Lie groupoid



\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



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∶−(X 1X 0)X \coloneq (X_1 \rightrightarrows X_0) is a groupoid such that both the space of arrows X 1X_1 and the space of objects X 0X_0 are smooth manifolds, all structure maps are smooth, and source and target maps s,t:X 1X 0s, t: X_1\rightrightarrows X_0 are surjective submersions.

A Lie groupoid XX 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× s,tX 1X_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 0X 1s,t : X_0 \to X_1 are submersions. This ensures the pullback exists to define said manifold of 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)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 XYX \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)Grpd(Diff).

See the section 2-Category of Lie groupoids below.

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

Examples for Lie groupoids

  • A Lie group GG is a Lie groupoid with X 1=GX_1=G and X 0=ptX_0=pt a point. Composition of XX is provided by the multiplication of GG.

  • A manifold MM is a Lie groupoid with X 1=MX_1=M and X 0=MX_0=M. Source and target maps are identities and we only have identity arrows in this example.

  • Given a manifold MM and an open cover {U i}\{U_i\}, we can form a Lie groupoid with X 1=U i× MU jX_1=\sqcup U_i\times_M U_j and X 0=U iX_0=\sqcup U_i. Then for an element x ij:=(x i,x j)U i× MU jX 1x_{ij}:=(x_i, x_j)\in U_i\times_M U_j \subset X_1, t(x ij)=x iU it(x_{ij})=x_i \in U_i, s(x ij)=x jU js(x_{ij})=x_j \in U_j, and x ijx jk=x ikx_{ij} \cdot x_{jk}= x_{ik}. This is sometimes called the Čech groupoid or covering groupoid.

  • Given a Lie group GG (right) action on a manifold MM, then we may form an associated action groupoid (or sometimes called transformation groupoid) as follows: X 1=M×GX_1 = M \times G and X 0=MX_0=M. For an element (x,g)X 1(x, g) \in X_1, we have t(x,g)=xt(x, g) = x, s(x,g)=xg 1s(x, g)=x\cdot g^{-1}, and (x,g)(y,h)=(x,gh)(x, g)\cdot (y, h) = (x, g\cdot h) (we must have y=xg 1y=x\cdot g^{-1} for the multiplication to happen). Action groupoid presents the quotient stack [M/G][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 MM, we may also form so-called pair groupoid: X 1=M×MX_1= M\times M and X 0=MX_0=M. Source and target are projections, and multiplication is given by (x,y)(y,z)=(x,z)(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 MM, we have also an associated fundamental groupoid or homotopy groupoid Π(M)\Pi(M): Π(M) 1={\Pi(M)_1=\{paths in M}/M\}/ homotopies, Π 0(M)=M\Pi_0(M)=M. Source and target are end points of a path. Multiplication is concatenation of paths (think why associative?).

  • Given a manifold MM with a foliation FF, we may form various groupoids associated with FF.

  1. FF-pair groupoid: X 1:={(x,y)|x,yare in the same leaf inF}X_1:=\{(x, y)| x, y \;\text{are in the same leaf in}\; F \}, X 0=MX_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 ).

  2. monodromy groupoid Mon F(M)Mon_F(M) (it is a foliation version of fundamental groupoid, thus it is also sometimes called FF-fundamental groupoid): X 1:={X_1:=\{ leaf-wise paths}/\}/ leaf-wise homotopy, X 0=MX_0=M and the rest is like in the case of fundamental groupoid.

  3. holonomy groupoid Hol F(M)Hol_F(M): X 1:={X_1:=\{ leaf-wise paths}/\}/ holonomy, X 0=MX_0=M and the rest is like in the case of fundamental groupoid. Here, the holonomy of a path γ\gamma is defined to the germ of diffeomorphisms induced by γ\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 XX, let X 1 isoX_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 1X 0X_1 \rightrightarrows X_0 is locally trivial if for every point pX 0p\in X_0 there is a neighbourhood UU of pp and a lift of the inclusion {p}×UX 0×X 0\{p\} \times U \hookrightarrow X_0 \times X_0 through (s,t):X 1 isoX 0×X 0(s,t):X_1^{iso}\to X_0 \times X_0.

Clearly for a Lie groupoid X 1 iso=X 1X_1^{iso} = X_1. It is simple to show from the definition that for a transitive Lie groupoid, (s,t)(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 anafunctors, i.e. spans of internal functors XX^YX \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)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 DiffDiff into stacks on DiffDiff. 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)(2,1)-category of Sh (2,1)(Diff)Sh_{(2,1)}(Diff) on differentiable stacks.

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

Lie algebroid

As the infinitesimal 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 AMA\to M together with a vector bundle morphism ρ:ATM\rho: A\to TM (called anchor map), and a Lie bracket [,][-,-] on the space of sections of AA, satisfying the Leibniz rule

[X,fY]=f[X,Y]+ρ(X)(f)Y.[X, fY]=f[X,Y]+\rho(X)(f) Y.


You would expect ρ\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 L_\infty-spaces (ask Owen Gwilliam for it)

Examples of Lie algebroids

  • A Lie algebra is a Lie algebroid with base space being a point.

  • 00-bundle over a manifold MM is certainly a Lie algebroid in a trivial way.

  • action Lie algebroid

  • Tangent bundle TMMTM\to M is a Lie algebroid with ρ=id\rho=id and [,][-,-] the usual Lie bracket for vector fields. See tangent Lie algebroid.

  • Given a Poisson manifold PP with Poisson bivector field π\pi, the cotangent bundle T *PT^*P is equipped with a Lie algebroid structure: ρ(ξ)=π(ξ)\rho(\xi)= \pi(\xi) and [ξ 1,ξ 2]=dπ(ξ 1,ξ 2)[\xi_1, \xi_2]=d\pi(\xi_1, \xi_2) (or you may have [df,dg]=d{f,g}[df, dg]=d\{ f, g\} if you prefer to think in Poisson bracket). See Poisson Lie algebroid.

  • Atiyah Lie algebroid

Morphisms of Lie groupoids

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 XX to YY is a functor from XX to YY as categories and preserving the smooth structures.

  • generalised morphism: a generalised morphism from XX to YY is a span of morphisms XX^YX \stackrel{\simeq}{\leftarrow} \hat X \to Y, where X^X\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:GHf : G\to H is a weak equivalence if it is

  1. essentially surjective; that is, tpr 2:G 0× H 0,sH 1H 0t \circ pr_2 : G_0 \times_{H_0,s} H_1 \to H_0 is a surjective submersion;

  2. fully faithful; that is, G 1H 1× t×s,H 0×H 0G 0×G 0G_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 X^Y\hat X\to Y and X^Y\hat X' \to Y, the weak pullback of X^Y\hat X\to Y along X^Y\hat X' \to Y is a groupoid X^× Y wX^\hat X \times_{Y}^w \hat X' with space of objects X^ 0× Y 0Y 1× Y 0X^ 0\hat X_0 \times_{ Y_0} Y_1 \times_{Y_0} \hat X'_0 and space of morphisms X^ 1× Y 0Y 1× Y 0X^ 1\hat X_1 \times_{Y_0} Y_1 \times_{Y_0} \hat X'_1. When X^Y\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?)

  • anafunctor: an anafunctor from XX to YY is a span of morphisms XX^YX \stackrel{\simeq}{\leftarrow} \hat X \to Y, where X^X\hat X \stackrel{\simeq}{\rightarrow} X is an acyclic fibration of Lie groupoids. That is, this map is a weak equivalence of Lie groupoids and X^ 0X 0\hat X_0 \to X_0 is a surjective submersion.

Composition of anafunctors is given through strong pullback of Lie groupoids, that is level-wise pullback.

  • bibundle functor (or H.S. bibundle, or Hilsum-Skandalis bibundle): a bibundle functor from GHG\to H is a groupoid principal bundle EE of HH (with right action) such that GG acts on EE from left and GG action commutes with HH action. If both GG and HH actions are principal, then EE gives arise to Morita equivalence between them.

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 XX^YX \stackrel{\simeq}{\leftarrow} \hat X \to Y to XX^YX \stackrel{\simeq}{\leftarrow} \hat X' \to Y is given by a morphism X^X^\hat X \to \hat X' such that the following diagram commutes

X^ X Y X^ \begin{matrix} & & \hat X \\ & \swarrow & & \searrow \\ X & & \downarrow & & Y \\ & \searrow & & \swarrow \\ & & \hat X' \end{matrix}

This forces the morphism X^X^\hat X \to \hat X' to be a weak equivalence by 2-out-of-3 for weak equivalences. A 2-morphism from XX^YX \stackrel{\simeq}{\leftarrow} \hat X \to Y to XX^YX \stackrel{\simeq}{\leftarrow} \hat X' \to Y is provided by a span of strict 2-morphisms:

X^ X X^ Y X^ \begin{matrix} & & \hat X \\ & \swarrow & \uparrow & \searrow \\ X & & \hat X'' & & Y \\ & \searrow & \downarrow & \swarrow \\ & & \hat X' \end{matrix}

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 X^X^\hat X \to \hat X'?)

Then these three (2,1)-categories, which we denote by GENGEN, ANAANA and BUNBUN, 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)(2,1)-categories between GENGEN, the (2,1)(2,1)-category made by generalised morphisms, and BUNBUN. The inverse is given by the following construction: given a bibundle functor E:GHE: G\to H, we pull back GG along the map EG 0E\to G_0 and obtain a Lie groupoid G| E:=G 1× G 0×G 0E×EEG|_E:=G_1\times_{G_0\times G_0} E \times E \Rightarrow E. Then the natural projection G| EGG|_E \to G is an acyclic fibration. Thus we obtain a generalised morphism which is also an anafunctor from GHG \to H.

Even though GENGEN contains more morphisms than ANAANA, a generalised morphism maybe equivalently replaced by an anafunctor. In fact a generalised morphism XX^YX \stackrel{\simeq}{\leftarrow} \hat X \to Y gives arise to an anafunctor XX× X wX^YX \stackrel{\simeq}{\leftarrow} X \times_{X}^w \hat X \to Y.

As a consequence of the universal property of the calculus of fractions, GENGEN and ANAANA are equivalent.

Morphisms of Lie algebroids

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 AA.

We consider AA to be a trivially graded vector bundle, i.e. concentrated in degree 00. Then A[1]A[1] is concentrated in degree 1-1. The functions on A[1]A[1] are given as

C(A[1])=C (M)Γ(A *)Γ( 2A *),C(A[1])=C^\infty(M)\oplus \Gamma(A^*)\oplus \Gamma(\wedge^2 A^*)\oplus \ldots ,

where C (M)C^\infty(M) is considered to be of degree 00, Γ(A *)\Gamma(A^*) to be of degree 11, and so forth.

Now we can define a degree-one derivation on C(A[1])C(A[1]) as follows: For ξΓ( nA *)\xi \in \Gamma(\wedge^n A^*) and X iΓ(A)X_i\in \Gamma(A), let

d A(ξ)(X 1,,X n):= 0i<jn(1) i+jξ([X i,X j] A,,X i^,,X i^,)+ i=0 n(1) iρ A(X i)ξ(,X i^,). d_A(\xi)(X_1,\,\ldots\,,\,X_n) := \sum_{0\leq i \lt j\leq n} (-1)^{i+j} \xi\bigl([X_i,\,X_j]_A,\, \ldots\,,\, \widehat{X_i},\,\ldots\,,\,\widehat{X_i},\,\ldots\bigr) + \sum_{i=0}^n (-1)^i \rho_A(X_i) \xi\bigl(\ldots\,,\,\widehat{X_i},\,\ldots\bigr).

The condition [d A,d A]=0[d_A,\,d_A] = 0 is not automatically fulfilled: since degd A=1\deg d_A = 1, we have [d A,d A]=d Ad A+d Ad A=2d Ad A[d_A,\,d_A] = d_A \circ d_A + d_A \circ d_A = 2 d_A \circ d_A. The condition d Ad A=0d_A \circ d_A = 0 is actually equivalent to (A,ρ A,[,] A)\bigl(A, \rho_A, [ - , - ]_A\bigr) being a Lie algebroid; that is, it is fulfilled if and only if

  • [,] A[- , - ]_A satisfies the Jacobi identity
  • and [,] A[ - , - ]_A and ρ A\rho_A together satisfy the Leibniz identity.

(Proof: calculation gives the restriction d A(f)=ρ *(df)d_A(f) = \rho^\ast(d f) and d A(ξ)(X 1,X 2)=ξ,[X 1,X 2]+ρ A(X 1)(ξX 2)ρ A(X 2)(ξX 1)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(ξ)(X 1,X 2,X 3)=ξ([X 1,X 2],X 3)+ξ([X 1,X 3],X 2)ξ([X 2,X 3],X 2)+ρ A(X 1)ξ(X 2,X 3)ρ A(X 2)ξ(X 1,X 3)+ρ A(X 3)ξ(X 1,X 2)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=TMA = T M, (C(TM[1]),d A)=(Ω *(M),d dR)\bigl(C(T M[1]), d_A\bigr) = \bigl(\Omega^\ast(M), d_{dR}\bigr).

Then a morphism from a Lie algebroid (A,ρ A,[,] A)(A, \rho_A, [-,-]_A) to (B,ρ B,[,] B)(B, \rho_B, [-,-]_B) is a morphism of the associated differential graded commutative algebras

(C(A[1]),d A)(C(B[1]),d B). (C(A[1]), d_A) \leftarrow (C(B[1]), d_B).

Such a morphism of c.d.g.a.‘s is determined by maps C (N)C (M)C^\infty(N) \to C^\infty (M) on degree 00 and a map Γ(B *)Γ(A *)\Gamma(B^*)\to \Gamma(A^*) on degree 11. Thus a morphism of vector bundles AfBA\xrightarrow{f} B give rise to a morphism f *f^* of c.g.a. For ff to be a Lie algebroid morphism, we further need ff to satisfy additional conditions so that f *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 MM, then a morphism from AA to BB can be described as a morphism of vector bundles that respects the anchor maps and the Lie bracket. If, however, BB is over a different manifold NN, this direct approach does not work. In this situation we have to pull back the Lie algebroid to MM (Note that this is not simply the vector bundle pullback of BB along ff, 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 II be an interval with the tangent bundle Lie algebroid (TI,id TI,[,])(TI, \id_{TI}, [-,-]) and (A,ρ A,[,] A)(A, \rho_A, [-,-]_A) an arbitrary Lie algebroid on MM. Then a path a:IAa\colon I \to A defines a map from φ:C(A[1])C(TI[1])Ω(I)\varphi\colon C(A[1]) \to C(TI[1]) \cong \Omega(I) which respects the commutative graded algebra structure. A function fC (M)f\in C^{\infty}(M) gets mapped to fγf\circ \gamma, where γ:IM\gamma\colon I \to M is the projection of aa. A section sΓ(A *)s\in\Gamma(A^*) gets mapped to (sγ(a)dt(s\circ \gamma (a) dt under φ\varphi, where dtdt is the canonical section of Ω 1(I)\Omega^1(I). Since C(TI[1])C(TI[1]) is concentrated in degree 00 and 11, the other degrees get mapped to 00.

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 (M)C^{\infty}(M):

φ(d Af)=d dRφ(f)fC (M). \varphi(d_A f) = d_{dR} \varphi(f) \qquad f\in C^{\infty}(M).

A quick calculation shows that this is true if and only if

ρ A(a(t))=ddtγ(t). \rho_A(a(t)) = \frac{d}{dt}\gamma(t).


Let MM the unit square in 2\mathbb{R}^2. Then a Lie algebroid morphism from (TM,id TM,[,])(TM, \id_{TM}, [-,-]) to (𝔤,T,[.] 𝔤)(\mathfrak{g}, T\cdot, [-.-]_{\mathfrak{g}}), where 𝔤\mathfrak{g} is a Lie algebra, is given by a morphism of c.d.g.a.

(CE(𝔤),d 𝔤)(Ω(M),d dR). (CE(\mathfrak{g}), d_{\mathfrak{g}}) \to (\Omega(M), d_{dR}).

Two smooth maps a,b:M𝔤a,b\colon M\to \mathfrak{g} give a map between c.g.a. spaces above. Here a section sΓ(𝔤 *)s\in\Gamma(\mathfrak{g}^*), i.e. an element of 𝔤 *\mathfrak{g}^* gets mapped to the one form

s(a)dt+s(b)ds, s(a) dt + s(b) ds,

where (t,s)(t,s) are the coordinates on MM. A quick calculation shows that this map respects the differential if and only if

dadsdbds=[a,b] 𝔤. \frac{da}{ds} - \frac{db}{ds} = [a,b]_{\mathfrak{g}}.

This formula also shows that adt+bdsa\cdot dt + b\cdot ds is a flat connection on the trivial principal GG-bundle on MM. Therefore Lie algebroid morphisms open a way to talk about higher connections and flat conditions.

Open problem

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.

Higher Lie groupoids




The notion of Lie categories, hence of Lie groupoids, goes back to

  • Charles Ehresmann, Catégories topologiques et categories différentiables, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 (pdf, zbMath:0205.28202)

Their understanding as internal categories/internal groupoids in SmoothManifolds is often attributed to

but the accepted definition of internal groupoids seems not to be contained in there.

Textbook accounts:

Historical review:

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

Last revised on July 27, 2021 at 10:18:14. See the history of this page for a list of all contributions to it.