Quasifolds, Diffeology and Noncommutative Geometry

After embedding the objects quasifolds into the category {Diffeology}, we associate a C*-agebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry (beginning with the today classical example of the irrational torus) which associates a Morita class of C*-algebras with a diffeomorphic class of quasifolds.

priori the sequence of categories: Then, we generalize to quasifolds the functor toward noncommutative geometry, developed in [IZL17] for orbifolds. In the same way, we associate with each atlas of a quasifold a structure groupoid, 3 in ( §5). The objects of this groupoid are the elements of the nebula of the strict generating family associated with the atlas. The arrows between the objects are the germs of the local di feomorphisms of the nebula that are absorbed by the evaluation map, that is, which project to the identity on the quasifold.
In parallel with the case of orbifolds, in ( §3) and in ( §4) we generalize to quasifolds the two fundamental results: T . Any local smooth map on R n that projects to the identity in the quotient R n /Γ, where Γ is a countable subgroup of Aff(R n ), is everywhere locally the action of some γ ∈ Γ.

T . Local di eomorphisms between quasifolds lift by local di eomorphisms on the level of the strict generating families. Pointed local di eomorphisms lift by pointed local di eomorphisms, where the source and the target can be chosen arbitrarily in the appropriate fibers over the quasifold.
The di culty here is to pass from the action of a nite group on a Euclidean domain to the action of a possibly in nite, but countable group, whose orbits can be dense. This has led us to a substantial revision of the methods, focusing on the countable nature of the groups, and has resulted in proofs that are minimal and essential.
As said above, in ( §5) we de ne the structure groupoid associated with an atlas of the quasifold. Then, thanks to the previous theorem, in ( §6) we prove the following: T . Two di erent atlases of a same quasifold give two equivalent groupoids, as categories [SML78]. Consequently, two di eomorphic quasifolds have equivalent structure groupoids.
In other words, the class of the structure groupoid is a di feological invariant of the quasifold. Then, in ( §7), we give a general description of the structure groupoids.
Next, in ( §9) we prove the following: T . The groupoids associated with two di erent atlases of a same quasifold satisfy the Muhly-Renault-Williams equivalence.
Then, having proved in ( §8) that the structure groupoids associated with the atlases of a quasifold are étale and Hausdor f, we show that they ful ll the conditions of Jean Renault's construction of an associated C * -algebra, by equipping the set of morphisms with the same counting measure as in the case of orbifolds. And in ( §10) we prove then, thanks to ( §9), the main result: T . The C * -algebras associated with di erent atlases of a same quasifold are Moritaequivalent. Therefore, di eomorphic quasifolds have Morita-equivalent C * -algebras.
Finally, we illustrate this construction with two simple examples: the traditional irrational torus T α and the Q-circle, quotient of R by Q. In these two examples, we observe that our construction gives the expected result. In work in progress, we apply these techniques also to the class of symplectic toric quasifolds [EP01,FBEP01].
From the very beginning, with the 1983 paper [PDPI83] on the irrational torus T α , it was clear that there existed some connection between di feology and noncommutative geometry. Beginning with the fact that two such tori T α and T β were di feomorphic if and only if α and β were equivalent modulo GL(2, Z), which is the same condition for their algebra to be Morita-equivalent [MR81]. That could not be just chance. This work, which began with the case of orbifolds [IZL17] and which continues here with quasifolds, shows and describes the logic behind this correspondence. We can reasonably expect wider links between the two theories, which will be addressed in the future. N 1. Unlike the categorical approach, which de nes its objects directly by means of higher structures (stacks, n-categories etc.), we induce the groupoid generating the C * -algebra of the quasifold via its singular geometry encoded in the di feology. So, to the current standard way {groupoid → C * -algebra}, we add a rst oor {di feology → groupoid}, which is not trivial and makes this construction non-tautological. N 2. The irrational tori in arbitary dimension, or quasitori, are particular quasifolds that are dual smooth geometric versions of quasilattices. Knowing how to associate a C * -algebra to a quasifold in a structural fashion can be viewed as a kind of geometric quantization. In fact, the study of the spectrum of the Hamiltonian in a quasicrystal was at the origin of Alain Connes' noncommutative geometry. It is obviously interesting to have a smooth version of this, which is what we are providing. N 3. We assume that the reader is familiar with the basic concepts in di feology and we refer to the textbook [PIZ13] for details. Let us just recall that a di feology on a set X is a set of smooth parametrizations, called plots, that satisfy three fundamental axioms: covering, locality and smooth compatibility. That said, there are a couple of important di feological constructions that we use in the following. First, the quotient di eology: every quotient of a di feological space inherits a natural di feology for which the plots are the parametrizations that can be locally lifted by plots in the source space. Then, the subset di eology: every subset of a di feological space inherits a di feology for which the plots are the plots of the ambient space, but with values in the subset. For example, in di feology a subset is discrete if the subset di feology is the discrete di feology, that is, the plots are locally constant. For example Q ⊂ R is discrete. Finally, the local di eology: 4 a map f , from a subset A of a di feological space X to a di feological space X , is a local smooth map if and only if its composite f • P with a plot P in X, de ned on P −1 (A), is a plot in X . That is equivalent to: A is an open subset for the D-topology 5 and f restricted to A is smooth for the subset di feology. With local smooth maps come local di feomorphisms, which are the fundamentals of modeling spaces in di feology [PIZ13,§4.19], on which many constructions of subcategories are based, like manifolds, manifolds with boundary and corners, orbifolds, quasifolds etc.

T
. It is a pleasure for one of the authors (PIZ) to thank Anatole Khelif for useful discussions on C * -algebras.

D Q
The notion of quasifold has been introduced in 1999 in the paper "On a generalization of the notion of orbifold" [EP99], see also [EP01]. The idea is that a n-quasifold is a smooth object which resembles locally everywhere a quotient R n /Γ, where Γ is some countable subgroup of di feomorphisms. The analogy with orbifolds, for which Γ is nite, is indeed clear. On the other hand, Di eology has been precisely developed, from the mid '80, to deal with this kind of situation, beginning with "Exemple de groupes di férentiels.. ." [PDPI83]. In particular, orbifolds have been later successfully included as a subcategory in {Di feology} in the paper "Orbifolds as Di feology" [IKZ10]. It was natural to try to include also quasifolds, and this is what we do now.

W
? -We have indeed a di feological version of quasifolds, formally de ned by:

D
. A n-quasifold is a di eological space X which is locally di eomorphic, everywhere, to some R n /Γ, where Γ is a countable subgroup, maybe infinite, of Aff(R n ). The group Γ maybe changing from place to place.
In more words, this de nition means precisely the following: for all x ∈ X, there exist a countable subgroup Γ ⊂ Aff(R n ), and a local di feomorphism φ from R n /Γ to X, de ned on some open subset U ⊂ R n /Γ, such that x ∈ φ(U). The subset U is open for the D-topology, that is in this case, the quotient topology [PIZ13, §2.12] by the projection map 6 class: R n → R n /Γ. That said:

D
. Any such di eomorphism is called a chart. A set of charts , covering X, is called an atlas.

N
. In the following we consider only quasifolds that support a locally finite atlas, that is, every point in the quasifold is covered by a nite number of charts. For example, a symplectic toric quasifold has a canonical atlas made of a nite number of charts [FBEP01, Thm. 3.2].

R
1. This approach to quasifolds considers spaces that are already equipped with a smooth structure, that is, a di feology, and then, checks if that di feology is generated by local di feomorphisms with some quotients R n /Γ. This is the standard construction of modeling di feology we mentioned above; it applies to manifolds, orbifolds... and now quasifolds. It is a reverse construction as the usual one, where the smooth structure is built after equipping the underlying set with a family of injections, compatible according to some speci c conditions. Recent works and results involving quasifolds in symplectic geometry can be found in [FBEP18], [FBEP19], and [BPZ19].

R
2. The group Γ is chosen inside the a ne group and not just the linear subgroup, as it is the case for orbifolds. In this way, one immediately has the well known example of the irrational torus T α = R/Z + αZ [PDPI83], where αx ∈ R − Q, as a quasifold. But, we can notice that Γ could be embedded in GL(n + 1, R) by considering R n as the subspace of height 1 in R n × R, and an element ( Hence, the a ne or linear nature for the subgroup Γ is not really discriminant.
R 3. In Example of Singular Reduction in Symplectic Di eology [PIZ16], an in nite dimensional quasi-projective space is built inside the category of di feology. That is, an example of an in nite dimensional analog of the present concept of quasifold. That leaves some space for a generalization of the kind of constructions explored in this paper.

S
-As an object of the category of di feological spaces, quasifolds inherit automatically the notion of smooth maps. A smooth map from a quasifold to another quasifold is just a map which is smooth when the quasifolds are regarded as di feological spaces. It follows immediately that the composite of smooth maps between quasifolds is again a smooth map. Hence, quasifolds form a full subcategory of {Di feology} we shall denote by {Quasifolds}.
A special phenomenon appearing in the case of orbifolds persists for quasifolds: smooth maps between di feological quasifolds may have no local equivariant lifting, as shown by the following example inspired by [IKZ10, Example 25].
Let α ∈ R − Q and C α be the irrational quotient: This di feological space 7 falls into the category of quasifolds.
Let now f : C → C be de ned by  where r = |z| 2 and ρ n is a function vanishing atly outside the interval ]1/(n + 1), 1/n[ and not inside, see Figure 1.
If we consider now τ ∈ U(1), one has: f (τz) = f (z) on the annulus where h z (τ) = 1 or h z (τ) = τ depending on whether z is in an even or odd annulus. Hence, class( f (γ z)) = class( f (z)) for all γ ∈ Γ. Then, the map f projects onto a smooth map φ : Next, assume that f is another lifting 14], this map is constant γ(z) = γ and f (z) = γ f (z) on C − {0}, and by continuity on C. Thus, two lifts of φ di fer only by a constant in Γ, which gives the same function h z = h z . Therefore, because the homomorphism h z ips from the trivial homomorphism to the identity on successive annuli, φ has no local equivariant smooth lifting.
3. L -Let = R n /Γ. Consider a local smooth map F from R n to itself, such that class • F = class. In other words, F is a local lifting of the identity on . Then, Proof. Let us assume rst that F is de ned on an open ball . Then, for all r in the ball, there exists a γ ∈ Γ such that F(r ) = γ · r . Next, for every γ ∈ Γ, let Let ∆ ⊂ R n × R n be the diagonal and let us consider There exist at least one γ ∈ Γ such that the interior ∆ γ is non-empty.
Indeed, since F γ is smooth (thus continuous), the preimage ∆ γ by F γ of the diagonal is closed in . However, the union of all the preimages F −1 γ (∆) -when γ runs over Γ -is the ball . Then, is a countable union of closed subsets. According to Baire's theorem, there is at least one γ such that the interior ∆ γ is not empty.
be an open ball. Let us denote with a prime the sets de ned above but for . Then, Thus, which is not empty for the same reason that ∪ γ∈Γ ∆ γ is not empty. Therefore, ∆ Γ is dense.
Hence, there exists a subset of Γ, indexed by a family , for which i = ∆ γ i ⊂ is open and non-empty, ∪ i∈ i is an open dense subset of , and F i : . Every local smooth liftingf of any local di eomorphism f of is necessarily a local di eomorphism. In particular n = n . Moreover, let x ∈ dom( f ), x = f (x), r, r ∈ R n be such that class(r ) = x and class(r ) = x . Then, the local liftingf can be chosen such thatf (r ) = r .
Note that n is also the di feological dimension of R n /Γ, see , r =f (r ), and then x = class (r ).
Next, letŨ = class −1 (U ). Since the composite f −1 • class is a plot in , there exists a smooth liftingf :Ṽ → R n , de ned on an open neighborhood of r , such that around r ,f is a local di feomorphism. Now, if we consider any another point r over , we get f (r ) = r , andf andf still remain inverse of each other.
Therefore, for any r ∈ R n over x and any r ∈ R n over x = f (x), we can locally lift f to a local di feomorphismf such thatf (r ) = r .
In this section, we associate a structure groupoid -or gauge groupoid -which is a di feological groupoid [PIZ13,8.3], with every atlas of a quasifold. Then we show that di ferent atlases give equivalent groupoids: as categories, according to the Mac Lane de nition [SML78], and in the sense of Muhly-Renault-Williams [MRW87]. We give a precise description of the structure groupoid in terms or the groupoid associated with the action of the structure groups Γ, and the connecting points of the charts. This construction is the foundation for a C * -algebra associated with the quasifold.

B
. -Let X be a quasifold, let be an atlas and let be the strict generating family over . We denote by the nebula 8 of , that is, the sum of the domains of its elements: The evaluation map is the natural subduction ev: → X with ev(F, r ) = F(r ).
Following the construction in the case of orbifolds [IZL17], the structure groupoid of the quasifold X, associated with the atlas , is de ned as the subgroupoid G of germs of F 2. The three levels of a quasifold.
local di feomorphisms of that project to the identity of X along ev. That is, The set Mor(G) is equipped with the functional di feology inherited by the full groupoid of germs of local di feomorphisms [IZL17,§2 & 3]. Note that, given Φ ∈ Diff loc ( ) and ν ∈ dom(Φ), there exist always two plots F and F in such that ν = (F, r ), with r ∈ dom(F), and a local di feomorphism φ of R n , de ned on an open ball centered in r , That is summarized by the diagram: According to the theorem in ( §3), the local di feomorphisms, de ned on the domain of a generating plot, and lifting the identity of the quasifold, are just the elements of the structure group associated with the plot. We can legitimately wonder what is the point of involving general germs of local di feomorphisms, if we merely end up with the structure group we could have began with. The reason is that the structure groups connect the points of the nebula that project on a same point of the quasifold, only when they are inside the same domain. They cannot connect the points of the nebula that project on the same point of the quasifold but belonging to di ferent domains, with maybe di ferent structure groups. This is the reason why we cannot avoid the use of germs of local di feomorphisms in the nebula, to begin with. That situation is illustrated in Figure 2. 6. E Let us recall that a functor S: A → C is an equivalence of categories if and only if, S is full and faithful, and each object c in C is isomorphic to S(a) for some object a in A [SML78, Chap. 4 § 4 Thm. 1]. If A and C are groupoids, the last condition means that, for each object c of C, there exist an object a of A and an arrow from S(a) to c.
In other words: let the transitivity-components of a groupoid be the maximal full subgroupoids such that each object is connected to any other object by an arrow. The functor S is an equivalence of groupoids if it is full and faithful, and surjectively projected on the set of transitivity-components. Now, consider an n-quasifold X. Let be an atlas, let be the associated strict generating family, let be the nebula of and let G the associated structure groupoid. Let us rst describe the morphology of the groupoid.

P
. The fibers of the subduction ev: Obj(G) → X are exactly the transitivitycomponents of G. In other words, the space of transitivity components of the groupoid G associated with any atlas of the quasifold X, equipped with the quotient di eology, is the quasifold itself.

T
. Di erent atlases of X give equivalent structure groupoids. The structure groupoids associated with di eomorphic quasifolds are equivalent.
In other words, the equivalence class of the structure groupoids of a quasifold is a di feological invariant.
Proof. These results are analogous to the results of [IZL17,§5]. They have the same kind of proof. The fact that the structure groups Γ of the quasifolds are countable instead of nite has no negative consequences, thanks to ( §4).
Let us start by proving the proposition. Let F: U → X and F : U → X be two generating plots from the strict family , and r ∈ U ⊂ R and r ∈ U ⊂ R . Assume that ev(F, r ) = ev(F , r ) = x, that is, to U , is a local di feomorphism that maps ξ = f (class(r )) to ξ = f (class (r )).
Then, according to ( §4), n = n and there exists a local di feomorphism ϕ of R n , lifting locally ψ and mapping r to r . Its germ realizes an arrow of the groupoid G connecting (F, r ) to (F , r ). Of course, when F(r ) = F (r ) there cannot be an arrow, by de nition.
Therefore, as in the more restrictive case of orbifolds, the bers of the evaluation map are the transitive components of the structure groupoid G of the quasifold. Now, the theorem follows the formal ow of (op. cit. §5): let and be two atlases of X and consider = .
With an obvious choice of notation: Obj(G ) = Obj(G) Obj(G ) and G contains naturally G and G as full subgroupoids. The question then is: how does the adjunction of the crossed arrows between G and G change the distribution of transitivity-components? According to the previous proposition, it changes nothing since, for G, G or G , the set of transitivity-components are always exactly the bers of the respective subductions ev. In other words, the set of groupoid components is always X, for any atlas of X. Thus G and G are equivalent to G , therefore G and G are equivalent.

G
. -The general description of the structure groupoid of a quasifold X follows exactly the description in the case of orbifolds (op. cit.). We remind it here for clarity. Let X be a quasifold. Let be an atlas, let be the associated strict generating family, and let G be the associated groupoid. We know from the previous paragraph that the groupoid components in Obj(G) are the bers of the projection ev: (F, r ) → F(r ). Then, the (algebraic) structure of the groupoid reduces to the algebraic structure of each full subgroupoid G x , x ∈ X, that is, more precisely, g = germ(ϕ) r where ϕ is a local di feomorphism de ned in the domain of F to the domain of F , mapping r to r and such that F • ϕ = loc F on an open neighborhood of r . In other words, Obj(G x ) = ev −1 (x) and Mor(G x ) = (ev •src) −1 (x).
Let f be a chart in , let U = dom( f ) and letŨ = class −1 (U) ⊂ R n be the domain of its strict lifting F = f • class Ũ , where class: R n → R n /Γ. Without loss of generality, we shall assume that the domains of all charts, and thus the domains of the strict liftings, are connected.
The subgroupoid G x is the assemblage of the subgroupoids G F x . For all F ∈ , That is, ). The assemblage is made rst by connecting the groupoid G F x to G F x with any arrow germ(ϕ) r , from (F, r ) to (F , r ) such that x = F(r ) = F(r ) and ϕ(r ) = r . Secondly, by spreading the arrows by composition. We can represent this construction by a groupoid-set-theoretical diagram: where the F i 's are the charts having x in their images and N x is the number of such charts (the atlas is assumed locally nite). The link between two groupoids: represents the spreading of the arrows by adjunction of one of them. Note that this is absolutely not a smooth representation of G, since the projection ev •src: Mor(G) → X is a subduction. Moreover, the order of assembly has no in uence on the result.
E . In the case of orbifolds, where the structure group is nite, this assemblage of groupoids can be completely visual: for example, the teardrop in [IZL17, Figure 3]. It is more di cult in the case of a strict quasifold, with dense structure group. For example, the irrational torus T α = R/(Z + αZ), which was described as a di feological space in [PDPI83] for the rst time. Now, with the identi cation of this new subcategory {Quasifolds} in {Di feology}, the irrational torus becomes a quasitorus. 9 For the generating family {class: R → T α }, the objects of the structure groupoid equal just R. Moreover, in this simple case, as we see in ( §11), the groupoid G α is the groupoid of the action of the subgroup Z + αZ by translation. Therefore, one has where the bold letter t denotes a translation. The source and target are given by src(x, t n+αm ) = x and trg(x, t n+αm ) = x + n + αm.
For example, at τ = 0 we get G class α,0 = {(n + αm, t n +αm ) | n, n , m, m ∈ Z}.  N . Since the atlas is assumed to be locally nite, the preimages of the objects of G by the source map, or the target map, are countable. 9 Or, in this case, a quasicircle.

Lifting local di feomorphisms
Proof. This proof is the same as in the case of orbifolds [IZL17,§7]. We just have to pay attention to the fact the structure group is now countable, and not just nite.
(1) Let us rst check that the groupoid G is étale. That is, src: Mor(G) → Obj(G) is everywhere a local di feomorphism.
Let us pick a germ g = germ(Φ) ν ∈ Mor(G), with ν = src(g) = (F, r ) and trg(g) = (F , r ). Thus, Φ is de ned by some ϕ ∈ Diff loc (R n ) with dom(ϕ) ⊂ dom(F), r = ϕ(r ) ∈ dom(F ) and such that F • ϕ = F . We choose ϕ: → dom(F ) to be de ned on a small ball centered at r . By abuse of notation we shall denote g = germ(ϕ) r , where ϕ ∈ Diff loc (dom(F), dom(F )). That is, ϕ now contains implicitly the data source and target. Now, let where f and f belong to , class: R n → R n /Γ and class : R n → R n /Γ are the projections. If ψ is the transition map f −1 • f , then class(r ) ∈ dom(ψ) and ψ(class(r )) = class (r ). This situation is illustrated by the diagram of Figure 3, where, except for the family ϕ s which will vary around ϕ, the vertices and arrows are xed as soon as the representant Φ of the germ g is chosen. Now, let Hence, src : germ(ϕ) t → t is smooth and injective, 10 as well as its inverse t → germ(ϕ) t , which is de ned on . Let us now show that is a D-open subset, That is, for each plot P: s → g s in Mor(G), the subset P −1 ( ) ⊂ dom(P) is open. Let s ∈ P −1 ( ), that is, g s ∈ , i.e. g s = germ(ϕ) r s , where r s = src(g s ), the discrete index F here is implicit.
Then, for all s ∈ dom(P), there exists a small ball centered at s and a plot s → (ϕ s , r s ), de ned on , such that g s = germ(ϕ s ) r s with germ(ϕ s ) r s = germ(ϕ) r s and r s ∈ . 10 Maybe we should recall that germ(ϕ) t = germ(ϕ ) t if and only if: t = t and there exists an open ball centered at t such that ϕ = ϕ .
Since s → ϕ s is smooth, by de nition the subset is necessarily open. Since it contains (s , r s ), it contains a product × , where is a small ball centered at s and is a small ball centered at r s . This implies that, for all s ∈ , ⊂ dom(ϕ s ). In particular, ⊂ dom(ϕ).
However, for all s , one has class • ϕ s = ψ • class, wherever it is de ned. This is shown by the above diagram, where the dots denote a local map.
(2) Next, let us check that Mor(G) is Hausdor f. As above, let g = germ(ϕ) r ∈ Mor(G). We can also represent g by a triple (F, r, germ(ϕ) r ), with ϕ ∈ Diff loc (dom(F), dom(F )). Then, let g = germ(ψ) s be another germ represented by (G, s, germ(ψ) s ), di ferent from g, with ψ ∈ Diff loc (dom(G), dom(G )). We separate the situation in three cases: In the rst two cases (F = G, and F = G but r = s), since the source map is étale and since the Nebula is Hausdor f, it is su cient to consider two small separated balls and , centered around r and s, to get two D-open subsets of Mor(G) that separate the two di ferent germs. Indeed, let = src −1 ( ) and = src −1 ( ) be the D-open subset on which the source map is a local di feomorphism. If there were a point g ∈ ∩ , then src(g ) would belong to ∩ , which is empty.
In the rst sub-case, when codom(ϕ) = codom(ψ), since the codomains are di ferent, we consider a small ball around r such that its images by ϕ and ψ are separated. Then = {germ(ϕ) t | t ∈ } and = {germ(ψ) t | t ∈ } are two open subsets in Mor(G) that separate g and g , since no germ in has the same codomain as any germ in .
In the second sub-case, when codom(ϕ) = codom(ψ), let us consider the composite f = ϕ • ψ −1 , de ned on an open neighborhood of ψ(r ). Thanks to the theorem of ( §3), f (s) = loc γ · s , for some γ ∈ Γ , which is the structure group of the quasifold for the plot F . Since we have assumed that germ(ϕ) r = germ(ψ) r , we have that γ = 1. Hence, there is a small ball around r on which ϕ = γ •ψ.
In conclusion, Mor(G) is Hausdor f for the D-topology.
According to the previous Note, the preimages of an object (F, r ) ∈ = Obj(G) are the germs of all the local di feomorphisms Φ : (F, r ) → (F , r ), such that F(r ) = r and F = loc F • ϕ around r , where ϕ is a local di feomorphism of R n . Since the atlas is locally nite, there are a nite number of F ∈ such that F(r ) = F (r ). Now, for such F the number of r ∈ domF such that F (r ) = F(r ) is at most equal to the number of elements of the structure group Γ , that is countable. Therefore, the preimages of (F, r ) by the source map is countable, and that works obviously in the same way for the preimages of the target map.
9. MRW-. -We consider a quasifold X and two atlases and , with associated strict generating families and . We shall show in this section that the associated groupoids are equivalent in the sense of Muhly-Renault-Williams [MRW87, 2.1]; this will later give Morita-equivalent C * -algebras.
This section follows [IZL17, §8]; we just check that the fact that the structure groups are countable and not just nite, does not change the result.
Let us recall what is an MRW-equivalence of groupoids. Let G and G be two locally compact groupoids. We say that a locally compact space Z is a (G, G )-equivalence if The G and G actions commute. (iv) The action of G on Z induces a bijection of Z/G onto Obj(G ).
(v) The action of G on Z induces a bijection of Z/G onto Obj(G).
Let src: Z → Obj(G) and trg: Z → Obj(G ) be the maps de ning the composable pairs associated with the actions of G and G . That is, a pair (g, z) is composable if trg(g) = src(z), and the composite is denoted by g · z. Moreover, a pair (g , z) is composable if src(g ) = trg(z), and the composite is denoted by z · g .
Let us also recall that an action is principal in the sense of Muhly-Renault-Williams, if it is free: g · z = z only if g is a unit, and the action map (g, z) → (g · z, z), de ned on the composable pairs, is proper [MRW87,§2]. Now, using the hypothesis and notations of ( §6), let us de ne Z to be the space of germs of local di feomorphisms, from the nebula of the family to the nebula of the family , that project on the identity by the evaluation map. That is, .
Then, the action of g ∈ Mor(G) on germ( f ) r is de ned by composition if trg(g) = r , that is, g · germ( f ) r = germ( f • ϕ) s , where g = germ(ϕ) s , ϕ ∈ Diff loc ( ) and ϕ(s ) = r . Symmetrically, the action of g ∈ Mor(G ) on germ( f ) r is de ned if src(g ) = f (r ) . Then, we have: T . The actions of G and G on Z are principal, and Z is a (G, G )-equivalence in the sense of Muhly-Renault-Williams.
Proof. First of all, let us point out that Z is a subspace of the morphisms of the groupoid G built in ( §6) by adjunction of G and G , and is equipped with the subset di feology.
All these groupoids are locally compact and Hausdor f ( §8).
Let us check that the action of G on Z is free. In our case, z = germ( f ) r and g = germ(ϕ) s , where f and ϕ are local di feomorphisms. If g · z = z, then obviously g = germ(1) r .
Next, let us denote by ρ the action of G on Z, de ned on This action is smooth because the composition of local di feomorphisms is smooth, and passes onto the quotient groupoid in a smooth operation, see [IZL17,§3]. Moreover, this action is invertible, its inverse being de ned on In detail, ρ −1 (germ(h) s , germ( f ) r ) = (germ( f −1 • h) s , germ( f ) r ), with f (r ) = h(s). Now, the inverse is also smooth, when Z Z ⊂ Z × Z is equipped with the subset di feology. In other words, ρ is an induction, that is, a di feomorphism from G Z to Z Z. However, since G Z and Z Z are de ned by closed relations, and G and Z are Hausdor f, G Z and Z Z are closed into their ambient spaces. Thus, the intersection of a compact subset in Z × Z with Z Z is compact, and its preimage by the induction ρ is compact. Therefore, ρ is proper. We notice that the fact that the structure groups are no longer nite but just countable does not play a role here.
It remains to check that the action of G on Z induces a bijection of Z/G onto Obj(G ). Let us consider the map class: Z → Obj(G ) de ned by class(germ( f ) r ) = f (r ). Then, let class(z) = class(z ), with z = germ( f ) r and z = germ( f ) r , that is, f (r ) = f (r ). However, since f and f are local di feomorphisms, ϕ = f −1 • f is a local di feomorphism with ϕ(r ) = r . Let g = germ(ϕ) r , then g ∈ Mor(G) and z = g · z. Hence, the map class projects onto an injection from Z/G to Obj(G ). Now, let (F , r ) ∈ Obj(G ), and let x = F (r ) ∈ X. Since is a generating family, there exist (F, r ) ∈ Obj(G) such that F(r ) = x. Let ψ and ψ be the charts of X de ned by factorization: F = ψ • class and F = ψ • class , where class: R n → R n /Γ and class : R n → R n /Γ . Let ξ = class(r ) and ξ = class (r ). Since ψ(ξ) = ψ (ξ ) = x, Ψ = loc ψ −1 • ψ is a local di feomorphism from R n /Γ to R n /Γ mapping ξ to ξ .
Hence, according to ( §4), there exists a local di feomorphism f from dom(F) to dom(F ), such that class • f = Ψ • class and f (r ) = r . Thus, z = germ( f ) r belongs to Z and class(z) = r (precisely the element (F , r ) of the nebula of ). Therefore, the injective map class from Z/G to Obj(G ) is also surjective, and identi es the two spaces. Obviously, what has been said for the side G can be translated to the side G ; the construction is completely symmetric. In conclusion, Z satis es the conditions of a (G, G )-equivalence, in the sense of Muhly-Renault-Williams.
We use the construction of the C * -Algebra associated with an arbitrary locally compact groupoid G, equipped with a Haar system, introduced and described by Jean Renault in [JR80, Part II, §1]. Note that, for this construction, only the topology of the groupoid is involved, and di feological groupoids, when regarded as topological groupoids, are equipped with the D-topology 12 .
We will denote by (G) the completion of the compactly supported continuous complex functions on Mor(G), for the uniform norm. And we still consider, as is done for orbifolds, the particular case where the Haar system is given by the counting measure. Let f and g be two compactly supported complex functions, the convolution and the involution are de ned by The sums involved are supposed to converge. Here, γ ∈ Mor(G), x = src(γ) and G x = trg −1 (x) is the subset of arrows with target x. The star in z * denotes the conjugate of the complex number z. By de nition, the vector space (G), equipped with these two operations, is the C * -algebra associated with the groupoid G.
10. T C * -. -Let X be a quasifold, let be an atlas and let G be the structure groupoid associated with . Since the atlas is locally nite, the convolution de ned above is well de ned. Indeed, in this case: P . For every compactly supported complex function f on G, for all ν = (F, r ) ∈ = Obj(G), the set of arrows g ∈ G ν such that f (g) = 0 is finite. That is, # Supp( f G ν ) < ∞. The convolution is then well defined on (G).
Then, for each atlas of the quasifold X, we get the C * -algebra A = ( (G), * ). The dependence of the C * -algebra on the atlas is given by the following theorem, which is a generalization of [IZL17,§9].
In other words, we have de ned a functor from the subcategory of isomorphic {Quasi-folds} in di feology, to the category of Morita-equivalent {C * -Algebras}.
Proof. Considering the proposition, G ν = trg −1 (ν) with ν ∈ Obj(G). The space of objects of G is a disjoint sum of Euclidean domains, thus {ν} is a closed subset. Now, trg: Mor(G) → Obj(G) is smooth then continuous, for the D-topology. Hence, G ν = trg −1 (ν) is closed and countable by ( §8). Now, Supp( f G ν ) = Supp( f ) ∩ G ν is the intersection of a compact and a closed countable subspace, thus it is compact and countable, that is nite.
11. T C * -A . -The rst and most famous example is the so-called Denjoy-Poincaré torus, or irrational torus, or noncommutative torus, or, more recently, quasitorus. It is, according to its rst de nition, the quotient set of the 2-torus T 2 by the irrational ow of slope α ∈ R−Q. We denote it by T α = T 2 /∆ α , where ∆ α is the image of the line y = αx by the projection R 2 → T 2 = R 2 /Z 2 . This space has been the rst example studied with the tools of di feology, in [PDPI83], where many non trivial properties have been highlighted. 13 Di feologically speaking, The composite summarizes the situation where = { f : R/(Z + αZ) → T α } is the canonical atlas of T α , containing the only chart f , and = {F = f • class} is the associated canonical strict generating family. According to the above ( §3), the groupoid G α associated with the atlas is simply Obj(G α ) = R and Mor(G α ) = {(x, t n+αm ) | x ∈ R and n, m ∈ Z}.
However, we can also identify T α with (R/Z)/[(Z + αZ)/Z], that is for all m ∈ Z and z ∈ S 1 . Moreover, the groupoid S of this action of Z on S 1 ⊂ C is simply Obj(S α ) = S 1 and Mor(S α ) = {(z, e 2iπαm ) | z ∈ S 1 and m ∈ Z}.
The groupoids G α and S α are equivalent, thanks to the functor Φ from the rst to the second: Φ Obj (x) = e 2iπx and Φ Mor (x, t n+αm ) = (e 2iπx , e 2iπαm ).
Moreover, they are also MRW-equivalent, by considering the set of germs of local di feomorphisms x → e 2iπx , everywhere from R to S 1 . Therefore, their associated C * -algebras are Morita equivalent. The algebra associated with S α has been computed numerous times and it is called irrational rotation algebra [MR81]. It is the universal C * -algebra generated by two unitary elements U and V, satisfying the relation VU = e 2iπα UV.  2. The converse of Rie fel's theorem is a di ferent matter altogether. We should recover a di feological groupoid G α from the algebra A α . Then, the space of transitive components would be the required quasifold, as stated by the proposition in ( §6). In the case of the irrational torus, it is not very di cult. The spectrum of the unitary operator V is the circle S 1 and the adjoint action by the operator U gives UVU −1 = e 2iπα V, which translates on the spectrum by the irrational rotation of angle α. In that way, we recover the groupoid of the irrational rotations on the circle, which gives T α as quasifold.

R
3. Of course, the situation of the irrational torus is simple and we do not exactly know how it can be reproduced for an arbitrary quasifold. However, this certainly is the way to follow to recover the quasifold from its algebra: nd the groupoid made with the Morita invariant of the algebra, which will give the space of transitivity components as the requested quasifold.

T
R/Q. -The di feological space R/Q is a legitimate quasifold. This is a simple example with a groupoid G given by Obj(G) = R and Mor(G) = {(x, t r ) | x ∈ R and r ∈ Q}.
The algebra that is associated with G is the set A of complex compact supported functions on Mor(G). Let us identify 0 (Mor(G), C) with Maps(Q, 0 (R, C)) by f = ( f r ) r ∈Q with f r (x) = f (x, t r ), and let Supp( f ) = {r | f r = 0}.
The convolution product and the algebra conjugation are, thus, given by: ( f * g ) r (x) = s f r −s (x + s)g s (x), and f * r (x) = f −r (x + r ) * . Now, the quotient R/Q is also di feomorphic to the Q-circle is the subgroup of rational roots of unity. As a di feological subgroup of S 1 , Q is discrete. The groupoid S of the action of Q on S 1 is given by: Obj(S) = S 1 and Mor(S) = (z, τ) z ∈ S 1 and τ ∈ Q .
The exponential x → z = e 2iπx realizes a MRW-equivalence between the two groupoids G and S. Their associated algebras are Morita-equivalent. The algebra S associated with S is made of families of continuous complex functions indexed by rational roots of unity, in the same way as before: The convolution product and the algebra conjugation are, then, given by: ( f * g ) τ (z) = σ fσ τ (σz)g σ (z) and f * τ (z) = fτ(τz) * , whereτ = 1/τ = τ * , the complex conjugate.
Now, consider f and let p be the subgroup in Q generated by Supp( f ); this is the group of some root of unity ε of some order p. Let M p (C) be the space of p × p matrices with complex coe cients. De ne f → M, from S to M p (C) ⊗ 0 (S 1 , C), by M(z) σ τ = fσ τ (σz), for all z ∈ S 1 and σ, τ ∈ p .
That gives a representation of S in the tensor product of the space of endomorphisms of the in nite-dimensional C-vector space Maps( Q , C) by 0 (S 1 , C), with nite support.