∞-Lie theory (higher geometry)
A Lie groupoid is a groupoid internal to smooth manifolds. This is a joint generalization of smooth manifolds and Lie groups to higher differential geometry.
Regarded in the more general context of smooth groupoids/smooth stacks, Lie groupoids present certain well-behaved such objects, often called differentiable stacks.
A Lie groupoid $X \coloneq (X_1 \rightrightarrows X_0)$ is a groupoid such that both the space of arrows $X_1$ and the space of objects $X_0$ are smooth manifolds, all structure maps are smooth, and source and target maps $s, t: X_1\rightrightarrows X_0$ are surjective submersions.
A Lie groupoid $X$ is an internal groupoid in the category Diff of smooth manifolds.
Since Diff does not have all pullbacks, to ensure that this definition makes sense, one needs to ensure that the space $X_1 \times_{s,t} X_1$ 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 $s,t : X_0 \to X_1$ are submersions. This ensures the pullback exists to define said manifold or composable pairs. Therefore a definition used in most modern differential geometry literature is as we see above.
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.
A Lie group $G$ is a Lie groupoid with $X_1=G$ and $X_0=pt$ a point. Composition of $X$ is provided by the multiplication of $G$.
A manifold $M$ is a Lie groupoid with $X_1=M$ and $X_0=M$. Source and target maps are identities and we only have identity arrows in this example.
Given a manifold $M$ and an open cover $\{U_i\}$, we can form a Lie groupoid with $X_1=\sqcup U_i\times_M U_j$ and $X_0=\sqcup U_i$. Then for an element $x_{ij}:=(x_i, x_j)\in U_i\times_M U_j \subset X_1$, $t(x_{ij})=x_i \in U_i$, $s(x_{ij})=x_j \in U_j$, and $x_{ij} \cdot x_{jk}= x_{ik}$. This is sometimes called the ?ech groupoid? or covering groupoid.
Given a Lie group $G$ (right) action on a manifold $M$, then we may form an associated action groupoid (or sometimes called transformation groupoid) as follows: $X_1 = M \times G$ and $X_0=M$. For an element $(x, g) \in X_1$, we have $t(x, g) = x$, $s(x, g)=x\cdot g^{-1}$, and $(x, g)\cdot (y, h) = (x, g\cdot h)$ (we must have $y=x\cdot g^{-1}$ for the multiplication to happen). Action groupoid presents the quotient stack $[M/G]$. Roughly speaking, it is a good replacement for quotient space even if the action is not as nice as you want.
Given a manifold $M$, we may also form so-called pair groupoid: $X_1= M\times M$ and $X_0=M$. Source and target are projections, and multiplication is given by $(x, y) \cdot (y , z)= (x, z)$. Pair groupoid may be interpreted as the global object of tangent bundle (think why? see the section below on Lie algebroid).
Given a manifold $M$, we have also an associated fundamental groupoid or homotopy groupoid $\Pi(M)$: $\Pi(M)_1=\{$paths in $M\}/$ homotopies, $\Pi_0(M)=M$. Source and target are end points of a path. Multiplication is concatenation of paths (think why associative?).
Given a manifold $M$ with a foliation $F$, we may form various groupoids associated with $F$.
$F$-pair groupoid: $X_1:=\{(x, y)| x, y \;\text{are in the same leaf in}\; F \}$, $X_0=M$. Source and target are obvious projections and multiplication is like in the case of pair groupoid. The problem for this groupoid is that it might not be a Lie groupoid. (why not? for counter example, we refer to Section 13.5 of Geometric Models for Noncommutative Algebras ).
monodromy groupoid $Mon_F(M)$ (it is a foliation version of fundamental groupoid, thus it is also sometimes called $F$-fundamental groupoid): $X_1:=\{$ leaf-wise paths$\}/$ leaf-wise homotopy, $X_0=M$ and the rest is like in the case of fundamental groupoid.
holonomy groupoid $Hol_F(M)$: $X_1:=\{$ leaf-wise paths$\}/$ holonomy, $X_0=M$ and the rest is like in the case of fundamental groupoid. Here, the holonomy of a path $\gamma$ is defined to the germ of diffeomorphisms induced by $\gamma$ between the transversals at the end points.
Among all possible Lie groupoids associated to a foliation, monodromy groupoid is the biggest and holonomy groupoid is the smallest.
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?)
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)
A topological groupoid $X_1 \rightrightarrows X_0$ is locally trivial if for every point $p\in X_0$ there is a neighbourhood $U$ of $p$ and a lift of the inclusion $\{p\} \times U \hookrightarrow X_0 \times X_0$ through $(s,t):X_1^{iso}\to X_0 \times X_0$.
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.
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 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 also diffeological 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.
As the infinitesimally approximation to a Lie group is a Lie algebra, so the infinitesimal approximation to a Lie groupoid is a Lie algebroid.
A Lie algebroid is a vector bundle $A\to M$ together with a vector bundle morphism $\rho: A\to TM$ (called anchor map), and a Lie bracket $[-,-]$ on the space of sections of $A$, satisfying the Leibniz rule
$[X, fY]=f[X,Y]+\rho(X)(f) Y.$
You would expect $\rho$ to preserve $[-,-]$, wouldn’t you? It is actually automatic! (see Y. Kosmann-Shwarzbah and F. Magri. Poisson-Nijenhuis strutures. Ann. Inst. H. Poinaré Phys. Théor., 53(1):3581, 1990.)
Recent progress: it turns out that one may link Lie algebroid with $L_\infty$-spaces (ask Owen Gwilliam for it)
A Lie algebra is a Lie algebroid with base space being a point.
$0$-bundle over a manifold $M$ is certainly a Lie algebroid in a trivial way.
Tangent bundle $TM\to M$ is a Lie algebroid with $\rho=id$ and $[-,-]$ the usual Lie bracket for vector fields. See tangent Lie algebroid.
Given a Poisson manifold $P$ with Poisson bivector field $\pi$, the cotangent bundle $T^*P$ is equipped with a Lie algebroid structure: $\rho(\xi)= \pi(\xi)$ and $[\xi_1, \xi_2]=d\pi(\xi_1, \xi_2)$ (or you may have $[df, dg]=d\{ f, g\}$ if you prefer to think in Poisson bracket). See Poisson Lie algebroid.
There are several versions of Lie groupoid morphisms, some of them are equivalent in a correct sense, some of them are not.
strict morphism: a strict morphism from Lie groupoid $X$ to $Y$ is a functor from $X$ to $Y$ as categories and preserving the smooth structures.
generalised morphism: a generalised morphism from $X$ to $Y$ is a span of morphisms $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$, where $\hat X \stackrel{\simeq}{\rightarrow} X$ is an weak equivalence of Lie groupoids, defined as below (see also at bibundle).
A Lie groupoid functor $f : G\to H$ is a weak equivalence if it is
essentially surjective; that is, $t \circ pr_2 : G_0 \times_{H_0,s} H_1 \to H_0$ is a surjective submersion;
fully faithful; that is, $G_1 \cong H_1\times_{t\times s, H_0\times H_0} G_0 \times G_0$.
Composition of generalised morphism is given by weak pullback of Lie groupoids (see also weak limit). Given (strict) morphisms $\hat X\to Y$ and $\hat X' \to Y$, the weak pullback of $\hat X\to Y$ along $\hat X' \to Y$ is a groupoid $\hat X \times_{Y}^w \hat X'$ with space of objects $\hat X_0 \times_{ Y_0} Y_1 \times_{Y_0} \hat X'_0$ and space of morphisms $\hat X_1 \times_{Y_0} Y_1 \times_{Y_0} \hat X'_1$. When $\hat X' \to Y$ is a weak equivalence, the weak pullback is a Lie groupoid thank to the property of essentially surjective. (Is this composition associative?)
Composition of anafunctors is given through strong pullback of Lie groupoids, that is level-wise pullback.
The last three morphisms are more or less equivalent, that is they give arise to equivalent 2-categories (in fact (2,1)-categories) of Lie groupoids. To make it explicit, we need to talk about 2-morphisms between them.
A 2-morphism between bibundle functors is simply a bibundle isomorphism (of course preserving all the structures of bibundles).
A strict 2-morphism from generalised morphism $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$ to $X \stackrel{\simeq}{\leftarrow} \hat X' \to Y$ is given by a morphism $\hat X \to \hat X'$ such that the following diagram commutes
This forces the morphism $\hat X \to \hat X'$ to be a weak equivalence by 2-out-of-3 for weak equivalences. A 2-morphism from $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$ to $X \stackrel{\simeq}{\leftarrow} \hat X' \to Y$ is provided by a span of strict 2-morphisms:
A 2-morphism between anafunctors are defined like above, however the left legs are required to be acyclic fibrations between Lie groupoids. (think this time what may you say about the morphism $\hat X \to \hat X'$?)
Then these three (2,1)-categories, which we denote by $GEN$, $ANA$ and $BUN$, are all equivalent to each other. For a nice survey on this statement, we refer to Section 1.5 of Du Li’s thesis.
The idea is that Bundlisation may extend to an equivalence of $(2,1)$-categories between $GEN$, the $(2,1)$-category made by generalised morphisms, and $BUN$. The inverse is given by the following construction: given a bibundle functor $E: G\to H$, we pull back $G$ along the map $E\to G_0$ and obtain a Lie groupoid $G|_E:=G_1\times_{G_0\times G_0} E \times E \Rightarrow E$. Then the natural projection $G|_E \to G$ is an acyclic fibration. Thus we obtain a generalised morphism which is also an anafunctor from $G \to H$.
Even though $GEN$ contains more morphisms than $ANA$, a generalised morphism maybe equivalently replaced by an anafunctor. In fact a generalised morphism $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$ gives arise to an anafunctor $X \stackrel{\simeq}{\leftarrow} X \times_{X}^w \hat X \to Y$.
As a consequence of the universal property of the calculus of fractions, $GEN$ and $ANA$ are equivalent.
Morphisms of Lie algebroids are counter-intuitive: they are not morphisms of vector bundles which preserve the algebroid structure. To define a Lie algebroid morphism, we first need to introduce the Chevalley-Eilenberg algebra associated to a Lie algebroid $A$.
We consider $A$ to be a trivially graded vector bundle, i.e. concentrated in degree $0$. Then $A[1]$ is concentrated in degree $-1$. The functions on $A[1]$ are given as
where $C^\infty(M)$ is considered to be of degree $0$, $\Gamma(A^*)$ to be of degree $1$, and so forth.
Now we can define a degree-one derivation on $C(A[1])$ as follows: For $\xi \in \Gamma(\wedge^n A^*)$ and $X_i\in \Gamma(A)$, let
The condition $[d_A,\,d_A] = 0$ is not automatically fulfilled: since $\deg d_A = 1$, we have $[d_A,\,d_A] = d_A \circ d_A + d_A \circ d_A = 2 d_A \circ d_A$. The condition $d_A \circ d_A = 0$ is actually equivalent to $\bigl(A, \rho_A, [ - , - ]_A\bigr)$ being a Lie algebroid; that is, it is fulfilled if and only if
(Proof: calculation gives the restriction $d_A(f) = \rho^\ast(d f)$ and $d_A(\xi)(X_1, X_2) = - \bigl\langle \xi, [X_1, X_2]\bigr\rangle + \rho_A(X_1)(\xi X_2) - \rho_A(X_2)(\xi X_1)$ and $d_A(\xi)(X_1, X_2, X_3) = - \xi([X_1, X_2], X_3) + \xi([X_1, X_3], X_2) - \xi([X_2, X_3], X_2) + \rho_A(X_1)\xi(X_2, X_3) - \rho_A(X_2)\xi(X_1, X_3) + \rho_A(X_3)\xi(X_1, X_2)$.)
This point of view also applies to higher Courant Lie algebroids and L-infinity-algebras.
Example: For the tangent Lie algebroid $A = T M$, $\bigl(C(T M[1]), d_A\bigr) = \bigl(\Omega^\ast(M), d_{dR}\bigr)$.
Then a morphism from a Lie algebroid $(A, \rho_A, [-,-]_A)$ to $(B, \rho_B, [-,-]_B)$ is a morphism of the associated differential graded commutative algebras
Such a morphism of c.d.g.a.‘s is determined by maps $C^\infty(N) \to C^\infty (M)$ on degree $0$ and a map $\Gamma(B^*)\to \Gamma(A^*)$ on degree $1$. Thus a morphism of vector bundles $A\xrightarrow{f} B$ give rise to a morphism $f^*$ of c.g.a. For $f$ to be a Lie algebroid morphism, we further need $f$ to satisfy additional conditions so that $f^*$ preserves the differential.
This way to explain morphisms of Lie algebroids is described in Kirill Mackenzie, chapter 4.3. If the Lie algebroids are over the same manifold $M$, then a morphism from $A$ to $B$ can be described as a morphism of vector bundles that respects the anchor maps and the Lie bracket. If, however, $B$ is over a different manifold $N$, this direct approach does not work. In this situation we have to pull back the Lie algebroid to $M$ (Note that this is not simply the vector bundle pullback of $B$ along $f$, but a more involved construction, see Kirill Mackenzie). Using the defintion of a morphism on a common base manifold one arrives at two conditions on the bundle morphism to be a morphism of Lie algebroids. For details see the linked book.
Let $I$ be an interval with the tangent bundle Lie algebroid $(TI, \id_{TI}, [-,-])$ and $(A, \rho_A, [-,-]_A)$ an arbitrary Lie algebroid on $M$. Then a path $a\colon I \to A$ defines a map from $\varphi\colon C(A[1]) \to C(TI[1]) \cong \Omega(I)$ which respects the commutative graded algebra structure. A function $f\in C^{\infty}(M)$ gets mapped to $f\circ \gamma$, where $\gamma\colon I \to M$ is the projection of $a$. A section $s\in\Gamma(A^*)$ gets mapped to $(s\circ \gamma (a) dt$ under $\varphi$, where $dt$ is the canonical section of $\Omega^1(I)$. Since $C(TI[1])$ is concentrated in degree $0$ and $1$, the other degrees get mapped to $0$.
For this map to be a morphism of Lie algebroids, it has to respect the differentials. As explained above this only needs to be checked for a smooth function in $C^{\infty}(M)$:
A quick calculation shows that this is true if and only if
Let $M$ the unit square in $\mathbb{R}^2$. Then a Lie algebroid morphism from $(TM, \id_{TM}, [-,-])$ to $(\mathfrak{g}, T\cdot, [-.-]_{\mathfrak{g}})$, where $\mathfrak{g}$ is a Lie algebra, is given by a morphism of c.d.g.a.
Two smooth maps $a,b\colon M\to \mathfrak{g}$ give a map between c.g.a. spaces above. Here a section $s\in\Gamma(\mathfrak{g}^*)$, i.e. an element of $\mathfrak{g}^*$ gets mapped to the one form
where $(t,s)$ are the coordinates on $M$. A quick calculation shows that this map respects the differential if and only if
This formula also shows that $a\cdot dt + b\cdot ds$ is a flat connection on the trivial principal $G$-bundle on $M$. Therefore Lie algebroid morphisms open a way to talk about higher connections and flat conditions.
It is unclear how to use the idea of bibundle or span to define a more general version of Lie algebroid morphisms so that they really correspond to the case of Lie groupoid morphisms.
See
Every smooth manifold $X$ is a 0-truncated Lie groupoid.
For every Lie group $G$ the one-object delooping groupoid $\mathbf{B}G$ is a Lie groupoid.
The Lie group $G$ itself is a 0-truncated group object in the 2-category or Lie groupoids.
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 $\mathbf{E}G \to \mathbf{B}G$.
For every $G$-principal bundle $P \to X$ there is its Atiyah Lie groupoid $At(P)$.
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 diffeological groupoid.
The Cech groupoid $C(U)$ of a cover $\{U_i \to X\}$ of a smooth manifold is a Lie groupoid.
Every foliation gives rise to its holonomy groupoid.
An orbifold 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
is a Lie groupoid equivalent to this principal 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).
Topological and differentiable (or smooth, “Lie”) groupoids (and more generally categories) were introduced in
Reviews and developments of the theory of Lie groupoids include
Pradines, ….
Kirill Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press, 2005, xxxviii + 501 pages (website)
Kirill 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)
Discussion in the context of foliation theory (foliation groupoids) is in
The relation to differentiable stacks is discussed/reviewed in section 2 of
Lie groupoids as a source for groupoid convolution C*-algebras are discussed in
Expository discussion of various kinds of groupoids is also in
Groupoids and their various morphisms between them in different categories, including in Diff, is also in