03.04.17 · modern-geometry / differential-forms

Lie functor: differentiating a Lie groupoid to its Lie algebroid

shipped3 tiersLean: none

Anchor (Master): Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §3 + Ch. IV §1; Crainic-Fernandes — Integrability of Lie brackets (Ann. Math. 157)

Intuition Beginner

A Lie group is a smooth family of symmetries, and you can shrink it down to its infinitesimal version, the Lie algebra: sit at the identity and look at the directions you can move away from it. The bracket of two such directions records how the two motions fail to commute. This shrinking is a one-way street that loses the large-scale shape but keeps the local rule.

A Lie groupoid is the version where symmetries run between pairs of points spread over a base space, rather than at one fixed object. It too has an infinitesimal version, called its Lie algebroid. The Lie functor is the machine that performs the shrinking: feed it a Lie groupoid, and it hands back the algebroid. The base space stays the same; over each point you now keep only the directions you can leave the "do-nothing" arrow in.

Why bother building one named machine for this? Because the same machine handles every example at once. A space of pairs becomes the tangent bundle. A group acting on a space becomes an action algebroid. A symmetry of a bundle becomes its Atiyah algebroid. Each global object differentiates by the very same rule, and the functor is the rule.

Visual Beginner

Picture a base surface with a "do-nothing" arrow sitting at each point. Floating above is the full space of arrows. To differentiate, zoom in near one do-nothing arrow and keep only the directions that move it while leaving its starting point pinned.

Those pinned directions, collected over every base point, form a new bundle: the algebroid. Each direction also casts a downward shadow onto the base, recording how it nudges the point it sits over. Two directions combine by spreading each one out into a flow of arrows, bracketing the flows, and zooming back in. The shadow map is the anchor, and the combining rule is the bracket.

Worked example Beginner

Take the base to be the line of real numbers, and let the arrows be all ordered pairs of points: an arrow from $x$ to $y$ for every choice. This is the pair groupoid of the line. The do-nothing arrow at $x$ is the pair $(x, x)$ , sitting on the diagonal.

Now differentiate. Pin the starting point and ask which directions move the pair $(x, x)$ . Moving the second slot up gives a direction; moving it down gives the opposite. So over each point $x$ you get one independent direction of motion, the change in the target. That single direction is just a tangent direction to the line at $x$ .

Collecting one tangent direction over every point of the line reproduces the tangent bundle of the line. The downward shadow of "raise the target by one unit" is exactly "move the base point up by one unit," so the anchor is the identity here.

What this tells us: the pair groupoid of a space differentiates to the tangent bundle of that space, with anchor the identity map. The biggest groupoid you can build over a space shrinks to the most basic algebroid, the tangent bundle. The Lie functor turns the global "all pairs" object into the local "all tangent directions" object.

Check your understanding Beginner

Formal definition Intermediate+

Let $G ⇉ M$ be a Lie groupoid 03.03.10 with source and target $s, t : G \to M$ and unit embedding $u : M \to G$ , $x \mapsto 1_{x}$ . Write $T^{s} G := ker (d s) \subseteq T G$ for the subbundle of vectors tangent to the source fibres; because $s$ is a submersion, $T^{s} G$ is a vector bundle over $G$ of rank $dim G - dim M$ .

The Lie algebroid of $G$ is the vector bundle $$ A(G) := \big(T^s G\big)\big|{u(M)} = \ker(ds)|{u(M)} \longrightarrow M, $$ the restriction of $T^{s} G$ to the unit submanifold, identified with $M$ through $u$ . Its fibre at $x$ is $A (G)_{x} = T_{1_{x}} (s^{- 1} (x))$ , the tangent space at the unit $1_{x}$ to the source fibre $G_{x} = s^{- 1} (x)$ . The anchor is $$ \rho := dt|_{A(G)} : A(G) \longrightarrow TM, $$ the restriction of the target differential, which makes sense because $t$ maps $G$ to $M$ and $d t$ carries $A (G)_{x} \subseteq T_{1_{x}} G$ into $T_{x} M$ ^{[Mackenzie Ch. III §3]}.

The bracket comes from right-invariant vector fields. For $g \in G$ with $t (g) = y$ , right translation $R_{g} : s^{- 1} (y) \to s^{- 1} (s (g))$ , $h \mapsto h g$ , is a diffeomorphism between source fibres. A vector field $X$ on $G$ tangent to the source fibres (so $X \in Γ (T^{s} G)$ ) is right-invariant when $(R_{g})_{*} X_{h} = X_{h g}$ for all composable $h, g$ . Each section $X \in Γ (A (G))$ extends to a unique right-invariant field $X \in Γ (T^{s} G)$ by $$ \overrightarrow X_g := (R_g)\ast, X{t(g)} \in T_g\big(s^{-1}(s(g))\big), $$ where the right-translate of the value $X_{t (g)} \in A (G)_{t (g)} = T_{1_{t (g)}} (s^{- 1} (t (g)))$ lands in $T_{g} (s^{- 1} (s (g)))$ . This is a bijection $Γ (A (G)) ≅ X_{inv}^{s} (G)$ between algebroid sections and right-invariant source-fibre fields. The bracket on $Γ (A (G))$ is defined by $$ \overrightarrow{[X, Y]} := \big[\overrightarrow X, \overrightarrow Y\big], $$ the ordinary Lie bracket of vector fields on $G$ , restricted to the units. The theorem below shows the right side is again right-invariant and source-fibre-tangent, so the bracket is well-defined.

A non-example clarifies why source-fibre tangency is needed: an arbitrary vector field on $G$ has no reason to be tangent to the source fibres, and its bracket with another need not restrict to $A (G)$ . The construction uses $ker (d s)$ , not all of $T G$ , precisely so that right translation along $t$ -related arrows acts and the bracket stays inside the fibre directions.

The convention here is right-invariance and $A (G) = ker (d s) ∣_{M}$ with anchor $d t$ , following Mackenzie. The mirror convention $ker (d t) ∣_{M}$ with left-invariant fields and anchor $d s$ produces an isomorphic algebroid through inversion $i : G \to G$ ; the two differ by the sign that inversion introduces, and this unit fixes the right-invariant convention throughout.

Key theorem with proof Intermediate+

Theorem (The bracket of right-invariant fields is right-invariant). Let $G ⇉ M$ be a Lie groupoid. If $X, Y \in Γ (T^{s} G)$ are right-invariant and tangent to the source fibres, then so is their Lie bracket $[X, Y]$ . Consequently the formula $[X, Y] := [X, Y]$ defines an $R$ -bilinear, antisymmetric bracket on $Γ (A (G))$ , and $(A (G), [\cdot, \cdot], ρ)$ is a Lie algebroid 03.04.16.

Proof. Fix a composable arrow datum: $g \in G$ with $t (g) = y$ , and right translation $R_{g} : s^{- 1} (y) \to s^{- 1} (s (g))$ , a diffeomorphism. Tangency to source fibres is preserved because $s \circ R_{g} = s \circ R_{g}$ is constant on each fibre image, so $R_{g}$ maps $T^{s} G ∣_{s^{- 1} (y)}$ to $T^{s} G ∣_{s^{- 1} (s (g))}$ ; thus $(R_{g})_{*}$ acts on source-fibre fields ^{[Mackenzie Ch. III §3]}.

The Lie bracket of vector fields is natural under diffeomorphisms: for any diffeomorphism $φ$ and fields $V, W$ on which $φ$ acts, $φ_{*} [V, W] = [φ_{*} V, φ_{*} W]$ . Right-invariance of $X$ means $(R_{g})_{*} X = X$ on the relevant fibres, and likewise for $Y$ . Apply naturality with $φ = R_{g}$ : $$ (R_g)\ast,[\overrightarrow X, \overrightarrow Y] = \big[(R_g)\ast \overrightarrow X,, (R_g)\ast \overrightarrow Y\big] = [\overrightarrow X, \overrightarrow Y]. $$ So $[X, Y]$ is right-invariant. It is tangent to the source fibres because the bracket of two fields tangent to a foliation (here the source-fibre foliation, integrable since the fibres of the submersion $s$ are its leaves) is again tangent to it: if $d s (X) = 0$ and $d s (Y) = 0$ then $d s ([X, Y]) = [d s X, d s Y] = 0$ , using that $s$ -relatedness to the zero field is preserved by brackets. Hence $[X, Y] \in Γ (T^{s} G)$ is right-invariant, so it equals $Z$ for a unique $Z \in Γ (A (G))$ , and we set $[X, Y] := Z = [\overrightarrow X, \overrightarrow Y]\big|{u(M)}$.

This bracket is $R$ -bilinear and antisymmetric because the vector-field bracket is, and it satisfies the Jacobi identity because the vector-field bracket does and the bijection $X \mapsto X$ is $R$ -linear: $$ \overrightarrow{[X, [Y, Z]]} = [\overrightarrow X, [\overrightarrow Y, \overrightarrow Z]], $$ and the Jacobi identity for vector fields transports back. For the Leibniz rule, let $f \in C^{\infty} (M)$ and pull it back along $t$ to $f \circ t \in C^{\infty} (G)$ . The right-invariant extension of $f Y$ is $f Y = (f \circ t) Y$ , since right translation $R_{g}$ preserves $t$ -fibres and $t \circ R_{g} = t$ . Then $$ \overrightarrow{[X, fY]} = [\overrightarrow X, (f\circ t)\overrightarrow Y] = (f\circ t)[\overrightarrow X, \overrightarrow Y] + \big(\overrightarrow X(f\circ t)\big)\overrightarrow Y. $$ Restricting to a unit $1_{x}$ , the coefficient $X (f \circ t)$ evaluated at $1_{x}$ is $X_{x} (f \circ t) = (d t X_{x}) f = (ρ (X) f) (x)$ , since $d t$ sends $X_{x} \in A (G)_{x}$ to $ρ (X)_{x} \in T_{x} M$ . Therefore $$ [X, fY] = f[X, Y] + (\rho(X)f),Y, $$ the Leibniz rule with anchor $ρ = d t ∣_{A (G)}$ . All axioms of 03.04.16 hold, so $(A (G), [\cdot, \cdot], ρ)$ is a Lie algebroid. $□$

Bridge. This construction builds toward the functoriality statement, where a smooth groupoid morphism differentiates to an algebroid morphism and the assignment $G \mapsto A (G)$ becomes a genuine functor; it appears again in the examples table, where the same right-invariant-field recipe produces $T M$ from the pair groupoid and $g$ from a Lie group, and it connects to the integrability question, where the failure of every algebroid to arise this way is the Crainic-Fernandes obstruction. The right-invariance lemma is the technical core: it is the single fact that converts the smooth global composition of arrows into a bracket on the infinitesimal bundle, and the rest of the algebroid structure follows from naturality of the vector-field bracket under the right translations.

Exercises Intermediate+

Exercise 6 (hard, proof).

Let $φ : G \to H$ be a morphism of Lie groupoids over a map $φ_{0} : M \to N$ (so $s_{H} \circ φ = φ_{0} \circ s_{G}$ , and $φ$ commutes with target, units, multiplication). Show that $d φ$ restricts to a vector-bundle morphism $A (φ) : A (G) \to A (H)$ over $φ_{0}$ that intertwines the anchors, $ρ_{H} \circ A (φ) = d φ_{0} \circ ρ_{G}$ .

Hint

$φ$ sends units to units and source fibres into source fibres because it commutes with $s$ ; differentiate at a unit.

Answer

Since $s_{H} \circ φ = φ_{0} \circ s_{G}$ , the map $φ$ carries each source fibre $s_{G}^{- 1} (x)$ into $s_{H}^{- 1} (φ_{0} (x))$ , and since $φ (1_{x}) = 1_{φ_{0} (x)}$ it sends units to units. Differentiating at $1_{x}$ , $d φ$ maps $A (G)_{x} = T_{1_{x}} s_{G}^{- 1} (x)$ into $A (H)_{φ_{0} (x)} = T_{1_{φ_{0} (x)}} s_{H}^{- 1} (φ_{0} (x))$ . Define $A (φ) := d φ ∣_{A (G)}$ ; it is fibrewise linear and smooth, hence a bundle morphism over $φ_{0}$ . For the anchors, $t_{H} \circ φ = φ_{0} \circ t_{G}$ gives $d t_{H} \circ d φ = d φ_{0} \circ d t_{G}$ ; restricting to $A (G)$ yields $ρ_{H} \circ A (φ) = d φ_{0} \circ ρ_{G}$ .

Exercise 7 (hard, proof).

Prove that the anchor $ρ = d t ∣_{A (G)}$ is a bracket homomorphism, $ρ ([X, Y]) = [ρ (X), ρ (Y)]$ , directly from the right-invariant-field construction (not by invoking the abstract algebroid theorem).

Hint

Show $X$ is $t$ -related to $ρ (X)$ , then use that brackets of $t$ -related fields are $t$ -related.

Answer

The right-invariant field $X$ is $t$ -related to the vector field $ρ (X)$ on $M$ : at $g$ , $d t (X_{g}) = d t (R_{g})_{*} X_{t (g)} = d (t \circ R_{g}) X_{t (g)} = d t X_{t (g)} = ρ (X)_{t (g)}$ , using $t \circ R_{g} = t$ . Likewise $Y$ is $t$ -related to $ρ (Y)$ . The Lie bracket of $t$ -related fields is $t$ -related to the bracket of their counterparts, so $[X, Y]$ is $t$ -related to $[ρ (X), ρ (Y)]$ . But $[X, Y] = [X, Y]$ is also $t$ -related to $ρ ([X, Y])$ . Two vector fields on $M$ that are both $t$ -images of the same field on $G$ along the surjective submersion $t$ agree, so $ρ ([X, Y]) = [ρ (X), ρ (Y)]$ .

Lean formalization Intermediate+

lean_status: none — Mathlib provides mfderiv for the differential of smooth maps and a developing TangentBundle and vector-bundle library, but it has neither a LieGroupoid object nor the LieAlgebroid target structure, so the Lie functor cannot yet be stated. The sketch below indicates the missing pieces; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: target construction, not in Mathlib.
variable {M G : Type*} [Manifold M] [Manifold G]
variable (𝒢 : LieGroupoid M G)   -- source s, target t (submersions), unit u, mul, inv

-- the algebroid bundle: ker(ds) restricted to the units u(M)
def LieAlgebroidOf : VectorBundle M :=
  (kernelSubbundle (mfderiv 𝒢.source)).restrict (𝒢.unit '' Set.univ)

def anchorOf : BundleMorphism (LieAlgebroidOf 𝒢) (TangentBundle M) :=
  (mfderiv 𝒢.target).restrict _      -- ρ = dt|_A

-- the bracket via right-invariant source-fibre fields
def rightInvariantExtend (X : Section (LieAlgebroidOf 𝒢)) : VectorField G := ...
theorem bracket_rightInvariant (X Y) :
    RightInvariant (vfBracket (rightInvariantExtend X) (rightInvariantExtend Y))
-- ⇒ defines bracketOf, then LieAlgebroidOf 𝒢 is a LieAlgebroid

The first genuine obstacle is rightInvariantExtend and the lemma bracket_rightInvariant: Mathlib has no notion of a vector field tangent to the fibres of a submersion that is invariant under partial right translation, and no lemma that the vector-field bracket of two such fields is again of that form. That lemma is the mathematical content of the construction.

Advanced results Master

The Lie functor. The assignment $G \mapsto A (G)$ extends to a functor from the category of Lie groupoids to the category of Lie algebroids. On objects it is the construction $A (G) = ker (d s) ∣_{M}$ with anchor $d t$ and bracket of right-invariant source-fibre fields. On morphisms, a smooth groupoid morphism $φ : G \to H$ over $φ_{0} : M \to N$ restricts $d φ$ to $A (φ) : A (G) \to A (H)$ , a base-map- $φ_{0}$ morphism of algebroids that intertwines anchors and brackets. Composition and identities are preserved because the differential is. This is Pradines' functor ^{[Pradines 1967]}, the groupoid-level realisation of the classical assignment of a Lie algebra to a Lie group.

The examples table. Differentiation acts uniformly across the standard families, and reading the table is the fastest route into algebroid theory. The pair groupoid $M \times M$ differentiates to the tangent algebroid $T M$ with anchor the identity, since $ker (d s) ∣_{Δ}$ at a diagonal point is the first-slot tangent space and $d t$ reads it off. The action groupoid $H ⋉ M$ of a $h$ -action differentiates to the action algebroid $h ⋉ M$ 03.03.11, the product bundle $M \times h$ whose anchor sends $ξ$ to its fundamental vector field; the right-invariant fields here are exactly the right-invariant fields of $H$ spread over $M$ . The gauge groupoid $(P \times P) / H$ of a principal bundle differentiates to the Atiyah algebroid $T P / H$ 03.05.22, with the anchor the descent of $d π$ and kernel the adjoint bundle; the source-fibre fields descend to $H$ -invariant vector fields on $P$ . The Lie group $G$ over a point differentiates to its Lie algebra $g$ 03.03.01, with zero anchor, recovering the classical Lie functor as the one-object case.

Lie I and Lie II for groupoids. Two of Lie's three theorems transfer. Lie I holds in the form that an $s$ -connected, $s$ -simply-connected Lie groupoid is determined up to isomorphism by its algebroid: if $G$ and $G^{'}$ have $s$ -simply-connected source fibres and isomorphic algebroids, they are isomorphic. Lie II holds as an integration of morphisms: given an algebroid morphism $A (G) \to A (H)$ and assuming $G$ has $s$ -simply-connected source fibres, it integrates to a unique groupoid morphism $G \to H$ , the analogue of integrating a Lie-algebra map to a group map between simply-connected groups. Both follow from path-lifting in the source fibres together with the monodromy of the source-simply-connected cover.

Lie III fails: the Crainic-Fernandes obstruction. Lie III — that every Lie algebra integrates to a Lie group — does not transfer. Not every Lie algebroid is $A (G)$ for a Lie groupoid $G$ . Crainic and Fernandes ^{[Crainic-Fernandes 2003]} constructed, for any algebroid $A$ , the Weinstein groupoid $G (A)$ of $A$ -paths modulo $A$ -homotopy, a topological groupoid that integrates $A$ exactly when it is smooth. Smoothness is governed by two monodromy groups $N_{x} (A) \subseteq Z (g_{x})$ sitting in the centres of the isotropy Lie algebras, measured by spherical periods of a transverse variation; $A$ is integrable if and only if these monodromy groups are locally uniformly discrete. When they accumulate at zero the quotient $G (A)$ is not a manifold and $A$ does not integrate. The standard example is the algebroid built from a closed $2$ -form with a dense group of spherical periods, which has no integrating groupoid.

Synthesis. The Lie functor $G \mapsto A (G)$ differentiates every Lie groupoid into a Lie algebroid by one recipe: take $ker (d s)$ along the units, anchor by $d t$ , and bracket by transporting the vector-field bracket through the bijection between algebroid sections and right-invariant source-fibre fields, the right-invariance of which is the single load-bearing lemma. The same recipe specialises across the entire example spectrum, sending the pair groupoid to $T M$ , the action groupoid to $h ⋉ M$ , the gauge groupoid to the Atiyah algebroid, and a one-object Lie group to its Lie algebra, so the classical group-to-algebra functor is the point-base case of one general construction. On morphisms the functor is the restricted differential $A (φ) = d φ ∣_{A (G)}$ , which preserves anchors and brackets, making $A (\cdot)$ a genuine functor rather than an object-level assignment. Of Lie's three theorems, the first two transfer once source fibres are taken simply connected, giving uniqueness and morphism-integration, but the third fails: the Crainic-Fernandes monodromy obstruction in the centres of the isotropy algebras blocks integration of some algebroids, a phenomenon with no Lie-group counterpart. The functor is therefore faithful but not essentially surjective, and the precise measure of its failure to be surjective is the monodromy of the Weinstein groupoid.

Full proof set Master

Proposition (Functoriality of $A (\cdot)$ ). Let $φ : G \to H$ and $ψ : H \to K$ be morphisms of Lie groupoids over $φ_{0} : M \to N$ and $ψ_{0} : N \to Q$ . Then $A (ψ \circ φ) = A (ψ) \circ A (φ)$ and $A (id_{G}) = id_{A (G)}$ , so $A (\cdot)$ is a functor; moreover each $A (φ)$ is a morphism of Lie algebroids, intertwining anchors and brackets.

Proof. On objects $A (G) = ker (d s_{G}) ∣_{u_{G} (M)}$ . A morphism $φ$ satisfies $s_{H} \circ φ = φ_{0} \circ s_{G}$ , $t_{H} \circ φ = φ_{0} \circ t_{G}$ , and $φ \circ u_{G} = u_{H} \circ φ_{0}$ , so $φ$ carries source fibres to source fibres and units to units, and $d φ$ restricts to $A (φ) : A (G) \to A (H)$ . The chain rule gives $d (ψ \circ φ) = d ψ \circ d φ$ , and restricting both sides to $A (G)$ gives $A (ψ \circ φ) = A (ψ) \circ A (φ)$ ; the identity morphism differentiates to the identity. Anchor intertwining $ρ_{H} \circ A (φ) = d φ_{0} \circ ρ_{G}$ follows from $d t_{H} \circ d φ = d φ_{0} \circ d t_{G}$ restricted to $A (G)$ .

For brackets, $φ$ carries the right-invariant extension $X$ of $X \in Γ (A (G))$ to a field $φ$ -related to the right-invariant extension of $A (φ) \circ X$ along $φ_{0}$ , because $φ$ commutes with right translation: $φ \circ R_{g} = R_{φ (g)} \circ φ$ on source fibres, so $d φ X = A (φ) X$ in the sense of $φ$ -relatedness when $X$ is $φ_{0}$ -projectable. The vector-field bracket of $φ$ -related fields is $φ$ -related, so $d φ [X, Y]$ is $φ$ -related to $[A (φ) X, A (φ) Y]$ ; restricting to units gives $A (φ) [X, Y] = [A (φ) X, A (φ) Y]$ wherever the sections are $φ_{0}$ -related, which is the algebroid-morphism condition. $□$

Proposition (The pair groupoid differentiates to $T M$ ). Let $Pair (M) = M \times M ⇉ M$ with $s (y, x) = x$ , $t (y, x) = y$ , units $1_{x} = (x, x)$ . Then $A (Pair (M)) ≅ T M$ as Lie algebroids, with anchor the identity.

Proof. The source map is the second projection, so $ker (d s)_{(y, x)} = T_{y} M \oplus 0 \subseteq T_{y} M \oplus T_{x} M = T_{(y, x)} (M \times M)$ . At a unit $(x, x)$ this is $T_{x} M \oplus 0$ , and the identification $A (Pair (M))_{x} = T_{x} M$ , $v \mapsto (v, 0)$ , is a bundle isomorphism $A (Pair (M)) ≅ T M$ . The anchor $ρ = d t$ is the first projection, so $ρ (v, 0) = v$ , giving $ρ = id_{T M}$ . For the bracket, the right-invariant extension of a constant-coefficient field is computed through $R_{(z, y)} : s^{- 1} (y) \to s^{- 1} (z)$ , $(w, y) \mapsto (w, z)$ , which is the identity on the first slot; so the right-invariant extension of a vector field $V \in X (M)$ (read into the first slot) is the field $(y, x) \mapsto (V_{y}, 0)$ , whose bracket reproduces $[V, W]$ in the first slot. Restricting to units gives the vector-field bracket on $X (M)$ , so the algebroid bracket is the Lie bracket of $T M$ . $□$

Proposition (Source-simply-connected uniqueness, Lie I). Let $G$ and $G^{'}$ be Lie groupoids over $M$ with $s$ -connected, $s$ -simply-connected source fibres. If $A (G) ≅ A (G^{'})$ as Lie algebroids over $M$ , then $G ≅ G^{'}$ as Lie groupoids.

Proof sketch. An isomorphism $A (G) \to A (G^{'})$ is, by the integration of algebroid morphisms (Lie II), integrated to a groupoid morphism $Φ : G \to G^{'}$ over $id_{M}$ because the source fibres of $G$ are simply connected; the path-lifting that defines $Φ$ on each source fibre is single-valued exactly because $s$ -simple-connectivity kills the monodromy of the lift. The inverse algebroid isomorphism integrates to $Φ^{'} : G^{'} \to G$ , and $Φ^{'} \circ Φ$ integrates the identity algebroid morphism, hence equals $id_{G}$ by uniqueness of integration; likewise $Φ \circ Φ^{'} = id_{G^{'}}$ . So $Φ$ is an isomorphism. The argument is the groupoid form of the classical statement that simply-connected Lie groups with isomorphic Lie algebras are isomorphic, with the source-fibre cover playing the role of the universal cover ^{[Mackenzie Ch. IV §1]}. $□$

Connections Master

A Lie groupoid 03.03.10 is the input to this functor: its source-fibre tangents at the units form the algebroid bundle, its target differential is the anchor, and the right translations that define its composition are exactly what makes the bracket of right-invariant fields well-defined. The isotropy Lie groups of the groupoid differentiate to the isotropy Lie algebras that form the kernel of the anchor, so the entire fibrewise group data passes to fibrewise algebra data under one map.

A Lie algebroid 03.04.16 is the output: the construction here produces precisely the anchor-bracket-Leibniz structure axiomatised in that unit, and the abstract theorem there that the anchor is automatically a bracket homomorphism is, in this concrete setting, the statement that brackets of $t$ -related right-invariant fields are $t$ -related. The functor is the bridge that makes the abstract algebroid the differentiated shadow of a global groupoid rather than free-standing data.

A Lie group 03.03.01 and its Lie algebra are the one-object case of this functor: when the base is a point the source map is constant, $ker (d s)$ is all of the tangent bundle, the units collapse to the identity, and the algebroid is the Lie algebra with zero anchor. The classical group-to-algebra functor is therefore not an analogy but a special case, recovered by setting $M = {*}$ .

The action Lie groupoid and action Lie algebroid 03.03.11 are matched by this functor: differentiating $H ⋉ M$ yields $h ⋉ M$ , with the group action's fundamental vector fields becoming the algebroid anchor. The Atiyah algebroid 03.05.22 arises the same way from the gauge groupoid of a principal bundle, so the functor sends the transitive-groupoid/principal-bundle correspondence to the Atiyah-sequence picture of connections.

Lie's third theorem 03.03.06 holds unconditionally for Lie algebras but fails for Lie algebroids, and the precise content of that failure — the Crainic-Fernandes monodromy obstruction — is what makes this functor faithful but not essentially surjective. The contrast between the integration of groups and the obstructed integration of groupoids is the sharpest place where the groupoid theory departs from the group theory.

Historical & philosophical context Master

The differentiation of a differentiable groupoid to its infinitesimal object was constructed by Jean Pradines in a sequence of Comptes Rendus notes between 1966 and 1968, where the Lie algebroid and the functor differentiating groupoids to algebroids first appear ^{[Pradines 1967]}. Pradines worked in the lineage of Charles Ehresmann's differentiable groupoids and stated both that every groupoid differentiates and that the converse integrability holds, the second of which was later found to fail. The right-invariant-vector-field construction he used is the direct generalisation of the right-invariant-field model of the Lie algebra of a Lie group.

Kirill Mackenzie systematised the construction, the functoriality, and the relation to connections in his 1987 lecture notes and 2005 monograph, where the Lie functor and the integrability problem are given their standard form ^{[Mackenzie Ch. III §3]}. Ieke Moerdijk and Janez Mrcun's 2003 text presents the differentiation in the language of foliations and étale groupoids ^{[Moerdijk-Mrcun 2003]}. The integrability question that Pradines's converse left open was settled by Marius Crainic and Rui Loja Fernandes in 2003, who built the Weinstein groupoid of $A$ -paths and identified the monodromy obstruction in the centres of the isotropy algebras, showing that the Lie functor is not essentially surjective ^{[Crainic-Fernandes 2003]}.

Bibliography Master

@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}
}

@book{moerdijk-mrcun2003,
  author    = {Moerdijk, Ieke and Mr\v{c}un, Janez},
  title     = {Introduction to Foliations and Lie Groupoids},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {91},
  publisher = {Cambridge University Press},
  year      = {2003}
}

@article{pradines1967,
  author    = {Pradines, Jean},
  title     = {Th\'eorie de Lie pour les groupo\"ides diff\'erentiables. Calcul diff\'erentiel dans la cat\'egorie des groupo\"ides infinit\'esimaux},
  journal   = {Comptes Rendus de l'Acad\'emie des Sciences, Paris, S\'er. A},
  volume    = {264},
  pages     = {245--248},
  year      = {1967}
}

@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}
}

Lie functor — the differentiation $G \mapsto A (G) = ker (d s) ∣_{M}$ with anchor $d t$ and bracket of right-invariant source-fibre vector fields, sending each Lie groupoid to its Lie algebroid; functorial, faithful, and not essentially surjective, with the Crainic-Fernandes monodromy obstruction measuring the failure of Lie's third theorem.

Prerequisites

03.03.10
03.04.16
03.03.01

Tier anchors

beginner: the rule that turns a global system of arrows into its infinitesimal shadow, the way a Lie group becomes its Lie algebra
intermediate: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §3; Moerdijk-Mrcun Ch. 6
master: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. III §3 + Ch. IV §1; Crainic-Fernandes — Integrability of Lie brackets (Ann. Math. 157)

References

Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. III §3 and Ch. IV §1, the Lie algebroid of a Lie groupoid and the Lie functor
Moerdijk-Mrcun — Introduction to Foliations and Lie Groupoids · Ch. 6, the Lie algebroid of a Lie groupoid
Mackenzie — Lie Groupoids and Lie Algebroids in Differential Geometry · LMS Lecture Notes 124 (1987), differentiation of groupoids
Pradines — Théorie de Lie pour les groupoïdes différentiables · C. R. Acad. Sci. Paris 264 (1967), the Lie functor for groupoids
Crainic-Fernandes — Integrability of Lie brackets · Annals of Mathematics 157 (2003), failure of Lie III for algebroids

Estimated time

beginner: 18m
intermediate: 50m
master: 85m