Pradines integration theorem and Mackenzie transitive integrability

Intuition Beginner

A Lie group can be shrunk to its Lie algebra by looking only at the directions you can move near the identity. The reverse trip also works: given a Lie algebra, you can always grow it back into a Lie group. Sophus Lie proved this around 1880, and it has felt like bedrock ever since. Shrink, then grow, and you return to where you started.

Lie groupoids and their infinitesimal versions, Lie algebroids, are the broader setting where the same arrows now run between many base points instead of sitting at one object. Shrinking a groupoid to its algebroid still works perfectly. The question this unit answers is whether the reverse trip still works: given an algebroid, can you always grow it back into a groupoid?

The answer is no, and the no is the whole story. Pradines announced in 1966 that growing-back always works, mirroring Lie's theorem. Nineteen years later Almeida and Molino found an algebroid that simply has no groupoid above it. So the surprise is real: some infinitesimal systems of arrows cannot be assembled into a global one, and a single number, called the monodromy, decides which.

Visual Beginner

Picture two staircases. The first goes down: a global groupoid on the top step, its infinitesimal algebroid on the bottom step. Everyone can walk down this staircase, because differentiating always works. The second staircase goes back up, and here some steps are missing.

Beside the broken staircase sits a dial called the monodromy. When its marks are evenly spaced and well separated, the step above is present and the algebroid grows back into a groupoid. When the marks crowd together and pile up at zero, the step is gone and no groupoid exists. The whole integrability question reduces to reading that one dial.

Worked example Beginner

Start with a base surface and take the simplest algebroid on it: at each point, all the tangent directions, with the rule that the shadow of a direction is the direction itself. This is the tangent algebroid. Can it grow back into a groupoid of arrows?

Yes, and there is even a smallest natural choice. Take as arrows all ordered pairs of points, one arrow from each point to each other point. The do-nothing arrow at a point is the pair of that point with itself, and shrinking this pair groupoid gives back exactly the tangent directions you started with. So the tangent algebroid grows back into the pair groupoid.

A second clean case: suppose over each base point you have a copy of a fixed Lie algebra, with no shadow at all, so the directions never move the base point. This is a bundle of Lie algebras. It grows back by replacing each Lie algebra with its Lie group, giving a bundle of Lie groups sitting over the base.

What this tells us: many algebroids do grow back, and they grow back into recognizable objects. The tangent algebroid becomes the pair groupoid; a bundle of Lie algebras becomes a bundle of Lie groups. The failures, when they come, are more delicate and are detected by the monodromy dial rather than by the obvious examples.

Check your understanding Beginner

Formal definition Intermediate+

Let $A\to M$ be a Lie algebroid 03.04.16 with anchor $\rho :A\to TM$ and bracket on $\Gamma (A)$. The algebroid is integrable when there exists a Lie groupoid $G\rightrightarrows M$ 03.03.10 whose Lie algebroid $A(G)$, produced by the Lie functor 03.04.17, is isomorphic to $A$. The integrability problem asks which $A$ are integrable and classifies the integrating groupoids.

Three statements organize the answer, the groupoid analogues of Lie's three theorems. Lie I: if $A$ is integrable, there is a canonical source-simply-connected integrating groupoid, unique up to isomorphism, and every integrating groupoid with connected source fibres is a quotient of it. Lie II: a morphism of integrable algebroids $\varphi :A\to B$ integrates to a unique groupoid morphism $G\to H$ whenever $G$ is source-simply-connected. Lie III: every Lie algebroid is integrable — and this statement is false ^{[Mackenzie Ch. IV §3]}.

The obstruction is read off paths. An $A$-path is a $C^1$ path $a:I=[0,1]\to A$ over a base path $\gamma :I\to M$ satisfying the admissibility condition $$ \rho\big(a(t)\big) = \frac{d}{dt},\gamma(t), \qquad \gamma = p \circ a, $$ where $p:A\to M$ is the projection. Two $A$-paths are $A$-homotopic when they are connected by a one-parameter family of $A$-paths whose endpoints stay fixed, in the precise sense of a flat variation in the algebroid. The Weinstein groupoid is the quotient $$ \mathcal G(A) := {A\text{-paths}},/,A\text{-homotopy}, $$ a topological groupoid over $M$ with source and target the endpoints of $\gamma$, multiplication by concatenation, and units the constant zero paths. Its isotropy at $x$ is a topological group with Lie algebra ${\mathfrak{g}}_x=\ker {\rho }_x$, the isotropy algebra at $x$.

The monodromy group at $x$ is the subgroup $$ \mathcal N_x(A) ;\subseteq; Z(\mathfrak g_x) ;\subseteq; \mathfrak g_x, $$ of the centre of the isotropy algebra, consisting of those central elements realized as $A$-homotopy classes of $A$-loops at $x$ that collapse to the unit in $\mathcal{G}(A)$; concretely ${\mathcal{N}}_x(A)=\{{\partial }_{\omega }(g):g\in {\pi }_2(\text{leaf through }x)\}$, the image of spherical periods of a transverse $2$-variation. The collection $({\mathcal{N}}_x{)}_{x\in M}$ is uniformly discrete when there is a continuous family of norms $\Vert \cdot {\Vert }_x$ on ${\mathfrak{g}}_x$ and a constant $r>0$ with $\inf \{\Vert v{\Vert }_x:0\ne v\in {\mathcal{N}}_x(A)\}\ge r$ locally in $x$. The convention here follows Crainic-Fernandes: $\mathcal{G}(A)$ is the source-simply-connected candidate, and discreteness is measured in the centre of the isotropy, not the whole isotropy algebra ^{[Crainic-Fernandes 2003]}.

Counterexamples to common slips

• The Weinstein groupoid $\mathcal{G}(A)$ always exists as a topological groupoid; integrability is precisely the failure or success of its smoothness, not its existence.
• The monodromy lives in the centre $Z({\mathfrak{g}}_x)$, not in all of ${\mathfrak{g}}_x$: non-central obstructions are absorbed by the isotropy group's own integration and do not block the algebroid.
• Uniform discreteness is a condition on a family over $M$, not pointwise discreteness: each ${\mathcal{N}}_x$ can be discrete while the family accumulates, and that accumulation is what breaks integrability.

Key theorem with proof Intermediate+

Theorem (Crainic-Fernandes integrability criterion). Let $A\to M$ be a Lie algebroid with Weinstein groupoid $\mathcal{G}(A)$ and monodromy groups ${\mathcal{N}}_x(A)\subseteq Z({\mathfrak{g}}_x)$. Then $A$ is integrable if and only if the monodromy groups are locally uniformly discrete. When this holds, $\mathcal{G}(A)$ is a smooth source-simply-connected Lie groupoid integrating $A$, and it is the canonical integrating groupoid of Lie I.

Proof. The space $P(A)$ of $A$-paths is a Banach manifold, and $A$-homotopy is the orbit equivalence of an integrable infinite-dimensional foliation on $P(A)$ whose leaves are the $A$-homotopy classes. The quotient $\mathcal{G}(A)=P(A)/\sim$ is therefore a topological groupoid, and the source map descends from the smooth submersion $P(A)\to M$ sending a path to the start of its base curve, so $s$-fibres of $\mathcal{G}(A)$ are quotients of the contractible fibres of $P(A)\to M$ and are connected and simply connected ^{[Crainic-Fernandes 2003]}.

Smoothness of $\mathcal{G}(A)$ is local near the units. Fix $x$ and restrict to the leaf $L$ through $x$ of the characteristic foliation $\mathrm{im}\rho$. The isotropy at $x$ is generated by $A$-loops, and the obstruction to a manifold chart at the unit $1_x$ is exactly that the isotropy group $G_x=s^{-1}(x)\cap t^{-1}(x)$ fail to be a Lie group. Its identity component integrates the isotropy algebra ${\mathfrak{g}}_x$, but the discrete part is the image of ${\pi }_1$ of the source fibre, computed by the boundary map of the long exact homotopy sequence of the fibration $G_x\to s^{-1}(x)\to L$: $$ \pi_2(L, x) \xrightarrow{\ \partial\ } \pi_1(G_x^0) \cong \mathcal N_x(A) \longrightarrow \pi_1(s^{-1}(x)) = 0, $$ the last term vanishing because $\mathcal{G}(A)$ is source-simply-connected. So ${\mathcal{N}}_x(A)$ is precisely the discreteness defect of the isotropy.

If the family $({\mathcal{N}}_x)$ is locally uniformly discrete, choose $r>0$ and the norm family bounding the nonzero monodromy below by $r$ on a neighbourhood $U$ of $x$. Then the union ${\bigcup }_{y\in U}{\mathcal{N}}_y$ avoids a uniform ball around $0$ in the bundle $Z(\mathfrak{g}){|}_U$, so the central exponential $\exp :Z(\mathfrak{g})\to G^0$ is injective on a uniform ball and the quotient $G_y^0/{\mathcal{N}}_y$ varies smoothly, giving a manifold chart for $\mathcal{G}(A)$ over $U$. Patching charts over a cover of $M$ makes $\mathcal{G}(A)$ a Lie groupoid; differentiating it returns $A$, so $A$ is integrable. Conversely, if some sequence $0\ne v_n\in {\mathcal{N}}_{x_n}$ has $\Vert v_n{\Vert }_{x_n}\to 0$ with $x_n\to x$, the central quotients $G_{x_n}^0/{\mathcal{N}}_{x_n}$ cannot converge to a manifold near $1_x$: distinct $A$-homotopy classes accumulate at the unit, so no chart exists, $\mathcal{G}(A)$ is not smooth, and — by Lie I, since any integrating groupoid would have $\mathcal{G}(A)$ as its source-simply-connected cover — $A$ is not integrable. $\square$

Bridge. This criterion builds toward the transitive case, where the monodromy is computed by a Chern-Weil curvature integral and uniform discreteness becomes a condition on the periods of a closed $2$-form, and it appears again in the Almeida-Molino counterexample, where a dense period group violates discreteness concretely; it grounds Lie I, since the smooth $\mathcal{G}(A)$ it produces is the canonical source-simply-connected integrator, and it sharpens Lie III, replacing the false "always integrable" with a precise measure of failure. The single load-bearing fact is the identification of ${\mathcal{N}}_x(A)$ with the discreteness defect of the isotropy through the homotopy sequence of the source fibration: every later computation, transitive or not, evaluates that one defect.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib provides mfderiv, a developing TangentBundle and vector-bundle library, and LieAlgebra, but it has neither a LieGroupoid object nor a LieAlgebroid structure, so integrability cannot be stated. The sketch below names the missing pieces; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: the integrability question, not in Mathlib.
variable {M : Type*} [Manifold M]
variable (A : LieAlgebroid M)     -- anchor ρ : A → TM, bracket on Γ(A)

structure APath (A : LieAlgebroid M) where
  base   : C1Path M
  lift   : ∀ t, A.fiber (base t)
  admiss : ∀ t, A.anchor (lift t) = deriv base t      -- ρ(a) = γ'

def AHomotopy (A) : APath A → APath A → Prop := ...     -- flat variation, fixed ends

def WeinsteinGroupoid (A) : TopGroupoid M :=
  Quotient (AHomotopy A)                                -- 𝒢(A), always topological

def monodromy (A) (x : M) : AddSubgroup (center (A.isotropy x)) := ...  -- 𝒩ₓ ⊆ Z(𝔤ₓ)

theorem integrable_iff_uniformly_discrete (A) :
    Integrable A ↔ LocallyUniformlyDiscrete (fun x => monodromy A x)

The first genuine obstacle is APath with its admissibility constraint and the Banach-manifold structure on the space of such paths; the second is monodromy, the spherical-period map into the centre of the isotropy. The theorem integrable_iff_uniformly_discrete is the Crainic-Fernandes criterion and is the mathematical content of the unit. None of these objects exists in Mathlib.

Advanced results Master

Lie I and Lie II. Two of Lie's theorems transfer without obstruction. Lie I: an integrable algebroid has a canonical source-simply-connected integrator $\mathcal{G}(A)$, unique up to isomorphism, and every connected-source integrator is a quotient $\mathcal{G}(A)/N$ by a discrete central subgroupoid $N\subseteq Z(\mathcal{G}(A))$. Lie II: a morphism $\varphi :A\to B$ of integrable algebroids integrates to a unique groupoid morphism $\mathcal{G}(A)\to G_B$ over the base map, by lifting $A$-paths through $\varphi$ and descending to $A$-homotopy classes. Both proofs run on the $A$-path model: Lie I is the universal property of the $A$-homotopy quotient, and Lie II is functoriality of $A$-paths under algebroid morphisms ^{[Crainic-Fernandes 2003]}.

The transitive case and the Chern-Weil obstruction (Mackenzie). For a transitive Lie algebroid the anchor is surjective and the algebroid sits in the Atiyah sequence 03.05.22 $$ 0 \longrightarrow L \longrightarrow A \xrightarrow{\ \rho\ } TM \longrightarrow 0, $$ with $L=\ker \rho$ the bundle of isotropy Lie algebras. A linear splitting $\sigma :TM\to A$ is a connection; its curvature $\Omega (X,Y)=[\sigma X,\sigma Y]-\sigma [X,Y]\in \Gamma (L)$ is an $L$-valued $2$-form. Restricting to the centre gives a closed $Z(L)$-valued $2$-form ${\Omega }_Z$, and the monodromy at $x$ is the group of spherical periods $$ \mathcal N_x(A) = \Big{ \int_{S^2} f^\ast \Omega_Z : f \in \pi_2(M, x) \Big} \subseteq Z(\mathfrak g_x). $$ Mackenzie's theorem: a transitive Lie algebroid integrates if and only if these period groups are uniformly discrete in $Z(L)$. This is a Chern-Weil statement — integrability is governed by the periods of the curvature of the adjoint sequence, exactly the prequantization-style condition that the curvature have a discrete period lattice ^{[Mackenzie Ch. IV §4]}.

The Almeida-Molino counterexample. The first non-integrable algebroid is transitive ^{[Almeida-Molino 1985]}. Take $M=S^2\times \Sigma$ with $\Sigma$ a surface carrying a closed $2$-form, and build a transitive algebroid with $L$ a rank-one central bundle whose curvature ${\Omega }_Z$ is a closed $2$-form whose spherical periods over the $S^2$-factor form a subgroup $\Gamma \subseteq \mathbb{R}$ that is dense — the periods are an irrational-ratio pair, so $\Gamma =\mathbb{Z}+\mathbb{Z}\alpha$ with $\alpha$ irrational accumulates at $0$. Then ${\mathcal{N}}_x(A)=\Gamma$ is dense, not discrete, uniform discreteness fails, and the algebroid does not integrate. This is the concrete refutation of Pradines' 1966 announcement.

Pradines' claim and its correction. Pradines stated in a 1966 Comptes Rendus note that every differentiable algebroid integrates, the groupoid analogue of Lie III ^{[Pradines 1966]}. The proof was never completed and the statement is false; the local version (integrability of local Lie groupoids, equivalently integrability near the units) does hold, and is the salvageable content. Crainic and Fernandes' monodromy criterion is the exact gap between the true local statement and the false global one: local integrability always holds, global integrability holds iff the local pieces patch, and the patching obstruction is the uniform discreteness of the monodromy.

Synthesis. Differentiation from Lie groupoids to Lie algebroids is always possible, but the reverse — integration — is obstructed, and the obstruction is a single discreteness condition on the monodromy groups ${\mathcal{N}}_x(A)\subseteq Z({\mathfrak{g}}_x)$. Lie I and Lie II survive the passage to algebroids unchanged: an integrable algebroid has a unique source-simply-connected integrator and morphisms integrate over it, both by the $A$-path model. Lie III does not survive: Pradines' 1966 claim that every algebroid integrates is false, and the precise replacement is the Crainic-Fernandes criterion that $A$ integrates exactly when its monodromy is locally uniformly discrete. In the transitive case this becomes a Chern-Weil statement on the Atiyah sequence — the monodromy is the spherical-period group of the centre-valued curvature of a connection — so integrability is the prequantization-style demand that the curvature periods form a discrete lattice. The Almeida-Molino algebroid violates exactly this, with a dense period group on an $S^2$-factor, giving the first transitive non-integrable example and turning Pradines' optimism into a sharp dichotomy: every algebroid integrates locally, and the global obstruction is measured, leaf by leaf, in the centre of the isotropy.

Full proof set Master

Proposition (Local integrability always holds). Every Lie algebroid $A\to M$ is integrable by a local Lie groupoid; equivalently, there is a neighbourhood of the units in $\mathcal{G}(A)$ carrying a smooth groupoid structure differentiating to $A$.

Proof. Work in a local trivialization $A{|}_U\cong U\times V$ over a chart $U\subseteq {\mathbb{R}}^n$, with anchor and bracket given by structure functions. The $A$-paths over $U$ form a Banach manifold, and near the constant zero paths the $A$-homotopy foliation has slices that are graphs of a smooth map, because the exponential of the anchor flow is a local diffeomorphism near $0$. Concretely, the time-one flow of the spray associated to a connection on $A$ gives a smooth local exponential ${\exp }_A:\mathcal{U}\subseteq A\to \mathcal{G}(A)$ defined on a neighbourhood $\mathcal{U}$ of the zero section, and on its image the groupoid multiplication is smooth by the Baker-Campbell-Hausdorff expansion of the bracket. This makes a neighbourhood of the units a smooth local Lie groupoid integrating $A$. The obstruction to extending globally is precisely whether these local exponentials patch to a single manifold structure on all of $\mathcal{G}(A)$, which is the monodromy condition; locally there is no obstruction. $\square$

Proposition (Transitive integrability via curvature periods). Let $A$ be a transitive Lie algebroid with Atiyah sequence $0\to L\to A\to TM\to 0$, connection $\sigma$, and centre-valued curvature ${\Omega }_Z\in {\Omega }^2(M;Z(L))$. Then $A$ is integrable if and only if the period groups ${\mathcal{N}}_x=\{{\int }_{S^2}{f}^{\ast }{\Omega }_Z:f\in {\pi }_2(M,x)\}$ are uniformly discrete in $Z(L)$.

Proof. The monodromy of $A$ equals the spherical-period group of ${\Omega }_Z$. By the homotopy-sequence computation, ${\mathcal{N}}_x=\partial ({\pi }_2(M,x))$ where $\partial :{\pi }_2(M)\to {\pi }_1(G_x^0)\subseteq Z({\mathfrak{g}}_x)$ is the boundary map of the source fibration. For a transitive algebroid the source fibre of $\mathcal{G}(A)$ fibres over $M$ with fibre the simply-connected isotropy group, and the boundary map is computed by integrating the connection's curvature over spheres: a class $f\in {\pi }_2(M)$ lifts to a $2$-sphere in $A$ whose holonomy around the boundary is $\exp ({\int }_{S^2}{f}^{\ast }{\Omega }_Z)$, central because ${\Omega }_Z$ is centre-valued. So ${\mathcal{N}}_x=\{{\int }_{S^2}{f}^{\ast }{\Omega }_Z\}$. Applying the Crainic-Fernandes criterion, $A$ integrates iff $({\mathcal{N}}_x)$ is uniformly discrete, which is the stated period condition. Independence of the connection $\sigma$ follows because changing $\sigma$ alters ${\Omega }_Z$ by an exact $Z(L)$-valued $2$-form, whose spherical periods vanish, so ${\mathcal{N}}_x$ is unchanged. $\square$

Proposition (Almeida-Molino: a non-integrable transitive algebroid). There is a transitive Lie algebroid whose monodromy groups are dense, hence non-integrable.

Proof. Let $M=S^2\times S^2$ and let $L=M\times \mathbb{R}$ be the product central rank-one bundle (so ${\mathfrak{g}}_x=\mathbb{R}$ is abelian and central). Choose two area forms ${\omega }_1,{\omega }_2$ on the two $S^2$-factors normalized to ${\int }_{S^2}{\omega }_1=1$ and ${\int }_{S^2}{\omega }_2=\alpha$ with $\alpha$ irrational, and set ${\Omega }_Z={\mathrm{pr}}_1^{\ast }{\omega }_1+{\mathrm{pr}}_2^{\ast }{\omega }_2\in {\Omega }^2(M;\mathbb{R})$, a closed $2$-form. Build the transitive algebroid $A$ with this Atiyah sequence and curvature, available because any closed centre-valued $2$-form is the curvature of some abelian-extension algebroid. Since ${\pi }_2(M)=\mathbb{Z}\oplus \mathbb{Z}$ generated by the two factor spheres, the period group is ${\mathcal{N}}_x=\mathbb{Z}\cdot 1+\mathbb{Z}\cdot \alpha =\mathbb{Z}+\mathbb{Z}\alpha \subseteq \mathbb{R}$. With $\alpha$ irrational this subgroup is dense in $\mathbb{R}$, so it is not discrete and the family $({\mathcal{N}}_x)$ is not uniformly discrete. By the integrability criterion $A$ is not integrable. This realizes Almeida-Molino's example in the cleanest period form ^{[Almeida-Molino 1985]}. $\square$

Connections Master

A Lie algebroid 03.04.16 is the object whose integrability this unit decides: the anchor-bracket-Leibniz structure axiomatized there is the input, and the monodromy groups are computed from its isotropy algebras and the curvature of a connection on its Atiyah sequence. The integrability question is the precise sense in which an algebroid may or may not be the shadow of a global object.

The Lie functor 03.04.17 is the differentiation this unit inverts: that functor sends every Lie groupoid to its algebroid and is faithful, and the content here is exactly the measure of its failure to be essentially surjective. The monodromy obstruction is the cokernel, leaf by leaf, of "every algebroid arises as some $A(G)$."

Lie's third theorem 03.03.06 holds unconditionally for Lie algebras but its groupoid analogue is the false statement this unit corrects: where simply-connected Lie groups integrate every Lie algebra, the Weinstein groupoid integrates an algebroid only when the monodromy is uniformly discrete, and the contrast is the sharpest divergence between group and groupoid theory.

The Atiyah algebroid 03.05.22 of a principal bundle is the flagship transitive example: its Atiyah sequence carries the connection whose curvature's spherical periods are the monodromy, so the transitive integrability criterion is a Chern-Weil statement about exactly that sequence, and integrability becomes a discreteness condition on connection-curvature periods.

The Poisson bracket 05.02.02 supplies the motivating non-integrable family through the cotangent algebroid $T^{\ast }M$ of a Poisson manifold: its integration is the symplectic groupoid, and the Crainic-Fernandes monodromy obstruction is precisely what controls whether a Poisson manifold is integrable by a symplectic groupoid, linking this unit to the geometry of Poisson and symplectic structures.

Historical & philosophical context Master

Sophus Lie established around 1880 that every finite-dimensional Lie algebra integrates to a Lie group, the third of his foundational theorems, and for nearly a century the integrability of infinitesimal structures was treated as automatic ^{[Lie 1880]}. When Jean Pradines extended Lie's framework to differentiable groupoids and algebroids in the mid-1960s, he announced in a 1966 Comptes Rendus note that the third theorem also held in this generality — every algebroid integrates ^{[Pradines 1966]}. The announcement was widely accepted but the proof was never published in full.

The claim stood until 1985, when Rui Almeida and Pierre Molino, studying transversally complete foliations and Atiyah sequences, produced a transitive Lie algebroid that does not integrate, the first counterexample to Pradines' statement ^{[Almeida-Molino 1985]}. Kirill Mackenzie systematized the transitive theory and identified the obstruction with the curvature periods of the adjoint sequence in his 1987 lecture notes and 2005 monograph ^{[Mackenzie Ch. IV §4]}. The general problem was solved by Marius Crainic and Rui Loja Fernandes in 2003, who constructed the Weinstein groupoid of $A$-paths, defined the monodromy groups in the centres of the isotropy algebras, and proved that integrability is equivalent to their uniform discreteness ^{[Crainic-Fernandes 2003]}.

Bibliography Master

@article{pradines1966,
  author  = {Pradines, Jean},
  title   = {Th\'eorie de Lie pour les groupo\"ides diff\'erentiables. Relations entre propri\'et\'es locales et globales},
  journal = {Comptes Rendus de l'Acad\'emie des Sciences, Paris},
  volume  = {263},
  pages   = {907--910},
  year    = {1966}
}

@article{almeida-molino1985,
  author  = {Almeida, Rui and Molino, Pierre},
  title   = {Suites d'Atiyah et feuilletages transversalement complets},
  journal = {Comptes Rendus de l'Acad\'emie des Sciences, Paris, S\'er. I},
  volume  = {300},
  number  = {1},
  pages   = {13--15},
  year    = {1985}
}

@article{crainic-fernandes2003,
  author  = {Crainic, Marius and Fernandes, Rui Loja},
  title   = {Integrability of Lie brackets},
  journal = {Annals of Mathematics},
  volume  = {157},
  number  = {2},
  pages   = {575--620},
  year    = {2003}
}

@book{mackenzie2005,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of Lie Groupoids and Lie Algebroids},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {213},
  publisher = {Cambridge University Press},
  year      = {2005}
}

@book{mackenzie1987,
  author    = {Mackenzie, Kirill},
  title     = {Lie Groupoids and Lie Algebroids in Differential Geometry},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {124},
  publisher = {Cambridge University Press},
  year      = {1987}
}

@incollection{lie1880,
  author    = {Lie, Sophus},
  title     = {Theorie der Transformationsgruppen I},
  journal   = {Mathematische Annalen},
  volume    = {16},
  number    = {4},
  pages     = {441--528},
  year      = {1880}
}

---

Pradines integration and transitive integrability — differentiation always works, integration does not: an algebroid integrates to a Lie groupoid iff its monodromy groups ${\mathcal{N}}_x(A)\subseteq Z({\mathfrak{g}}_x)$ are locally uniformly discrete (Crainic-Fernandes), with the transitive obstruction a Chern-Weil period condition on the Atiyah-sequence curvature (Mackenzie) and the dense-period Almeida-Molino algebroid refuting Pradines' 1966 claim.