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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Lie groupoid} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples_for_lie_groupoids}{Examples for Lie groupoids}\dotfill \pageref*{examples_for_lie_groupoids} \linebreak \noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak \noindent\hyperlink{specialisations}{Specialisations}\dotfill \pageref*{specialisations} \linebreak \noindent\hyperlink{2CatOfGrpds}{The (2,1)-category of Lie groupoids}\dotfill \pageref*{2CatOfGrpds} \linebreak \noindent\hyperlink{lie_algebroid}{Lie algebroid}\dotfill \pageref*{lie_algebroid} \linebreak \noindent\hyperlink{examples_of_lie_algebroids}{Examples of Lie algebroids}\dotfill \pageref*{examples_of_lie_algebroids} \linebreak \noindent\hyperlink{MorphismsOfLieGroupoids}{Morphisms of Lie groupoids}\dotfill \pageref*{MorphismsOfLieGroupoids} \linebreak \noindent\hyperlink{morphisms_of_lie_algebroids}{Morphisms of Lie algebroids}\dotfill \pageref*{morphisms_of_lie_algebroids} \linebreak \noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak \noindent\hyperlink{example_2}{Example}\dotfill \pageref*{example_2} \linebreak \noindent\hyperlink{open_problem}{Open problem}\dotfill \pageref*{open_problem} \linebreak \noindent\hyperlink{higher_lie_groupoids}{Higher Lie groupoids}\dotfill \pageref*{higher_lie_groupoids} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{Lie groupoid} is a [[groupoid]] [[internal 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 \emph{[[differentiable stacks]]}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{ExplicitDefinition}\hypertarget{ExplicitDefinition}{} A \emph{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. \end{defn} A \emph{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 [[morphism]]s 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 [[submersion]]s. 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 functor]]s and internal [[natural transformation]]s is \emph{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 functor|essentially surjective]] and [[full and faithful functor|full and faithful]] may nevertheless not be an equivalence, in that it may not have a weak inverse in $Grpd(Diff)$. See the section \hyperlink{2CatOfGrpds}{2-Category of Lie groupoids} below. A bit more general than a Lie groupoid is a [[diffeological groupoid]]. \hypertarget{examples_for_lie_groupoids}{}\subsubsection*{{Examples for Lie groupoids}}\label{examples_for_lie_groupoids} \begin{itemize}% \item 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 $G$. \item 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. \item 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 [[Čech groupoid]] or \textbf{covering groupoid}. \item Given a Lie group $G$ (right) action on a manifold $M$, then we may form an associated [[action groupoid]] (or sometimes called \textbf{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. \item 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). \item Given a manifold $M$, we have also an associated [[fundamental groupoid]] or \textbf{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?). \item Given a manifold $M$ with a [[foliation]] $F$, we may form various groupoids associated with $F$. \end{itemize} \begin{enumerate}% \item \textbf{$F$-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 \href{https://math.berkeley.edu/~alanw/Models.pdf}{Section 13.5 of Geometric Models for Noncommutative Algebras} ). \item \textbf{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. \item \textbf{[[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. \end{enumerate} Among all possible Lie groupoids associated to a foliation, monodromy groupoid is the biggest and holonomy groupoid is the smallest. \hypertarget{terminology}{}\subsubsection*{{Terminology}}\label{terminology} Originally Lie groupoids were called (by Ehresmann) \emph{differentiable groupoids} (and also one considered differentiable \emph{categories}). Sometime in the 1980s there was a change of terminology to \emph{Lie groupoid} and [[differentiable stack]]s. (reference?) \hypertarget{specialisations}{}\subsubsection*{{Specialisations}}\label{specialisations} One definition which Ehresmann introduced in his paper \emph{Cat\'e{}gories topologiques et cat\'e{}gories diff\'e{}rentiables} (see below) is that of [[locally trivial category|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 - [[David Roberts|DR]]) \begin{defn} \label{}\hypertarget{}{} A topological groupoid $X_1 \rightrightarrows X_0$ is \textbf{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 $(s,t):X_1^{iso}\to X_0 \times X_0$. \end{defn} 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. \hypertarget{2CatOfGrpds}{}\subsubsection*{{The (2,1)-category of Lie groupoids}}\label{2CatOfGrpds} As usual for internal categories, the naive 2-category of internal groupoids, [[internal functor]]s and internal [[natural transformation]]s 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 [[anafunctor]]s, i.e. [[span]]s of internal functors $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$. Such generalized morphisms -- called \emph{[[Morita morphisms]]} or \emph{generalized morphisms} in the literature -- are sometimes modeled as [[bibundle]]s and then called [[Hilsum-Skandalis morphism]]s. Another equivalent approach is to embed Lie groupoids into the context of [[2-topos]] theory: The [[2-topos|(2,1)-topos]] $Sh_{(2,1)}(Diff)$ of [[stack]]s/[[2-sheaves]] on [[Diff]] may be understood as a nice [[2-category]] of general groupoids \emph{modeled on} [[smooth manifold]]s. The degreewise [[Yoneda embedding]] allows to emebed groupoids internal to $Diff$ into stacks on $Diff$. this wider context contains for instance also [[diffeological groupoid]]s. Regarded inside this wider context, Lie groupoids are identified with [[differentiable stack]]s. The [[(n,r)-category|(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]]. \hypertarget{lie_algebroid}{}\subsection*{{Lie algebroid}}\label{lie_algebroid} As the [[infinitesimal space|infinitesimally]] approximation to a [[Lie group]] is a [[Lie algebra]], so the infinitesimal approximation to a Lie groupoid is a [[Lie algebroid]]. \begin{defn} \label{}\hypertarget{}{} 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 $[X, fY]=f[X,Y]+\rho(X)(f) Y.$ \end{defn} \begin{defn} \label{}\hypertarget{}{} 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. Poinar\'e{} Phys. Th\'e{}or., 53(1):3581, 1990.) Recent progress: it turns out that one may link Lie algebroid with $L_\infty$-spaces (ask Owen Gwilliam for it) \end{defn} \hypertarget{examples_of_lie_algebroids}{}\subsubsection*{{Examples of Lie algebroids}}\label{examples_of_lie_algebroids} \begin{itemize}% \item A Lie algebra is a Lie algebroid with base space being a point. \item $0$-bundle over a manifold $M$ is certainly a Lie algebroid in a trivial way. \item [[action Lie algebroid]] \item Tangent bundle $TM\to M$ is a Lie algebroid with $\rho=id$ and $[-,-]$ the usual Lie bracket for vector fields. See [[tangent Lie algebroid]]. \item 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]]. \item [[Atiyah Lie algebroid]] \end{itemize} \hypertarget{MorphismsOfLieGroupoids}{}\subsection*{{Morphisms of Lie groupoids}}\label{MorphismsOfLieGroupoids} There are several versions of Lie groupoid morphisms, some of them are equivalent in a correct sense, some of them are not. \begin{itemize}% \item 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. \item 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 \emph{[[bibundle]]}). \end{itemize} A Lie groupoid functor $f : G\to H$ is a \textbf{weak equivalence} if it is \begin{enumerate}% \item essentially surjective; that is, $t \circ pr_2 : G_0 \times_{H_0,s} H_1 \to H_0$ is a surjective submersion; \item fully faithful; that is, $G_1 \cong H_1\times_{t\times s, H_0\times H_0} G_0 \times G_0$. \end{enumerate} 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 \textbf{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?) \begin{itemize}% \item [[anafunctor]]: an anafunctor 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 [[acyclic fibration]] of Lie groupoids. That is, this map is a [[weak equivalence]] of Lie groupoids and $\hat X_0 \to X_0$ is a surjective submersion. \end{itemize} Composition of anafunctors is given through strong [[pullback]] of Lie groupoids, that is level-wise pullback. \begin{itemize}% \item [[bibundle]] functor (or H.S. bibundle, or Hilsum-Skandalis bibundle): a bibundle functor from $G\to H$ is a [[groupoid principal bundle]] $E$ of $H$ (with right action) such that $G$ acts on $E$ from left and $G$ action commutes with $H$ action. If both $G$ and $H$ actions are principal, then $E$ gives arise to [[Morita equivalence]] between them. \end{itemize} 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-morphism]]s 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 \begin{displaymath} \begin{matrix} & & \hat X \\ & \swarrow & & \searrow \\ X & & \downarrow & & Y \\ & \searrow & & \swarrow \\ & & \hat X' \end{matrix} \end{displaymath} 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: \begin{displaymath} \begin{matrix} & & \hat X \\ & \swarrow & \uparrow & \searrow \\ X & & \hat X'' & & Y \\ & \searrow & \downarrow & \swarrow \\ & & \hat X' \end{matrix} \end{displaymath} A [[2-morphism]] between [[anafunctor]]s are defined like above, however the left legs are required to be [[acyclic fibration]]s 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 \textbf{[[equivalent]]} to each other. For a nice survey on this statement, we refer to Section 1.5 of \href{http://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0022-5F4F-A}{Du Li's thesis}. The idea is that \href{http://ncatlab.org/nlab/show/bibundle#bundlisation}{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. \hypertarget{morphisms_of_lie_algebroids}{}\subsection*{{Morphisms of Lie algebroids}}\label{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 \emph{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 \begin{displaymath} C(A[1])=C^\infty(M)\oplus \Gamma(A^*)\oplus \Gamma(\wedge^2 A^*)\oplus \ldots , \end{displaymath} 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 \begin{displaymath} 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). \end{displaymath} 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 \begin{itemize}% \item $[- , - ]_A$ satisfies the Jacobi identity \item and $[ - , - ]_A$ and $\rho_A$ together satisfy the Leibniz identity. \end{itemize} (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-algebra]]s. 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 \begin{displaymath} (C(A[1]), d_A) \leftarrow (C(B[1]), d_B). \end{displaymath} 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. \hypertarget{example}{}\subsubsection*{{Example}}\label{example} 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 $C^{\infty}(M)$: \begin{displaymath} \varphi(d_A f) = d_{dR} \varphi(f) \qquad f\in C^{\infty}(M). \end{displaymath} A quick calculation shows that this is true if and only if \begin{displaymath} \rho_A(a(t)) = \frac{d}{dt}\gamma(t). \end{displaymath} \hypertarget{example_2}{}\subsubsection*{{Example}}\label{example_2} 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. \begin{displaymath} (CE(\mathfrak{g}), d_{\mathfrak{g}}) \to (\Omega(M), d_{dR}). \end{displaymath} 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 \begin{displaymath} s(a) dt + s(b) ds, \end{displaymath} where $(t,s)$ are the coordinates on $M$. A quick calculation shows that this map respects the differential if and only if \begin{displaymath} \frac{da}{ds} - \frac{db}{ds} = [a,b]_{\mathfrak{g}}. \end{displaymath} 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. \hypertarget{open_problem}{}\subsubsection*{{Open problem}}\label{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. \hypertarget{higher_lie_groupoids}{}\subsection*{{Higher Lie groupoids}}\label{higher_lie_groupoids} See \begin{itemize}% \item [[internal ∞-groupoid]] \item [[∞-Lie groupoid]] \end{itemize} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item Every [[smooth manifold]] $X$ is a 0-[[0-groupoid|truncated]] Lie groupoid. \item For every [[Lie group]] $G$ the one-object [[delooping]] groupoid $\mathbf{B}G$ is a Lie groupoid. \item The Lie group $G$ itself is a 0-[[truncated]] [[group object]] in the 2-category or Lie groupoids. \item Every [[Lie 2-group]] is in particular a Lie groupoid: a [[group object]] in the category of Lie groupoids. \item The [[inner automorphism 2-group]] $\mathbf{E}G = INN(G) = G//G$ is a Lie groupoid. There is an obvious morphism $\mathbf{E}G \to \mathbf{B}G$. \item For every $G$-[[principal bundle]] $P \to X$ there is its [[Atiyah Lie groupoid]] $At(P)$. \item The [[fundamental groupoid]] $\Pi_1(X)$ of a smooth manifold is naturally a Lie groupoid. \item The [[path groupoid]] of a smooth manifold is naturally a [[diffeological groupoid]]. \item The [[Cech groupoid]] $C(U)$ of a [[cover]] $\{U_i \to X\}$ of a smooth manifold is a Lie groupoid. \item Every [[foliation]] gives rise to its [[holonomy groupoid]]. \item An [[orbifold]] is a Lie groupoid. \item An [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} C(U) \to \mathbf{B}G$ from a smooth manifold $X$ to $\mathbf{B}G$ is a [[Cech cohomology|Cech cocycle]] in degree 1 with values in $G$, classifying $G$-[[principal bundle]] $P$. \item The (1-categorical) [[pullback]] \begin{displaymath} \itexarray{ P &\stackrel{\simeq}{\leftarrow}& \mathbf{P} &\to& \mathbf{E}G \\ && \downarrow && \downarrow \\ && C(U) &\stackrel{}{\to}& \mathbf{B}G \\ && \downarrow^{\mathrlap{\simeq}} \\ && X } \end{displaymath} 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]].) \item Similarly an anafunctor from $P_1(X)$ to $\mathbf{B}G$ is a [[connection on a bundle]] (see there for details). \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[smooth groupoid]] \item [[LieGrpd]], [[SmoothGrpd]] \item [[bisection of a Lie groupoid]] \item [[orbifold]] \item [[effective Lie groupoid]], [[proper Lie groupoid]], [[etale groupoid]] \item [[homotopy groups of a Lie groupoid]] \item [[Morita morphism]], [[Hilsum-Skandalis morphism]] \item [[Tannaka duality for Lie groupoids]] \item [[double Lie groupoid]] \item [[higher differential geometry]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Topological and differentiable (or smooth, ``Lie'') groupoids (and more generally categories) were introduced in \begin{itemize}% \item [[Charles Ehresmann]], \emph{Cat\'e{}gories topologiques et cat\'e{}gories diff\'e{}rentiables}, Colloque de G\'e{}ometrie Differentielle Globale (Bruxelles, 1958), 137--150, Centre Belge Rech. Math., Louvain, 1959; (p. 263- in \href{http://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_I-1_et_I-2.pdf}{Ouvres pdf}) \end{itemize} Reviews and developments of the theory of Lie groupoids include \begin{itemize}% \item Pradines, \ldots{}. \item [[Kirill Mackenzie]], \emph{General Theory of Lie Groupoids and Lie Algebroids,} Cambridge University Press, 2005, xxxviii + 501 pages (\href{http://kchmackenzie.staff.shef.ac.uk/gt.html}{website}) \item [[Kirill Mackenzie]], \emph{Lie groupoids and Lie algebroids in differential geometry}, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (\href{http://www.ams.org/mathscinet-getitem?mr=896907}{MathSciNet}) \end{itemize} Discussion in the context of [[foliation theory]] ([[foliation groupoids]]) is in \begin{itemize}% \item [[Ieke Moerdijk]], [[Janez Mrčun]] \emph{Introduction to Foliations and Lie Groupoids}, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, 2003 (\href{https://doi.org/10.1017/CBO9780511615450}{doi:10.1017/CBO9780511615450}) \end{itemize} The relation to [[differentiable stacks]] is discussed/reviewed in section 2 of \begin{itemize}% \item [[Christian Blohmann]], \emph{Stacky Lie groups}, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082 (\href{http://arxiv.org/abs/math/0702399}{arXiv:math/0702399}) \end{itemize} Lie groupoids as a source for [[groupoid convolution algebras|groupoid convolution]] [[C\emph{-algebras]] are discussed in} \begin{itemize}% \item [[Alain Connes]], \emph{[[Noncommutative Geometry]]} \end{itemize} Expository discussion of various kinds of groupoids is also in \begin{itemize}% \item [[John Baez]] \emph{\href{http://math.ucr.edu/home/baez/TWF.html}{TWF} \href{http://math.ucr.edu/home/baez/week256.html}{256}}. \end{itemize} Groupoids and their various morphisms between them in different categories, including in Diff, is also in \begin{itemize}% \item Ralf Meyer and [[Chenchang Zhu]], \emph{Groupoids in categories with pretopology}, \href{http://arxiv.org/pdf/1408.5220.pdf}{arXiv:math/1408.5220} \end{itemize} [[!redirects Lie groupoids]] \end{document}