(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 87).
Bibundles also appear as transition bundles of nonabelian bundle gerbes.
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
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 $\mathbb{R}^n, n \in \mathbb{N}$ and smooth functions between them, equipped with the standard coverage of good open covers.
We write
$\;\;\;$SmoothGrpd $\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd)$
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 $\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx)$.
We often write $\mathbf{H} \coloneqq$ 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
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object we write
for its (∞,1)-colimiting cocone, hence $\mathcal{G} \in \mathbf{H}$ (without subscript decoration) denotes the realization of $\mathcal{G}_\bullet$ as a single object in $\mathbf{H}$.
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object, we write $\mathcal{G} \coloneqq {\lim}_{n} \mathcal{G}_n \in \mathbf{H}$ for its realization and call the canonical 1-epimorphism
the canonical atlas of this realization.
For $\mathcal{G}_\bullet \in Grpd(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd)$ a Lie groupoid, we have that
$\mathcal{G}_0 \in SMoothMfd \hookrightarrow Smooth\infty Gprd$ is its smooth manifold of objects
$\mathcal{G} \in$ SmoothGrpd $\hookrightarrow$ Smooth∞Grpd is the realization of the Lie groupoid as a differentiable stack, hence as a smooth groupoid
$\mathcal{G}_0 \to \mathcal{G}$ is the canonically induced atlas in the traditional sense of geometric stack-theory.
By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos $\mathbf{H}$ this morphism $\mathcal{G}_0 \to \mathcal{G}$ is a 1-epimorphism and its construction establishes is an equivalence of (∞,1)-categories $Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}$, hence morphisms $\mathcal{G}_\bullet \to \mathcal{K}_\bullet$ in $Grpd_\infty(\mathbf{H})$ are equivalently diagrams in $\mathbf{H}$ of the form
This is evidently more constrained than just morphisms
by themselves. The latter are the generalized morphisms or Morita morphisms between the groupoid objects $\mathcal{G}_\bullet$, $\mathcal{K}_\bullet$.
Given groupoid objects $\mathcal{G}_\bullet, \mathcal{K}_\bullet \in Grpd_\infty(\mathbf{H})$, a Morita morphism between them is a morphism $\mathcal{G} \to \mathcal{K}$ in $\mathbf{H}$ between their realizations. A Morita morphism that is an equivalence in $\mathbf{H}$ is called a Morita equivalence of groupoid objects in $\mathbf{H}$.
Here we want to express these Morita morphisms $\mathcal{G} \to \mathcal{K}$ in terms of bibundle objects $\mathcal{P} \in \mathbf{H}$ on which both $\mathcal{G}_\bullet$ and $\mathcal{K}_\bullet$ act.
For $X \in \mathbf{H}$ any object, its pair groupoid $Pair(X)_\bullet \in Grpd_\infty(\mathbf{H})$ is
The realization of this is equivalent to the point
Hence all Morita morphisms, def. , to the pair groupoid are equivalent. As a groupoid object $Pair(X)_\bullet$ is non-trivial, but it is Morita equivalent to the terminal groupoid object.
For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object, $X \in \mathbf{H}$ any object equipped with a morphism $a \colon X \to \mathcal{G}_0$ to the object of objects of $\mathcal{G}$, a $\mathcal{G}_\bullet$-groupoid ∞-action on $X$ with anchor $a$ is a groupoid $(X//\mathcal{G})_\bullet$ over $\mathcal{G}_\bullet$ of the form
where the homotopy fiber products on the left are those of the anchor $a$ with the leftmost 0-face map $\mathcal{G}_{(\{0\} \hookrightarrow \{0, \cdots, n\})}$ and the horizontal morphisms are the corresponding projections on the second factor.
We call $(X//\mathcal{G})_\bullet$ also the action groupoid of the action of $\mathcal{G}_\bullet$ on $(X,a)$ and call its realization $X \to (X//\mathcal{G})$ the homotopy quotient of the action.
For $\mathcal{G}_\bullet = (\mathbf{B}G)_\bullet$ the delooping of a group object, def. reduces to the definition of an ∞-action of the ∞-group $G$.
Under this relation, the discussion of ∞-groupoid-principal ∞-bundles proceeds in direct analogy with that of $G$-principal ∞-bundles:
For $X \in$ Smooth∞Grpd any object, a morphism $f \colon X \to \mathcal{G}$ in $\mathbf{}H$ induces (“modulates”) a $\mathcal{G}_\bullet$-groupoid action, def. , on the homotopy pullback $f^\ast \mathcal{G}_0$
of the atlas of $\mathcal{G}$: the corresponding action groupoid is the Cech nerve of the projection $p \colon f^*\mathcal{G}_0 \to X$ (which as the (∞,1)-pullback of a 1-epimorphism is itself a 1-epimorphism):
Let $f_\bullet \colon X_\bullet \to \mathcal{G}_\bullet$ be a morphism of 1-groupoid objects, say of Lie groupoids. Then, as discussed at homotopy pullback, the (∞,1)-pullback of the atlas $\mathcal{G}_0 \to \mathcal{G}$ along the realization $f$ is computed as the 1-categorical pullback
in $Sh(CartSp)^{\Delta^{op}}$. Schematically the groupoid on the right has morphisms $\gamma_0 \to \gamma_1$ which are commuting diagrams in $\mathcal{G}$ of the form
Therefore the pullback is the sheaf of groupoids which is schematically of the form
In this presentation now
and the target map $\mathcal{G}_1 \to \mathcal{G}_0$ 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
from
The left one just forgets the top vertical morphism, the right one composes it with the diagonal morphisms. This composion is the $\mathcal{G}_\bullet$-action on $f^\ast \mathcal{G}_0$.
Finally then for $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ two Lie groupoids and $f \;\colon\; \mathcal{X} \to \mathcal{G}$ a morphism in Smooth∞Grpd between the corresponding differentiable stacks, we obtain first the $\mathcal{G}$-groupoid principal bundle $f^* \mathcal{G}_0 \stackrel{p}{\to} \mathcal{X}$ and then by further homotopy pullback also the left $\mathcal{X}$-groupoid principal bundle $p^* \mathcal{X}_0$:
For $\mathcal{X}_\bullet, \mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ two groupoid objects and $f \colon \mathcal{X} \to \mathcal{G}$ a Morita morphism between them, def. , we say that the corresponding $\mathcal{X}_\bullet-\mathcal{G}_\bullet$-bibundle $\mathcal{P}(f)$ is the $\mathcal{G}_\bullet$-groupoid-principal bundle $f^\ast \mathcal{G}_0$ pulled back to the canonical atlas of $\mathcal{X}$ and equipped with the induced $\mathcal{X}_\bullet$-groupoid action:
Here the $\mathcal{G}_\bullet$-action on $\mathcal{P}(f)$ is principal over $\mathcal{X}_0$, in that the quotient map is
since $\mathcal{P}(f)$ is the pullback of a $\mathcal{G}_\bullet$-principal bundle (modulated by the bottom composite map in the above diagram).
On the other hand the $\mathcal{X}_\bullet$-action on $\mathcal{P}(f)$ is not principal over $\mathcal{G}_0$ – unless $f$ is an equivalence in an (infinity,1)-category (hence a (Morita equivalence) from $\mathcal{X}_\bullet$ to $\mathcal{G}_\bullet$.) It is instead always principal over $f^\ast \mathcal{G}_0$.
Thus we arrive at an equivalent, however more basic definition of Lie groupoid bibundle:
Given Lie groupoids $G:=G_1\Rightarrow G_0$ and $H:=H_1\Rightarrow H_0$, a $G$-$H$-bibundle is a principal $H$-bundle $E \xrightarrow{\pi_G} G_0$ over $G_0$ with anchor $E\xrightarrow{\pi_H} H_0$ together with a left $G$-action (see here ) with anchor $\pi_G$, such that the two actions commute. If the $G$-action also gives rise to a principal bundle over $H_0$, then $E$ induces a Morita equivalence between $G$ and $H$ and it is sometimes called a Morita bibundle in this case.
Given a manifold $M$, and two open covers $\{U_i\}$ and $\{V_\alpha\}$, we may form two Cech groupoids (see here ) $\sqcup U_{ij} \Rightarrow \sqcup U_i$ and $\sqcup V_{\alpha \beta} \Rightarrow \sqcup V_\alpha$. Then $\sqcup_{i, \alpha} U_i \times_{M} V_\alpha$ (which is a common refinement of $\{U_i\}$ and $\{V_\alpha\}$) is a Morita bibundle. The actions are
Obviously these actions are free. Moreover, it is also not hard to see that $\sqcup U_i\times_M V_\alpha/\sqcup V_{\alpha \beta} = \sqcup U_i$ and $\sqcup U_i\times_M V_\alpha/\sqcup U_{ij} = \sqcup V_\alpha$. When a free action has representible quotient, it must automatically be proper.
Given a bibundle functor $E: G\to H$ and a bibundle functor $F: H\to K$ between Lie groupoids, the composition $E\circ F: G\to K$ is the quotient manifold $E\times_{H_0} F/H_1$ equipped with the remaining $G$ and $K$ action. Here $H$ acts on $E\times_{H_0} F$ from right by $(x, y)\cdot h_1=(x\cdot h_1^{-1}, y \cdot h_1)$. It is free and proper because the right action of $H$ on $E$ is so. Then $G$ action and $K$ action descend to the quotient $E\times_{H_0} F/H_1$. Moreover, those who free and proper is (are), remains so.
Thus bibundle functors compose to a bibundle functor, and Morita bibundles compose to a Morita bibundle.
Then we see that there is a $(2,1)$-category $BUN$ with objects Lie groupoids, 1-morphisms bibundle functors, and 2-morphisms isomorphisms of bibundles. It is $(2,1)$-category because 2-morphisms are obviously invertible. This $(2,1)$-category is equivalent to the one obtained by generalised morphism or by anafunctors.
Given a strict morphism $G\xrightarrow{f} H$, then we may form a bibundle $E:= G_0\times_{f_0, H_0, t} H_1$ with right $H$ action induced by $H$-multiplication and with left $G$ action induced by $G$-action on $G_0$. Bundlisation preserves composition.
If both atlases are 0-truncated objects (smooth spaces) $\mathcal{X}_0, \mathcal{G}_0 \in Sh(CartSp) \simeq \tau_1 \mathbf{H} \hookrightarrow \mathbf{H}$, then by the pasting law for homotopy pullbacks we have that $\mathcal{P}(f)$ is (n-1)-truncated if $\mathcal{G}$ is n-truncated.
In particular therefore the total space of a smooth 1-groupoid bibundle is 0-truncated hence is a smooth space.
In order to discuss Lie-groupoid bibundles we continue the discussion in example 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 $\mathcal{P}$ of the bibundle is presented by the sheaf of groupoids whose schematic depiction is
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 $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ both are groupoids, these morphisms are unique if they exist, and hence, as predicted by remark , $\mathcal{P}(f)$ 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 $\mathcal{X}_\bullet$ and $\mathcal{G}_\bullet$ are Lie groupoids).
This then recovers the definition of bibundles for Lie groupoids as often found in the literature.
The right $\mathcal{G}_\bullet$-action is by precomposition of these diagram with morphisms in $\mathcal{G}$, while the left $\mathcal{X}$-action is by postcomposition with morphisms in $\mathcal{X}$.
Conversely, given a $\mathcal{X}_\bullet$-$\mathcal{G}_\bullet$-Lie groupoid bibundle which is principal on the left
we recover the Morita morphism $f$ that it coresponds to by the Giraud-Rezk-Lurie axioms: first $p$ is the induced map between the homotopy colimits of the Cech nerves of the two left horizontal maps
and then $f$ is similarly the map between the homotopy colimits of the Cech nerves of the two right vertical maps.
(…)
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.
For $p \colon E \to X$ a smooth function between smooth manifolds, we write $T^p E \hookrightarrow T E$ for the bundle of vertical vector fields, the sub-bundle of the tangent bundle of $E$ on those vectors in the kernel of the differentiation maps $d p|_{e} \colon T_e E \to T_{\tau(e)} X$.
We write ${\vert \Lambda\vert^{1/2}}(T^\tau E)$ for the bundle of half-densities on vertical vector fields.
Let $\mathcal{G}_\bullet$ be a Lie groupoid and let ($E \stackrel{\tau}{\to} \mathcal{G}_0, \rho)$ be a $\mathbb{G}_\bullet$-groupoid-principal bundle $E \to E//\mathcal{G}$ (with anchor $\tau$ and action map $\rho$).
Then the bundle of vertical vector fields $T^\tau E$ equipped with the anchor map $T^\tau E \stackrel{d \tau}{\to} T \mathcal{G}_0 \to \mathcal{G}_0$ inherits a canonical $\mathcal{G}_\bullet$-action itself.
The quotient map
exists and is naturall a vector bundle again.
In the situation of remark , write
$C^\infty_{c/G}(E, {\vert\Lambda\vert}^{1/2}(T^\tau) E)^G$
for the space of smooth sections of the half-density-bundle of $T^\tau E$ which are $\mathcal{G}$-equivariant and which have compact support up to $\mathcal{G}$-action;
$C^\infty_c(E/\mathcal{G}, {\vert \Lambda\vert}^{1/2}(T^\tau E))$
for the space of smooth sections with compact support of the quotient bundle.
The following constructions work by repeatedly applying the following identification:
In the situation of def. , there is a natural isomorphism
The central definition here is now:
For $(E_1, \tau_1)$, $(E_2, \tau_2)$ two principal $\mathcal{G}_\bullet$ manifolds, set
And the central fact is:
Given 3 $\mathcal{G}_\bullet$-manifolds $(E_i, \tau_i)$, $i \in \{1,2,3\}$, there is a smooth function
given on sections $\sigma_1, \sigma_2$ and points $(e_1, e_3)$ by
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 $\mathcal{G}_\bullet$-principal manifolds and whose spaces of morphisms are as in def. .
In particular one has the following identifications.
For $\mathcal{G}_1 \to \mathcal{G}_0$ regarded as a $\mathcal{G}_\bullet$-principal action space, there is a natural isomorphism
and the algebra structure on this by prop. is isomorphic to the groupoid convolution algebra of smooth sections over $\mathcal{G}_\bullet$.
More generally:
For $E \stackrel{\tau}{\to} \mathcal{G}_0$ any $\mathcal{G}$-principal manifold, we have a natural isomorphism
We consider completion of all this to the C-star-algebra context (…)
Now we can put the pieces together and sends groupoid-bindunles to $C^\ast$-bimodules over the two groupoid convolution algebras.
Given two Lie groupoids $\mathcal{G}_\bullet$ and $\mathcal{K}_\bullet$ and given a Morita equivalence groupoid bibundle $E$ between them, we have
and this identification makes $N$ into a $C^\ast(\mathcal{G}_\bullet)-C^\ast(\mathcal{K}_\bullet)$-pre-Hilbert bimodule as follows:
The identification $N \simeq (E, \mathcal{K}_1)_{\mathcal{K}}$ defines the right $C^\ast(\mathcal{K}_\bullet)$-action by example ; and similarly the identification $N \simeq (\mathcal{G}_1, E)_{\mathcal{G}}$ defines a left $C^\ast(\mathcal{G}_\bullet)$-action.
The $C^\ast(\mathcal{K})$-valued inner product on $N$ is that induced by the composite
Groupoid bibundles were first considered for foliation groupoids in
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 (numdam:CTGDC_1989__30_3_229_0)
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 $C^\ast$-algebras, J. Operator Th. 17 (1987), 3–22.
Klaas Landsman, The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart, Lett. Math. Phys. 54 (2000), no. 1, 43–59. (arXiv:math-ph/0008005)
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
Michael Murray, Bispaces and bibundles (pdf slides)
See also
For groupoid bibundles between étale Lie groupoids the assignment of the groupoid convolution algebra-bimodule to them is shown to be functorial in
For more references along these lines see for the moment at groupoid convolution algebra – Extension to bibundles and bimodules