Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $\Sigma$ of groupoids $G$ by bundles $A$ of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid $G$ by a given bundle $A$. There is a natural action of $\Sigma$ on the dual of $A$, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of $A$ with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of $G$ on the dual of $A$. We prove that the full $C^*$-algebra of this twist is isomorphic to the full $C^*$-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.


Introduction
There is a significant body of literature regarding the C * -algebras of extensions of groupoids by group bundles. The main goal of this paper is to introduce a pushout construction for extensions of groupoids by abelian group bundles and explore its applications.
Specifically, we consider a locally compact Hausdorff groupoid G together with an abelian group bundle p A : A → G (0) where p A a continuous, open map. Then we consider unit space fixing extensions where Σ is a locally compact Hausdorff groupoid, both ι and p are groupoid homomorphisms, p is a continuous, open surjection inducing a homeomorphism of Σ (0) and G (0) , ι is a homeomorphism of A onto ker p.
More recently, we considered more general extensions in [IKSW19] and [IKR + 21] as in ( †) where it is assumed that A is endowed with an action of G and that the extension is compatible in the sense that σι(a)σ −1 = ι(σ · a) for all a ∈ A and σ ∈ Σ such that p A (a) = s(σ).
As a consequence of the main result in [IKR + 21], we showed that if Σ has a Haar system, then C * (Σ) can be realized as the C * -algebra of a twist. Specifically, the action of G on A induces a natural action of G onÂ (regarded as a space). We constructed a T-groupoid Σ of the form We proved ([IKR + 21, Theorem 3.4]) that C * (Σ) is isomorphic to the restricted C *algebra C * (Â ⋊ G; Σ) of this T-groupoid. (In [IKR + 21] the T-groupoid is denoted Σ, but here we use Σ to avoid possible confusion in our examples.) The T-groupoid Σ is at the heart of the Mackey obstruction which appears in the classical "Mackey machine" of [Mac58]. The chief motivation for this article is the observation that the T-groupoid Σ above-which was based on the construction of [MRW96,Proposition 4.3]-is derived from a natural and functorial "pushout" construction based on the second author's work in [Kum88] forétale groupoids (there called "sheaf groupoids"). Specifically, suppose we are given and extension as in ( †), and abelian group bundle B admitting a G-action, and a equivariant groupoid homomorphism f : A → B. Then there is a similar sort of extension inducing the given G-action on B. In Theorem 1.5, we show that the construction producing f * Σ has good functorial properties that characterize the extension up to a suitable notion of isomorphism. Using these properties, we show in Theorem 2.5 that the collection T G (A) of isomorphism classes of extensions of A by G forms an abelian group (see also [Tu06,§5.3

]).
We close by illustrating how the pushout construction clarifies and interacts with our work in [IKSW19] and [IKR + 21]. In Theorem 3.2 we prove that the extension ( ‡) employed in [IKR + 21] arises from our pushout construction. Specifically, the natural pairing (χ, a) → χ(a) fromÂ * A to T yields a groupoid homomorphism f :Â * A → A × T given by f (χ, a) = (χ, χ(a)) (see Section 3.1). There is a natural action of Σ onÂ (compatible with that of G as above) and we prove that Σ ∼ = f * (Â ⋊ Σ). This allows us to realise the C * -algebra of an extension of a groupoid G by an abelian group bundle A as the C * -algebra of a T-groupoid over the resulting transformation groupoidÂ ⋊ G.
Several consequences flow from this observation. First suppose that A is an abelian group and that A = G (0) × A, carrying the action of G that is trivial in the second coordinate, so that Σ is a generalised twist. Each χ ∈Â defines a homomorphism f χ : A → T × G (0) , so we can form the resulting pushout f χ * (Σ). We prove in Proposition 3.6 that C * (Σ) is the section algebra of an upper-semicontinuous C *bundle overÂ with fibres C * (G, f χ * (Σ)). When A is compact, this yields a direct sum decomposition which remains valid for the corresponding reduced C * -algebras (see Proposition 3.7). In Corollary 3.10 we extend [IKR + 21, Theorem 3.4] to the case that Ω is a T-groupoid extension of Σ such that its restriction to ι(A) is abelian. When G isétale, this enables us to generalize [IKR + 21, Theorem 4.6] to this case (see Corollary 3.11) thereby providing criteria that guarantee that the natural abelian subalgebra of C * r (Σ; Ω) is Cartan (see also [DGN + 20, Theorem 5.8] and [DGN20, Theorem 4.6]).
In Subsection 3.2, we consider the case where the extension Σ is determined by an A-valued 2-cocycle defined on G and show that the pushout construction is compatible with the natural change of coefficients map on cocycles. We describe the explicit construction of Σ in terms of 2-cocycles at the beginning of Subsection 3.3, and then consider various examples of this construction.

Pushouts of groupoid extensions
We fix a locally compact Hausdorff groupoid G. In our applications, G will have a Haar system, but this is not required for the pushout construction itself. However, we do assume that G has open range and source maps. We call a locally compact abelian group bundle p A : A → G (0) a G-bundle if p A is open and G acts on the left of A by automorphisms. For compatibility with [IKSW19]-and other examples we have in mind-we will write the group operations in the fibres of such A additively. An extension Σ of A by G is determined by a diagram ( †) as in the introduction. Recall that Σ is a locally compact Hausdorff groupoid, p is continuous and open surjection inducing a homeomorphism from Σ (0) onto G (0) , and ι is a continuous open injective homomorphism onto ker p = { σ ∈ Σ : p(σ) ∈ G (0) }. We call such an extension compatible if the action of G on A induced by conjugation is the given G-action on A; that is, σι(a)σ −1 = ι(σ · a). Definition 1.1. If Σ 1 and Σ 2 are compatible extensions by a locally compact abelian group G-bundle A, then we say that they are properly isomorphic if there is a groupoid isomorphism f : Σ 1 → Σ 2 such that the diagram Of course, given G and a G-bundle A, we would like to know that T G (A) is not empty. To provide a basic example, we follow [Kum88, Definition 2.1].
Example 1.4. For i = 1, 2 let A i be a locally compact abelian group G-bundle. Note that A 1 * A 2 = { (a, a ′ ) : p A 1 (a) = p A 2 (a ′ ) } is also a locally compact abelian group G-bundle. Let Σ i be a compatible groupoid extension of G by A i . Then as in [Kum88,§2], we may form the fibered product It is straightforward to check that Σ 1 * G Σ 2 is a compatible groupoid extension of G by A 1 * A 2 .
Assume now that B is another abelian group G-bundle, and that f : A → B is a G-equivariant map. Following [Kum88, Proposition 2.6], we prove that we can "pushout" Σ in a unique way to an extension of G by B.
Theorem 1.5 (Pushout Construction). Let A and B be locally compact abelian group G-bundles. Let f : A → B be a continuous G-equivariant map. Assume that Σ is a compatible extension of G by A. Then there is a compatible extension f * Σ of G by B and a homomorphism f * : Σ → f * Σ such that the following diagram commutes Moreover, f * and f * Σ are unique up to proper isomorphism in the sense that if Σ ′ is a compatible extension such that the diagram Proof. Consider the fibred-product groupoid of Example 1.4. Define θ : A → D via θ(a) = (−f (a), p A (a)), ι(a) . Since ι is a homeomorphism onto its closed range, θ(A) is a closed wide subgroupoid of D.
Let d = ((b, γ), σ) ∈ D. We claim that dθ(A) = θ(A)d. To see this, note that Let f * Σ := D/θ(A). As usual, we denote the class of ( We can identify the unit space with G (0) and then To see that f * Σ is a compatible extension by B, let It is not hard to verify that this satisfies the algebraic requirements for an extension. The most difficult one might be the inclusion p −1 * (G (0) ) ⊆ ι * (B) for which we provide an outline of the proof: take [(b, γ), σ] ∈ f * Σ such that p * ([(b, γ), σ]) = u ∈ G (0) . Then γ = u, givingσ = u. Since Σ is an extension, there exists a ∈ A(u) such that ι(a) = σ. It follows that [((b, u), ι(a))] = [((b + f (a), u), u)] = ι * (b + f (a)). It is easy to check that b + f (a) is independent of the choice of the representative of [(b, γ), σ].
Since ι * and p * are clearly continuous and since ι * is easily seen to be a homeomorphism onto its range, we just need to see that p * is open. For this, we apply Fell's Criterion (see [IKR + 21, Lemma 3.1]). Suppose that γ n → γ = p * [(b, σ), γ] . Since p : Σ → G is open, we can pass to a subnet, relabel, and assume that there are σ n → σ in Σ such thatσ n = γ n . Since p B is open, we can pass to subnet, relabel, and assume The map f * is the composition of the embedding of Σ into D and the quotient map and since p ′ (f ′ (σ 1 )) =σ 1 , it follows thatg is a groupoid homomorphism. On the other hand, Henceg factors through a homomorphism g : To see that g is a proper isomorphism, we still need to see that g is an isomorphism with a continuous inverse. For Thus g is injective.
To see that g is an isomorphism of topological groupoids, it suffices to see that g is open. We use Fell's criterion. So suppose that g( Since p is open, we can pass to a subnet, relabel, and assume there exist a i ∈ A such that ι(a i )σ i → σ. But and then It follows that Since ι ′ is a homeomorphism onto its range, α i → α as required.
Corollary 1.6. Let A, B and C be locally compact abelian group G-bundles. Let f : A → B and g : B → C be continuous G-equivariant maps. Assume that Σ is a compatible extension of G by A.
Proof. This follows from the uniqueness of (g • f ) * Σ up to proper isomorphism guaranteed by Theorem 1.5. .

The Extension Group T G (A)
As in [Kum88,§2], we can use our pushout construction to introduce a binary operation on The uniqueness assertion of Theorem 1.5 then yields a proper isomorphism The uniqueness assertion in Theorem 1.5 implies that In Examples 2.1-2.4, we have proved much of the following theorem, which is patterned on [Kum88, Theorem 2.7].
Theorem 2.5. Let G be a locally compact groupoid with open range and source maps, and let A be a locally compact abelian group G-bundle. Then the binary operation into an abelian group with neutral element given by the class [A ⊳ G] of the semidirect product of Example 1.3, and Then T G is a functor from the category of G-bundles to the category of abelian groups.
Proof. By considering diagrams similar to that in Example 2.4, we see that the operation in (2.1) is well-defined and associative. We saw that [A ⊳ G] acts as an identity in Example 2.1 and the statement about inverses follows from Example 2.3. The computation in Example 2.4 shows that T G (A) is an abelian group.
By Corollary 1.6 we have

Applications and Examples
In this section we consider a unit space fixing extension Σ of G by the group bundle A as illustrated in the diagram ( †) from the introduction. We review the basic details. We assume that all groupoids considered in this section are second-countable locally compact Hausdorff groupoids with Haar systems. The Haar system on Σ is denoted λ = {λ u } u∈Σ (0) and we further assume that p A : A → Σ (0) is a bundle of abelian groups that is a closed subgroupoid of Σ. It is equipped with a Haar system denoted β = {β u } u∈Σ (0) and the fibers are denoted A(u) for u ∈ Σ (0) . The existence of a Haar system on A implies that p A is open. It follows by [IKR + 21, Lemma 2.6(c)] that there is a Haar system α = {α u } u∈Σ (0) on G such that for all f ∈ C c (Σ) and u ∈ Σ (0) we have Moreover, there is a natural action of Σ, and therefore G, on A.
Note that p : Σ → G is a continuous, open surjection inducing a homeomorphism from Σ (0) onto G (0) , and ι : A → Σ is a homeomorphism onto ker p. (Both p and ι are assumed to be groupoid morphisms).
Recall that if Σ is a T-groupoid over G then is a * -algebra under the operations described in [MW92,§2], and C * (G; Σ) is its closure in the norm obtained by taking the supremum of the operator norm under all * -representations.
We may also view C c (G; Σ) as compactly supported continuous sections of the onedimensional Fell line bundle over G associated to Σ. One can then construct the associated (right) Hilbert C 0 (G (0) )-module (see [IKR + 21, §1.3]) as the completion of C c (G; Σ) in the norm arising from the C 0 (G (0) )-valued pre-inner product given by f, g := (f * * g)| G (0) for all f, g ∈ C c (G; Σ). We denote the Hilbert module by H(G; Σ) and observe that left multiplication induces a natural * -homomorphism λ : C c (G; Σ) → L(H(G; Σ)). We may define the reduced norm of an element f ∈ C c (G; Σ) to be the operator norm of its image: f r := λ(f ) . Then C * r (G; Σ) is the closure of C c (G; Σ) in the reduced norm.
Lemma 3.1. With notation as above, let F ⊂ G (0) be a G-invariant clopen subset. Then F is also Σ-invariant and the reduction Σ| F is a twist over the reduction G| F . Moreover, the characteristic function of F determines a central multiplier projection p F such that Proof. Observe that H(G; Σ) decomposes as the direct sum of a Hilbert C 0 (F )-module and a Hilbert C 0 (F c )-module in the following way Note that multiplication by the characteristic function of F , which we denote by p F is the projection onto the first component, that p F is in the center of the multiplier algebra of C * r (G; Σ), and C c (G| F ; Σ| F ) acts trivially on the second component. Hence the operator norm of C c (G| F ; Σ| F ) acting on H(G| F ; Σ| F ) coincides with that of its action on H(G; Σ). . Then H is a normal subgroupoid of D and we can form the locally compact Hausdorff groupoid Σ := D/H (we use the notation Σ, rather than the notation Σ of [IKR + 21], to avoid clashing with classical notational conventions when Σ is a group, for example in Remark 3.3).
Theorem 3.2. Let Σ be the extension of G by the group bundle A as in the diagram ( †) and adopt the notation established above. Let f :Â * A →Â × T be the canonical map given by Then Σ is properly isomorphic to the pushout f * (Â ⋊ Σ). Moreover, Proof. Theorem 1.5 implies that there is a unique (up to proper isomorphism) extension f * (Â ⋊ Σ) ofÂ ⋊ G byÂ × T and a twist morphism that is compatible with f .
In particular, f * (Â ⋊ Σ) is a T-groupoid. We get a natural map g :Â ⋊ Σ to Σ given It follows immediately that if Σ is properly isomorphic to the semidirect product Remark 3.3. As mentioned in the introduction, the twist Σ appearing in Theorem 3.2 is responsible for the Mackey obstruction of the classical normal subgroup analysis of [Mac58]. Indeed, let us apply the theorem when Σ is a locally compact group and A is a closed normal abelian subgroup. Then Σ and G = Σ/A act on A by conjugation and give right actions on the space of charactersÂ. The corresponding twist Σ is the quotient of the groupoid (Â ⋊ Σ) × T where (χ, aσ, θ) is identified with (χ, σ, θχ(a)) for all a ∈ A. We let [χ, σ, θ] be the class of (χ, σ, θ) in Σ. If χ ∈Â, then let Σ(χ) and G(χ) be the stabilizers at χ for the actions onÂ, and let Σ(χ) be the isotropy group of Σ at χ. We observe that Σ(χ), up to an obvious identification, is the pushout of the group extension by the homomorphism χ : A → T. Indeed, this pushout χ * (Σ(χ)) is the quotient of Σ(χ) × T by the equivalence relation identifying (aσ, θ) with (σ, θχ(a)) for all a ∈ A. Thus we just identify [χ, σ, θ] ∈ Σ(χ) with [σ, θ] ∈ χ * (Σ(χ)). The class of Σ(χ) in H 2 (G(χ), T) is the classical Mackey obstruction. More precisely, let L be an irreducible unitary representation of Σ. According to Theorem 3.2, we may view it as a representation of the twisted groupoid (Â ⋊ G, Σ). Its restriction toÂ defines a measure class which is invariant and ergodic under the action of G. If this measure class is transitive, which will be always the case if A is regularly embedded, then we have a representation of a twisted transitive measured groupoid (O ⋊ G, Σ| O ), where O ⊂Â is an orbit of the action and Σ| O is the reduction of Σ to O. We pick χ ∈ O.
Example 3.4. Let H be a locally compact abelian group and let A ⊂ H be a closed subgroup. Then applying the above theorem with Σ = H and A = A, we conclude that Σ is a bundle of abelian groups over Σ (0) ∼ =Â where each fiber is an extension of H/A by T. Each of these extensions is abelian because H is abelian (and the action of H onÂ is trivial). Hence, each extension is determined by a symmetric T-valued Borel 2-cocycle and any such 2-cocycle is necessarily trivial by [Kle65, Lemma 7.2]. But the twist is not trivial in general: for example, if H = R and A = Z ≤ R, then triviality of the twist would imply C * (R) ∼ = C 0 (T × Z), which is nonsense.
Example 3.5 (Generalized Twists). We now consider the case where A is a locally compact abelian group, A = G (0) × A, and G acts on A by translation on the first factor. Since this simply gives us a twist when A = T, we will say that Σ is a generalized twist in this case. Note that even for twists, Σ need not be a trivial extension. Generalized twists were studied in [IKSW19].
ViewÂ :=Â × G (0) as a locally compact space. (We put the factor of G (0) on the right, as a reminder that we are thinking ofÂ as a space rather than as a group, and to line up with the natural identification ofÂ * A withÂ × G (0) × A, which we make without further comment). Then G acts on the second factor ofÂ. This means we can replaceÂ ⋊ G andÂ ⋊ Σ with the productsÂ × G andÂ × Σ, respectively. Under these identifications, Equation (3.2) becomes f (χ, u, a) = (χ, u, χ(a)). Moreover we may assume that the Haar system β on A = G (0) × A is constant in the sense that there is a fixed Haar measure µ on A such β u = µ for all u ∈ G (0) .
Proposition 3.7. With notation as in Example 3.5, suppose that A compact. Then the dualÂ is discrete and Proof. To prove the first isomorphism, note that by Proposition A.1 is a C 0 (Â)-algebra. That is, letting ZM(C * (Σ)) denote the center of M(C * (Σ)), there is a σ-unital *-homomorphism ρ : C 0 (Â) → ZM(C * (Σ)). SinceÂ is discrete, the images of the characteristic functions of singleton sets under ρ give rise to a family {q χ } χ∈Â of mutually orthogonal central projections in M(C * (Σ)) which sum to unity in the strict topology. Moreover, the summands coincide with the fibers of the upper-semicontinuous C * -bundle overÂ given in Proposition 3.6 and hence For the second isomorphism, let π : C * (Σ) → C * r (Σ) be the canonical quotient map. An argument like that of the preceding paragraph using the family {π(q χ )} χ∈Â of mutually orthogonal central projections in M(C * r (Σ)) gives C * r (Σ) ∼ = χ∈Â π(q χ )C * r (Σ). Lemma 3.1 gives π(q χ )C * r (Σ) ∼ = C * r (G; f χ * (Σ)), and the result follows. Remark 3.8. If A = T and Σ is a twist, thenÂ = Z, and we have [f n * (Σ)] = n[Σ] for n ∈ Z. It follows that the central summand corresponding to n = 1 is isomorphic to C * (G; Σ) and thus there is central projection q = q 1 ∈ M(C * (Σ)) such that C * (G; Σ) ∼ = qC * (Σ) and C * r (G; Σ) ∼ = qC * r (Σ) Now suppose that G = G (0) so that Σ = A is itself an abelian group bundle regarded as a groupoid with unit space G (0) and let Λ be a T-twist over A. Then since A is amenable C * (A; Λ) = C * r (A; Λ) (see, for example [SW13, Thm 1]). We shall say that such a twist is abelian if Λ is also an abelian group bundle-that is if Λ(u) is abelian for each u ∈ G (0) . Then Λ is abelian if and only if C * (Λ) is abelian and in that case C * (Λ) ∼ = C 0 (Λ). Arguing as in Example 3.4, we see that such extensions must be pointwise trivial but need not be globally trivial. If Λ is determined by a continuous T-valued 2-cocycle c, then Λ is abelian if and only if c is symmetric (cf., [DGN + 20, Lemma 3.5]). Suppose now that Λ is abelian. For n ∈ Z, let V n := {χ ∈Λ : χ(z, u) = z n for all z ∈ T and u ∈ G (0) }. Then C * (Λ) ∼ = C 0 (Λ) decomposes as a direct sum with summands of the form C 0 (V n ). Note that each V n is clopen. The projection q in Remark 3.8 may then be identified with the characteristic function of U Λ := V 1 and hence See [DGN20, Section 3] for a related construction.
In the case that Λ ∼ = T × A and thusΛ ∼ = Z ×Â, we have We return now to the more general situation where Σ is a unit space fixing extension of G by the group bundle A as in the diagram ( †) from the introduction. Suppose that, in addition, Ω is a T-groupoid extension of Σ ιp such that Λ Ω :=p −1 (A), its restriction to A, is an abelian group bundle over G (0) . We may thus regard Ω as an extension of G by Λ Ω . We assume that A, Σ and G are endowed with Haar systems that satisfy (3.1), the Haar system in G (0) × T is given by the Haar measure on T, and the Haar system on Ω is the one naturally defined by the Haar systems on G (0) × T and Σ. To declutter notation a little, we write Λ Ω for the dual bundle (Λ Ω ) ∧ .
By arguing as in Remark 3.8 and Corollary 3.9 we may conclude that C * (Σ; Ω) is isomorphic to the corner associated to the central projection q Ω in corresponding to the characteristic function of Observe that U Ω is an invariant clopen set under the action of both G and Ω and thus both groupoids act on U Ω . Corollary 3.10. With notation as above define g : U Ω * Λ Ω → U Ω × T by g(χ, a) = (χ, χ(a)). Then C * (Σ; Ω) ∼ = C * (U Ω ⋊ G; g * (U Ω ⋊ Ω)) and C * r (Σ; Ω) ∼ = C * r (U Ω ⋊ G; g * (U Ω ⋊ Ω)). Proof. Observe that Hence, by Remark 3.8 and Corollary 3.9 The case for the reduced C * -algebras follows by a similar argument.
Recall that anétale groupoid G is said to be effective if the interior of the isotropy groupoid is G (0) and topologically principal if the set of points with trivial isotropy is dense in G (0) . These notions are equivalent if theétale groupoid G is second countable (see [BCFS14,Lemma 3.1]). The above corollary allows us to generalize [IKR + 21, Theorem 4.6] (see also [DGN + 20, Theorem 5.8] and [DGN20, Theorem 4.6]).
Corollary 3.11. With notation as above, suppose that G isétale and that the action groupoid U Ω ⋊ G is second countable and effective. Then the image of C * r (A, Λ Ω ) under the natural embedding into C * r (Σ; Ω) is a Cartan subalgebra with Weyl twist g * (U Ω ⋊ Ω).
Example 3.12. Let H be a discrete abelian group and let E be a T-twist over H-that is, a central extension by T. Since H is discrete, there is a T-valued skew-symmetric bicharacter ̟ on H and a set of generating unitaries {u h | h ∈ H} in C * (H; E) such that for all g, h ∈ H u g u h = ̟(g, h)u h u g . By [Kle65, Lemma 7.2] the extension E is trivial if and only if ̟(g, h) = 1 for all g, h ∈ H. Let A be a subgroup of H which is maximal amongst subgroups on which ̟(·, ·) is identically 1. It is shown in [Kum86, Example 1.12] that the C * -subalgebra B generated by {u a | a ∈ A} is a diagonal subalgebra of C * (H; E). We now show that this also follows from Corollary 3.11 with Σ := H, A := A, G = H/A and Ω := E.
Since the restriction of ̟ to A is trivial the extension E is trivial on A and thus Λ is trivial as a T-twist. Hence, B ∼ = C * (A) and U Λ ∼ =Â. There is a continuous homomorphism ̟ A : H →Â such that for all h ∈ H, a ∈ A (̟ A (h))(a) = ̟(h, a).
Example 3.14. Let ϕ be a continuous normalized T-valued 2-cocycle and let Σ ϕ be the T-twist associated to ϕ. Then by Proposition 3.7 and Remark 3.8, and the fact that Σ ϕ n ∼ = n * (Σ ϕ ) for all n ∈ Z, we have
commutes. Therefore the lemma follows from Theorem 1.5.
Example 3.17. The following example was studied in [IKSW19]. Let X be a secondcountable locally compact Hausdorff space, and G a second-countable locally compact abelian group. Let G denote the sheaf of germs of continuous G-valued functions on X, and let c ∈ Z 2 (U , G ) be a normalizedČech two cocycle for some locally finite cover U = {U i } i∈I of X by precompact open sets. The blow-up groupoid G U with respect to the natural map from i U i into X is with (i, x, j)(j, x, k) = (i, x, k) and (i, x, j) −1 = (j, x, i). As noted in [IKSW19, Remark 3.3], theČech 2-cocycle c defines a groupoid 2-cocycle ϕ c : G Let Σ c be the extension of G U by the 2-cocycle ϕ c . Definê Thenφ is a groupoid 2-cocycle, and the pushout groupoid Σ is isomorphic to the T-groupoid that is the extension of (Ĝ × i U i ) ⋊ G U defined byφ. Let V = {Ĝ × U i } i∈I be the locally finite cover ofĜ × X, let S be the sheaf of germs of continuous T-valued functions, and define ν c = { ν c ijk } ∈ Z 2 (V , S ) by ν c (τ, (i, x, j)), (τ, (j, x, k)) = τ (c ijk (x)).
That is, ν c is the normalized 2-cocycle considered in [IKSW19, Equation (3.4)]. Hence the generalized Raeburn-Taylor C * -algebra A(ν) studied in [IKSW19] is isomorphic to the restricted C * -algebra of the T-groupoid defined by the 2-cocycle ν c .
Example 3.18. This example is an expansion of [IKR + 21, Example 4.10]. Let Γ = Z act on T via rotation by α ∈ Q: z · k := ze 2πikα . If α = m/n with m and n relatively prime, then nZ fixes the action. We have a short exact sequence of groups The action on T leads to an extension of groupoids i π Thus, using the notation from the previous section, A = T × nZ, Σ = T ⋊ Z, and G = T ⋊ Z n . The C * -algebra C * (T ⋊ Z) is the rational rotation C * -algebra A α (see, for example, [DB84]). The groupoid D is the cartesian product T × T n × T × Z, where T n = T/Z n is the dual of nZ. The extension Σ is the quotient of D where we identify (ω, χ, z, nl + k) with (ω, χ nl , z, k). Therefore the rational rotation algebra A α is the completion of continuous functions F on T×T n ×Z such that F (ω, χ, nl +k) = χ nl F (ω, χ, k) for all l ∈ Z.
The extension Σ is properly isomorphic to the one defined by a 2-cocycle. Indeed, let σ = e 2πiα ∈ T and view σ as a character on Z. Thus we can identify Z n with σ(Z) and then the map p in the short exact sequence (3.5) equals σ. Choose s ∈ Z such that sm = 1 (mod n). Then the map τ : Z n → Z defined by τ (k) = sk defines a crosssection of σ. In particular, Z is properly isomorphic to the extension nZ× ω Z n by a two cocycle ω : Z n ×Z n → nZ defined by τ . Using the proof of [IKSW19, Proposition A.6], ω(k 1 ,k 2 ) = τ (k 1 ) + τ (k 2 ) − τ (k 1 +k 2 ). A quick computation shows that ω(k 1 ,k 2 ) = 0 ifk 1 +k 2 < n ns ifk 1 +k 2 ≥ n, which recovers the 2-cocycle used in Step 2 of the proof of [DB84, Proposition 1].
Then Proposition 3.6 implies that A α is the section algebra of an upper-semicontinuous C * -bundle over T n with fiber at χ ∈ T n isomorphic to C * (T ⋊ Z n ; Σ χ * (ϕ) ).

Appendix A. Bundles of Twists
Let Σ be a twist over G. Alternatively, Σ is a T-groupoid so that we have the following diagram where as usual we have identified Σ (0) and G (0) . In particular, if F ⊂ G (0) is Ginvariant, then it is Σ-invariant and the reduction Σ| F is also a twist over the reduction G| F . Suppose that p : G (0) → T is a continuous map such that p • r = r • s. Then we say that Σ is a groupoid bundle over T . 1 Then p −1 (t) is invariant for all t ∈ T . We write Σ(t) and G(t) for the restrictions to p −1 (t), respectively. Then Σ(t) is a twist over G(t).
Proposition A.1. Suppose that G is a second countable locally compact Hausdorff groupoid with a Haar system and that Σ is a twist over G. If p : G (0) → T is a continuous map such that p • r = p • s, then C * (G; Σ) is a C 0 (T )-algebra. Let Σ(t) be the twist over G(t) defined above. Then C * (G; Σ) is (isomorphic to) the section algebra of an upper-semicontinuous C * -bundle over T . The fibre C * (G; Σ)(t) is isomorphic to C * (G(t); Σ(t)).