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bibundle

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Idea

A bibundle is a (groupoid-)principal bundle which is equipped with a compatible second (groupoid-)action “from the other side”.

In particular, Lie groupoid bibundles serve to exhibit “generalized morphisms”/Morita morphisms between Lie groupoids. This is in generalization of how the differentiable stack/smooth groupoid represented by a Lie groupoid is the moduli stack for groupoid-principal bundles.

Therefore groupoid bibundles play a role in geometry analogous to the role played by bimodules in algebra. In this role they were originally introduced in (Haefliger 84, Hilsum-Skandalis 87, Pradines 89) and accordingly they are also called Hilsum-Skandalis maps. Independently they were seen in topos theory (Bunge 90, Moerdijk 91). Historically, a central motivation for their study has been that the groupoid convolution algebra construction sends smooth bibundles between Lie groupoids to (Hilbert-)bimodules of the corresponding C-star convolution algebras, such that Morita equivalence is respected (Muhly-Renault-Williams 87, Landsman 00, Mrčun 05). This is of relevance notably for KK-theory of Lie groupoids (Hilsum-Skandalis 87b).

Bibundles also appear as transition bundles of nonabelian bundle gerbes.

Properties

Lie groupoid bibundles and Morita/stack morphisms

We discuss how Lie groupoid bibundles correspond to Morita morphism (morphisms of differentiable stacks/smooth stacks) between the Lie groupoids.

First we set up the relevant definitions and establish our notation in

Then we discuss smooth groupoid-principal bundles and how a Lie groupoid moduli stack for the bundles principal over it in

Finally we consider the corresponding smooth bibundles and how they correspond to their modulating stack morphisms in

Lie groupoids and smooth stacks

A smooth stack or smooth groupoid is a stack on the site SmoothMfd of smooth manifolds or equivalently (and often more conveniently) on its dense subsite CartSp of just Cartesian spaces n,n\mathbb{R}^n, n \in \mathbb{N} and smooth functions between them, equipped with the standard coverage of good open covers.

We write

\;\;\; SmoothGrpd Sh (2,1)(CartSp)L lheFunc(CartSp op,Grpd)\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd)

for the (2,1)-category of stacks on this site, equivalently the result of taking groupoid-valued presheaves and then universally turning local (as seen by the coverage) equivalences of groupoids into global equivalence in an (infinity,1)-category.

By generalizing here groupoids to general Kan complexes and equivalences of groupoids to homotopy equivalences of Kan complexes, one obtains smooth ∞-stacks or smooth ∞-groupoids, which we write

\;\;\; Smooth∞Grpd Sh (,1)(CartSp)L lheFunc(CartSp op,KanCplx)\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx) .

We often write H\mathbf{H} \coloneqq Smooth∞Grpd for short.

By the (∞,1)-Yoneda lemma there is a sequence of faithful inclusions

\;\;\; SmoothMfd \hookrightarrow SmoothGrpd \hookrightarrow Smooth∞Grpd.

This induces a corresponding inclusion of simplicial objects and hence of groupoid objects

LieGrpdGrpd (SmoothMfd)Grpd (SmoothGrpd). LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd) \,.

For 𝒢 Grpd (H)\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H}) a groupoid object we write

𝒢 0𝒢lim n𝒢 n \mathcal{G}_0 \to \mathcal{G} \coloneqq \underset{\longrightarrow}{\lim}_{n} \mathcal{G}_n

for its (∞,1)-colimiting cocone, hence 𝒢H\mathcal{G} \in \mathbf{H} (without subscript decoration) denotes the realization of 𝒢 \mathcal{G}_\bullet as a single object in H\mathbf{H}.

Definition

For 𝒢 Grpd (H)\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H}) a groupoid object, we write 𝒢lim n𝒢 nH\mathcal{G} \coloneqq {\lim}_{n} \mathcal{G}_n \in \mathbf{H} for its realization and call the canonical 1-epimorphism

𝒢 0𝒢 \mathcal{G}_0 \to \mathcal{G}

the canonical atlas of this realization.

Example

For 𝒢 Grpd(SmoothMfd)Grpd (SmoothGrpd)\mathcal{G}_\bullet \in Grpd(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd) a Lie groupoid, we have that

  1. 𝒢 0SMoothMfdSmoothGprd\mathcal{G}_0 \in SMoothMfd \hookrightarrow Smooth\infty Gprd is its smooth manifold of objects

  2. 𝒢\mathcal{G} \in SmoothGrpd \hookrightarrow Smooth∞Grpd is the realization of the Lie groupoid as a differentiable stack, hence as a smooth groupoid

  3. 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G} is the canonically induced atlas in the traditional sense of geometric stack-theory.

Remark

By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos H\mathbf{H} this morphism 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G} is a 1-epimorphism and its construction establishes is an equivalence of (∞,1)-categories Grpd (H)H 1epi Δ 1Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}, hence morphisms 𝒢 𝒦 \mathcal{G}_\bullet \to \mathcal{K}_\bullet in Grpd (H)Grpd_\infty(\mathbf{H}) are equivalently diagrams in H\mathbf{H} of the form

𝒢 0 𝒦 0 𝒢 𝒦. \array{ \mathcal{G}_0 &\to& \mathcal{K}_0 \\ \downarrow &\swArrow& \downarrow \\ \mathcal{G} &\to& \mathcal{K} } \,.

This is evidently more constrained than just morphisms

𝒢𝒦 \mathcal{G} \to \mathcal{K}

by themselves. The latter are the generalized morphisms or Morita morphisms between the groupoid objects 𝒢 \mathcal{G}_\bullet, 𝒦 \mathcal{K}_\bullet.

Definition

Given groupoid objects 𝒢 ,𝒦 Grpd (H)\mathcal{G}_\bullet, \mathcal{K}_\bullet \in Grpd_\infty(\mathbf{H}), a Morita morphism between them is a morphism 𝒢𝒦\mathcal{G} \to \mathcal{K} in H\mathbf{H} between their realizations. A Morita morphism that is an equivalence in H\mathbf{H} is called a Morita equivalence of groupoid objects in H\mathbf{H}.

Here we want to express these Morita morphisms 𝒢𝒦\mathcal{G} \to \mathcal{K} in terms of bibundle objects 𝒫H\mathcal{P} \in \mathbf{H} on which both 𝒢 \mathcal{G}_\bullet and 𝒦 \mathcal{K}_\bullet act.

Example

For XHX \in \mathbf{H} any object, its pair groupoid Pair(X) Grpd (H)Pair(X)_\bullet \in Grpd_\infty(\mathbf{H}) is

Pair(X) nX × n+1. Pair(X)_n \coloneqq X^{\times^{n+1}} \,.

The realization of this is equivalent to the point

Pair(X)lim nPair(X) n*. Pair(X) \coloneqq \underset{\longrightarrow}{\lim}_n Pair(X)_n \simeq * \,.

Hence all Morita morphisms, def. 2, to the pair groupoid are equivalent. As a groupoid object Pair(X) Pair(X)_\bullet is non-trivial, but it is Morita equivalent to the terminal groupoid object.

Smooth groupoid-principal bundles

Definition

For 𝒢 Grpd (H)\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H}) a groupoid object, PHP \in \mathbf{H} any object equipped with a morphsim a:P𝒢 0a \colon P \to \mathcal{G}_0 to the object of objects of 𝒢\mathcal{G}, a 𝒢 \mathcal{G}_\bullet-groupoid ∞-action on XX with anchor aa is a groupoid (X//𝒢) (X//\mathcal{G})_\bullet over 𝒢 \mathcal{G}_\bullet of the form

X×𝒢 0𝒢 2 𝒢 2 X×𝒢 0𝒢 1 𝒢 1 X a 𝒢 0, \array{ \vdots && && \vdots \\ \downarrow \downarrow \downarrow \downarrow && && \downarrow \downarrow \downarrow \downarrow \\ X \underset{\mathcal{G}_0}{\times} \mathcal{G}_2 && \to && \mathcal{G}_2 \\ \downarrow \downarrow \downarrow && && \downarrow \downarrow \downarrow \\ X \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 && \to && \mathcal{G}_1 \\ \downarrow \downarrow && && \downarrow \downarrow \\ X && \stackrel{a}{\to} && \mathcal{G}_0 } \,,

where the homotopy fiber products on the left are those of the anchor aa with the leftmost 0-face map 𝒢 ({0}{0,,n})\mathcal{G}_{(\{0\} \hookrightarrow \{0, \cdots, n\})} and the horizontal morphisms are the corresponding projections on the second factor.

We call (X//𝒢) (X//\mathcal{G})_\bullet also the action groupoid of the action of 𝒢 \mathcal{G}_\bullet on (X,a)(X,a) and call its realization X(X//𝒢)X \to (X//\mathcal{G}) the homotopy quotient of the action.

Example

For 𝒢 =(BG) \mathcal{G}_\bullet = (\mathbf{B}G)_\bullet the delooping of a group object, def. 3 reduces to the definition of an ∞-action of the ∞-group GG.

Under this relation, the discussion of ∞-groupoid-principal ∞-bundles proceeds in direct analogy with that of GG-principal ∞-bundles:

Proposition

For XX \in Smooth∞Grpd any object, a morphism f:X𝒢f \colon X \to \mathcal{G} in H\mathbf{}H induces (“modulates”) a 𝒢 \mathcal{G}_\bullet-groupoid action, def. 3, on the homotopy pullback f *𝒢 0f^\ast \mathcal{G}_0

f *𝒢 0 𝒢 0 pb X f 𝒢. \array{ f^* \mathcal{G}_0 &\to& \mathcal{G}_0 \\ \downarrow &pb_\infty& \downarrow \\ X &\stackrel{f}{\to}& \mathcal{G} \,. }

of the atlas of 𝒢\mathcal{G}: the corresponding action groupoid is the Cech nerve of the projection p:f *𝒢 0Xp \colon f^*\mathcal{G}_0 \to X (which as the (∞,1)-pullback of a 1-epimorphism is itself a 1-epimorphism):

(f *𝒢 0)×𝒢 0𝒢 1 𝒢 1 f *𝒢 0 a 𝒢 0 pb (f *𝒢 0)//𝒢 X f 𝒢 lim n𝒢 n. \array{ && \vdots && \vdots \\ && \downarrow \downarrow \downarrow && \downarrow \downarrow \downarrow \\ && (f^\ast \mathcal{G}_0) \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 &\to& \mathcal{G}_1 \\ && \downarrow \downarrow && \downarrow \downarrow \\ && f^* \mathcal{G}_0 &\stackrel{a}{\to}& \mathcal{G}_0 \\ && \downarrow &pb_\infty& \downarrow \\ (f^\ast \mathcal{G}_0)//\mathcal{G} &\simeq& X &\stackrel{f}{\to}& \mathcal{G} &\simeq& \underset{\longrightarrow}{\lim}_n \mathcal{G}_n \,. }
Example

Let f :X 𝒢 f_\bullet \colon X_\bullet \to \mathcal{G}_\bullet be a morphism of 1-groupoid objects, say of Lie groupoids. Then as discussed at homotopy pullback the (∞,1)-pullback of the atlas 𝒢 0𝒢\mathcal{G}_0 \to \mathcal{G} along the realization ff is computed as the 1-categorical pullback

(𝒢 0×𝒢𝒢 Δ 1) pb 𝒳 𝒢 \array{ &\to& (\mathcal{G}_0 \underset{\mathcal{G}}{\times} \mathcal{G}^{\Delta^1})_\bullet \\ \downarrow &pb& \downarrow \\ \mathcal{X}_\bullet &\to& \mathcal{G}_\bullet }

in Sh(CartSp) Δ opSh(CartSp)^{\Delta^{op}}. Schematically the groupoid on the right has morphisms γ 0γ 1 \gamma_0 \to \gamma_1 which are commuting diagrams in 𝒢\mathcal{G} of the form

g γ 0 γ 1 g 0 g 1. \array{ && g \\ & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} \\ g_0 && \to && g_1 } \,.

Therefore the pullback is the sheaf of groupoids which is schematically of the form

f *𝒢 0{ g γ 0 γ 1 𝒢 f(x 0) f(ξ) f(x 1) x 0 ξ x 1 𝒳}. f^\ast \mathcal{G}_0 \;\simeq\; \left\{ \array{ && g \\ & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} & && \in \mathcal{G} \\ f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1) \\ x_0 && \stackrel{\xi}{\to}&& x_1 && \in \mathcal{X} } \right\} \,.

In this presentation now

𝒢 1{ g 0 γ g 1 γ 0 γ 1 g 10 g 11}. \mathcal{G}_1 \;\simeq\; \left\{ \array{ && g_0 \\ && \downarrow^{\gamma} \\ && g_1 \\ & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} \\ g_{1 0} && \to && g_{11} } \right\} \,.

and the target map 𝒢 1𝒢 0\mathcal{G}_1 \to \mathcal{G}_0 is given by forgetting the top vertical morphism in this diagram, while the source map is given by composing (!) the top vertical morphism with the two diagonal morphism.

Pullback of these two maps induces the left and right vertical map in

f *𝒢 0×𝒢 0𝒢 1 𝒢 1 f *𝒢 0 a 𝒢 0. \array{ f^\ast \mathcal{G}_0 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 &\to& \mathcal{G}_1 \\ \downarrow\downarrow && \downarrow \downarrow \\ f^\ast \mathcal{G}_0 &\stackrel{a}{\to}& \mathcal{G}_0 } \,.

from

f *𝒢 0×𝒢 0𝒢 1{ g 0 γ g 1 𝒢 γ 0 γ 1 f(x 0) f(ξ) f(x 1) x 0 ξ x 1 𝒳}. f^\ast \mathcal{G}_0 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \;\simeq\; \left\{ \array{ && g_0 \\ && \downarrow^{\gamma} \\ && g_1 && && \in \mathcal{G} \\ & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} & \\ f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1) \\ x_0 && \stackrel{\xi}{\to} && x_1 & & \in \mathcal{X} } \right\} \,.

The left one just forgets the top vertical morphism, the right one composes it with the diagonal morphisms. This composion is the 𝒢 \mathcal{G}_\bullet-action on f *𝒢 0f^\ast \mathcal{G}_0.

Smooth groupoid-principal bibundles

Finally then for 𝒳 \mathcal{X}_\bullet and 𝒢 \mathcal{G}_\bullet two Lie groupoids and f:𝒳𝒢f \;\colon\; \mathcal{X} \to \mathcal{G} a morphism in Smooth∞Grpd between the corresponding differentiable stacks, we obtain first the 𝒢\mathcal{G}-groupoid principal bundle f *𝒢 0p𝒳f^* \mathcal{G}_0 \stackrel{p}{\to} \mathcal{X} and then by further homotopy pullback also the left 𝒳\mathcal{X}-groupoid principal bundle p *𝒳 0p^* \mathcal{X}_0:

Definition

For 𝒳 ,𝒢 Grpd (H)\mathcal{X}_\bullet, \mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H}) two groupoid objects and f:𝒳𝒢f \colon \mathcal{X} \to \mathcal{G} a Morita morphism between them, def. 2, we say that the corresponding 𝒳 𝒢 \mathcal{X}_\bullet-\mathcal{G}_\bullet-bibundle 𝒫(f)\mathcal{P}(f) is the 𝒢 \mathcal{G}_\bullet-groupoid-principal bundle f *𝒢 0f^\ast \mathcal{G}_0 pulled back to canonical atlas of 𝒳\mathcal{X} and equipped with the induced 𝒳 \mathcal{X}_\bullet-groupoid action:

𝒫(f) p *𝒳 0 f *𝒢 0 𝒢 0 pb p pb 𝒳 0 𝒳 f 𝒢. \array{ \mathcal{P}(f)& \coloneqq& p^* \mathcal{X}_0 &\to& f^* \mathcal{G}_0&\to& \mathcal{G}_0 \\ && \downarrow &\pb\infty& \downarrow^{\mathrlap{p}} &pb_\infty& \downarrow \\ && \mathcal{X}_0 &\to& \mathcal{X} &\stackrel{f}{\to}& \mathcal{G} } \,.
Remark

Here the 𝒢 \mathcal{G}_\bullet-action on 𝒫(f)\mathcal{P}(f) is principal over 𝒳 0\mathcal{X}_0, in that the quotient map is

𝒫(f)𝒫(f)//𝒢𝒳 0, \mathcal{P}(f) \to \mathcal{P}(f)//\mathcal{G} \simeq \mathcal{X}_0 \,,

since 𝒫(f)\mathcal{P}(f) is the pullback of a 𝒢 \mathcal{G}_\bullet-principal bundle (modualted by the bottom composite map in the above diagram).

On the other hand the 𝒳 \mathcal{X}_\bullet-action on 𝒫(f)\mathcal{P}(f) is not principal over 𝒢 0\mathcal{G}_0 – unless ff is an equivalence in an (infinity,1)-category (hence a (Morita) from 𝒳 \mathcal{X}_\bullet to 𝒢 \mathcal{G}_\bullet.) It is instead always principal over f *𝒢 0f^\ast \mathcal{G}_0.

Remark

If both atlases are 0-truncated objects (smooth spaces) 𝒳 0,𝒢 0Sh(CartSp)τ 1HH\mathcal{X}_0, \mathcal{G}_0 \in Sh(CartSp) \simeq \tau_1 \mathbf{H} \hookrightarrow \mathbf{H}, then by the pasting law for homotopy pullbacks we have that 𝒫(f)\mathcal{P}(f) is (n-1)-truncated if 𝒢\mathcal{G} is n-truncated.

In particular therefore the total space of a smooth 1-groupoid bibundle is 0-truncated hence is a smooth space.

Example

In order to discuss Lie-groupoid bibundles we continue the discussion in example 4 of Lie-groupoid principal bundles. Proceeding for the second homotopy pullback diagram as discussed there for the first one, one finds that the total space 𝒫\mathcal{P} of the bibundle is presented by the sheaf of groupoids whose schematic depiction is

f *𝒢 0{ g γ 0 γ 1 𝒢 f(x 0) f(ξ) f(x 1) x 0 ξ x 1 𝒳 x}. f^\ast \mathcal{G}_0 \;\simeq\; \left\{ \array{ && g \\ & {}^{\mathllap{\gamma_0}}\swarrow && \searrow^{\mathrlap{\gamma_1}} & && \in \mathcal{G} \\ f(x_0) && \stackrel{f(\xi)}{\to} && f(x_1) \\ x_0 && \stackrel{\xi}{\to}&& x_1 \\ & \searrow && \swarrow & && \in \mathcal{X} \\ && x } \right\} \,.

Here the vertically-running morphisms are the objects and two such are related by a morphism if they fit into a commuting diagram complete by horizontal morphisms as indicated. Since 𝒳 \mathcal{X}_\bullet and 𝒢 \mathcal{G}_\bullet both are groupoids, these morphisms are unique if they exist, and hence, as predicted by remark 3, 𝒫(f)\mathcal{P}(f) is 0-truncated, hence is a smooth space. Moreover, since the isomorphism equivalence relation here is free, the quotient smooth space is actually a smooth manifold (since 𝒳 \mathcal{X}_\bullet and 𝒢 \mathcal{G}_\bullet are Lie groupoids).

This then recovers the definition of bibundles for Lie groupoids as often found in the literature.

The right 𝒢 \mathcal{G}_\bullet-action is by precomposition of these diagram with morphisms in 𝒢\mathcal{G}, while the left 𝒳\mathcal{X}-action is by postcomposition with morphisms in 𝒳\mathcal{X}.

Conversely, given a 𝒳 \mathcal{X}_\bullet-𝒢 \mathcal{G}_\bullet-Lie groupoid bibundle which is principal on the left

𝒫 𝒢 0 𝒳 0 𝒳 𝒢 \array{ \mathcal{P} &\to& \mathcal{B} &\to& \mathcal{G}_0 \\ \downarrow && && \downarrow \\ \mathcal{X}_0 &\to& \mathcal{X} && \mathcal{G} }

we recover the Morita morphism ff that it coresponds to by the Giraud-Rezk-Lurie axioms: first pp is the induced map between the homotopy colimits of the Cech nerves of the two left horizontal maps

𝒫 𝒢 0 p 𝒳 0 𝒳 𝒢 \array{ \mathcal{P} &\to& \mathcal{B} &\to& \mathcal{G}_0 \\ \downarrow && \downarrow^{\mathrlap{p}} && \downarrow \\ \mathcal{X}_0 &\to& \mathcal{X} && \mathcal{G} }

and then ff is similarly the map between the homotopy colimits of the Cech nerves of the two right vertical maps.

(…)

Relation to groupoid convolution bimodules

There should be a 2-functor from Lie groupoids to C-star-algebras and Hilbert C-star-bimodules between them given by forming groupoid convolution algebras and naturally exhibited by Lie groupoid bibundles: the groupoid convolution algebra of the total space of the bibindle becomes a bimodule over the two other groupoid convolution algebras.

Some aspects of this are in the literature, e.g. (Mrčun 99) for étale Lie groupoids and (Landsman 00) for general Lie groupoids. The follwing is taken from the latter article.

Definition/Notation

For p:EXp \colon E \to X a smooth function between smooth manifolds, we write T pETET^p E \hookrightarrow T E for the bundle of vertical vector fields, the sub-bundle of the tangent bundle of EE on those vectors in the kernel of the differentiation maps dp| e:T eET τ(e)Xd p|_{e} \colon T_e E \to T_{\tau(e)} X.

We write |Λ| 1/2(T τE){\vert \Lambda\vert^{1/2}}(T^\tau E) for the bundle of half-densities on vertical vector fields.

Remark

Let 𝒢 \mathcal{G}_\bullet be a Lie groupoid and let (Eτ𝒢 0,ρ)E \stackrel{\tau}{\to} \mathcal{G}_0, \rho) be a 𝔾 \mathbb{G}_\bullet-groupoid-principal bundle EE//𝒢E \to E//\mathcal{G} (with anchor τ\tau and action map ρ\rho).

Then the bundle of vertical vector fields T τET^\tau E equipped with the anchor map T τEdτT𝒢 0𝒢 0T^\tau E \stackrel{d \tau}{\to} T \mathcal{G}_0 \to \mathcal{G}_0 inherits a canonical 𝒢 \mathcal{G}_\bullet-action itself.

The quotient map

|Λ| 1/2(T τE)/𝒢E/𝒢 {\vert \Lambda\vert^{1/2}}(T^\tau E)/\mathcal{G} \to E/\mathcal{G}

exists and is naturall a vector bundle again.

Definition

In the situation of remark 4, write

  • C c/G (E,|Λ| 1/2(T τ)E) GC^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G

    for the space of smooth sections of the half-density-bundle of T τET^\tau E which are 𝒢\mathcal{G}-equivariant and which have compact support up to 𝒢\mathcal{G}-action;

  • C c (E/𝒢,|Λ| 1/2(T τE))C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E))

    for the space of smooth sections with compact support of the quotient bundle.

The following constructions work by repeatedly applying the following identification:

Proposition

In the situation of def. 6, there is a natural isomorphism

C c/G (E,|Λ| 1/2(T τ)E) GC c (E/𝒢,|Λ| 1/2(T τE)). C^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G \simeq C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E) ) \,.

The central definition here is now:

Definition

For (E 1,τ 1)(E_1, \tau_1), (E 2,τ 2)(E_2, \tau_2) two principal 𝒢 \mathcal{G}_\bullet manifolds, set

(E 1,E 2) 𝒢C c (E 1×𝒢 0)E 2,|Λ| 1/2(T τE 1)|Λ| 1/2(T τE 2) (E_1, E_2)_{\mathcal{G}} \coloneqq C^\infty_c( E_1 \underset{\mathcal{G}_0}{\times}) E_2, {\vert\Lambda\vert^{1/2}(T^\tau E_1)} \otimes {\vert\Lambda\vert^{1/2}(T^\tau E_2)}

And the central fact is:

Proposition

Given 3 𝒢 \mathcal{G}_\bullet-manifolds (E i,τ i)(E_i, \tau_i), i{1,2,3}i \in \{1,2,3\}, there is a smooth function

:(E 1,E 2) 𝒢×(E 2,E 3) 𝒢(E 1,E 3) 𝒢 \star \;\colon\; (E_1, E_2)_{\mathcal{G}} \times (E_2, E_3)_{\mathcal{G}} \to (E_1, E_3)_{\mathcal{G}}

given on sections σ 1,σ 2\sigma_1, \sigma_2 and points (e 1,e 3)(e_1, e_3) by

σ 1σ 2:(e 1,e 3) τ 2 1(τ 1e 1)σ 1(e 1,)σ 2(,e 3), \sigma_1 \star \sigma_2 \colon (e_1, e_3) \mapsto \int_{\tau_2^{-1}(\tau_1 e_1)} \sigma_1(e_1, -) \otimes \sigma_2(-,e_3) \,,

where the integration is against the measure that appears by tensoring two (of the four) half-densities in the integrand.

This operation is an associative and invoutive partial composition operation and hence defines a star-category whose objects are 𝒢 \mathcal{G}_\bullet-principal manifolds and whose spaces of morphisms are as in def. 7.

In particular one has the following identifications.

Example

For 𝒢 1𝒢 0\mathcal{G}_1 \to \mathcal{G}_0 regarded as a 𝒢 \mathcal{G}_\bullet-principal action space, there is a natural isomorphism

(𝒢 1,𝒢 1) 𝒢C c (𝒢 1,|Λ| 1/2(T s𝒢 1)|Λ| 1/2(T t𝒢 1)) (\mathcal{G}_1, \mathcal{G}_1)_{\mathcal{G}} \simeq C^\infty_c(\mathcal{G}_1, {\vert\Lambda\vert}^{1/2}(T^s \mathcal{G}_1) \otimes {\vert\Lambda\vert}^{1/2}(T^t \mathcal{G}_1))

and the algebra structure on this by prop. 3 is isomorphic to the groupoid convolution algebra of smooth sections over 𝒢 \mathcal{G}_\bullet.

More generally:

Example

For Eτ𝒢 0E \stackrel{\tau}{\to} \mathcal{G}_0 any 𝒢\mathcal{G}-principal manifold, we have a natural isomorphism

(𝒢 1,E) 𝒢C c (E 1,|Λ| 1/2(T GE)|Λ| 1/2(TτE)). (\mathcal{G}_1, E)_{\mathcal{G}} \simeq C^\infty_c(E_1, {\vert\Lambda\vert}^{1/2}(T^G E) \otimes {\vert\Lambda\vert}^{1/2}(T\tau E)) \,.

We consider completion of all this to the C-star-algebra context (…)

Now we can put the pieces together and sends groupoid-bindunles to C *C^\ast-bimodules over the two groupoid convolution algebras.

Proposition

Given two Lie groupoids 𝒢 \mathcal{G}_\bullet and 𝒦 \mathcal{K}_\bullet and given a Morita equivalence groupoid bibundle EE between them, we have

N(𝒢 1,E) 𝒢(E,𝒦) 𝒦 N \coloneqq (\mathcal{G}_1, E)_{\mathcal{G}} \simeq (E, \mathcal{K})_{\mathcal{K}}

and this identification makes NN into a C *(𝒢 )C *(𝒦 )C^\ast(\mathcal{G}_\bullet)-C^\ast(\mathcal{K}_\bullet)-pre-Hilbert bimodule as follows:

  1. The identification N(E,𝒦 1) 𝒦N \simeq (E, \mathcal{K}_1)_{\mathcal{K}} defines the right C *(𝒦 )C^\ast(\mathcal{K}_\bullet)-action by example 6; and similarly the identification N(𝒢 1,E) 𝒢N \simeq (\mathcal{G}_1, E)_{\mathcal{G}} defines a left C *(𝒢 )C^\ast(\mathcal{G}_\bullet)-action.

  2. The C *(𝒦)C^\ast(\mathcal{K})-valued inner product on NN is that induced by the composite

    (E,𝒦 1) 𝒦 *×(E,𝒦 1) 𝒦(𝒦 1) 𝒦,E×(E,𝒦 1) 𝒦(𝒦 1,𝒦 1) 𝒦C *(𝒦 ). (E,\mathcal{K}_1)_{\mathcal{K}}^\ast \times (E,\mathcal{K}_1)_{\mathcal{K}} \stackrel{\simeq}{\to} (\mathcal{K}_1)_{\mathcal{K}, E} \times (E,\mathcal{K}_1)_{\mathcal{K}} \to (\mathcal{K}_1, \mathcal{K}_1)_{\mathcal{K}} \hookrightarrow C^\ast(\mathcal{K}_\bullet) \,.

References

General

Groupoid bibundles were first considered for foliation groupoids in

  • Michel Hilsum and Georges Skandalis, Morphismes K-orientes d’espaces de feuilles et functorialite en theorie de Kasparov. Ann. Scient. Ec. Norm. Sup. 20 (1987), 325–390. (numdam)

The generalization to arbitrary topological groupoids was considered in

  • André Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1984), 70–97.
  • Jean Pradines, Morphisms between spaces of leaves viewed as fractions. Cahiers Top. Géom. Diff. Cat. XXX-3 (1989), 229–246
  • M. Hilsum and Georges Skandalis, Morphismes K-orientes d’espaces de feuilles et functorialite en theorie de Kasparov. Ann. Scient. Ec. Norm. Sup. 20 (1987), 325–390.

and independently in topos theory in

  • Marta Bunge, An application of descent to a classification theorem. Math. Proc. Cambridge Phil. Soc. 107 (1990), 59–79.
  • Ieke Moerdijk, Classifying toposes and foliations. Ann. Inst. Fourier, Grenoble 41, 1 (1991), 189–209.

Groupoid bibundles are used in the context of groupoid convolution algebras as geometric analogs of bimodules in

  • Paul Muhly, Jean Renault, and D. Williams, Equivalence and isomorphism for groupoid C∗-algebras, J. Operator Th. 17 (1987), 3–22.

A review of Lie groupoid-bibundles and maps of differentiable stacks is in section 2 of

Discussion of Lie group cohomology and the string 2-group infinity-group extension in terms of Lie groupoid bibundles is in

Talk notes on bibundles include

See also

Convolution to C *C^\ast-bimodules

For groupoid bibundles between étale Lie groupoids the assignment of the groupoid convolution algebra-bimodule to them is shown to be functorial in

  • Janez Mrčun, Functoriality of the bimodule associated to a Hilsum-Skandalis map. K-Theory 18 (1999) 235–253.

For more references along these lines see for the moment at groupoid convolution algebra – Extension to bibundles and bimodules

Revised on May 30, 2013 20:36:07 by Urs Schreiber (89.204.138.103)