Cluster algebras arising from the infinity-gon

We introduce a handy construction of cluster algebras of type $\mathbb{A}_{\infty}$, we give a complete classification of the cluster algebras arising from the infinity-gon, and finally we construct the category of the diagonals of the infinity-gon and show that it is triangle-equivalent to the infinite cluster category of type $\mathbb{A}_{\infty}$ described by Holm and J{\o}rgensen.

A cluster algebra is a commutative ring with a distinguished set of generators, called cluster variables. The set of all cluster variables is constructed recursively from an initial set using a procedure called mutation. Generators are organized into clusters and each cluster contains exactly n clusters variables. The study of cluster structures for 2-Calabi-Yau categories in [4] led however to the introduction of cluster algebras with countable clusters. The infinite cluster category D of Dynkin type A ∞ was constructed by P. Jorgensen in [17]. The cluster tilting subcategories of D were classified by using the triangulations of infinity-gon in [16]. The Caldero-Chapoton map has been introduced in [6] and [8] by Caldero, Chapoton and Keller to formalize the connection between the Fomin-Zelevinsky cluster algebras and the cluster category of Buan, Marsh, Reineke, Reiten and Todorov. The analogue of this map was constructed between infinite cluster algebras of type A ∞ and the infinite cluster category D by Jorgensen and Palu, see [18]. This paper is devoted to the study of cluster algebras of type A ∞ , cluster algebras arising from the triangulations and the categorification of the infinity-gon. In this paper we first give a handy construction of each cluster algebra B of type A ∞ . More specifically we prove that the cluster algebra B is a particular subalgebra of the projective limit algebra A of a particular projective system (A i , p i,j ) i,j≥1 , where A n is a cluster algebra of type A n , and p i,j : A j −→ A i is a surjective cluster morphism in the sense of Assem, Dupont and Schiffler. We next continue the study of cluster algebras arising from the infinity-gon started by Grabowski and Gratz in [15]. In this study, we associate to each triangulation T of the infinity-gon, the cluster algebra A(T ) as done by Fomin, Shapiro and Thurston in [11] for the case of the marked surfaces with a finite marked points. For the case of marked surfaces with a finite number of marked points, Fomin, Shapiro and Thurston have shown that the cluster algebra associated to a triangulated surface does not depend upon the choice of triangulation. We shall show that this result does not hold for cluster algebras arising from the triangulations of the infinity-gon. However, we give a complete classification of the clusters algebras arising from the infinity-gon using the notion of congruence between triangulations inside the set of all triangulations. In [13] Fomin and Zelevinsky have considered the notion of strong isomorphisms, by which they mean an isomorphism of the cluster algebras which maps cluster to cluster. Theorem 1.2. Let T and T ′ be two triangulations of the infinity-gon, and A(T ) and A(T ′ ) the associated cluster algebras. Then T and T ′ are congruent if and only if the clusters algebras A(T ) and A(T ′ ) are strongly isomorphic.
We also define the category of diagonals of the infinity-gon as the one of the (n+3)-gon constructed by Caldero, Chapoton and Schiffler in [7]. It is well-known, that the category of diagonals of the (n+3)-gon is equivalent to the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov [5] for the quiver of type A n . Here we show that the category of diagonals of the infinity-gon C is equivalent to the infinite cluster category D of type A ∞ of Jørgensen [17]. Note that the category C appears implicitly in [17]. Theorem 1. 3. The categories C and D are triangle-equivalent.
As a consequence of this result, we give a description of the Auslander-Reiten triangles in geometric terms inspired by [3]. Our paper is organized as follows.
In section 2, we introduce a special projective system of clusters algebras of type A n and give the relation between the projective limit of this system and the corresponding cluster algebra of type A ∞ ; this gives rise to a handy construction of clusters algebras of type A ∞ . This construction is handy because the algebra B is expressed as a classical sub-algebra of a finite product of Z-algebras.
In section 3, we give a complete classification of the cluster algebras arising from the infinity-gon.
Finally, in section 4, we construct the category C of diagonals of the infinity-gon and show that it is triangle-equivalent to the category D . Therefore, inspired by [3], and we give a description of Auslander-Reiten triangles of C using the diagonals of the infinity-gon. ACKNOWLEDGMENTS I would like to thank my supervisors Ibrahim Assem and Vasilisa Shramchenko for their patience and availability. I thank Grégoire Dupont for interesting and helpful discussions, David Smith for valuable remarks. I also would like to thank Yann Palu and Peter Jørgensen for some clarifications of the infinite cluster category of type A ∞ .

2.
Cluster algebras of type A ∞ 2.1. Basic construction. We recall that a quiver is a quadruple Q = (Q 0 , Q 1 , s, t) consisting of two sets, Q 0 (whose elements are called points) and Q 1 (whose elements are called arrows) and two functions s, t : Q 1 −→ Q 0 associating to each arrow α ∈ Q 1 its so-called source s(α) and target t(α). If i = s(α) and j = t(α), we denote this situation by i α −→ j. Given a point i, we set i + = {α ∈ Q 1 |s(α) = i} and i − = {α ∈ Q 1 |t(α) = i}. We say that a quiver Q is locally f inite if for each i ∈ Q 0 , the sets i + and i − are finite.
Let Q be a countably infinite, but locally finite quiver without cycles of length at most two, and let X = {x n |n ≥ 1} be a countable set of undeterminates. where we agree that the point i of the quiver Q corresponds to the variable x i . We define the mutation µ k in k ∈ Q 0 exactly as in the case of a finite quiver, that is µ k (X, Q) = (X ′ , Q ′ ), where Q ′ is the quiver obtained from Q by performing the following operations: -for any path of i −→ j −→ k of length two having k as midpoint, we insert a new arrow i −→ j. -all arrows incident to the point k are reversed, -all newly occurring cycles of length two are deleted. Clearly, Q ′ is still locally finite. On the other hand, X ′ is a countable set of variables defined as follows: These operations are performed inside the field F = Q(X) of rational functions over the undeterminates x n , called the ambient f ield. One verifies exactly as in the case of a finite quiver that µ 2 k (X, Q) = (X, Q). From now on, let Q be a quiver having as underlying graph the infinite tree A ∞ 1 2 3 ... n − 1 n ... and let X = {x n |n ∈ N} be a countable set of undeterminates. Each pair (X ′ , Q ′ ) obtained from (X, Q) by a finite sequence of mutations is called a seed, and the set X ′ is called a cluster. The elements of X ′ are called cluster variables. The pair (X, Q) is called the initial seed, and X is called the initial cluster.
Definition 2.1. The cluster algebra B = A(X, Q) is the Z-subalgebra of F generated by the set X which is the union of all possible sets of variables obtained from X by finite sequences of mutations.
Gekhtman, Shapiro and Vainshtein showed in [14] that for every seed (X,Q) of a given cluster algebra, the quiverQ is uniquely defined by the clusterX. Because mutation is a local operation, this result remains true for the countable seeds above. Our objective in this first section is to give a handy construction of the cluster algebra B = A(X, Q).

2.2.
A projective system of cluster algebras. In this subsection we denote by − → A n the linearly oriented quiver of type A n , 1 −→ 2 −→ 3 −→ · · · −→ n and by X n = {x 1 , x 2 , ..., x n } an associated set of variables. Let F n = Q(x 1 , x 2 , ..., x n ) be the field of rational functions on the x i for 1 ≤ i ≤ n (with rational coefficients) and X n be the union of all possible sets of variables obtained from X n by successive mutations. This data defines a cluster algebra A n = A(X n , − → A n ) having (X n , − → A n ) as initial seed. We recall that the Laurent phenomenon asserts that each cluster variable in A n can be expressed as a Laurent polynomial in the x i , with 1 ≤ i ≤ n, that is, such a variable is of the form where p ∈ Z[x 1 , x 2 , ..., x n ] and d l ≥ 0 for all l, with 1 ≤ l ≤ n. The positivity theorem asserts that all coefficients of the polynomial P are non-negative integers. The positivity theorem holds because the cluster algebra A n is of type A n . Now let i, j be positive integers with i ≤ j, we define the map p i,j : A j −→ A i on the generators of A j as follows Since p i,j is an evaluation, it is a morphism of Z-algebras. Clearly, we have Proposition 2.1. With the above notation, Proof. Because of the above equalities, it suffices to show that, for each n ≥ 2, we have A n−1 = Z[p n−1,n (X n )]. This is done by induction on n. Assume first that n = 2. In this case ]. The morphism p 1,2 : A 2 −→ A 1 is defined on the generators as follows: p 1,2 (x 1 ) = x 1 , p 1,2 (x 2 ) = 1, p 1,2 ( 1+x2 x1 ) = 2 x1 , p 1,2 ( 1+x1+x2 x1 ) = 1+ 2 x1 , and p 1,2 ( 1+x1 x2 ) = 1 + x 1 . Thus, clearly, A 1 = Z[p 1,2 (X 2 )] so that p 1,2 : A 2 −→ A 1 is a surjective morphism of Z-algebras.
We now assume that, for every j < n, we have A j−1 = Z[p j−1,j (X j )] and show that A n−1 = Z[p n−1,n (X n )]. For this purpose, we use the categorification of the cluster algebras A n , and A n−1 , as in [5]. The Auslander-Reiten quiver Γ n of the cluster category attached to A n is of the form  where we agree to identify each point of Γ n with the corresponding cluster variable and y 0,i = x i for each i such that 1 ≤ i ≤ n; and we denote by y i,j with 0 ≤ i ≤ n, 1 ≤ j ≤ n and i + j ≤ n + 1, the clusters variables of A n . Because the quiver − → A n is of Dynkin type A n , the cluster algebra A n is of finite type and the Auslander-Reiten quiver Γ n lies on a Moebius strip. Thus y 0,i = y i,s , where i + s = n + 1 and 1 ≤ i, s ≤ n.
We denote by Γ n−1 the Auslander-Reiten quiver of the cluster category attached to A n−1 . It is of the form where again each point is identified with the corresponding cluster variable. Therefore y ′ 0,i = x i for all i such that 1 ≤ i ≤ n − 1; and we denote by y ′ i,j with 0 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1 and i + j ≤ n, the clusters variables of A n−1 . Because the quiver − → A n−1 is of Dynkin type A n−1 , the cluster algebra A n−1 is of finite type and the Auslander-Reiten quiver Γ n−1 lies on a Moebius strip. Thus, y ′ 0,i = y ′ i,s , where r + s = n and 1 ≤ i, s ≤ n.
We say that a point y i,j of Γ n is stable provided p n−1,n (y i,j ) = y ′ i,j . Clearly, all the points y 0,i with 1 ≤ i ≤ n − 1 are stable. It then follows from the definition of mutation that, for every pair (i, j), such that i + j ≤ n − 1, the point y i,j is stable. Now it remains to consider the points {y i,j |n ≤ i + j ≤ n + 1} of Γ n . Because p n−1,n is an evaluation, we may write p n−1,n (y i,j ) = y i,j (1) for brevity.
The proof is completed in the following three steps (a) (b) (c). (a) We first claim that, if i + j = n and n ≥ 1, then y i,j (1) = y ′ i,j . This is done by induction on i.
Assume now the result valid for all j ≤ i. We have y i+1,n−i−1 = 1+yi+1,n−i−2yi,n−i yi,n−i−1 . The evaluation of y i+1,n−i−1 at 1 is given by: where we used the induction hypothesis and the stability of y i+1,n−i−2 and y i+1,n−i−1 . This establishes our claim for the step (a).
We want to show that p i,j is a morphism of cluster algebras in the sense of Assem, Dupont and Schiffler in [1]. In order to define cluster morphisms, we recall the definitions of rooted cluster algebras and rooted cluster morphisms due to I. Assem, G. Dupont and R. Schiffler.

Definition 2.2.
A seed is a triple Σ = (X, ex, B) such that: (1) X is a countable set of undeterminates over Z, called the clusters of Σ; (2) ex ⊂ X is a subset of X whose elements are the exchangeable variables of Σ; The elements of X\ex are called the f rozen variables. Note that in the above definition, the matrix B can be replaced by a (locally finite) quiver without loops and 2-cycles.
Let Σ = (X, ex, Q) be a seed. We say that ( The mutations are made along finite admissible sequences of variables.
A rooted cluster algebra is defined similarly as the Fomin-Zelevinsky cluster algebras, but the definition of rooted cluster algebras authorises seeds whose clusters are empty. Such seeds are called empty seeds and by convention the rooted cluster algebra corresponding to an empty seed is Z. The rooted cluster algebra is always viewed with its initial seed. To know more about the different points of view between Fomin-Zelevinsky cluster algebras and rooted cluster algebras, we refer to [1,Remark 1.7].
Let Σ = (X, ex, Q) and Σ ′ = (X ′ , ex ′ , Q ′ ) be two seeds and let f : The following definition is due to Assem, Dupont and Schiffler.
The rooted cluster algebras and the rooted cluster morphisms forms a category, see [1].
Let Σ n = (X n , ex n , Q n ), with ex n = X n , then Σ n is a seed of the rooted cluster algebra A(Σ n ); the cluster algebra A(Σ n ) coincides with the cluster algebra A n . We also have Σ n−1 = Σ n \{x n }, where the seed Σ n \{x n } is defined in [1, Section 6.2].
Proof. By the Proposition 2.1, the map p n−1,n is a Z-morphism induced by the specialisation of x n to 1. Because of [1, Proposition 6.10], the Z-morphism p n−1,n is a surjective rooted cluster morphism. Since p i,j = p i,i+1 • p i+1,i+2 • ... • p j−1,j and each p n−1,n is a surjective rooted cluster morphism, by [1, Proposition 2.5] p i,j is also a surjective rooted cluster morphism.
Corollary 2.2. The family (A i , p i,j ) i,j≥1 forms a projective system of cluster algebras.
We denote by A = lim ←− A n the corresponding projective limit in the category of We also denote by p i : A −→ A i the canonical morphisms induced by the projective limit. Let a l be a cluster variable of A i , then the element (a 1 , a 2 , ..., a l , a l , ...) with a j = a l for j ≥ l ≥ i, is an element of A and p i (a 1 , a 2 , ..., a l , a l , ...) = a i ; therefore p i is a surjective homomorphism of Z-algebras. The p i are called the canonical projections morphisms. We thus have a commutative diagram It was shown that the category of rooted cluster algebras admits countable coproducts [1, Lemma 5.1] and does not generally admit products [1,Proposition 5.4].
A n is ultimately constant if there exists j ∈ N such that a n = a j for all n ≥ j.
Let Q be a linearly oriented quiver of type A ∞ having 1 as unique source. Here (X, Q) is a seed of the cluster algebra B. We want to understand the relation between the cluster algebra B and the Z-algebras A. Because the algebra A is a projective limit of cluster algebras of finite type, it allows to express the cluster algebra B in term of A. Our first result is the following. Proof. Assume that Q is the quiver having as underlying graph the infinite tree A ∞ with linear orientation and for unique source the vertex 1. Recall that B = A(X, Q) is a cluster algebra of seed (X, Q), where X = {x n , n ≥ 1}. Let Y = {y n , n ≥ 1} be a new set of undeterminates whose elements are defined by y 1 = (x 1 , x 1 , x 1 , ...), By definition, all y i are elements of A. LetF = Q(Y ) be the field of rational functions over y i (with rational coefficients), we callF the ambient f ield.
The cluster algebraÃ = A(Y, Q) is the Z-subalgebra ofF generated by the set Y which is the union of all possible sets of variables obtained from Y by successive mutations. We define the map ϕ : B −→Ã by setting ϕ(x i ) = y i and we extend it to all cluster variables of B by respecting mutations, that is if . We extend again ϕ to an injective morphism of Z-algebras. Thus ϕ is a monomorphism of Z-algebras.
Let x = p(x 1 , x 2 , ..., x k ) be an element of X then y = p(y 1 , y 2 , ..., y k ) is an element of Y. By definition we have ϕ(x) = y.
This shows that the morphism ϕ is an isomorphism between B and ϕ(B) =Ã.
Each ultimately constant element of A belongs toÃ. But the element a = ( is an element of A which does not belong tõ A. ThereforeÃ is a proper Z-subalgebra of the algebra A. Now let x = {q n |n ≥ 1} be a cluster of B, by the definition of ϕ we have ϕ(µ q k (x)) = µ ϕ(q k ) (ϕ(x)) and ϕ(x) is a cluster ofÃ. It follows that ϕ is a cluster isomorphism. For more details about cluster isomorphisms, we refer to [1]. This completes the proof.
Remark 2.1. Let Q be a quiver of type A ∞ and Q n the full sub-quiver of Q whose set of vertices is Q n 0 = {1, 2, ..., n}. We denote by B ′ the cluster algebra of seed (X, Q) and A ′ n the cluster algebra of seed (X n , Q n ). We reproduce the above construction with B ′ playing the role of B and A ′ n playing the role of A n . Then the Theorem 2.1 remains true for any cluster algebra of type A ∞ . Proof. Let x be a cluster variable of the cluster algebra B. Then there exists a nonnegative integer n such that x is identified to a cluster variable a m of A n . We have x = (a 1 , a 2 , ..., a m , a m , ...) with m ≤ n. Thus by Theorem 2.1, a m = q(x 1 , x 2 , ..., x n ) if and only if x = q(y 1 , y 2 , ..., y n ), where Q is a Laurent polynomial. Since a m is a cluster variable of A n , and it is well-known that the Laurent phenomenon and the positivity theorem hold for the cluster algebra of type A n , then the Laurent phenomenon and the positivity theorem hold for the cluster algebra B ′ .

Cluster algebras arising from infinity-gon
Fomin, Shapiro and Thurston initiated a study of the cluster algebras arising from triangulations of a surface with boundary and finitely many marked points in [11]. In this approach, it was shown that the cluster algebra associated to a triangulation of a marked surface (S, M ) depends only on the surface (S, M ) and not on the choice of triangulation. As we shall see this is not true in the case of the infinity-gon. Our objective in this section is to classify the cluster algebras arising from the infinity-gon.

3.1.
Triangulations of the infinity-gon. In this subsection, we classify the triangulations of the infinity-gon S using the notions of connected component and frozen arc which will be defined later.
We adopt the same philosophy as that of [16], that is, we view the integers as the vertices of the infinity-gon and the pairs of integers as the arcs.
Let (m, n) be an arc of the infinity-gon, with m < n. If n − m = 1, we say that the arc (m, n) is a boundary arc, and if n − m ≥ 2, we say that (m, n) is a internal arc of the infinity-gon. The In the following internal arcs will simply called arcs. Two arcs (m, n) and (p, q) are said to cross if we have either m < p < n < q or p < m < q < n. A triangulation of S is a maximal set of non-crossing arcs.
The following definition is due to Holm and Jørgensen in [16]. It is shown in [16] that if a triangulation of S has a right-fountain, then it also has a left-fountain and vice versa. The following result in [16,Lemma 3.3] characterizes the triangulations of infinity-gon. Following Jørgensen and Palu in [18], we say that the arc ω = (s, t) spans the arc δ = (u, v) if s ≤ u < v < t or s < u < v ≤ t. We denote by B(ω) the set of all arcs spanned by the given arc ω. Definition 3.3. Let T be a triangulation, and τ an arc of T . We say that an arc γ belongs to the connected component of τ if there exists a finite sequence of flips µ 1 , µ 2 , ..., µ k such that γ = µ k ...µ 2 µ 1 (τ ).
We denote by C τ the connected component of τ. The connected components of the arcs of T are called simply the connected components of T .
An arc γ of S is reachable by T if it belongs to a connected components of T . If this is not the case then we say that γ is unreachable by T . An arc of T which cannot be flipped to any other arc is called a f rozen arc. Proof. Let T be a triangulation of S. Assume that T is locally finite. Let γ be an arc of S, then there exists an arc ζ such that γ is an arc of the polygon P ζ bounded by ζ. The restriction T ζ of the triangulation T to P ζ is a triangulation. Because γ is an arc of P ζ , it is joined by a finite sequence of flips of arcs of T ζ . Since T ζ ⊂ T , then γ is joined by a finite sequence of flips of arcs of T . Thus, each arc of S is reachable; hence T has no frozen arc. If T has a left-fountain m 0 and a right-fountain n 0 such that n 0 − m 0 = 1, then T has two connected components and no frozen arc. If T has a left-fountain m 0 and a right-fountain n 0 such that n 0 − m 0 ≥ 2, then T has three connected components and the arc (m 0 , n 0 ) is a frozen arc. Assume that T possesses another frozen arc (m 1 , n 1 ), then (m 1 , n 1 ) crosses an infinity of arcs of T incident to m 0 or an infinity of arcs of T incident to n 0 . Assume now that T is a triangulation with two frozen arcs ω 1 and ω 2 . The arc ω 1 does not span ω 2 and vice versa, because if not, then one of the two frozen arcs can be flipped to another arc. Each frozen arc bounds a finite connected component, and then it is finite. Therefore, the triangulation T has more than one left-fountain or more than one right-fountain. This is a contradiction to Lemma 3.1.
Let T be a triangulation of S with a frozen arc τ , we say that T is of Type (III) k with k = |B(τ )|, where B(τ ) denotes the number of arcs spanned by the frozen arc ω.  Proof. Let T be a triangulation of S. If T is locally finite, then every arc of S can be reached by a sequence of flips of arcs of T. Then T is of type (I). If T has a fountain or a left-fountain m 0 and a right-fountain n 0 with n 0 − m 0 = 1, then T has two connected components, and any arc of each component is reachable. In this case T has no frozen arc, hence T is of type(II).
If T has a left-fountain m 0 and a right-fountain n 0 with n 0 − m 0 ≥ 2, then T has three connected components. The arc (m 0 , n 0 ) is an arc of T. Assume that (m 0 , n 0 ) is not in T ; then one of the arcs (m 0 + 1, n 0 ), (m 0 , n 0 − 1) belongs to T and one of the arcs (m 0 − 1, n 0 ), (m 0 , n 0 + 1) belongs to T. If the arc (m 0 + 1, n 0 ) belongs to T, then (m 0 − 1, n 0 ) ∈ T and (m 0 − 1, n 0 ) crosses an infinite number of arcs of T incident to a left fountain m 0 . This is a contradiction because T is a triangulation.
Similarly, if (m 0 , n 0 − 1) belongs to T , we get a contradiction. Hence the arc (m 0 , n 0 ) is a frozen arc, and it is unique by the lemma 4.1.1. Thus T is of type (III) k .
Assume now that T has l connected components, where l ≥ 4. Then only one of the l components is finite, because if not, T would have more than one frozen arc, and this is a contradiction to Lemma 3.2. Therefore, the triangulation T has at least three infinite connected components. Each of the infinite connected components of T contains either a right-fountain or a left-fountain, thus T has more than two fountains, this contradicts Lemma 3.
Fomin, Shapiro and Thurston in [11] associated to a triangulation of a marked surface (S, M ) a finite quiver without cycles of length at most two. Similarly, we associate to each triangulation of S an infinite quiver without cycles of length at most two.
Let T be a triangulation of the infinity-gon S, we associate to T a quiver Q T . The classification of quivers Q T is given in [15,Theorem 3.11]. We associate to the triangulation T the cluster algebra A(T ) of seed (X T , Q T ) in the same way as the one for a marked surface (S, M ). The following remark gives a relationship between the quiver of type A ∞ and the quivers associated to the triangulations on the infinity-gon.

Remark 3.2.
If R is a quiver of type A n , then there exists a triangulation Γ of the (n + 3)-gon such that R ∼ = Q Γ ; but if we reproduce the assumption above with A ∞ playing the role of A n and the infinity-gon playing the role of (n + 3)-gon, then this is not true. For example let Q be a linearly oriented quiver of type A ∞ and assume that there exists a triangulation T of the infinity-gon such that Q ∼ = Q T . Hence Q is a triangulation with left fountain and without right fountain or Q is a triangulation with right fountain and without left fountain. This is a contradiction with [16].
An isomorphism f between two Z-algebras is called a strong isomorphism of clusters algebras if f maps each cluster to a cluster and preserves mutations. For more details we refer to [13]. The quiver associated to the triangulation T is the quiver Q T given by: .. and the quiver associated to the triangulation T ′ is the non connected quiver Q T ′ given by:

There is no strong isomorphism between the cluster algebras A(T ) and A(T ′ ).
This shows that the theorem of Fomin, Shapiro and Thursten mentioned above does not hold for the infinity-gon.

3.2.
The specificity of cluster algebras arising from the infinity-gon. In this section we find a criterion that allows us to decide whether two triangulations give rise to isomorphic cluster algebras. The bijection θ is called an admissible map. If T and T ′ are congruent, we denote this situation by T ≃ T ′ . Congruence is an equivalence relation.
Definition 3.6. A sequence of arcs (γ n ) n≥1 in S, is said to span S if for any arc γ there exists an integer k such that γ is spanned by γ k . Lemma 3.3. Let T be a triangulation of type (I), then there exist a zigzag triangulation Z and a sequence of common arcs (γ n ) n≥1 in T and Z that spans S.
Proof. Let T be a triangulation of type (I), we shall show that there exists a sequence of distinct arcs (γ kn ) n≥1 of T such that γ kn = (s kn , s kn ), where s kn+1 < s kn<0 and t kn+1 > t kn>0 .
We assume first that any arc γ = (s, t), with with s < 0 and t > 0 does not belong to T.
Because (s, t) does not belong to T and T is a triangulation, there is an arc γ 1 = (s 1 , t 1 ) of T which crosses (s, t) and γ 1 is closer to a vertex t and t 1 > t. Analogously, there exists an arc γ 2 = (s 2 , t 2 ) of de T which crosses (s, t) and γ 2 is closer to a vertex s and s 2 < s. The connected components C γ1 and C γ2 respectively of γ 1 and γ 2 are distinct, this is a contradiction because T is of type (I). Hence γ ∈ T.
We show that T has an infinite sequence of arcs (γ kn ) n≥1 such that γ kn = (s kn , s kn ), where s kn+1 < s kn<0 and t kn+1 > t kn>0 . Assume that any sequence of arcs (γ kn ) n≥1 such that γ kn = (s kn , s kn ), where s kn+1 < s kn<0 and t kn+1 > t kn>0 is finite. Let γ k l = (s k l , t k l ), with s k l < 0 and t k l > 0 be an arc of T such for all arcs of (γ kn ) n≥1 , we have s k l < s kn and t k l > t kn . The same argument used for γ to γ k l gives rise to a contradiction. Thus there exists a sequence of arcs (γ kn ) n≥1 of T which can be chosen such that s kn+1 < s kn<0 and t kn+1 > t kn>0 . Now we construct a zigzag triangulation having infinitely many common arcs with T . Because (γ kn ) n≥1 is a sequence of infinitely many non-crossing arcs, we complete it in each polygon bounded by γ kn and γ kn+1 . We choose one zigzag triangulation of the polygon bounded by γ kn and γ kn+1 . We apply this process for all non-negative integers n, thus for all polygons bounded by γ kn and γ kn+1 . This process gives rise to two subsets of S as follows.
The first is composed by the arcs γ kn , for all n.
The second is the union of the new arcs of the zigzag triangulations of each polygon bounded by γ kn and γ kn+1 .
Let Z be the union of the two subsets defined above. In fact, Z is a triangulation of S by the construction, and each γ k l is a common arc of T and Z.
Finally, we show that (γ kn ) n≥1 spans S. Let δ = (u, v) be an arc of S, since (γ kn ) n≥1 is infinite, there is an integer l such that the arc δ is spanned by γ k l . This completes the proof.
Lemma 3.4. Let T and T ′ be two triangulations of type (II) or T and T ′ be two triangulations of type (III) k , and let C T and C T ′ their connected components respectively. Then there is a sequence of common arcs (γ n ) n≥1 in T and T ′ that spans C T and C T ′ .
Proof. (i) Let T and T ′ be two triangulations of type (II), suppose that T has a left-fountain m 0 and a right-fountain n 0 , n 0 − m 0 = 1. There is a triangulation Γ with one fountain such that its associated quiver Q Γ is isomorphic to the associated quiver Q T of T .
Because of the above argument, it is sufficient to give a proof just for the case where each triangulation has a left-fountain and a right-fountain. Let T and T ′ be two triangulations of type (II) such that T has a left-fountain m 0 and a rightfountain n 0 and T ′ has a left-fountain m ′ 0 and a right-fountain n ′ 0 . Because n 0 and n ′ 0 are integers, we can assume that n 0 ≤ n ′ 0 ; and there is a non-negative integer l such that l = n ′ 0 − n 0 . We define the map σ : S −→ S by σ(m, n) = (m + l, n + l). The map σ is a bijection and the image σ(T ) of T is a triangulation. Moreover, σ preserves the flips of arcs. Thus σ is an admissible map. Therefore, we can suppose without loss of generality that T and T ′ have the same left-fountain m 0 and the same right-fountain n 0 . Because the triangulations T and T ′ have the same leftfountain m 0 and the same right-fountain n 0 , then T and T ′ have infinitely many common arcs of the form (n 0 , n) and so, T and T ′ have infinitely many common arcs of the form(m, m 0 ). Thus there is a sequence of distinct common arcs (γ kn ) n≥1 which spans C T and C T ′ .
(ii) Now we suppose that T and T ′ are two triangulations of type (III) k . By using the argument above, we can assume without loss of generality that T and T ′ have the same left-fountain m 0 and the same right-fountain n 0 .
If k is equal to zero, the proof is similar to the case (i). If k ≥ 1, the frozen arc ω bounds a polygon P ω ; and by the definition of k, each arc of the restriction of T to P ω is spanned by ω and each arc of the restriction of T ′ to P ω is spanned by ω. Combining this argument and the one used in (i) to define the common arcs, we find that there exists a sequence of distinct arcs (γ kn ) n≥1 which spans C T and C T ′ . Proof. Let T and T ′ be two triangulations of S, and assume that T ≃ T ′ . Since T ≃ T ′ , there exists an admissible map θ which maps T to T ′ . Let C γ be a connected component of T, then θ(C γ ) is a connected component of T ′ . Because θ(γ) ∈ θ(C γ ), we have θ(C γ ) = C θ(γ) . If C γ and C δ are two distinct connected components of T , then θ(C γ ) and θ(C δ ) are distinct connected components of T ′ . The number of connected components is invariant by θ. Moreover the connected components C γ and C θ(γ) are both either finite, or infinite. Moreover, if T has a finite component C γ , then |C γ | = |C θ(γ) | = k. Hence T and T ′ are of the same type.
Conversely let T and T ′ be two triangulations of S, we have three cases.
(i) If T and T ′ are of type (I), then by Lemma 3.3 there exists a zigzag triangulation Z and a sequence of common distinct arcs (γ kn ) n≥1 in T and Z that spans S.
Let P kn be the polygon bounded by the arc γ kn . The restriction T kn and Z kn of the triangulations T and Z are the triangulations of P kn . We want to show that T = n≥1 T kn and Z = n≥1 Z kn . Because T kn ⊂ T , then n≥1 T kn ⊂ T . For now let us show that n≥1 T kn is a triangulation. Assume that n≥1 T kn is not a triangulation.
We have n≥1 T kn ⊂ T and T is a maximal set of arc, then there exists an arc ς ∈ T which does not belong to n≥1 T kn . Because the sequence (γ kn ) n≥1 spans S, there exists an integer l such that the arc ς is spanned by γ k l . Hence ς is an arc of T k l this is a contradiction. Thus T = n≥1 T kn . Analogously, one can show that Z = n≥1 Z kn .
We know that T kn and Z kn are related by a sequence of flips. This sequence of flips induces an admissible map θ n which maps T kn to Z kn . Let P kn be the set of all arcs of P kn , we have P kn ⊂ P kn+1 . We have also n≥1 P kn = S.
We define the map θ : S −→ S by the following: let γ be an element of S, then there is a minimal integer n such that γ ∈ P kn , we set θ(γ) = θ n (γ). By construction, θ is a bijection which maps T to Z and preserves the flips of arcs. Hence T ≃ Z.
We reproduce the same reasoning above with T ′ playing the role of T and Z ′ playing the role of Z.
If (m 0 , n 0 ) is the arc of Z and (m ′ 0 , n ′ 0 ) is the arc of Z ′ such that n 0 − m 0 = 2 = n ′ 0 −m ′ 0 . We set l = n ′ 0 −n 0 and define the map σ : S −→ S by σ(m, n) = (m+l, n+l). Then σ is an admissible map which maps Z to Z ′ . Then Z ≃ Z ′ and thus T ≃ T ′ (ii) If T and T ′ are of the type (II), because C T = C T ′ we use Lemma 3.4 and we have a bijection θ in C T which maps T to T ′ and preserves the flips of arcs. We define θ on each unreachable arc γ by θ(γ) = γ. Thus we have extended the bijection θ to S. Hence θ is an admissible map, thus T ≃ T ′ .
(iii) If T and T ′ are of the type (III k ), the restrictions of the triangulations T and T ′ to the polygon bounded by the frozen arc are congruent. Because C T = C T ′ , by using the same principle as in (b), we construct an admissible map θ which maps T to T ′ .  Proof. Let T and T ′ be two triangulations of S, assume that T ≃ T ′ . Let A(T ) be a cluster algebra of seed (X T , Q T ) := T , where Q T is a quiver of T and X T = {x γ |γ ∈ T } the set of undeterminates. Let A(T ′ ) be a cluster algebra of seed (X T , Q T ′ ) := T where Q T ′ is a quiver of T ′ and X T ′ = {u λ |λ ∈ T } the set of undeterminates.
Since T ≃ T ′ , there exists an admissible bijection θ, such that θ(T ) = T ′ . We denote by f γ the flip of the arc γ. We define ϕ θ : A(T ) −→ A(T ′ ) on the cluster variable by ϕ θ (x γ ) = u θ(γ) . We have on one hand On the other hand, we have We extend ϕ θ to an isomorphism of Z-algebras from A(T ) to A(T ′ ). Therefore according to [1], A(T ) and A(T ′ ) are strongly isomorphic.
Conversely assume that A(T ) and A(T ′ ) are strongly isomorphic and that T and T ′ are not congruent. Then there exists a strong isomorphism ψ : A(T ) −→ A(T ′ ). According to Proposition 3.2, T and T ′ are not of the same type. We have four cases to consider.
(a) T is of type (I) and T ′ is of type (II). Because T ′ is of type (II), it has two disjoint connected components. Let u λ1 and u λ2 be two cluster variables such that λ 1 and λ 2 do not belong to the same connected component. Since ψ is a strong isomorphism, there are two arcs γ 1 and γ 2 such that ψ(x γ1 ) = u λ1 and ψ(x γ2 ) = u λ2 . The two arcs γ 1 and γ 2 are related by a sequence of flips and then the two variables x γ1 and x γ2 are related by a sequence of mutations, because T is of type (I). In fact, ψ is a strong isomorphism, then u λ1 and u λ2 are related to a sequence of mutations. Thus λ 1 and λ 2 are related by a sequence of flips. This is a contradiction, because λ 1 and λ 2 do not belong to the same connected component.
(b) T is of type (I) and T ′ is of type (III) k . The proof is analogous of the one in the case (a).
(c) T is of type (II) and T ′ is of type (III) k . The triangulation T ′ has a frozen arc ω, then the cluster algebra A(T ′ ) has a frozen cluster variable u ω in the sense of [1]. The map ψ is an isomorphism, then it maps a frozen variable x γ of A(T ) to a cluster variable u ω = ψ(x γ ). Because ψ is a strong isomorphism, it maps frozen variable to frozen variable while A(T ) has no frozen variable. This is a contradiction.
(d) T is of type (III) k and T ′ is of type (III) k ′ with k = k ′ . We assume without loss of generality that k > k ′ . Let ω and ω ′ be respectively the frozen arcs of T and T ′ . Because ψ is a strong isomorphism , we have ψ(x ω ) = u ω ′ . Since k > k ′ , there is an arc γ spanned by ω such that ψ(x γ ) = u λ , and λ is not spanned by ω ′ . The connected component C γ is finite, because γ is spanned by ω and the connected component C λ = Cψ( γ ) is infinite. This is a contradiction, because ψ is a strong isomorphism.
Now we want to show that each cluster algebras of type A ∞ can be embedded in a cluster algebra arising from S. Lemma 3.5. Let Q be a quiver mutation equivalent to a quiver of type A ∞ . Then Q is not a quiver associated to a triangulation of S if and only if Q has a subquiver of type A ∞ with linear orientation.
Proof. Assume that Q has a subquiver of type A ∞ with orientation not necessarily linear. Q is a connected quiver, Q has a subquiver of type A ∞ with orientation not necessarily linear and Q is mutation equivalent to a quiver of type A ∞ . There is a triangulation T of S such that Q T ∼ = Q.
Conversely, assume that Q has a subquiver of type A ∞ with linear orientation. It is sufficient to show that the quiver R: 1 −→ 2 −→ ... is not the quiver associated to any triangulation. Suppose that there exists a triangulation T such that Q T = R. We denote by τ i the arc of T corresponding to the vertex i. All τ i , where i is a non-negative integer, have the same origin and are the arcs of the same half-line. T = {τ i |i ≥ 1} is a triangulation of S with left-fountain, but no right-fountain. This is a contradiction see [16]. Proof. Assume that Q is a mutation equivalent to a quiver of type A ∞ . If Q has a subquiver with orientation not necessarily linear, then Q is the quiver associated to a triangulation of S; in this case η is the identity morphism. Hence the result.
Assume now that Q has a subquiver of type A∞ with linear orientation. By Lemma 3.5, Q is not the quiver of any triangulation of S. We define the quiver R of type A ∞ with linear orientation distinct to the one of Q. The quiver Q ∪ R is the quiver associated to a triangulation T of S. Since the quivers Q and R are nonintersecting connected quivers, the inclusion of the quiver η 0 : Q ⊂ Q∪R induces an embedding of cluster algebras η : A(u, Q) ֒→ A(Σ T ). Because the corestriction η 0 : Q −→ η 0 (Q) = Q is the identity map, the Z-morphism η is a cluster morphism. 4. The cluster category of associated to S 4.1. The infinite cluster category of type A ∞ . We recall the description of the infinite cluster category given in [17,16]. Let K be a field and R = K[T ] be the polynomial algebra. We view R as a differential graded algebra with zero differential and T placed in homological degree 1. Then we set D f (R) be the derived category of differential graded R-modules with finite dimensional homology over K, then D = D f (R) is the infinite cluster category of type A ∞ . The suspension and the Serre functor of D are denoted by Σ and S respectively. The category D is a Klinear, Hom-finite, Krull-Schmidt, triangulated and 2-Calabi-Yau category whose Auslander-Reiten quiver is of the form ZA ∞ , we refer to [17]. The Auslander-Reiten translation of D is τ = SΣ −1 = Σ. For a given integer r ≥ 0, we have a differential graded R-module X r = R/(T r+1 ) which is concentrated in homological degrees from 0 to r. The indecomposable objects of D are Σ j X r for j, r integers, r ≥ 0 and Σ the shift of D. The Auslander-Reiten quiver Γ(D) of D is of the form . . .
The following proposition in [16] characterizes the morphisms of D. The forward morphisms have an easy model: up to multiplication by a nonzero scalar, they are induced by certain canonical morphisms of differential graded modules. the backward morphisms cannot be seen in the Auslander-Reiten quiver; they are in the infinite radical of D.

4.2.
The category of diagonals of the infinity-gon. In this section we provide a geometric realization of the category D.
We adopt the same philosophy as that of [16], that is, the integers can be viewed as the vertices of the infinity-gon and the pairs of integers can be viewed as the arcs of the infinity-gon. Let (m, n) be an arc of infinity-gon, with m < n. If n − m = 1, we say that the arc (m, n) is a boundary arc, and if m ≤ n − 2, we say that (m, n) is a diagonal of the infinity-gon. Our construction is similar to that of [7] in the case of the (n + 3)-gon.
One can define a combinatorial K-linear category C as follows: The indecomposable objects are the arcs (m, n) of S, with m, n ∈ Z and m ≤ n − 2; the objects of C are the linear combinations of the arcs and each arc is stable by the product of the scalars of K. The boundary arcs are identified to zero. The space of morphisms between two arcs (m, n) and (p, q) of this section is given by: The morphisms between two objects are direct sums of morphisms between arcs. The composition of morphisms between arcs is given by the product of scalars in K. The construction of C is inspired by the standard coordinates used in [16]. The category C is a category generated by all the diagonals of S. Therefore by construction C is K-linear, Hom-finite and Krull-Schmidt. Our main result of this section is the following. Proof. Let F 0 : indC −→ indD be such that, for (m, n) ∈ indC we have F 0 (m, n) = Σ −n X n−m−2 . According to [16], F 0 is a bijection. One can define the additive functor F : C −→ D as follows: F (m, n) = F 0 (m, n), and we extend F by additivity and K-linearity to all objects of C. Let u α : (m, n) −→ (p, q) be a morphism of C which is identified with the scalar α of K.
We recall that, via the standard coordinates defined above, if F (m, n) = x and F (p, q) = y then (p, q) ∈ F On the one hand, if (p, q) ∈ F (m,n) R then y ∈ H + (Σx); let f : x −→ y be a forward morphism of D that is f is induced by a canonical morphism of DGmodules. Then each morphism from x to y is of the form λf where λ ∈ K and we set F (u α ) = αf = f α . On the other hand, if (p, q) ∈ F (m,n) L then y ∈ H − (Σx); let g : x −→ y be a backward morphism, because the category D is 2-Calabi-Yau that is Hom D (x, y)=DHom D (y, S(x)), where D=Hom(−, K) is the usual duality. The morphismḡ is the isomorphic image of a forward morphism g : y −→ Σ 2 x. We set F (u α ) = αḡ =ḡ α and F (1 (m,n) ) = 1 x . Let us show now that F is a functor. Let u α : (m, n) −→ (p, q) and u β : (p, q) −→ (r, s) where F (m, n) = x, F (p, q) = y and F (r, s) = z. The proof is completed in three steps (a)(b)(c).
(a) If (p, q), (r, s) ∈ F (m,n) R and (r, s) ∈ F (p,q) R then y, z ∈ H + (Σx) and z ∈ H + (Σy). We have F (u α ) = αf and F (u β ) = βg, where f : x −→ y and g : y −→ z are forward morphisms. The morphism u β u α = u βα is a morphism from (m, n) to (p, q). Then F (u βα ) = βαh, where h : x −→ z is a forward morphism of D. According to [16,Lemma 2.5], the morphism g is nonzero and we have the following commutative triangle where f ′ is the morphism induced by a canonical morphism of differential graded modules. By uniqueness of the canonical morphism between two indecomposables objects, we have f ′ = f and thus F (u βα ) = F (u β )F (α). (b) If (p, q), (r, s) ∈ F (m,n) g and (r, s) ∈ F (p,q) d then y, z ∈ H − (Σx) and z ∈ H + (Σy). We have F (u α ) = αf and F (u β ) = βg where g : y −→ z is a morphism induced by a canonical morphism of differential graded modules. So,f : x −→ y is the isomorphic image of a morphism f : y −→ Σ 2 x induced by a canonical morphism of differential graded modules. Since g is a nonzero morphism, in accordance with [16, Lemma 2.7], we have the following commutative triangle whereh is the isomorphic image of a forward morphism h andf ′ is the image of the morphism f ′ : y −→ Σ 2 x which is induced by the canonical morphism of DG-modules from y to Σ 2 x. By uniqueness of the canonical morphism between two indecomposables objects, we havef ′ =f and hence F (u βα ) = F (u β )F (α). For all other cases not mentioned above, the composition of morphisms are equal to zero see [16,Corollary 2.3]. This shows that F is a functor.
(c) F is essentially surjective because by the definition, each indecomposable module of D is the image of an arc of C under F.
The map F :Hom C ((m, n), (p, q)) −→Hom D (x, y) which associates to u α , the function F (u α ) is a bijection because of the step (a) and (b). Therefore F is full and faithful.
Finally, it follows from (a), (b), (c) that F is an equivalence.
We can give now the description of the category C, via the equivalence established above; clearly, the category C is triangulated, 2-Calabi-Yau and has Auslander-Reiten triangles. In addition, the suspension is given by (m, n)[1] = (m − 1, n − 1) and the Serre functor is given by S(m, n) = (m − 2, n − 2); this situation was predictable from Holm and Jorgensen in [16].
We have also the following operations between the arcs of S defined by: s (m, n) = (m + 1, n) and (m, n) e = (m, n + 1). These operations are defined for the n + 3-gon in [7] and for marked surfaces without punctures in [3]. The operations s (m, n) and (m, n) e can be extended as functors in the category C.
Proof. It is shown in [17] that the following triangle
Assuming now that (m, n) −→ n i=1 (m i , n i ) −→ (p, q) −→ (m, n) [1] is an Auslander-Reiten triangle of C. Since F is an equivalence of categories, is an Auslander-Reiten triangle of D; that is is an Auslander-Reiten triangle of D. The form of the Auslander-Reiten triangle of D is well known; by identification, we have Σ −q X q−p−2 = Σ −n−1 X n−m−2 . There exist r, s with 1 ≤ r, s ≤ l such that F (m r , n r ) = Σ −n X n−m−3 , F (m s , n s ) = Σ −n X n−m−1 and F (m i , n i ) = 0, for all i different from r and s. This completes the proof of our assertion.
Let E be a subcategory of C. Perpendicular subcategories of E are defined by: E ⊥ is the set of all objects x ∈ C such that Hom C (b, x) = 0 for b ∈ B} and ⊥ E is the set of all objects x ∈ C such that Hom C (x, b) = 0 for b ∈ B} A subcategory H of C is a weak cluster tilting if it satisfies (Σ −1 H) ⊥ = ⊥ (ΣH).
For a weak cluster tilting subcategory H of C we can consider the set H of indecomposable objects of H, whence H=addH. The following corollary is a geometric interpretation of the Theorem 4.3 in [16]. Proof. Let T be a triangulation of S. Let U be the image of T by the equivalence of Theorem 3.1 and [16,Theorem 4.3], then U=addU is a weak cluster tilting subcategory of D. Hence T =addT is a weak cluster tilting subcategory of C. Conversely, according to the equivalence of Theorem 3.1, if U is a cluster tilting subcategory of C, then the set of distinct arcs U such that U=addU is a triangulation of S.