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Idea

A Lie groupoid is a groupoid with smooth structure The notion of Lie groupoid is the groupoid analog of Lie group.

Definition

As an object, a Lie groupoid $X$ is understood to be an internal groupoid in the category Diff of smooth manifold.

To ensure that this definition makes sense, one needs to ensure that the space $Mor(X) \times_{s,t} Mor(X)$ of composable morphisms is an object of Diff. This is achieved either by adopting the definition of internal groupoid in the sense of Ehresmann?, which includes as data the smooth manifold of composable pairs, or by taking the conventional route and demanding that the source and target maps $s,t : Mor(X) \to Obj(X)$ are submersions. This ensures the pullback exists to define said manifold or composable pairs.

But for most practical purposes, the apparently evident 2-category $Grpd(Diff)$ of such internal groupoids, internal functors and internal natural transformations is not the correct 2-category to consider. One way to see this is that the axiom of choice fails in Diff, which means that an internal functor $X \to Y$ of internal groupoids which is essentially surjective and full and faithful may nevertheless not be an equivalence, in that it may not have a weak inverse in $Grpd(Diff)$.

See the section 2-Category of Lie groupoids below.

A bit more general than a Lie groupoid is a diffeological groupoid.

Terminology

Originally Lie groupoids were called (by Ehresmann) differentiable groupoids (and also one considered differentiable categories). Sometime in the 1980s there was a change of terminology to Lie groupoid and differentiable stacks. (reference?)

Specialisations

One definition which Ehresmann introduced in his paper Catégories topologiques et catégories différentiables (see below) is that of locally trivial groupoid. It is defined more generally for topological categories, and extends in an obvious way to topological groupoids, and Lie categories and groupoids. For a topological (resp. Lie) category $X$, let $X_1^{iso}$ denote the subspace (resp. submanifold) of invertible arrows . (This always exists, by general abstract nonsense - I should look up the reference, it’s in Bunge-Pare I think - DR)

Clearly for a Lie groupoid $X_1^{iso} = X_1$. It is simple to show from the definition that for a transitive Lie groupoid, $(s,t)$ has local sections. Ehresmann goes on to show a link between smooth principal bundles and transitive, locally trivial Lie groupoids. See locally trivial category for details.

The (2,1)-category of Lie groupoids

As usual for internal categories, the naive 2-category of internal groupoids, internal functors and internal natural transformations is not quite “correct”. One sign of this is that the axiom of choice fails in Diff so that an internal functor which is an essentially surjective functor and a full and faithful functor may still not have an internal weak inverse.

One way to deal with this is to equip the 2-category with some structure of a homotopical category and allow morphisms of Lie groupoids to be 2-anafunctors, i.e. spans of internal functors $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$.

Such generalized morphisms – called Morita morphisms or generalized morphisms in the literature – are sometimes modeled as bibundles and then called Hilsum-Skandalis morphisms.

Another equivalent approach is to embed Lie groupoids into the context of 2-topos theory:

The (2,1)-topos $Sh_{(2,1)}(Diff)$ of stacks/2-sheaves on Diff may be understood as a nice 2-category of general groupoids modeled on smooth manifolds. The degreewise Yoneda embedding allows to emebed groupoids internal to $Diff$ into stacks on $Diff$ . this wider context contains for instance alsodiffeological groupoids.

Regarded inside this wider context, Lie groupoids are identified with differentiable stacks. The (2,1)-category of Lie groupoids is then the full sub-$(2,1)$-category of $Sh_{(2,1)}(Diff)$ on differentiable stacks.

For more comments on this, see also the beginning of ∞-Lie groupoid.

Lie algebroids

As the infinitesimally approximation to a Lie group is a Lie algebra, so the infinitesimal approximation to a Lie groupoid is a Lie algebroid.

Higher Lie groupoids

See

• internal ∞-groupoid

• ∞-Lie groupoid

Examples

• For every Lie group $G$ the one-object delooping groupoid $\mathbf{B}G$ is a Lie groupoid.

• Every Lie 2-group is in particular a Lie groupoid: a group object in the category of Lie groupoids

• The inner automorphism 2-group $\mathbf{E}G = INN(G) = G//G$ is a Lie groupoid. There is an obvious morphism $\mathbf{E}G \to \mathbf{B}G$.

  The Lie group itself is a 0-truncated group object in the 2-category or Lie groupoids.

• For every $G$-principal bundle $P \to X$ there is its Atiyah Lie groupoid $At(P)$.

• Every smooth manifold $X$ is a 0-truncated Lie group.

• The fundamental groupoid $\Pi_1(X)$ of a smooth manifold is naturally a Lie groupoid.

• The path groupoid of a smooth manifold is naturally a groupoid internal to diffeological space groupoid s . (a “diffeological groupoid”).

• The Cech groupoid $C(U)$ of a cover $\{U_i \to X\}$ of a smooth manifold is a Lie groupoid.

• An anafunctor $X \stackrel{\simeq}{\leftarrow} C(U) \to \mathbf{B}G$ from a smooth manifold $X$ to $\mathbf{B}G$ is a Cech cocycle in degree 1 with values in $G$, classifying $G$-principal bundle $P$.

• The (1-categorical) pullback

  $$\array{ P &\stackrel{\simeq}{\leftarrow}& \mathbf{P} &\to& \mathbf{E}G \\ && \downarrow && \downarrow \\ && C(U) &\stackrel{}{\to}& \mathbf{B}G \\ && \downarrow^{\mathrlap{\simeq}} \\ && X }$$

  is a Lie groupoid equivalent to this pricipal bundle $P$.

  (For more on the general phenomenon of which this is a special case see principal ∞-bundle and universal principal ∞-bundle.)

• Similarly an anafunctor from $P_1(X)$ to $\mathbf{B}G$ is a connection on a bundle (see there for details).

References

Topological and differentiable (or smooth, “Lie”) groupoids (and more generally categories) were introduced in

• Charles Ehresmann, Catégories topologiques et catégories différentiables Colloque de Géometrie Differentielle Globale (Bruxelles, 1958), 137–150, Centre Belge Rech. Math., Louvain, 1959;

Reviews and developments of the theory of Lie groupoids include

• Pradines, ….

• K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005, xxxviii + 501 pages (website)

• K. C. H. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (MathSciNet)

• John Baez talks about various kinds of Lie groupoids in TWF 256.