then we can nd a morphism from

X α to X β [k], for some k ∈ {1, • • • , m}.
By the geometric realization of Torkildsen in [START_REF] André | A geometric realization of the m-cluster category of type[END_REF], the m-diagonal α is situated at the bottom of the rst tube. As β crosses α, we can shift it k < m times so that one end point of β is in common with one of α. There are two cases:

First case: β corresponds to an m-rigid in a tube. Then, by the proof of Proposition 7.2 in [START_REF] André | A geometric realization of the m-cluster category of type[END_REF], there exists a nonzero morphism from X α to X β [k] (see gure 13 of the article for a clear picture of this map).

Second case: β corresponds to a preinjective arc. Then, by the paragraph 4.1 of the article written by Baur and Torkildsen [START_REF] Baur | A geometric realization of tame categories[END_REF], as α and β[k] share an oriented angle, there is a so-called "long move", hence a nonzero morphism in the module category from X α to X β [k].

In any case, we have found a nonzero morphism in the higher cluster category from X α to X β [k]. Then the arcs which cross α exactly correspond to the rigid which do not lie in U. Then the Iyama-Yoshino reduction corresponds to cutting along an arc.

We are now able to prove theorem 2.2.1: Proof of theorem 2.2.1. If α and β are two crossing m-diagonals in the geometric realization of a quiver of type à (an external polygon P with p sides together with an internal polygon R with q sides). There are two cases:

1. First case: The m-diagonal α links two vertices of P , and is homotopic to the boundary path (or in the same way, α lies in the inner polygon). If α is an m-ear, then the result is shown. Else:

-If it is impossible to draw an m-ear between both end points of α without crossing β, then it means that α is an m-ear, and the result is shown. It suces to count the vertices to make sure of it: if α is not an m-ear, it cuts the polygon into a u-gon, with u ≥ 2m + 2. Then, at one side of β, there is at least m + 1 free vertices, where we can draw an m-ear.

110Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations -Else we can draw an m-ear γ between an end point of α and an end point of β, then it suces to cut along γ and repeat the operation as many times as necessary, in order to reduce to the previous case. See gure 2.13 for an illustration. (a) If β is homotopic to the boundary path of one of the polygons (let us say for instance that β is homotopic to the boundary path of the external polygon). Then, we use the same type of argument.

-If we cannot draw an m-ear lying inside the end points of β which does not cross α, then it suces to shift β k < m times in order to hang one end point of β to one end point of α. This corresponds to a long move, then to a morphism in the module category in the sense of Baur and Torkildsen in [START_REF] Baur | A geometric realization of tame categories[END_REF].

-Else, we cut along this m-ear, and repeat the operation as many times as necessary to reduce to the rst case. -Else, there exists an m-ear γ which does not cross α nor β. It suces to cut along γ and repeat as many times as necessary in order to reduce to the previous case.

Case

D

Denition 2.2.13. Let P be a polygon with (n -2)m sides with two m -1-gons inside of it, associated with a quiver of type Dn . Then, an m-ear is an m-diagonal linking a vertex i to the vertex i + m + 1 homotopic to the boundary of P . Lemma 2.2.14. Let P be a polygon with (n -2)m sides associated with a quiver Q of type Dn and let α be an m-ear. Then the Iyama-Yoshino reduction of C m Dn applied on X α corresponds to cutting along α. More precisely, let C be the mcluster category associated with a quiver of type Dn , and let C = U/X α , where X α is the m-rigid object associated with α, and

U = {Y, Ext l C ( X α , Y ) = 0 ∀l ∈ {1, • • • , m}}.
Let Q/α be the quiver Q where the vertex corresponding to α and all the incident arrows have been removed. Then we have the following result:

C C m Q/α .
Let us begin by illustrating this fact with the Gabriel quivers. The mutation at vertex k leads to the following quiver:

1 0 0 n -1 | | 3 G G • • • G G k -1 T T A A k 2 d d n
Using the Iyama-Yoshino reduction at vertex k corresponds to forget this vertex and all incident arrows. By doing this, we are ensured to be reduced to a quiver of type Dn-1 :

1 0 0 n -1 3 G G • • • G G k -1 T T @ @ 2 d d n
Proof. Here again, we use theorem 3.3.8 of Keller and Reiten.

We are going to use this theorem, by building a new m-cluster-tilting object respecting the hypotheses.

Let T = P i be the sum of all projective modules. We know that T is an m-cluster-tilting object. This object is naturally associated with the initial (m + 2)-angulation.

112Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations Let P k be the following projective module:

K 2 2 0 K G G • • • G G K G G K c c 1 1 K b b 0
From Iyama and Yoshino in [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF], we have an exchange triangle:

P k → Y → X → P k [1] (2.1)
where Y ∈ add j =k P j . We mutate the object T as Wraalsen and Zhou, Zhu in [Wr a09] and [START_REF] Zhou | Cluster combinatorics of d-cluster categories[END_REF] at P k in order to study the new m-cluster-tilting object

T = j =k P j ⊕ X.
Let us rst show that X corresponds to the arc α, which is the arc obtained by ipping the arc of type 1 corresponding to the vertex k of the Auslander-Reiten quiver (see gure 2.14). We have to show that τ X = S k , the simple module in k, which is situated at the bottom of the tube of size n -2 as we set in the previous section.

Let

us nd C(X[-1], T ). For all i = k, C(X[-1], P i ) C(X, P i [1]) = 0 since T is an m-cluster-tilting object. Now we focus on C(X[-1], P k ). Wa have that C(X[-1], P k ) C(X, P k [1]
) K from Iyama and Yoshino in [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF].

From the (m + 1)-Calabi-Yau property, we have

C(X[-1], T ) C(X, T [1]) DC(T, τ X)
Then τ X S k and X = X α corresponds to the arc α.

It now remains to check the hypotheses of Keller-Reiten's theorem. First, C is a Hom-nite algebraic (m + 1)-Calabi-Yau category. The object T is our candidate. It is still an m-cluster-tilting object. First of all, from [KR08, Lemma 4.1], the morphisms C(P j , P l [-i]) are zero for any j and l. Moreover, as X and X[-i] are not in the sale tube, we have C(X, X[-i]) = 0. In addition, C(X[-i], P j ) = 0 as there is no morphism from a regular object to a preprojective object. It now remains to show that C(P j , X[-i]) = 0 for any j = k and any i ∈ {1, • • • , m}. We then have the following diagram:

P k [-i] G G U [-i], U ∈ add ⊕ l =k P l G G X α [-i] G G P k [-i + 1] P j ∃ g i i f y y 0 V V .
and we conclude in the same way as in type Ã.

If i = 0, we remark that there can be no relations or factorizations in the slice taken, which means , we have that C(T, T ) K Dn-1 (see illustration at the beginning of the proof), and this permits to apply Keller-Reiten's Theorem and nishes the proof. Lemma 2.2.15. Let α be an m-ear. Then cutting along α corresponds to applying the Iyama-Yoshino reduction at X α . More precisely, the application which, with an indecomposable rigid object of the higher cluster category, associates an mdiagonal, induces a map {m-diagonals which do not cross α} → {rigid indecomposable modules of U}. Proof. Here again, it suces to show that, if β is an arc cutting α, then there exists some k ∈ {1, • • • , m} such that Ext k C (X α , X β ) = 0. Let β be an arc crossing α. Then we can shift β k < m times in order for both of them to share a common ending vertex a.

If β[k] is in a tube of size n -2, then it is situated in the same tube as α, higher than it. Then there exists a morphism from

X α to X β [k].
Else, β[k] is situated in the preinjective part of the Auslander-Reiten quiver. Whatever the type of β, is it in a slice of the preinjective part. Then it suces to show that there is a morphism from X α to the head of this slice, it means in our orientation, to A 1 in the following quiver. 114Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

A8 A6 b b G G A7 Xβ[k] À4 À1 b b G G 2 2 A2 A3
We note that β[k -1] is exactly the arc corresponding to τ -1 P l for an l ∈ {1, • • • , n + 1}. We can prove the existence of a morphism in the module category from the simple regular X α to τ -1 P l for any l.

To draw an example, in case n = 7, let us give the dimension vectors of τ -1 P l , for each l: They are

1 0 0 0 Ð Ð 2 1 o o 1 o o 1 d d 1 ; 1 0 0 1 Ð Ð 2 1 o o 1 o o 1 d d 0 ; 1 0 0 1 Ð Ð 3 2 o o 2 o o 1 d d 1 ; 1 0 0 0 Ð Ð 2 1 o o 1 o o 1 d d 0 ; 1 0 0 0 Ð Ð 2 1 o o 0 o o 1 d d 0 ; 1 0 0 0 Ð Ð 1 1 o o 0 o o 0 d d 0 ; 0 0 0 0 Ð Ð 1 1 o o 0 o o . 1 d d 0
In any case there is a morphism from the simple

0 0 0 0 Ð Ð 1 0 o o 0 o o 0 d d 0
to any of the τ -1 P l . We can have a deeper analyse in the article of Dlab and Ringel in [START_REF] Dlab | Representations of graphs and algebras[END_REF].

We now generalize this result to all arcs excepted the one in the tubes of size 2.

Lemma 2.2.16. Let P be a polygon with (n -2)m sides associated with a quiver Q of type Dn and let α be an arc which corresponds to a regular module in a tube of size n -2. Then α cuts the gure into a polygon T on the one hand and another gure of type D n for some n < n on the other hand. Let α 1 , • • • , α k be arcs lying in T , such that, if we cut along α i , then α i+1 becomes an m-ear. Then the successive Iyama-Yoshino reduction of C m Dn applied on the X α i corresponds to cutting successively along the α i . More precisely, let C be the m-cluster category associated with a quiver of type Dn , and let C = U/ X α i , where X α i is the mrigid object associated with α i , and

U = {Y, Ext l C ( X α i , Y ) = 0 ∀l ∈ {1, • • • , m}}.
Let Q/α 1 , • • • , α n be the quiver Q where the vertex corresponding to α 1 , • • • , α n and all the incident arrows have been removed. Then we have the following result:

C C m Q/α 1 ,••• ,αn .
Proof. We have that α is an arc linking two dierent vertices i and j and α is homotopic to the boundary path.

If α is an m-ear, this is exactly the previous lemma. Else, it means that j > i + m + 1. Then there exists an m-ear from i to i + m + 1 which does not cut α. We use Iyama-Yoshino reduction in order to cut along this m-ear. We do this operation again as many times as necessary, to reduce n until α becomes an m-ear. We are ensured that the process stops since α cuts the polygon into a km-gon with both m -1 gons inside of it on the rst side, and into a km + 2-gon of type A on the other side. This shows the result if α is in a tube of size n -2.

Lemma 2.2.17. Let P be a polygon with (n -2)m sides associated with a quiver Q of type Dn and let α be an arc which is associated to an m-rigid object lying in the transjective component of the Auslander-Reiten quiver of C m Q . Then the Iyama-Yoshino reduction of C m Dn applied on X α corresponds to cutting along α. More precisely, let C be the m-cluster category associated with a quiver of type Dn , and let C = U/X α , where X α is the m-rigid object associated with α, and

U = {Y, Ext l C ( X α , Y ) = 0 ∀l ∈ {1, • • • , m}}.
Let Q/α be the quiver Q where the vertex corresponding to α and all the incident arrows have been removed. Then we have the following result:

C C m Q/α .
Moreover, the following diagram is commutative.

{β does not cross α, β = α} G G y y {X ∈ U; X X α }/ y y {β ∈ S/α} G G {X ∈ C }/
where the rst vertical bijection is given in the way of Marsh and Palu in [START_REF] Marsh | Coloured quivers for rigid objects and partial triangulations: the unpunctured case[END_REF], and the second is given by the Iyama-Yoshino reduction. The horizontal arrows are maps sending β to X β . Proof. We have that α is an arc situated in the preprojective (or preinjective) part of the Auslander-Reiten quiver of Q.

Then, we complete α into an (m + 2)-angulation ⊕α i composed by the slice containing α in the Auslander-Reiten quiver of Q. Then all the arcs are preprojective (or preinjective), and we can use the theorem of Keller and Reiten in [KR08, Theorem 4.2]. Indeed, let T = ⊕X α i , where X α i corresponds to the arcs α i . We 116Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations know that T is an m-cluster-tilting object (because, by applying τ several times if necessary, we go back to the rst slice, which contains the projective modules, and is actually an m-cluster-tilting object). Moreover, Ext -i C (T , T ) = 0. Moreover, we can check that End C (T ) = KQ T as in the previous lemma. Then we can apply the theorem of Keller and Reiten, and this shows the result.

Moreover, the diagram is commutative since the application β → X β is the same, no matter if we consider the surface S or the surface cut S/α (this applications does not depends on the type of the surface). Then the upper arrow is the same as the lower arrow.

We now state a technical lemma which helps us to nd morphisms between two m-rigid objects.

Lemma 2.2.18. Let α and β be two m-diagonals. Suppose that there exists an (m + 2)-angulation ∆ which contains α and not β such that µ α (∆) contains β (it means that there exists ∆ such that

β = κ i ∆ (α), for an i ∈ {1, • • • , m -1}). Then Ext i C (X α , X β ) = 0 where X α (respectively X β )
is the m-rigid object associated with α (respectively β) thanks to the bijection between m-rigid objects and m-diagonals. Proof. We number the arcs in ∆ and consider that α corresponds to k. We use Calabi-Yau reduction in order to prove the statement. Let us introduce C = U/ j =k X j , where

U = {Y, Ext l C (⊕X j , Y ) = 0 ∀l ∈ {1, • • • , m}}.
By Iyama and Yoshino in [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF], we know that C is triangulated and

(m + 1)- Calabi-Yau. If X → Y → Z → X[1] is a triangle in C, where X → Y is a ⊕X j -left approximation, then Z is isomorphic to the shift of X in C .
From the previous lemma, we know that the Iyama-Yoshino reduction corresponds to cutting along an arc. We can suppose that ∆ does not contain any arc lying in a tube of size 2. In this way we have cut along all the arcs of the (m + 2)angulation excepted α. Then, as β is the i-th twist of α, it becomes the i-th shift in the reduced category. From Iyama and Yoshino in [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF],

X β = Σ i C X α . Then Ext i C (X α , X β ) Ext i C (X α , X β ) = 0 (the rst equivalence is due to the fact that 1 ≤ i ≤ m).
Before showing the main lemma of this section, we show that we can reduce to the case D4 . The following lemma show that we can reduce to cases where n ≤ 6, and the next remark treats cases n = 5 and n = 6. Lemma 2.2.19. Suppose that n > 6. Let α and β be two crossing arcs in the (n -2)m-gon realizing Dn . Then there exists at least (n -4) m-ears which do not cut α neither β.

Proof. The case where α and β are of type one is the most dicult. The arcs cut the polygon P into 4 parts. If we cannot draw an m-ear between one of the parts, it means that the number of vertices strictly contained in a part is at most m -1 in each part. Then the total number of vertices is at most 4(m -1) + 4. Then (n -2)m ≤ 4m this means n ≤ 6.

Remark 2.2.20. If n = 5 or n = 6, then the only case where we cannot reduce to D4 is when α and β are of type 1. But at this moment there exists k < m such that α = β[k], then there exists a nonzero extension between α and β. Lemma 2.2.21. Let α and β be two arcs in an (m + 2)-angulation ∆. Let X α and X β be their associated m-rigid object.

If ∀i ∈ {1, • • • , m}, Ext i C (X α , X β ) = 0,
then α and β do not cross. Proof. We recall that we are in the case where n = 4, it means that we study a 2mgon We show that if α crosses β, then Ext i C (X α , X β ) = 0, for some i ∈ {1, 118Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

As we have seen previously, we only have to show the result for D4 . Then P has 2m vertices. Remark 2.2.22. We need to note that the cases are symmetric. Indeed, to shift β k times in order to nd a morphism from α to β[k] is the same as to shift α k times, which gives a morphism from α to β[m + 1 -k] thanks to the (m + 1)-Calabi-Yau property.

First, we have to notice that cases 4,7,9 are already treated from lemmas 2.2.15 and 2.2.16.

Case 1: α and β are of type 1 (cf gure 2.15). Let i and j be the closest vertices of the 2m-gon, where i is an end of α and j is an end of β.

We have either j ≤ i+m+1 or i ≤ j +m+1, then β can be shifted k < m times in order to hang to one end point of α. Then α and β[k] share an oriented angle. As they are of type 1, and share an oriented angle, they are on the same slice of the Auslander-Reiten quiver. Then, this is a nonzero composition of arrows. In this way, we have found a nonzero morphism from Σ k α to β. Then Ext k C (X α , X β ) = 0.

Case 2: α is of type 1 and β is of other type (cf gure 2.16). This case is similar to that of the rst one. It suces to shift α k < m times in order to hang both arcs to the same vertex. Consequently, they do not cross a mesh in the Auslander-Reiten quiver. Then, there is a Hom-hammock from one to another. Then there is a nonzero extension from α to β. In this case, it is more dicult to see morphisms in the Auslander-Reiten quiver of Q because one arc is in the transjective component and the other is in a tube. Nonetheless, if we can nd an (m + 2)-angulation where β is the i-th twist of α, then from lemma 2.2.18, there is an extension which is nonzero. We have to complete α to an (m + 2)-angulation containing this arc (see gure 2.19):

As β is the i-twist of α, then there exists i ∈ {1, • • • , m} such that

Ext i C (X α , X β ) = 0.
The case where α is in the tube and β is of type 1 is similar.

Case 5: If both α and β are of other type (cf gure 2.20). We can move β k < m times in order to hang its end to α. Then the composition of elementary moves in gure 2.21 is not zero since it follows a slice of the Auslander-Reiten quiver (so do not cross a mesh).

Then there is a nonzero extension between α and β. The same arguments as in case 3 lead to nd an (m + 2)-angulation containing these arcs in gure 2.23.

Here again, there exists a nonzero extension between X α and X β . The inverse case is similar.

Case 8: If we are in a tube of size 2 (cf gure 2.24) If they cross each other, it means that they are in the same tube. Then one is situated higher than the other and there exists a Hom-hammock between them. In any case, we have shown that if α crosses β, then there exists k such that Ext k C (X α , X β ) = 0.

Compatibility ip/bijection between m-clustertilting objects and (m + 2)-angulations

With theorem 2.2.1, we are able to dene an (m + 2)-angulation from an mcluster-tilting object.

Denition 2.3.1. Let T = ⊕T i be an m-cluster-tilting object, and T i its m-rigid components. With each T i we associate α i the corresponding m-diagonal. We know that the α i do not cross each other from the previous section. Then the set {α i , i ∈ {1, • • • , m}} form a maximal set of noncrossing m-diagonals, then an (m + 2)-angulation, called ∆ T , the (m + 2)-angulation associated with T .

We rst show the theorem of compatibility between the ip of an (m + 2)angulation, and the mutation of an m-cluster-tilting object.

Theorem 2.3.2. Let ∆ be an (m + 2)-angulation. Let X be its associated object. Let µ i be the ip at the arc α i in ∆ as well as the mutation of the m-cluster-tilting object X at summand i. Then we have:

µ i (∆) = ∆ µ i (X)
Proof. The ip at vertex α i aects only α i itself. In the same way, the mutation of X aects only the i-th component of the m-cluster-tilting object.

By Buan and Thomas in [BT09], we know that there is a triangle

X i → B (0) i → X
(1) i →, where X i is the m-rigid object corresponding to α i and B (0) i ∈ addT . The aim is to show that X

(1) i X ĩ, where X ĩ is the m-rigid corresponding to the arc αi which is the twist of α i .

Let X = X/X i be the almost m-cluster-tilting object. Then from Wraalsen ([Wr a09]) and Zhou,Zhu ([ZZ09]), X has m + 1 complements. Let ∆ be the 122Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations "almost" (m + 2)-angulation, containing all arcs of ∆ except α i . Then by 1.4.1, ∆ corresponds to X.

Let U = {Y ∈ C m Q , ∀k ∈ {1, • • • , m}, Ext k C (X, Y ) = 0}.
Then by lemma 2.2.21, an object in U corresponds to an arc which does not cross ∆, it means that there are m + 1 possibilities of remaining arcs in order to have an (m + 2)-angulation. In a way similar to that of Marsh and Palu in [START_REF] Marsh | Coloured quivers for rigid objects and partial triangulations: the unpunctured case[END_REF], studying C = U/X corresponds to cut along the arcs of ∆ Then by Keller, and Iyama and Yoshino in [START_REF] Keller | On triangulated orbit categories[END_REF] and [START_REF] Iyama | Mutation in triangulated categories and rigid Cohen-Macaulay modules[END_REF], C is a triangulated, hom-nite, algebraic and (m + 1)-Calabi-Yau category. Moreover, each arc which does not cross ∆ is an m-cluster-tilting object in C . In addition,

Ext k C (α i , α i ) = Ext -k C (α i , α i ) = 0 for all k ∈ {1, • • • , m} since α i does not cross itself. The algebra End(α i ) = K is hereditary since it is of global dimension 0.
Then, by [START_REF] Keller | Acyclic Calabi-Yau categories[END_REF], theorem 4.2, we have an equivalence

C

C m A 1 . Therefore we have a distinguished triangle

α i → E i → Σ C α i → Σα i ,
where E i is the set of arcs which follow α i in the sense of its quiver and where Σ C α i is the shift in the category C m A 1 , which means the shift in the remaining (2m + 2)-gon. Then it follows that Σ C α i = αi . Then we have two distinguished triangles:

X i G G ∼ B (0) i G G ∼ X (1) i G G ΣX i ∼ α i G G E i G G αi G G Σα i
By TR3, the third axiom of triangulated categories, we have a morphism

X (1) i → αi X i G G ∼ B (0) i G G ∼ X (1) i G G ΣX i ∼ α i G G E i G G αi G G Σα i
By the ve lemma applied to triangulated categories, we have an isomorphism

X i G G ∼ B (0) i G G ∼ X (1) i G G ∼ ΣX i ∼ α i G G E i G G αi G G Σα i 2.
3. Compatibility ip/bijection between m-cluster-tilting objects and (m+2)-angulations123

Then we have shown that X

(1) i X ĩ.

Lemma 2.3.3. Let α and β be any m-diagonals in the the category C. Then ∀i ∈ {1, • • • , m}, Ext i C (X α , X β ) = 0 ⇔ α and β do not cross. Proof. Associating an (m + 2)-angulation with an m-cluster-tilting object in the natural way above is the same thing as associating an (m + 2)-angulation with an m-cluster-tilting object in the following way:

Take X an m-cluster-tilting object. From Buan and Thomas in [BT09, Proposition 7.1], we can introduce f as a sequence of ip necessary to bring X to the sum of the projective modules P i . We associate with P i the initial (m + 2)angulation, and ip back to ∆ X via the inverse sequence of ips.

From theorem 2.3.2, these two ways of dening an (m + 2)-angulation from an m-cluster-tilting object are the same. Then we can show the result:

If α and β do not cross each other, we can complete them into an (m + 2)angulation ∆. We associate with ∆ an m-cluster-tilting object by applying ips on ∆ as told at the beginning of this proof, and this shows that the associated object is m-cluster-tilting. Then ∀i ∈ {1, • • • , m}, Ext i C (X α , X β ) = 0. Finally, we show the bijection between (m+2)-angulations and m-cluster-tilting objects.

Theorem 2.3.4. The natural application from (m + 2)-angulations to m-clustertilting objects induces a bijection between these two notions. Proof. By lemma 2.3.3, with an (m + 2)-angulation, we associate a unique mcluster-tilting object. Therefore the application is well-dened. Let us call by Φ this function.

First, Φ is injective since, if we take X an m-cluster-tilting object, we can associate a unique (m + 2)-angulation. Indeed, X = X i , where the X i are mrigid. With each summand X i , we associate the corresponding arc α i . By theorem 2.2.1, as the X i are m-rigid, we know that the α i do not cross, and they are n + 1, so they form a maximal set of noncrossing arcs, thus an (m + 2)-angulation. It is uniquely dened. So Φ is injective.

Finally, we show that Φ is surjective. If X is an m-cluster-tilting object, then it exists f , a sequence of ips, such that f (X) = P i . Moreover, P i = Φ(∆ init ), then f (X) = Φ(∆ init ). Let g be the inverse sequence of ips of f . Then, X = g(Φ(∆ init )) and by theorem 2.3.2,

X = Φ(g(∆ init )).
124Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations This nishes to show the bijection between m-cluster-tilting objects and (m + 2)angulations.

We can summarize all the important properties between m-cluster-tilting objects, colored quivers, and (m + 2)-angulations in the following diagram: We now nish this section with a direct consequence of this diagram.

Theorem 2.3.5. Let ∆ be an (m+2)-angulation. Let Q ∆ be the associated colored quiver. Let X ∆ be the m-cluster-tilting object associated with ∆, and let Q X ∆ be the quiver associated with X ∆ in the sense of Buan and Thomas in [START_REF] Bakke | Coloured quiver mutation for higher cluster categories[END_REF]. Then

Q ∆ = Q X ∆
Note here that theorem 2.1.12 is a direct consequence of theorems 2.3.2 and 3.4.5.

Part II

Model categories structures from rigid objects in exact categories Lemma 1.1.1. Assume that the rigid subcategory M, contains all injectives and all projectives. Then, for any X in E, there exist A in prM and a right prM approximation A → X. Proof. Let X ∈ E. Let M 0 → X be an M-approximation of X. Since E is weakly idempotent complete, with enough projectives and M contains the projective objects, the morphism M 0 → X is a deation. Let 0 → K 0 → M 0 → X → 0 be the associated short exact sequence. Similarly, let

0 → K 1 → M 1 → K 0 → 0
be a short exact sequence coming from an M-approximation of K 0 . Then we have the following diagram:

0 K 1 β M 1 α M 1 0 G G K 0 b G G M 0 a G G X G G 0 0 We have a • b • α = 0.
Let A be the push-out of the square:

M 1 bα G G M 0 r a I M 1 G G 0 G G A ∃ϕ 2 2
X Then there exists a morphism ϕ : A → X such that ϕ • r = a and the other triangle commutes.

We have A ∈ prM. Indeed, we have the following short exact sequence:

0 → M 1 → I M 1 ⊕ M 0 → A → 0.

Weak equivalences and brations 129

Moreover, A → X is an approximation. Indeed, let i : B → X be a morphism, with B ∈ prM. Let us show that there exists B → A which makes the triangle commute.

Let 0 → M 1 → M 0 → B → 0 be a short exact sequence with M 0 , M 1 ∈ M. We have the following diagram:

M 1 } } ∃δ t t M 1 ι M 1 abα @ @ M-app G G K 0 b M 0 M-app j } } ∃ε t t I M 1 G G 0 @ @ A l 2 2 M 0 o o a B h h i } } M 1 X
As a is an M-approximation, then there exists a morphism j : M 0 → M 0 which makes the lower-right square commute. Then, there exists a morphism M 1 → K 0 which makes it a morphism of short exact sequences.

As M 1 → K 0 is an M-approximation, then there exists δ : M 1 → M 1 which makes the upper triangle commute. Since I M 1 is injective and M 1 → M 0 is an ination, there exists a morphism ε : M 0 → I M 1 which leads to a morphism of short exact sequences.

All the conditions are required to build a morphism k : B → A such that l • k = i + 0 since I M 1 → A → X = 0. Then we have shown the result.

Weak equivalences and brations

Denition 1.2.1. We recall that G is the functor

G : E → ModM X → E(-, X)|M
We call by W, the weak equivalences, the class of morphisms f for which Gf is an isomorphism. Denition 1.2.2. Let f : X → Y and g : A → B be two morphisms. We say that f g when, for any commutative square

A G G X B G G b b
Y there exists a morphism B → X such that both triangles commute. For a class of morphisms A, we call by A = {g, ∀f ∈ A, f g} and A = {f, ∀g ∈ A, f g}.

Let J = {f : 0 → M, M ∈ M}.
The morphisms of J are called brations and compose the class Fib.

The next lemma shows that the prM-approximation constructed in lemma 1.1.1 is actually a weak equivalence. This permits to take cobrant replacements as we will see later.

Lemma 1.2.3. Let h : A → X be the prM-approximation constructed in lemma 1.1.1. Then h ∈ J ∩ W. Proof. We rst show that h ∈ J . Let M → X be a morphism. Since M ∈ prM, and A → X is a prM-approximation, there automatically exists a lift as wanted.

Next, we have to show that the morphism E(-, A)|M → E(-, X)|M is an isomorphism. It is surjective, since if we take M → X a morphism, since M ∈ prM, there exists a lift as we have seen in order to show that h is a bration.

Then, if a : M → A is a morphism such that h • a = 0 (we will see later the case where this morphism factorizes through an injective module). Using the same notations as in lemma 1.1.1, as M is rigid, there exists b

1 b 2 : M → I 1 ⊕ M 0 such that π • b 1 + r • b 2 = a where π : I 1 → A. As α : M 1 → K 0 is an M-approximation, there exists c : M → M 1 such that b • α • c = b 2 . Then we have a = π • b 1 + r • b 2 so a = π • b 1 + r • b • α • c. By the pushout of lemma 1.1.1, r • b • α = π • ι 1 where ι 1 : M 1 → I 1 . Then a = π • b 1 + π • ι 1 • c.
This shows that a factorizes through an injective module.

Finally, if we suppose that h • a factorizes through an injective J, for example h • a = µ • ν, as J ∈ prM, there exists ã such that h • ã = µ. We then proceed as above with the morphism a -ã • ν. This nishes to show the result.

We recall that trivial brations are those morphisms that are both brations and weak equivalences. Lemma 1.2.4. Let f : X → Y be a trivial bration. If α is a morphism from an element M of M to Y , then there exists β : M → X such that f • β = α. Proof. As f is a weak equivalence, there exists β : M → X, ι M : M → I M and

γ : I M → Y such that α + γ • ι M = f • β. As f ∈ J and I M ∈ M, there exists δ : I M → X such that f • δ = γ. Then we have α = f • ( β -δ • ι M ).
Lemma 1.2.5. Let f : X → Y be a trivial bration. Then it is automatically a deation. Proof. Let P Y be a projective cover of Y . As P Y ∈ M, and f is a trivial bration, from the previous lemma, there exists a lift from P Y → X. From Bühler in [Büh10, Proposition 7.6, (ii)], as P Y → Y is a deation, then f is a deation.

Factorization

Let us now show a characterization of the morphisms of (J ).

Lemma 1.3.1. Suppose that M → Y is a right M-approximation. A mor- phism f : X → Y is in (J ) if and only if it is a retract of the canonical injection X → X ⊕ M . Proof. Let f : X → Y ∈ (J ). Let α : M → Y be a M-approximation of Y .
Then, we have the following commutative square:

X ( 1 0 ) G G f X ⊕ M (f α) Y s X X Y The morphism (f α) ∈ J . Indeed, if M → Y is a morphism,
as α is a Mapproximation, there exist a lift as wanted (which is zero on X). As f ∈ (J ), there exist s : Y → X ⊕ M which makes both triangles commute. Then, f is a retract of the canonical injection X → X ⊕ M .

Conversely, it is well-known that the lifting property is stable under retract.

Lemma 1.3.2. Under the assumption that there exist some M-approximations, any morphism can be factorized through a morphism in (J ) followed by a morphism in J . Proof. Let f : X → Y be a morphism. It factorizes through X → X ⊕ M → Y by 1 0 and f α , where α is a M-approximation (this morphism is a retract of itself). The rst morphism is in (J ). As f α is a M-approximation, it satises the lifting property of J . Then f α ∈ J .

1.4 Cobrant objects and homotopy

Cobrant objects

Cobrant objects are objects which have a lift along trivial brations.

In this subsection, we characterize brant and cobrant objects.

Lemma 1.4.1. Any object is brant. Proof. For any X ∈ E, the map X → 0 is a bration. Lemma 1.4.2. Suppose that the subcategory M contains the projective objects.

Let C ∈ E. Then C is cobrant if and only if C ∈ prM. Proof. Let C ∈ prM. We introduce 0 G G M 1 h G G M 0 h G G C G G 0 . Let f :
X → Y be a trivial bration and b : C → Y . As f ∈ W, Gf is an isomorphism, and there exists from lemma 1.2.4 a morphism a : M 0 → X such that

f • a = b • h.
Since M contains all the projective objects, the lemma 1.2.5 shows that f is a deation. Let k : K → X be the kernel of f . We then have a morphism of short exact sequences:

M 1 c G G h K k M 0 h a G G X f C b G G Y As k ∈ M ⊥ , there exists I an injective object, α : M 1 → I and β : I → X such that k • c = β • α.
As h is an ination, there exists β : M 0 → I such that

β • h = α. M 1 c G G h α 2 2 K k I β 1 1 M 0 h a G G X f C γ a a b G G Y So, h • a = β • β • h and there exists γ : C → X such that γ • h = β • β + a. Then, f • γ • h = f • a + f • β • β and (f • γ -b) • h = f • β • β .
Then, in modM, we have the good lifting. By Demonet and Liu in [DL13], we have that modM prM/ M.

As C ∈ prM, there exists M ∈ M such that the morphism b factorizes through M , let us say

M ε 4 4 C b G G δ a a Y such that ε • δ = b. As f ∈ J , there exists ι : M → X such that f • ι = ε. Then ι • δ : C → X is the good candidate to lift b, it means that f • ι • δ = ε • δ = b.
On the other hand, let X be a cobrant object. We can say that the prMapproximation is a trivial bration from lemma 1.2.3, then a retraction, and X is a direct summand of A.

Let

0 → Y → M 0 → X → 0
be an M-approximation of X. We have the following diagram:

Homotopy

We dene here cylinder objects and left homotopies.

Denition 1.4.3. Let X ∈ E. A cylinder object for X is a factorization of the morphism ∇ : X ⊕ X → X (which is the identity on each copy of X) through X , where X → X is a weak equivalence. Let f, g : X → Y be two morphisms. A left homotopy from f to g is a morphism h : X → Y , where X is a cylinder object for X, such that h Let f, g : X → Y be two morphisms. A right homotopy from f to g is a morphism

• (∇ 1 ∇ 2 ) = (f g) where (∇ 1 ∇ 2 ) is the morphism X ⊕ X → X in the factorization of ∇.
k : X → Y , where Y is a path object for Y , such that ∆ 1 ∆ 2 •k = f g
where ∆ 1 ∆ 2 is the morphism Y → Y in the factorization of ∆. Lemma 1.4.5. For two morphisms f and g from an object X to Y ,f and g are homotopic if and only if f -g factorizes through M ⊥ . Proof. We begin by noting this fact: in the next diagram, a factorization of ∆ is a path object if and only if it is isomorphic to Y

( 1 0 ) G G Y ⊕ V ( 1 1 c d ) G G Y ⊕ Y for a V ∈ M ⊥ . Indeed, let Y r G G Y G G Y ⊕ Y be a factorization of ∆. As r is a section, it is isomorphic to some Y → Y ⊕ V . Then r ∈ W if and only if V ∈ M ⊥ .
Now, let us suppose that f -g factorizes in the following way:

X f -g G G α 2 2 Y A β y y .
The object Y ⊕ A is a path object. Then we have

X (g α) G G (f g) @ @ Y ⊕ A = Y ( 1 1 β 0 ) Y (1 0) o o ∆ v v Y ⊕ Y
and the morphism 1 1 β 0 gives a homotopy between f and g. Moreover, s•a = Id,

thus s Y ⊕ A → Y and s ∈ W, so A ∈ M ⊥ . Then f is homotopic to g.
Conversely, if f is homotopic to g, then there exists k :

X → Y such that ∆ 1 ∆ 2 • k = g . Then, f -g = (∆ 1 -∆ 2 ) • k. As Y = Y ⊕ A, and A ∈ M ⊥ ,
this nishes to show the result.

Remark 1.4.6. The notions of left and right homotopy are the same for the cobrant objects.

The following lemma is a corollary of the theorem of Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF]. However, we give a direct proof here.

Lemma 1.4.7. With the above notations, we have

prM ∩ M ⊥ M.
Proof. The indirect inclusion is obvious since M is rigid. Now, let X ∈ prM∩M ⊥ . As X ∈ prM, we have a short exact sequence

0 → M 1 → M 0 → X → 0, with M 1 , M 0 ∈ M. We have the following diagram: M 1 G G M 0 c G G α X k M 1 G G I ψ h G G β Y Y M 1 ξ M 0 ϕ Y Y M 0
In order to show that X ∈ M, we will show that the short exact sequence 0 → X → M 1 → M 0 → 0 splits. As X ∈ M ⊥ , the morphism c : M 0 → X factorizes through I. We call by α : M 0 → I and β :

I → X such that β • α = c. Let k : X → M 1 and h : I → M 1 . Then, k • β • α = k • c = h • α. Then, h -k • β factorizes through M 0 . Let us call ϕ : M 0 → M 1 and ψ : I → M 0 . Then, we have h = k • β + ϕ • ψ.
Let ξ : M 1 → M 0 in the short exact sequence (see diagram for sake of clearance).

Then we have

ξ • ϕ • ψ = ξ • (h -k • β) = ξ • h -ξ • k • β = ψ -0.
As ψ is a deation, we can conclude that ξ • ϕ = 1. Then, there is a section to the short exact sequence and X ∈ M (which is stable under direct summands since M is stable under direct summands.

Prebration structures from rigid subcategories

In this section, we show that an exact category E is nearly equipped with a structure of prebration category in the sense of Anderson-Brown-Cisinski (for more details, see the book of Radulescu-Banu, [START_REF] Radulescu-Banu | Ams-latex, 158 pages[END_REF]).

We recall that G : E → ModM X → E(-, X)|M induces the following equivalence of categories from Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF] prM/ M mod M.

Theorem 1.5.1. Let E be a weakly idempotent complete exact category with enough injective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective objects, and stable under taking direct sums and summands. Suppose moreover that M is contravariantly nite. Let J = {f : 0 → M, M ∈ M} and J be the class of brations. Let W be the class of morphisms whose image under functor G are isomorphisms.

Then E is almost equipped with a structure of prebration category, it means that there exist two spaces of morphisms, W, the weak equivalences, and Fib the brations, such that:

(i) The space W is stable under retracts and satises the two out of three axiom.

(ii) The space Fib is stable under composition, and all the isomorphisms are brations. (iii) Pullbacks exist along brations, and the pull-back of a bration is a bration.

Moreover, if 0

G G A i G G B p G G Y G G 0 is a short exact sequence, if Gi is a monomorphism and f : X → Y is a trivial bration (f ∈ Fib ∩ W),
then h dened by the following pull-back is a trivial bration.

E h a G G X f A i G G B p G G Y
(iv) There exist path objects such that for any object X, the diagonal map X → X ⊕ X can be factorized through X , where the rst morphism is a weak equivalence. (v) For any object B, the morphism 0 → B is a bration.

Let us rst show the following lemma: Lemma

1.5.2. Let E h a G G X f A i G G B p G G Y
where (i, p) is a short exact sequence, Gi monomorphism and f ∈ J ∩ W. Then h ∈ J ∩ W.

Proof. First, we have, without using the fact that Gi is a monomorphism, that

h ∈ J . Indeed, let E h a G G X f B p G G Y be a commutative square, with f ∈ J . Let us show that h ∈ J . Let M ∈ M and b : M → B. As f ∈ J , there exists α : M → X such that f • α = p • b.
We use the pullback property to claim that there exists ϕ : M → E such that h • ϕ = b. Then h ∈ J . Second, let us show that Gh is a monomorphism. As h is a deation (since it belongs to J , and from lemma 1.2.5), the morphisms f and h have the same kernel. Then we have the following diagram:

K K A k G G E h a G G X f A i G G B p G G Y We have a short exact sequence 0 → A → E → X → 0. Let us show that Gh is a monomorphism. Let β : D → GE such that Gh • β = 0. We show that β = 0. We have Gp • Gh • β = 0. Then G(p • h) • β = 0 and G(f • a) • β = 0. Thus, we have Gf • Ga • β = 0.
As Gf is a monomorphism since f ∈ W, we have

Ga • β = 0.
The fact that G is left exact shows that there exists c : D → GA such that

β = Gk • c. (1.1) Moreover, by hypothesis Gi • c = Gh • β = 0.
As Gi is a monomorphism, c = 0 and from 1.1 we have β = 0. This shows that Gh is a monomorphism. Now we show that Gh is an epimorphism. Let b : M → B be a morphism, where M ∈ M. Then we have p

• b : M → Y . From lemma 1.2.4, there exists α : M → X such that f • α = p • h.
Then, from the pullback property, there exists ϕ : M → E such that h • ϕ = b and this shows that Gh is an epimorphism.

where ∼ is the right homotopy relation (which is also left taken into account the denition of the homotopy).

Proof. This functor is well-dened because the relation ∼ behaves well with the right composition. Let now f : A → B. We factor f through a morphism g : A → C ∈ (J ) followed by a morphism p : C → B ∈ J . As g ∈ W and f ∈ W, then by the two out of three property, p is also a weak equivalence. In addition, B is cobrant, then there exists w : B → C such that p • w = 1. Then, we have

1 = F (1) = F (p • w) = F (p) • F (w).
Then F (p) ∈ W. This shows by the two out of three property, that F (f ) ∈ W, and F f is surjective.

Let us now show that it is injective. If α : B → X is such that F α = 0, then α • f ∼ 0. As A is cobrant, we have α • f ∈ ( M) (from the equivalence of Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF]). This shows that G(α

• f ) = 0 since M is rigid. But f ∈ W, then Gf is an isomorphism, then Gα = 0. As B is cobrant, α ∈ ( M).
Then α ∼ 0. Then F f is injective and this shows the lemma. Now we can show the following result: Lemma 1.6.4. If A and B are two cobrant objects, then

Ho E(A, B) = E(A, B)/ ∼ .
Proof. Let A and B be two cobrant objects.

Step 1: We show that, f : A → B is a weak equivalence if and only if it is a homotopy equivalence.

Suppose f is a weak equivalence. We use the previous lemma with X = A. From the surjectivity of F f , there exists g : B → A such that g • f ∼ 1. Then, f • g • f ∼ f . Now we apply the result to X = B and have that

F f (f • g) = F f (1).
However, F f is injective, then f • g ∼ 1 and f is a homotopy equivalence.

Suppose now that f is a homotopy equivalence. Let f be a homotopy inverse for f . We have f

•f ∼ 1 and f •f ∼ 1. Then f •f -1 ∈ M and f •f -1 ∈ M. Then G(f • f ) = 1 and G(f • f ) = 1. Then Gf is an isomorphism and f ∈ W.
This shows that f = g in HoE(A, B) if and only if f ∼ g.

Step 2: We now show the surjectivity of E(A, B) → Ho(A, B). We are going to use the book of Radulescu (see [RB06, Theorem 6.4.2]). Let us check the hypothesis with the pair of categories (E/{f, Gf = 0}, W), where W is the image of W in the quotient of E by {f, Gf = 0}.

The two out of three property is automatically checked.

If we have a pair of morphisms

A a B p G G A
such that a ∈ W, then there exists B , h : B → B and k : B → A such that the following square commutes:

B k G G h A a B p G G A
Indeed, we introduce the factorization of A → A by b ∈ (J ) followed by c ∈ J . As a, b ∈ W , we also have c ∈ W . Then c is a trivial bration. Let B be a prM-approximation of B. We lift the morphism p • h to c, let us say h 1 h 2 : B → A ⊕ M (which is the shape of the factor in the factorization we have). Then h 1 is a lift from B to A . We can check that the square commutes and h ∈ W .

Suppose that we have

A f G G g G G B t G G B with t ∈ W and t • f = t • g. Then Gf = Gg, so, if t : A → A ∈ W , we have G(f • t ) = G(g • t ), then f • t -g • t ∈ M ⊥ . Then f • t = g • t in E/{f, Gf = 0}.
We can now apply the theorem of Radulescu 6.4.2 and any morphism in Ho(E(A, B)) can be written as f • s -1 with s ∈ W. As A and B are cobrant, we factor through A → A in this way:

A f ' ' s Ó Ó A ⊕ M α∈J z z 6 6 A a G G B
As s ∈ W, then α ∈ J ∩ W. Then we can lift A ⊕ M to B by (f 0).

From the theorem of Radulescu, if α ∈ Ho(A, B), then there exists s ∈ W and f

∈ E(A, B) such that α = f • s -1 .
Then α = a (since a • s = f ).

Then we have shown the surjectivity and then the lemma.

Proof of the theorem of Quillen. First of all, the functor is well-dened, since

0 → M ∈ W for any M ∈ M (because Ext 1 (-, M )|M = 0 implies E(-, M )|M = 0).
Then, the functor is essentially surjective, since there exist some prM-approximations, which are weak equivalences.

Next, from lemma 1.6.4, we have that f = g in HoE(A, B) if and only if f ∼ g which immediately shows that the functor is faithful.

Finally, the functor is full, from the surjectivity of lemma 1.6.4. The theorem of Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF] nishes to show the result.

Proof. Let a : M → ΩX be an M-approximation of ΩX. We can introduce Y as the push-out of the pair of morphisms:

M ι M a G G ΩX c I M b G G Y in particular c • a = b • ι M . (2.1) Let 0 G G ΩX ι X G G P X π X G G X G G 0
be a short exact sequence expressing the fact that ΩX is a sizygy of X. We then have the following diagram:

M ι M a G G ΩX c ΩX ι X I M b G G π M Y e α 4 4 P X π X M M β 5 5 X Note that we have e • b = π M . (2.2)
As ι X • a factorizes through the module I M , there exists α : Y → P X such that

ι X = α • c and α • b = d. (2.3)
Then there exists β, which makes a morphism of short exact sequences:

π X • α = β • e (2.4)
Then we have a short exact sequence

0 → Y → P X ⊕ M → X → 0.
We now show that (π X β) is a M-approximation (we note that, E being Frobenius, M contains the projective modules). Let γ : N → X be a morphism. As I N is also projective, there exists δ :

I N → P X such that π X • δ = γ • π N (2.5) M ι M a G G ΩX c ΩX ι X N ε o o ι N I M b G G π M Y e α 4 4 P X π X I N δ o o π N M M β 5 5 X N γ o o
We complete into a morphism of short exact sequences by ε, it means that (2.7)

ι X • ε = δ • ι N . ( 2
As I M is injective, and ι N is an ination, there exists η :

I N → I M such that η • ι N = ι M • ζ. (2.8)
Then there exists θ : N → M which makes a morphism of short exact sequences:

θ • π N = π M • η.
(2.9)

Then we have:

δ • ι N = ι X • ε by 2.6 = ι X • a • ζ by 2.7 = α • c • a • ζ by 2.3 = α • b • ι M • ζ by 2.1 = α • b • η • ι N by 2.8.
(2.10)

Then, the morphism α • b • η -δ factorizes through π N . Thus, there exists κ : N → P X such that

κ • π N = δ -α • b • η. (2.11) Let us nally show that the morphism κ θ satises (π X β) κ θ = γ.
As π N is a deation, is suces to show that

π X • κ • π N + β • θ • π N = γ • π N .
We have:

π X • κ • π N + β • θ • π N = π X (δ -α • b • η) + β • θ • π N by 2.11 = π X (δ -α • b • η) + β • π M • η by 2.9 = γ • π N -π X • α • b • η + β • π M • η by 2.5 = γ • π N -β • e • b • η + β • π M • η by 2.4 = γ • π N -β • π M • η + β • π M • η by 2.2 = γ • π N .
(2.12) Denition 2.1.2. We call by copr M the class of objects X ∈ E such that there exist

M, M ∈ M such that 0 → X → M → M → 0 is a short exact sequence. copr M = {X ∈ E, 0 → X → M → M → 0}.
Lemma 2.1.3. We have the following equality:

copr M = prM.

Proof. Let C ∈ copr M. Let us show that we then have a short exact sequence

0 → M 1 → M → C → 0. We have the diagram: M 1 G G Ω M 0 G G C M 1 G G I G G M 1 M 0 M 0
We have Ω M M . Indeed, we have a two-ways short exact sequence

M G G I G G M Ω M y y G G P y y G G M
Then Ω M M ⊕ I ∈ M. The other inclusion is proved in part 1, lemma 1.4.7. Lemma 2.1.4. The class M is rigid. Proof. Let M, M ∈ M. We show that a morphism f : M → 2 M is zero. Let us introduce the following diagram:

M h G G ι M M ι M I M g G G π M α X X I M π M M f G G β Y Y 2 M
As I M is a lso projective and π M is a deation, there exists g which makes the lower square commute. Then, there exists h such that the upper square commutes. Now, h factorizes through an injective module since M is rigid. Let us say that there exists α : I M → M such that α•ι M = h. Then, there exists β :

M → I M such that g = ι M • α + β • π M .
Then,

f • π M = π M • g = π M • (ι M • α + β • π M ) = π M • β • π M
As π M is a deation, this shows that f = π M • β and thus factorizes through an injective. Then M is rigid.

Lemma 2.1.5. Let A, C be objects of E. Let M ∈ M. Suppose that we have a short exact sequence

0 → A → C → M → 0.
Then C ∈ prM if and only if A ∈ prM.

Proof. First, suppose that A ∈ prM. Then, we have the following diagram:

A G G M 1 G G M 0 C G G E G G M 0 M 2 M 2
Where E is the pullback of the morphisms A → M 1 and A → C. Then there is a section to the short exact sequence

0 → M 1 → E → M 2 → 0,
and E M 1 ⊕ M 2 M . Then we have a short exact sequence 0 → C → M → M 0 → 0. Then the previous lemma concludes this implication.

Conversely, if C ∈ prM. Then, by the previous lemma, C ∈ copr M, so there exists a short exact sequence

0 → C → M 0 → M 1 → 0.
Let us take the push-out:

A G G C G G M A G G M 0 G G E M 1 M 1
As M is rigid from the previous lemma, then E ∈ M. Then, A ∈ copr M then A ∈ prM.

Remark 2.1.6. We have the dual lemma: If there is a short exact sequence 0 → M → C → A → 0, then A ∈ prM if and only if C ∈ prM.

Weak equivalences

We start with a denition.

Denition 2.2.1. Let A be a subcategory of E. Let g : Y → Z be a morphism. We say that g ∈ A ⊥ if, for any morphism b : A → Y , with A ∈ A, the morphism g • b factorizes through an injective module. Remark 2.2.2. We note that A ⊥ is not an ideal.

This lemma characterizes weak equivalences.

Lemma 2.2.3. Let X, Y ∈ E. Let f : X → Y . Then Gf is an isomorphism if and only if, for

X f G G ι X Y g I X u G G π X Z X X
we have I X ⊕ Y (u g) G G Z ∈ M ⊥ and, for

X f G G Y Z G G g y y P Y y y Y y y Y y y we have Z G G X ⊕ P Y ∈ M ⊥ . Proof. If Gf is an isomorphism. Let X f G G ι X Y g I X u G G π X Z X X
be the diagram which induces the short exact sequence

0 G G X G G I X ⊕ Y (u g) G G Z G G 0 .
Let us show that (u g) ∈ M ⊥ . Since u : I X → Z, with I X injective we already have that E(M, u) = 0. Let E(M, g) :

E(M, Y ) → E(M, Z) h : M ∈ M → Y → g • h .
As Gf is an isomorphism, there exists h ∈ E(M, X) such that f

• h = h. Then h → f • h = h → g • f • h = g • h = 0,
and E(M, g) = 0.

In the same way, let

0 G G Z G G X ⊕ P Y G G Y G G 0 .
Let us show that g ũ ∈ M. We already have that E(M, ũ) = 0. Let E(M, g) :

E(M, Z) → E(M, X) h : M ∈ M → Z → g • h .
The map g • h is sent on 0 by Gf . As Gf is an isomorphism, we have that g • h = 0.

Conversely, let us suppose that

I X ⊕ Y (ũ g) G G Z ∈ M ⊥ and Z G G X ⊕ P Y ∈ M ⊥ .
Let us introduce the short exact sequences

0 G G X G G I X ⊕ Y (ũ g) G G Z G G 0 (2.13) and 0 G G Z G G X ⊕ P Y G G Y G G 0 (2.14)
We take the image of equation 2.13 by the functor G = E(A, -) for an A ∈ M. As G is left exact, we have the following long exact sequence:

0 → E(A, X) → E(A, Y ⊕ I A ) → E(A, Z) → Ext 1 (A, X) → • • •
and g ∈ M ⊥ , where g is a cone of f , dened by

X f G G ι X Y g I X u G G π M Z X X .
Proof. Let f : X → Y be a deation. Let g be a cone of f . Suppose that

-α 1 • ι α 2 • ι = β • f ι X .
As I is also projective since E is a Frobenius category, and f is a deation, there exists α1 : I → X such that

α 1 = f • α1 . Then, we have f • (β -α1 • ι) = b
and this shows the rst implication.

On the other hand, let g be a cone of f . If f ∈ J : X → Y , then it is a deation. Indeed, let π Y : P Y → Y be a projective pre-cover of Y (we can equivalently introduce a M-approximation of Y ). There exists a morphism ϕ : P Y → X which make the diagram commutative. As f • ϕ = π Y is a deation and E is a weakly idempotent complete category, from the article of Bühler, [Büh10, Theorem 7.6], then f is also a deation. Let us now show that g ∈ M ⊥ . Let h : B → Y , where B ∈ M. We have to show that g • h = 0. As f ∈ J , there exists ϕ : B → X such that f • ϕ = h. Then g • h = g • f • ϕ. As we know g • f factorize through an injective, and so g • h. Then g ∈ M ⊥ .

Factorizations

In addition to the rst factorization, we also have a second factorization:

Lemma 2.4.1. Any morphism f : X → Y , where X ∈ prM can be factorized through a morphism in (I ) followed by a morphism in I .

Proof. Let us show that any morphism f : X → Y , where X ∈ prM can be factorized through a morphism in (I ) followed by a morphism in I . We are going to use item (iv) in the next theorem (whose proof does not need this lemma) We have made a proof of point (iv) lower for sake of clarity (in order to have a one-piece-proof for the main theorem).

Let f : X → Y be a morphism, such that X ∈ prM. Let 0 → M 1 → M 0 → X → 0 be a short exact sequence, and let c : M 0 → X. From lemma 1.1.1, there exist a : A → Y a prM-approximation, with A ∈ prM. Let ε : X → M 1 be the induced morphism in the short exact sequence

0 → M 0 → X ⊕ I M 0 → M 1 → 0.
As A → Y is a prM-approximation, and X ∈ prM, there exists r : X → A such that f = a • r.

We are going to show that

X f G G ( r ε ) 6 6 Y A ⊕ M 1 (a 0) X X
is the good factorization. It is immediate that (a 0) is a trivial bration. So it is an element of I . Let us show that r ε ∈ (I ). We introduce the following commutative square:

X α G G ( r ε ) U h A ⊕ M 1 (β 1 β 2 ) G G V
with h ∈ I . The object A ⊕ M 1 is in prM, then from lemma 1.4.2, part 1, it is cobrant. Then, there exists a morphism (ϕ 1 ϕ 2 ) :

A ⊕ M 1 → U such that h • ϕ 1 = β 1 and h • ϕ 2 = β 2 .
(2.15)

Unfortunately, this morphism is not the good candidate, because it does not make the upper triangle commute. We have to modify it.

As h ∈ J ∩ W, then it is a deation, we introduce its kernel u : K → X. There exists γ : X → K such that

α = u • γ + ϕ 1 • r + ϕ 2 • ε.
(2.16) Moreover, we know that u ∈ M ⊥ . So the morphism u • γ • c factorizes through an injective module, let us say I M 1 . Then we have the following diagram:

M 0 c G G X ε u•γ I M 1 G G G G M 1 ∃a 2 4 4

C

From the push-out property, there exists a morphism a 2 :

M 1 → C such that u • γ = a 2 • ε.
(2.17) Then, from equations 2.16 and 2.17, we have (ϕ

1 ϕ 1 + a 2 ) • r ε = α.
Unfortunately again, this morphism is not the good candidate. Now, the upper triangle commutes, but we have lost the commutativity of the lower one. We have to modify it one last time.

Moreover, the morphism r ε is an ination. We can introduce its cokernel C.

Let (c 1 c 2 ) : A ⊕ M 1 → C. We have

h • (ϕ 1 ϕ 2 + a 2 ) • r ε = h • α = (β 1 β 2 ) • r ε .
Then the morphism (ϕ 1 ϕ 2 + a 2 ) -(β 1 β 2 ) factorizes through the cokernel C of r ε . Then there exists b : C → V such that

(β 1 β 2 ) = h • (ϕ 1 ϕ 2 + a 2 ) -b • (c 1 c 2 ).
(2.18)

Moreover, C ∈ prM from lemma 2.1.5. Then it is cobrant, as h ∈ J ∩ W, there exists d :

C → U such that h • d = b.
(2.19)

We now have enough information in order to choose the good candidate for the morphism A ⊕ M 1 → U which makes both triangles commute. It is:

(ϕ 1 -d • c 1 ϕ 2 + a 2 -d • c 2 ).
Indeed, for the upper triangle:

(ϕ 1 -d • c 1 ϕ 2 + a 2 -d • c 2 ) • r ε = ϕ 1 • r -d • c 1 • r + ϕ 2 • ε + a = α -u • γ -ϕ 2 • ε -d • c 1 • r + ϕ 2 • ε + a 2 • ε -d • c 2 • ε from 2.16 = α -a 2 • ε -d • c 1 • r + a 2 • ε -d • c 2 • ε from 2.17 = α -d • (c 1 c 2 ) • r ε = α.
(2.20) The upper triangle commutes. Now let us show the commutativity of the lower triangle:

h • (ϕ 1 -d • c 1 ϕ 2 + a 2 -d • c 2 ) = (h • ϕ 1 -h • d • c 1 h • ϕ 2 + h • a 2 -h • d • c 2 ) = (h • ϕ 1 -b • c 1 h • ϕ 2 + h • a 2 -b • c 2 ) from 2.19 = (h • ϕ 1 + β 1 -h • ϕ 1 β 2 + b • c 2 -c • c 2 ) from 2.18 = (β 1 β 2 ) (2.21)
Then, the lower triangle commutes, so r ε ∈ (I ), and we have the factorization when the domain of the morphism is cobrant.

The almost model structure on Frobenius categories

We are going to show the following theorem:

Theorem 2.5.1. Let E be a weakly idempotent complete Frobenius category. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective objects, and stable under taking direct sums and summands. Let Let G be the functor

G : E → ModM X → E(-, X)/M Let W = {f,
Gf is an isomorphism}. Let J be the class of brations. The cobrations are given by the left-lifting property from trivial brations. Then, (F ib, Cof, W) nearly form a model structure for the category E. Indeed, the second factorization is found only when the domain is cobrant. Proof. We show the conditions required in order to apply theorem ?? of Hovey in [Hov99, Theorem 2.1.19].

(i) W is stable under retracts. Let f be a retract of w ∈ W. Let us show that Gf is an isomorphism. We have:

A i G G f Id 9 9 X r G G w A f B j G G Id U U Y s G G B
As Gf is a retract of an isomorphism, it is itself en isomorphism. Indeed,

GA G G Gf Id A A GX β G G Gw GA Gf GB γ G G Id S S GY G G Gw -1
y y

GB

The morphism β • Gw -1 • γ is an inverse of Gf . Then Gf is an isomorphism.

(ii) "2 out of 3" property.

If f, g ∈ W. Then G(f • g) = G(f ) • G(g), and we have that f • g ∈ W.

If f, f • g ∈ W. Then (G(f • g)) -1 • G(f • g) = Id. So (G(f • g)) -1 • G(f ) • G(g) = Id.
Let G(g) -1 be the morphism (G(f • g)) -1 • G(f ). It is a left inverse. Let us show that it is also a right inverse: (iii) (J ) ⊆ W ∩ (I ).

G(f • g) • (G(f • g)) -1 = Id. Then G(f ) • G(g) • (G(f • g)) -1 = Id,
As J ⊆ I, we automatically have that (J ) ⊆ (I ). From lemma 1.3.1 it is immediate that any morphism of (J ) is a weak equivalence.

(iv) I = J ∩ W.

First, let us show that J ∩ W ⊆ I . Let f ∈ J ∩ W. We introduce the following commutative square:

M 0 a G G ( h ι 0 ) X f A ⊕ I 0 (b 1 b 2 ) G G Y Then we have b 1 • h + b 2 • ι 0 = f • a. (2.22)
As f is a deation, there exists K such that 0 → K → X → Y → 0 is a short exact sequence. If we consider the diagram:

K K K ⊕ I 0 G G X I 0 G G Y
as f is a deation, the morphism K ⊕ I 0 → I 0 is a deation. They moreover have the same kernel (as K ⊕ I 0 is a pullback), thus the short exact sequence

0 → K ⊕ I 0 → K ⊕ I 0 → Y → 0
exists. As h ι 0 ∈ I, then there exist M 1 , M 0 ∈ M such that

0 → M 1 → M 0 → A → 0
is a short exact sequence. We build the following morphism of short exact sequences:

0 0 M 1 k ( c µ) G G K ⊕ I 0 ( g α 0 1 ) M 0 h ( a ι 0 ) G G X ⊕ I 0 (f -b 2 ) A b 1 G G Y 0 0 From equation 2.22, we have that b 1 • h = (f -b 2 ) • a ι 0 .
Then, there exists c µ such that

a • k ι 0 • k = g • c + α • µ mu .
As g ∈ M ⊥ , the morphism M 1 → K → X factorizes through an injective.

We can suppose without loss of generality, that it factorizes through I 0 . Then we have:

M 1 k ( c µ) G G α A 2 2 K ⊕ I 0 ( g α 0 1 ) I 0 β 1 β 2 5 5 M 0 ( a ι 0 ) G G X ⊕ I 0 with β 1 β 2 • α A = a • k ι 0 • k .
As I 0 is injective, and k is an ination, we can lift α A to ι 0 : M 0 → I 0 such that ι 0 • k = α A . Then,

M 0 h ( a ι 0 ) G G ι 0 2 2 X ⊕ I 0 (f -b 2 ) I 0 β 1 β 2 Y Y A b 1 G G Y .
Then, there exists α 1 α 2 : A → X ⊕ I 0 such that

a ι 0 = β 1 β 2 • ι 0 + α 1 α 2 • h.
(2.23)

Then, we have a morphism A ⊕ I 0 → X which makes the upper triangle commute:

M 0 a G G ( h ι 0 ) X f A ⊕ I 0 (b 1 b 2 ) G G (α 1 β 1 ) Y Y Y .
We have

(α 1 β 1 ) • h ι 0 = α 1 • h + β 1 • ι 0 = a from equation 2.23.
The morphism M 0 → A ⊕ I 0 is an ination. Let us introduce its cokernel C.

Then we have the push-out:

M 0 h G G A γ 1 f •α 1 -b 1 I 0 γ 2 G G f •β 1 -b 2 G G C ∃ψ 1 1 Y
Then there exists a unique ψ : C → Y such that:

ψ • γ 2 = f • β 1 -b 2 (2.24) and ψ • γ 1 = f • α 1 -b 1 (2.25)
The push-out C is exactly the cokernel of the morphism M 1 → M 0 → I 0 . Then C M 1 . As f ∈ J , if we still denote by ψ the morphism from M 1 to Y , then there exists ζ : M 1 → X such that f • ζ = ψ. Then the equations 2.24 and 2.25 give respectively

f • ζ • γ 1 = f • α 1 -b 1 so f • (α 1 -ζ • γ 1 ) = b 1 and f • ζ • γ 2 = f • β 1 -b 2 then f • (β 1 -ζ • γ 2 ) = b 2
and, the morphism

N ⊕ I M G G X C ⊕ J G G Y
and as f ∈ I , there exist (ϕ 1 ϕ 2 ) : C ⊕ J → X such that f • ϕ 1 = r and f • ϕ 2 = 0 and

(ϕ 1 • k + ϕ 2 • ι 1 ϕ 1 • u + ϕ 2 • ι 2 ) = (a 0).
Then we have

g • b = a • h = (a 0) • h ι M = (ϕ 1 ϕ 2 ) • k u ι 1 ι 2 • h_ι M = ϕ 1 • k + ϕ 2 • ι 1 ϕ 1 • u + ϕ 2 • ι 2 • h ι M = (ϕ 1 ϕ 2 ) • 0 ι 1 • h + ι 2 • ι M .
(2.26) This shows that g • b factorizes through an injective, thus g ∈ M ⊥ .

Then we have shown that I = J ∩ W.

(v) The rst required factorization is exactly corollary 1.3.2, part 1.

(vi) The second required factorization is exactly lemma 2.4.1.

As a consequence, we have directly the theorem of Quillen for Frobenius categories.

Theorem 2.5.2. Let Ho E be the localization of the quasi-isomorphisms of E at the class W. Let mod M be the class of M-modules. There is an equivalence of categories Ho E mod M.

Dans cette thèse, nous traitons le cas Dn en profondeur et démontrons quelques résultats additionnels aux travaux de Baur, Marsh et Torkildsen sur les réalisations géométriques des catégories amassées supérieures. Pour le cas Dn , on trouve la gure qui correspond à l'étude, on démontre la compatibilité entre le ip d'une (m+ 2)-angulation, et la mutation de carquois coloré. On trouve une bijection entre les objets m-rigides et chaque arc dit admissible, puis entre les objets amas-basculants et les (m+2)-angulations au théorème. De plus, on démontre la compatibilité entre la réduction d'Iyama-Yoshino, et le fait de couper le long d'un arc, qu'on dénira formellement.

Nous démontrons ensuite qu'une catégorie exacte est une catégorie de prébration au sens de Anderson-Brown-Cisinski, qui vérie le théorème de Quillen, et une catégorie de Frobenius est munie d'une structure de modèle, compatible avec le passage à la catégorie stable, qui est triangulée.

We show that a subcategory of the m-cluster category of type Dn is isomorphic to a category consisting of arcs in an (n -2)m-gon with two central (m -1)-gons inside of it. We show that the mutation of colored quivers and m-cluster-tilting objects is compatible with the ip of an (m + 2)-angulation.

In this thesis, we study the geometric realizations of m-cluster categories of Dynkin types A, D, Ã and D. We show, in those four cases, that there is a bijection between (m+2)-angulations and isoclasses of basic m-cluster tilting objects. Under these bijections, ips of (m + 2)-angulations correspond to mutations of m-cluster tilting objects. Our strategy consists in showing that certain Iyama-Yoshino reductions of the m-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. This allows to infer results from small cases to the general ones.

Let E be a weakly idempotent complete exact category with enough injective and projective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective and projective objects, and stable under taking direct sums and summands. In this paper, E is equipped with the structure of a prebration category with cobrant replacements. As a corollary, we show, using the results of Demonet and Liu in [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF], that the category of nite presentation modules on the costable category M is a localization of E. We also deduce that E → modM admits a calculus of fractions up to homotopy. These two corollaries are analogues for exact categories of results of Buan and Marsh in [START_REF] Aslak | From triangulated categories to module categories via localisation[END_REF], [START_REF] Aslak | From triangulated categories to module categories via localization II: calculus of fractions[END_REF] (see also [START_REF] Beligiannis | Rigid objects, triangulated subfactors and abelian localizations[END_REF]) that hold for triangulated categories.

If E is a Frobenius exact category, we enhance its structure of prebration category to the structure of a model category (see the article of Palu in [?] for the case of triangulated categories). This last result applies in particular when E is any of the Hom-nite Frobenius categories appearing in relation to cluster algebras.

  chere d is the dierential. The homotopy category K(C) is dened in the following way: Objects: same as the ones of C. Morphisms: classes of morphisms of C(C) modulo the null-homotopic relation. Natural composition. Now we can dene the derived category of C. Denition 3.1.10. The morphisms which induce isomorphisms in homology are called quasi-isomorphisms. The derived category D(C) is the localization of K(C) with respect to the class of quasi-isomorphisms. Let us explain the morphisms of D(C): a morphisms from X to Y in D(C) is the class of diagrams of the type

  . Let B = I 4 ⊕ I 1 ⊕ Y [1] be an almost m-cluster-tilting object. Then it has three complements which are given by the exchange triangles. They are X, P 2 [1] and I 3 [1].

A

  geometric realization of the m-cluster categories of type Dn 1.1 How to associate a colored quiver with an (m+

Figure 1 . 1

 11 Figure 1.1 The (n -2)m-gon with two digons. Here m = 3 and n = 4.

Figure 1 . 2

 12 Figure 1.2 Both gures on the left represent the same m-diagonal, which is dierent from the m-diagonal on both gures on the right.

Figure 1 . 3

 13 Figure 1.3 Some examples of arcs in the hexagon. For example, if m = 1, the second arc is not admissible because it cuts the polygon into a 3-gon and a 5-gon.

Figure 1 . 4

 14 Figure 1.4 These two arcs represent the same m-diagonal.

Figure 1 . 6

 16 Figure 1.6 Exception of 1.1.10: the initial 4-angulation of Q for n = 4 and m = 3

Figure 1 . 7

 17 Figure 1.7 The initial 4-angulation for m = 2 and n = 7.

Figure 1

 1 Figure 1.8 Denition of a twist

Figure 1 . 9

 19 Figure1.9 The arc tangent to the left and the loop belong to the same class of m-diagonals. The gure on the right is a 5-angle (going twice through the same arc).

→Figure 1

 1 Figure 1.11 Another example of a ip

Figure 1

 1 Figure 1.14 A triangulation ∆

Figure 1 .

 1 Figure 1.15 The triangulation associated with ∆

Figure 1 .

 1 Figure 1.19 The mutation at k in the third case

Figure 1 .→Figure 1 .

 11 Figure1.21 The translation of the arc on the left picture becomes the arc on the middle picture, which is the same as the arc on the right picture.

  By denition, l(a, b) = m. If a is the end point of the m-diagonal τ β, then l(a , a) = m. So a = b and τ β and α share an oriented angle. Thus there exists f : τ β → α (b) If β is of type 1. There are two sub-cases: i. If α is of type dierent from 1. Let a and b be the vertices of β, where a is also the one of α. Then l(a, b) = m. So, if c and d are the vertices of τ β then d = a (and l(c, d) = m). Thus there exists f : τ β → α. ii. If α is of type 1. Let a and b be the vertices of β and a and c the ones of α. We have l(c, b) = m. So, if d and e are the vertices of τ β, we have e = c and l(d, a) = m. Thus there exists f : τ β → α. 2. Now, if α and β are not in the transjective part of the Auslander-Reiten quiver of Q. (a) If α is homotopic to the boundary path. Then β must be also homotopic to the boundary path. Let a and b be the vertices of β, where a is also the one of α. Then, l(c, b) = m. So, if d and e are the vertices of τ β then d = c (and l(a, e) = m). Thus there exists f : τ β → α. (b) If α is from a thick vertex of R to a thick vertex of S, then β is of the same type. Let a and b be the vertices of β, where a is also the one of α. Then, l(c, b) = m. So, if d and e are the vertices of τ β then d = c (and l(a, e) = m). Thus there exists f : τ β → α

Figure 1 .

 1 Figure 1.23 A small part of the Auslander-Reiten quiver of C (m) Q with vertices m-diagonals, for m = 3 and n = 4

  Figure 1.24

Figure 1 .

 1 Figure 1.25 The three dierent types of arcs in the tube, d t 1 , d t 2 and d t 3 for m = 2 and n = 7

Figure 1 .

 1 Figure 1.26 We ip β in order to dene α in the base of the rst tube

Figure 1 .

 1 Figure 1.28 The initial quadrangulation for n = 7 and m = 2

Figure 1 .

 1 Figure 1.30 Arc of order 5

Figure 1 .

 1 Figure 1.32 A tube of size 5. We can see that from the 5th layer, the arcs cross themselves and do not correspond to an m-rigid object anymore

Figure 2 . 1

 21 Figure 2.1 The rst is a 2-diagonal, the second is not

Figure 2 . 5

 25 Figure 2.5 Examples of m-diagonals

Figure 2 . 6

 26 Figure 2.6 The (n -2)m-gon with two digons. Here m = 3 and n = 4.

Figure 2

 2 Figure 2.8 A 4-angulation for m = 2 and n = 7.

Figure 2 . 9

 29 Figure 2.9 Example of an m-ear in case A

αFigure 2

 2 Figure 2.10 Example of an m-ear in case Ã

Figure 2 .

 2 Figure 2.13 Illustration of the rst case for p = 12, q = 6 and m = 2

  (b) If β is a transjective arc. -If we can shift β k < m times in order to hang one end point of β to one end point of α, then both arcs α and β[k] belong to the same slice in the Auslander-Reiten quiver of Q. We can show that there exists a morphism in the module category from X α to X β [k] with the article of Baur and Torkildsen [BT15, Paragraphs 3.3 and 3.4].

Figure 2 .

 2 Figure 2.14 We ip the arc corresponding to P k . The new arc is called by α.

Figure 2 .

 2 Figure 2.15 First case, α and β are of type 1

Figure 2 .

 2 Figure 2.16 Case where α and β are not of the same type

Figure 2 .

 2 Figure 2.17 The arrows are elementary move. The nonzero extension corresponds to the composition of the arrows

120Chapter 2 .Figure 2 . 21 mmFigure 2 . 23 Case 6 :

 22212236 Figure 2.20 Case where both α and β are of other type

Figure 2 .

 2 Figure 2.24 Case of a tube of size 2, for example for m = 3

  Colored quiver Q ∆ T h e o r e m 2 . 1 . 1 2 Theorem 3.4.5 m-cluster-tilting object X ∆ h h o o

  Dually, we can dene path objects and right homotopies. Denition 1.4.4. Let Y ∈ E. A path object for Y is a factorization of the morphism ∆ : Y → Y ⊕ Y (which is the identity on each copy of Y ) through Y , where Y → Y is a weak equivalence.

. 6 )

 6 As a : M → ΩX is an M-approximation, there exists ζ :N → M such that a • ζ = ε.

,

  be the diagram which induces the short exact sequence

Z

  g ∈ M ⊥ . Let us show that f ∈ J . Let b : M → Y , where M ∈ M. As (g u) ∈ M ⊥ , then the morphism (g u) • b 0 factorizes through an injective. We have g u b -α 1 • ι 0 -α 2 • ι = 0.We have the following diagram:Then, there exists β : M → X such that b

J

  = {f : 0 → M, M ∈ M} and I = {f : M 0 → X ⊕ I 0 , X ∈ prM} ∪ {0 → M, M ∈ M}.

  andG(g) • (G(f • g)) -1 • G(f ) = Id and it is indeed a right inverse. If g, f • g ∈ W.Then we use the same hint to show that Gf is an isomorphism.

(α 1

 1 -ζ • γ 1 β 1 -ζ • γ 2 ) : A ⊕ I 0 → X make both triangles commute. Now, let us show the inverse inclusion. We have immediately I ⊆ J since J ⊆ I. Let f ∈ I . Let us show that, following the natural notations, g ∈ M ⊥ and g ∈ M ⊥ . First, let b : M → Y be a morphism, with M ∈ M.As f ∈ I , there exists ϕ :M → X such that f • ϕ = b then g • b = g • f •ϕ and this morphism factorizes through an injective. This shows that g ∈ M ⊥ . Second, Let K be the kernel of f . Let b : M → K. Let N be an M-approximation of X. We denote it by a : N → X. As a is an M-approximation, there exists h :M → N such that g • b = a • h. Let ι M :M → I M be the canonical injection. As h ι M is an ination, we introduceC its cokernel. Let (k u) : N ⊕ I M → C. If we put (a 0) : N ⊕ I M → X,then there exists r : C → Y such that there is a morphism of short exact sequences:We are going to build an element of I from N ⊕ I M → C. We add J the injective envelope of N ⊕ I M (J = I M ⊕ I N ). Then the morphismk u ι 1 ι 2 : N ⊕ I M → C ⊕ J is in I.Then the following square is commutative

  Vers les catégories amassées supérieures . . . . . . . . . . . . . . . . 1.2 Réalisation géométrique . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Catégories triangulées, algèbre homotopique et généralisation aux catégories exactes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Higher cluster categories . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Denition of higher cluster categories . . . . . . . . . . . . . 3.2.2 Example of a 2-cluster category . . . . . . . . . . . . . . . . 3.3 Cluster-tilting subcategories . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Denition of cluster-tilting categories . . . . . . . . . . . . . 3.3.2 Some properties of m-cluster-tilting objects . . . . . . . . . 3.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Quivers and colored quivers . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Denition of a colored quiver . . . . . . . . . . . . . . . . . 3.4.2 Main theorem about colored quivers . . . . . . . . . . . . . . 31 3.4.3 Example of the mutation of colored quivers . . . . . . . . . . 32 3.5 Geometric realization of cluster categories . . . . . . . . . . . . . . 32 3.5.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5.2 Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5.3 Case à . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5.4 General case of the marked surface . . . . . . . . . . . . . . 34 3.6 Exact and Frobenius categories . . . . . . . . . . . . . . . . . . . . 35 3.6.1 Axioms of exact categories . . . . . . . . . . . . . . . . . . . 35 3.6.2 Some properties of exact categories . . . . . . . . . . . . . . 35 3.6.3 Examples of exact categories . . . . . . . . . . . . . . . . . . 36 3.6.4 The particular case of Frobenius categories . . . . . . . . . . 37 3.7 Weakly idempotent complete categories . . . . . . . . . . . . . . . . 37 3.8 Quotient of exact categories by cluster-tilting subcategories as module categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.9 Localization of triangulated categories . . . . . . . . . . . . . . . . 39 3.10 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.13 A weak model structure on triangulated categories . . . . . . . . . . 46 3.14 Some examples of such categories . . . . . . . . . . . . . . . . . . . 46 1.3 The case of regular modules . . . . . . . . . . . . . . . . . . . . . . 1.4 About non-self-crossing arcs and m-rigid objects . . . . . . . . . . . 1.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Quadrangulations of the decagon and colored quiver . . . . . 1.5.2 2-diagonals and preprojective component of the Auslander-Reiten quiver . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 2-diagonals in the tubes of the Auslander-Reiten quiver of D7 2 Bijection between m-cluster-tilting objects and (m+2)-angulations 2.1 Geometric realizations . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Case à . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Noncrossing arcs and extensions . . . . . . . . . . . . . . . . . . . . Prebration structures from rigid subcategories . . . . . . . . . . . 1.6 Theorem of Quillen . . . . . . . . . . . . . . . . . . . . . . . . . . . Dans cette thèse, nous décrivons une réalisation géométrique des carquois de type Dynkin, et certains carquois euclidiens. Nous traitons le cas Dn en profondeur et démontrons quelques résultats complémentaires aux travaux de Baur, Marsh et Torkildsen sur les réalisations géométriques des catégories amassées supérieures.

	Table des matières
	1 Introduction
	1.1

2 Summary of results 2.1 Higher cluster categories of type A n , D n , Ãn and Dn . . . . . . . . 2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 On the homotopy category of an exact category . . . . . . . . . . . 2.2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Preliminaries 3.1 Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Axioms of a triangulated category . . . . . . . . . . . . . . . 3.1.2 Some properties of triangulated categories . . . . . . . . . . 3.1.3 Examples of triangulated categories . . . . . . . . . . . . . . 3.2 3.10.1 Denition of a model category . . . . . . . . . . . . . . . . . 39 3.10.2 Some properties of model categories . . . . . . . . . . . . . . 40 3.10.3 Examples of model categories . . . . . . . . . . . . . . . . . 41 3.11 Prebration and precobration categories . . . . . . . . . . . . . . . 42 3.11.1 Denition of prebration and precobration categories . . . 42 3.11.2 Properties of such categories . . . . . . . . . . . . . . . . . . 43 3.12 The homotopy category of a model category . . . . . . . . . . . . . 44 3.12.1 Denition of the homotopy category . . . . . . . . . . . . . 44 3.12.2 Theorem of Quillen . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.1 Cases A and D . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Case à . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Case D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Compatibility ip/bijection between m-cluster-tilting objects and (m + 2)-angulations . . . . . . . . . . . . . . . . . . . . . . . . . . . II Model categories structures from rigid objects in exact categories 1 The case of exact categories 1.1 Study of the properties of prM . . . . . . . . . . . . . . . . . . . . 1.2 Weak equivalences and brations . . . . . . . . . . . . . . . . . . . 1.3 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Cobrant objects and homotopy . . . . . . . . . . . . . . . . . . . . 1.4.1 Cobrant objects . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5

  • • • , n}. Suppose that α and β cross. Then both arcs can be of dierent type. Let us sum up all the cases to treat in the following tabular:

	m	m	m	m
	α, β
	m
	Case 1	Case 2	Case 3	Case 4
	m
	Case 2	Case 5	Case 6	Case 7
	m
	Case 3	Case 6	Case 8	Impossible
	m
	Case 4	Case 7	Impossible	Case 9

94Chapter 2. Bijection between m-cluster-tilting objects and (m + 2)-angulations

1.5. Prebration structures from rigid subcategories

Remerciements

Chapter 1

The case of exact categories

Study of the properties of prM

We show some preliminary lemmas which will be used in section 1.5 in order to associate a prebration structure with a given rigid subcategory.

We note that Lemma 1.1, which is an analogue of lemma 3.3 shown by Buan and Marsh in [START_REF] Aslak | From triangulated categories to module categories via localisation[END_REF] for exact categories, might be of independent interest.

Let E be a weakly idempotent complete exact category with enough injective and projective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective objects, and stable under taking direct sums and summands. Let

which induces the following equivalence of categories prM/ M mod M.

For more details, see the article of Demonet and Liu, [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF]. In the following lemma, we prove that if M is contravariantly nite, then so is prM, provided that M also contains all projective objects.

This lemma tells us that we can replace each object by a cobrant replacement.

127

Proof of the theorem. Some of the properties are trivial because of the results on Frobenius categories which did not use the Frobenius condition.

(i) Let f be a retract of w ∈ W. Let us show that Gf is an isomorphism. We have:

As Gf is a retract of an isomorphism, it is itself en isomorphism. Indeed,

Then Gf is an isomorphism. Now let us show that W has the 2-out-of-3 property.

Let us show that it is also a right inverse:

Then we use the same hint to show that Gf is an isomorphism.

(ii) This result is easy, because brations are dened by a lifting property.

(iii) Fibrations are deations (this is because E is weakly idempotent complete, see lemma 1.2.5), then pullbacks exist along brations. The rest of the item is the previous lemma.

(iv) We have the factorization from lemma 1.3.2. It means that any morphism can be factorised through a morphism of (J ) followed by a morphism of J . Indeed, it only uses the fact that there exist some M-approximations, which we suppose in the hypotheses of this theorem. Then for any X, the diagonal X → X ⊕ X can be factorised X → X → X ⊕ X, where the rst morphism is a weak equivalence (we have seen that the morphisms of (J ) are weak equivalences from lemma 1.3.1).

(v) By denition of J, any object is brant, then 0 → B is a bration for any object B of E.

Theorem of Quillen

We recall that the objects of prM are exactly the same as cobrant objects (dened here by satisfying the lifting property along trivial brations). Now, we show that the theorem of Quillen is satised. We note that we still have the same notion of homotopy.

Theorem 1.6.1. Let E be a weakly idempotent complete exact category with enough injective objects. Assume that M ⊆ E is a rigid, contravariantly nite subcategory of E containing all the injective objects, and stable under taking direct sums and summands. Suppose moreover that M is contravariantly nite. Let

Let Ho E be the localization of E at the class W of weak equivalences. Let mod M be the category of nitely presented M modules. There is an equivalence of categories Ho E mod M.

Remark 1.6.2. In order to prove this theorem, we need the following lemmas. The proof of these is well-known, but we give here some details in order to show that the restriction of (iii) in theorem does not aect the results. Then, some parts of the proofs simplify due to the particular shape of the relation of homotopy. Lemma 1.6.3. Let A, B, X be three cobrant objects. Let F be the functor dened by

Suppose moreover that f is a weak equivalence. Then F induces an isomorphism

The particular case of Frobenius categories We recall the notations: Let E be a Frobenius category. Let M be a full rigid subcategory which contains the injective objects. Let

which induces the following equivalence of categories prM/ M mod M.

For more details, see the article of Demonet and Liu, [START_REF] Demonet | Quotients of exact categories by cluster tilting subcategories as module categories[END_REF]. Let M is a subcategory of E which is contravariantly nite. As the injective objects are also projective, note that M contains the projective objects.

A deeper study of prM

We have this rst lemma: Lemma 2.1.1. Let X ∈ E. If there is an M-approximation of ΩX, then there exists a M-approximation of X. In particular, if M is contravariantly nite, then so is M.

If we factorize through the injective modules, then we get:

where Ext corresponds to the extension morphisms in the category E. Moreover, we have E(A, g) = 0 because A ∈ M and g ∈ M ⊥ . Then Gf is an epimorphism. Now let us apply the functor G to equation 2.14 and factorize through projective modules. We then have:

Then Gf is a monomorphism, thus an isomorphism.

Fibrations and cobrations

Let us recall the notations: Let W be the class of morphisms f such that Gf is an isomorphism. Such morphisms are called weak equivalences. Let

where 0 → M 1 → M 0 → X → 0 is a short exact sequence and I 0 appears in the short exact sequence 0 → M 0 → I 0 → M 0 . The morphisms of J are called brations and compose the class Fib. The following lemma characterizes brations. We recall that g ∈ M if, for any morphism b : M → Y , with M ∈ M, the morphism g • b factorizes through an injective module. Denition 2.3.1. For a morphism f : X → Y , we call a cone of f , any morphism g, obtained from a push-out of an ination X → I for some injective I along f . Then, we notice that g is unique up to isomorphism. Lemma 2.3.2. Suppose that E is a weakly idempotent complete category (see the article of Bühler, [START_REF] Bühler | Exact categories[END_REF]). Then f is a bration if and only if f is a deation,