Classes of bundles
Examples and Applications
Special and general types
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.
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 and smooth functions between them, equipped with the standard coverage of good open covers.
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
We often write Smooth∞Grpd for short.
By the (∞,1)-Yoneda lemma there is a sequence of faithful inclusions
SmoothMfd SmoothGrpd Smooth∞Grpd.
This induces a corresponding inclusion of simplicial objects and hence of groupoid objects
For a groupoid object we write
for its (∞,1)-colimiting cocone, hence (without subscript decoration) denotes the realization of as a single object in .
For a groupoid object, we write for its realization and call the canonical 1-epimorphism
the canonical atlas of this realization.
For a Lie groupoid, we have that
is its smooth manifold of objects
SmoothGrpd Smooth∞Grpd is the realization of the Lie groupoid as a differentiable stack, hence as a smooth groupoid
is the canonically induced atlas in the traditional sense of geometric stack-theory.
This is evidently more constrained than just morphisms
by themselves. The latter are the generalized morphisms or Morita morphisms between the groupoid objects , .
Given groupoid objects , a Morita morphism between them is a morphism in between their realizations. A Morita morphism that is an equivalence in is called a Morita equivalence of groupoid objects in .
Here we want to express these Morita morphisms in terms of bibundle objects on which both and act.
For any object, its pair groupoid is
The realization of this is equivalent to the point
Hence all Morita morphisms, def. 2, to the pair groupoid are equivalent. As a groupoid object is non-trivial, but it is Morita equivalent to the terminal groupoid object.
Smooth groupoid-principal bundles
For a groupoid object, any object equipped with a morphsim to the object of objects of , a -groupoid ∞-action on with anchor is a groupoid over of the form
where the homotopy fiber products on the left are those of the anchor with the leftmost 0-face map and the horizontal morphisms are the corresponding projections on the second factor.
We call also the action groupoid of the action of on and call its realization the homotopy quotient of the action.
Under this relation, the discussion of ∞-groupoid-principal ∞-bundles proceeds in direct analogy with that of -principal ∞-bundles:
For Smooth∞Grpd any object, a morphism in induces (“modulates”) a -groupoid action, def. 3, on the homotopy pullback
of the atlas of : the corresponding action groupoid is the Cech nerve of the projection (which as the (∞,1)-pullback of a 1-epimorphism is itself a 1-epimorphism):
Let be a morphism of 1-groupoid objects, say of Lie groupoids. Then as discussed at homotopy pullback the (∞,1)-pullback of the atlas along the realization is computed as the 1-categorical pullback
in . Schematically the groupoid on the right has morphisms which are commuting diagrams in 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 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
The left one just forgets the top vertical morphism, the right one composes it with the diagonal morphisms. This composion is the -action on .
Smooth groupoid-principal bibundles
Finally then for and two Lie groupoids and a morphism in Smooth∞Grpd between the corresponding differentiable stacks, we obtain first the -groupoid principal bundle and then by further homotopy pullback also the left -groupoid principal bundle :
For two groupoid objects and a Morita morphism between them, def. 2, we say that the corresponding -bibundle is the -groupoid-principal bundle pulled back to canonical atlas of and equipped with the induced -groupoid action:
Thus we arrive at an equivalent, however more basic definition of Lie groupoid bibundle:
Given Lie groupoids and , a --bibundle is a principal -bundle over with anchor together with a left -action (see here ) with anchor , such that the two actions commute. If the -action also gives arise to a principal bundle over , then induces a Morita equivalence between and and it is sometimes called a Morita bibundle in this case.
Given a manifold , and two open covers and , we may form two Cech groupoids (see here ) and . Then (which is a common refinement of and ) is a Morita bibundle. The actions are
Obviously these actions are free. Moreover, it is also not hard to see that and . When a free action has representible quotient, it must automatically be proper.
Given a bibundle functor and a bibundle functor between Lie groupoids, the composition is the quotient manifold equipped by remained and action. Here acts on from right by . It is free and proper because the right action of on is so. Then action and action descend to the quotient . 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 -category with objects Lie groupoids, 1-morphisms bibundle functors, and 2-morphisms isomorphisms of bibundles. It is -category because 2-morphisms are obviously invertible. This -category is equivalent to the one obtained by generalised morphism or by anafunctors.
Given a strict morphism , then we may form a bibundle with right action induced by -multiplication and with left action induced by -action on . Bundlisation preserves composition.
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 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 and both are groupoids, these morphisms are unique if they exist, and hence, as predicted by remark 3, 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 and are Lie groupoids).
This then recovers the definition of bibundles for Lie groupoids as often found in the literature.
The right -action is by precomposition of these diagram with morphisms in , while the left -action is by postcomposition with morphisms in .
Conversely, given a --Lie groupoid bibundle which is principal on the left
we recover the Morita morphism that it coresponds to by the Giraud-Rezk-Lurie axioms: first is the induced map between the homotopy colimits of the Cech nerves of the two left horizontal maps
and then 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.
For a smooth function between smooth manifolds, we write for the bundle of vertical vector fields, the sub-bundle of the tangent bundle of on those vectors in the kernel of the differentiation maps .
We write for the bundle of half-densities on vertical vector fields.
In the situation of remark 4, write
for the space of smooth sections of the half-density-bundle of which are -equivariant and which have compact support up to -action;
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. 7, there is a natural isomorphism
The central definition here is now:
For , two principal manifolds, set
And the central fact is:
Given 3 -manifolds , , there is a smooth function
given on sections and points 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 -principal manifolds and whose spaces of morphisms are as in def. 8.
In particular one has the following identifications.
For regarded as a -principal action space, there is a natural isomorphism
and the algebra structure on this by prop. 3 is isomorphic to the groupoid convolution algebra of smooth sections over .
For any -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 -bimodules over the two groupoid convolution algebras.
Given two Lie groupoids and and given a Morita equivalence groupoid bibundle between them, we have
and this identification makes into a -pre-Hilbert bimodule as follows:
The identification defines the right -action by example 7; and similarly the identification defines a left -action.
The -valued inner product on 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
- 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
Convolution to -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