03.03.10 · modern-geometry / lie

Lie groupoid: source, target, smooth composition

shipped3 tiersLean: none

Anchor (Master): Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. I–II

Intuition Beginner

A group collects the symmetries of one object: every symmetry can be undone, and any two can be done in sequence. But many situations have symmetries that only make sense between specific points. A path in a space joins one point to another; it cannot be glued to a second path unless the second one starts where the first ends. A groupoid is the structure that handles exactly this: a collection of reversible arrows, each one running from a starting point to an ending point, that compose only when the endpoints match.

A Lie groupoid adds smoothness. The points live on a smooth space, the arrows live on a second smooth space, and everything in sight — reading off where an arrow starts, where it ends, reversing an arrow, and gluing two matching arrows — varies smoothly. So a Lie groupoid is to a group what a family of symmetries spread smoothly over a space is to a single symmetry.

Why bother? Because geometry is full of objects that are "the same" only locally or only along paths. Two fibres of a bundle, two points joined by a path, two leaves of a foliation: each carries a symmetry that lives between a pair of points rather than at one point. The groupoid packages all of these point-to-point symmetries into one smooth object you can do calculus on.

Visual Beginner

Picture a curved base surface with several marked points. Above it floats a second surface of arrows. Each arrow drops down to two marked points on the base: the one it starts at and the one it ends at. Two arrows can be joined head-to-tail only when the head of one lands on the tail of the next.

The two downward projections are the source map (where an arrow starts) and the target map (where it ends). Reversing every arrow swaps its start and end. Each point of the base has a "do-nothing" arrow that starts and ends there.

Worked example Beginner

Take a base set of three points, ${1, 2, 3}$ , and let the arrows be all ordered pairs: an arrow from $i$ to $j$ for every choice of $i$ and $j$ . Write the arrow from $i$ to $j$ as $(j, i)$ , read "from $i$ to $j$ ".

The source of $(j, i)$ is $i$ and the target is $j$ . To compose, the arrow $(k, j)$ followed by the arrow $(j, i)$ produces the arrow $(k, i)$ — start at $i$ , pass through $j$ , end at $k$ . The composition $(k, j) \cdot (j, i) = (k, i)$ works only because the middle label matches.

Count the arrows that start and end at the same point: $(1, 1)$ , $(2, 2)$ , $(3, 3)$ . These are the do-nothing arrows, one per point. The reverse of $(2, 1)$ is $(1, 2)$ , and composing them gives $(1, 1)$ , the do-nothing arrow at $1$ .

What this tells us: even with just three points, the arrows organize into a system where source, target, composition, units, and reverses all fit together. This pair-of-points example is the simplest groupoid that is not a group, and the smooth version of it is the first example in the formal theory.

Check your understanding Beginner

Formal definition Intermediate+

A groupoid is a small category in which every morphism is invertible. Spelled out: it is a set $G$ of arrows together with a set $M$ of objects, two maps $s, t : G \to M$ (the source and target), a partially defined multiplication, an inclusion of objects as identity arrows, and an inversion, satisfying the category axioms with all arrows invertible. Two arrows $g, h$ are composable when $s (g) = t (h)$ , and then $g h$ is an arrow with $s (g h) = s (h)$ and $t (g h) = t (g)$ .

A Lie groupoid is a groupoid $G ⇉ M$ — the double arrow denotes the pair $(s, t)$ of structure maps from the arrow manifold to the object manifold — in which ^{[Mackenzie Ch. I]}:

$G$ (arrows) and $M$ (objects) are smooth manifolds 03.02.02, with $M$ Hausdorff but $G$ permitted to be non-Hausdorff in some applications;
the source and target $s, t : G \to M$ are surjective submersions 03.02.03;
the set of composable pairs $G \times_{M} G := {(g, h) \in G \times G : s (g) = t (h)}$ is a smooth submanifold of $G \times G$ (it is the fibred product of $s$ and $t$ , and is a manifold because $s$ and $t$ are submersions);
the multiplication $m : G \times_{M} G \to G$ , $m (g, h) = g h$ , is smooth and associative whenever defined;
the unit map $u : M \to G$ , $x \mapsto 1_{x}$ , is a smooth embedding with $s (1_{x}) = t (1_{x}) = x$ , and $1_{t (g)} g = g = g 1_{s (g)}$ ;
the inversion $i : G \to G$ , $g \mapsto g^{- 1}$ , is smooth, with $s (g^{- 1}) = t (g)$ , $t (g^{- 1}) = s (g)$ , and $g^{- 1} g = 1_{s (g)}$ , $g g^{- 1} = 1_{t (g)}$ .

Here $G \times_{M} G$ denotes the fibred product just defined; $1_{x}$ denotes the unit arrow at $x$ ; and $G ⇉ M$ is the standard shorthand for the whole structure. The symbol $⋉$ , introduced below for action groupoids, denotes the semidirect-product-style arrow space $G \times M$ .

For each $x \in M$ the source fibre is $G_{x} := s^{- 1} (x)$ and the target fibre is $G^{x} := t^{- 1} (x)$ ; both are embedded submanifolds since $s$ and $t$ are submersions. The isotropy group at $x$ is $$ G_x^x := s^{-1}(x) \cap t^{-1}(x), $$ the arrows that start and end at $x$ . The orbit of $x$ is $t (s^{- 1} (x)) \subseteq M$ , the set of objects reachable from $x$ by an arrow. A Lie groupoid is transitive when it has a single orbit, equivalently when the anchor $(t, s) : G \to M \times M$ is a surjective submersion.

A non-example worth keeping in mind: dropping the submersion requirement on $s$ and $t$ gives a topological or merely set-theoretic groupoid in which $G \times_{M} G$ need not be a manifold and $m$ need not be smooth. The submersion condition is precisely what makes the fibred product and the partial multiplication live in the smooth category.

Key theorem with proof Intermediate+

Theorem (Isotropy groups are Lie groups). Let $G ⇉ M$ be a Lie groupoid and $x \in M$ . Then the isotropy group $G_{x}^{x} = s^{- 1} (x) \cap t^{- 1} (x)$ is an embedded Lie subgroup of the source fibre $G_{x} = s^{- 1} (x)$ ; in particular it is a Lie group under the restriction of the groupoid multiplication.

Proof. Because $s : G \to M$ is a submersion, $x$ is a regular value, so the source fibre $G_{x} = s^{- 1} (x)$ is an embedded submanifold of $G$ of dimension $dim G - dim M$ 03.02.03. Restrict the target map to this fibre: let $t_{x} := t ∣_{G_{x}} : G_{x} \to M$ . Then $$ G_x^x = s^{-1}(x) \cap t^{-1}(x) = t_x^{-1}(x). $$ The map $t_{x}$ is a submersion. Indeed, fix $g \in G_{x}^{x}$ and consider right translation by $g$ , $$ R_g : G^x = t^{-1}(x) \to G_x = s^{-1}(x), \qquad R_g(h) = hg, $$ which is defined because $s (h) = x = t (g)$ , is smooth as a restriction of $m$ , and has smooth inverse $h^{'} \mapsto h^{'} g^{- 1}$ . So $R_{g}$ is a diffeomorphism between the target fibre through $x$ and the source fibre through $x$ . Composing, $t_{x} = t \circ R_{g^{- 1}} \circ (inclusion)$ near $g$ realises $t_{x}$ locally as the submersion $t$ followed by a diffeomorphism, hence $t_{x}$ is a submersion at $g$ . As $g \in G_{x}^{x}$ was arbitrary, $x$ is a regular value of $t_{x}$ , and $G_{x}^{x} = t_{x}^{- 1} (x)$ is an embedded submanifold of $G_{x}$ .

The set $G_{x}^{x}$ is closed under multiplication and inversion: if $g, h$ both start and end at $x$ then $s (g h) = s (h) = x$ and $t (g h) = t (g) = x$ , and $s (g^{- 1}) = t (g) = x$ , $t (g^{- 1}) = s (g) = x$ . Multiplication restricted to $G_{x}^{x} \times G_{x}^{x}$ is total (every such pair is composable, since the middle object is $x$ ), and it is smooth and associative as a restriction of $m$ . Inversion restricts to a smooth involution. The unit $1_{x}$ lies in $G_{x}^{x}$ and is a two-sided identity. Thus $G_{x}^{x}$ is a smooth manifold with smooth, associative, total multiplication, a smooth two-sided unit, and smooth inverses: a Lie group 03.03.01. $□$

Bridge. This construction builds toward the linearised theory, where differentiating the source fibres at the units produces the Lie algebroid that plays the role the Lie algebra plays for a Lie group; it appears again in the gauge groupoid of a principal bundle 03.05.01, where the isotropy at each point recovers the structure group, and it connects to the group action picture 03.03.02, since the isotropy of an action groupoid is exactly the stabilizer subgroup. Putting these together, the isotropy groups are the pointwise symmetry data that the groupoid threads smoothly across the base, and reading them off is the first step in passing from the global groupoid to its infinitesimal model.

Exercises Intermediate+

Exercise 6 (hard, proof).

Let $H$ be a Lie group acting smoothly on $M$ . Prove that $s : H \times M \to M$ , $(h, x) \mapsto x$ , and $t : H \times M \to M$ , $(h, x) \mapsto h \cdot x$ , are both surjective submersions, so that $H ⋉ M$ is a Lie groupoid.

Hint

$s$ is a projection. For $t$ , fix $h$ and vary $x$ , or fix $x$ and vary $h$ along the orbit.

Answer

The source $s$ is the projection $H \times M \to M$ , a surjective submersion. For the target, write $t = a \circ (id_{H} \times id_{M})$ where $a : H \times M \to M$ is the action map; $t = a$ . For fixed $h$ , $x \mapsto h \cdot x$ is a diffeomorphism of $M$ (its inverse is $h^{- 1} \cdot$ ), so $t$ restricted to ${h} \times M$ is already a diffeomorphism onto $M$ . Hence $d t$ is surjective at every point and $t$ is a submersion; it is surjective because for any $y$ , $t (e, y) = y$ . The composable set ${((h_{2}, x_{2}), (h_{1}, x_{1})) : x_{2} = h_{1} \cdot x_{1}}$ is the graph of a smooth map, hence a submanifold diffeomorphic to $H \times H \times M$ , and composition $(h_{2} h_{1}, x)$ is smooth. So $H ⋉ M ⇉ M$ is a Lie groupoid.

Lean formalization Intermediate+

lean_status: none — Mathlib has CategoryTheory.Groupoid as a purely algebraic structure (a category with all morphisms invertible) and an evolving smooth-manifold library (ContMDiff, submersions via mfderiv surjectivity), but no internalisation of a groupoid object in the smooth category. The intended statement is sketched below in pseudo-Lean to indicate the missing structure; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: target structure, not in Mathlib.
structure LieGroupoid
    (M : Type*) [Manifold M] (G : Type*) [Manifold G] where
  source  : G → M
  target  : G → M
  source_submersion : Submersion source
  target_submersion : Submersion target
  -- composable pairs form the fibred product manifold G ×_M G
  mul     : {p : G × G // source p.1 = target p.2} → G
  mul_smooth  : Smooth mul
  unit    : M → G
  unit_embedding : Embedding unit
  inv     : G → G
  inv_smooth  : Smooth inv
  -- associativity, unit, and inverse axioms ...

The first genuine obstacle is the fibred product: Mathlib lacks a packaged "fibred product of manifolds along a submersion is a manifold" lemma, which the smoothness of mul depends on.

Advanced results Master

Transitive Lie groupoids and the gauge groupoid. Let $P \to M$ be a principal $H$ -bundle 03.05.01 with structure group $H$ acting on the right, freely and properly. The gauge groupoid (Ehresmann's groupoïde des automorphismes) has arrow manifold $$ \frac{P \times P}{H}, \qquad H \cdot (p, q) = (p \cdot h, q \cdot h), $$ with source $s [p, q] = π (q)$ and target $t [p, q] = π (p)$ , where $π : P \to M$ is the bundle projection and $[p, q]$ denotes the $H$ -orbit. An arrow $[p, q]$ from $π (q)$ to $π (p)$ is precisely an $H$ -equivariant map between the fibres $P_{π (q)} \to P_{π (p)}$ . Composition is composition of equivariant maps; it is smooth because the quotient by the free proper $H$ -action is a smooth manifold. The gauge groupoid is transitive, and its isotropy at each point is a copy of $H$ . The construction is reversible: every transitive Lie groupoid arises as the gauge groupoid of a principal bundle, namely the bundle $t : s^{- 1} (x) \to M$ over any chosen base point $x$ , with structure group the isotropy $G_{x}^{x}$ . This is the groupoid form of the statement that transitive symmetry over a base is the same data as a principal bundle.

Fundamental groupoid. For a manifold $M$ , the fundamental groupoid $Π (M) ⇉ M$ has as arrows the homotopy classes (rel endpoints) of paths, with source the initial point and target the terminal point 03.12.08. When $M$ is a smooth manifold $Π (M)$ carries a smooth structure for which it is a Lie groupoid; it is transitive precisely when $M$ is connected, and its isotropy at $x$ is the fundamental group $π_{1} (M, x)$ , a discrete Lie group. The source fibres are copies of the universal cover $M$ .

Bundle of groups and the units. At the opposite extreme from a transitive groupoid sits a groupoid in which $s = t$ . Then every arrow is an automorphism of a single object, and the groupoid is a smoothly varying family of Lie groups over $M$ — a bundle of Lie groups. The identity (unit) groupoid $M ⇉ M$ , with only the units $1_{x}$ as arrows and $s = t = id_{M}$ , is the extreme case where each isotropy is the one-element group; it is the groupoid that remembers only the base manifold.

Orbits, the anchor, and the orbit foliation. The anchor $(t, s) : G \to M \times M$ has image an equivalence relation on $M$ (the orbit relation). For a Lie groupoid the orbits are immersed submanifolds and partition $M$ ; when the groupoid has locally constant orbit dimension they assemble into a foliation. The tangent foliation groupoid of a foliation $F$ — the holonomy and monodromy groupoids — realises this picture, with arrows recording holonomy of paths within leaves; these are the Lie groupoids whose orbits are the leaves of $F$ .

Differentiation to a Lie algebroid. Restricting the tangent spaces of the source fibres to the units, $A := ker (d s) ∣_{M} = ⨆_{x} T_{1_{x}} (s^{- 1} (x))$ , produces a vector bundle over $M$ with a bracket on its sections and an anchor $ρ = d t ∣_{A} : A \to T M$ . This is the Lie algebroid of $G$ , the groupoid analogue of the Lie algebra of a Lie group; the symbol $⨆$ denotes the disjoint union assembling the fibrewise tangent data into a bundle. Lie's theorems partially transfer: every Lie groupoid differentiates to a Lie algebroid, but not every Lie algebroid integrates to a Lie groupoid (the integrability obstruction of Crainic-Fernandes).

Synthesis. A Lie groupoid $G ⇉ M$ is a single smooth object that carries a symmetry between each pair of related points rather than at one point, and its anatomy is read off from the source and target submersions: the source fibres are smooth manifolds, the composable pairs form the fibred product manifold $G \times_{M} G$ , the partial multiplication is smooth and associative, and the isotropy groups $G_{x}^{x}$ are honest Lie groups varying over the base. The examples organise the whole theory along one axis from the unit groupoid $M ⇉ M$ (every isotropy the one-element group, only the diagonal of relations) through the action groupoid $H ⋉ M$ (isotropy the stabilizers) and the fundamental groupoid $Π (M)$ (isotropy $π_{1}$ , source fibres the universal cover) to the transitive case; the pair groupoid $M \times M$ and the gauge groupoid $(P \times P) / H$ exhaust transitivity and show that transitive Lie groupoids are exactly principal bundles seen as groupoids. Differentiation sends each Lie groupoid to its Lie algebroid, the bracket-and-anchor structure on the source-fibre tangents at the units, and the failure of every algebroid to integrate is the precise sense in which Lie's third theorem breaks for groupoids. Across these constructions the groupoid is the geometric home for symmetries that are intrinsically relational — fibres of bundles, points joined by paths, leaves of foliations — and the smoothness of source, target, and composition is what lets the entire relational symmetry be differentiated, quotiented, and glued.

Full proof set Master

Proposition (Composable pairs form a manifold and multiplication descends smoothly). Let $G ⇉ M$ be a Lie groupoid with $s, t : G \to M$ surjective submersions. Then $G \times_{M} G = {(g, h) : s (g) = t (h)}$ is an embedded submanifold of $G \times G$ of dimension $2 dim G - dim M$ , and the partial multiplication is a smooth map $m : G \times_{M} G \to G$ .

Proof. Consider the smooth map $$ \Phi : G \times G \to M \times M, \qquad \Phi(g, h) = (s(g), t(h)). $$ Its differential at $(g, h)$ is $d s_{g} \oplus d t_{h}$ on $T_{g} G \oplus T_{h} G \to T_{s (g)} M \oplus T_{t (h)} M$ . Since $s$ and $t$ are submersions, $d s_{g}$ and $d t_{h}$ are each surjective, so $d Φ_{(g, h)}$ is surjective onto $T_{s (g)} M \oplus T_{t (h)} M$ ; thus $Φ$ is a submersion. The diagonal $Δ = {(a, a) : a \in M} \subseteq M \times M$ is an embedded submanifold of codimension $dim M$ . The preimage of a submanifold under a submersion is a submanifold of the same codimension 03.02.03, so $$ G \times_M G = \Phi^{-1}(\Delta) $$ is an embedded submanifold of $G \times G$ of codimension $dim M$ , hence of dimension $2 dim G - dim M$ .

For smoothness of $m$ : by the groupoid axioms $m$ is defined on $G \times_{M} G$ , and the defining clause of a Lie groupoid requires $m : G \times_{M} G \to G$ to be smooth. The content here is that the domain $G \times_{M} G$ is a manifold, established above, so "smooth" is meaningful: $m$ is the restriction to the submanifold $G \times_{M} G$ of a map, and smoothness is checked in charts adapted to the submanifold. The associativity identity $m (m (g, h), k) = m (g, m (h, k))$ holds wherever both sides are defined, i.e. on the iterated fibred product $G \times_{M} G \times_{M} G$ , which by the same submersion argument is an embedded submanifold of $G^{3}$ . $□$

Proposition (Transitive Lie groupoids are gauge groupoids). Let $G ⇉ M$ be a transitive Lie groupoid and fix $x_{0} \in M$ . Set $P := s^{- 1} (x_{0})$ and $H := G_{x_{0}}^{x_{0}}$ . Then $t ∣_{P} : P \to M$ is a principal $H$ -bundle, and $G$ is isomorphic as a Lie groupoid to the gauge groupoid $(P \times P) / H$ .

Proof sketch. By transitivity the anchor $(t, s)$ is a surjective submersion, so $t ∣_{P} : P = s^{- 1} (x_{0}) \to M$ is a surjective submersion. The isotropy $H = G_{x_{0}}^{x_{0}}$ acts on $P$ on the right by $p \cdot h = p h$ (defined since $s (p) = x_{0} = t (h)$ , and $s (p h) = s (h) = x_{0}$ keeps $p h \in P$ ); the action is free because $p h = p$ forces $h = 1_{x_{0}}$ , and proper because $t ∣_{P}$ is a submersion with the orbits exactly its fibres. So $P \to M$ is a principal $H$ -bundle. Define $Ψ : (P \times P) / H \to G$ by $[p, q] \mapsto p q^{- 1}$ , well defined since $(p h) (q h)^{- 1} = p h h^{- 1} q^{- 1} = p q^{- 1}$ , and smooth because it factors the smooth map $(p, q) \mapsto p q^{- 1}$ through the quotient. It intertwines sources and targets: $s (p q^{- 1}) = s (q^{- 1}) = t (q) = (t ∣_{P}) (q)$ and $t (p q^{- 1}) = t (p) = (t ∣_{P}) (p)$ . It is a bijection on arrows by transitivity and freeness, with smooth inverse $g \mapsto [g q, q]$ for any $q \in P$ with $t (q) = s (g)$ . Hence $Ψ$ is an isomorphism of Lie groupoids. $□$

Proposition (Inversion is a diffeomorphism). The inversion $i : G \to G$ is a diffeomorphism with $i \circ i = id_{G}$ , interchanging $s$ and $t$ .

Proof. Inversion is smooth by axiom, and $i \circ i = id_{G}$ because $(g^{- 1})^{- 1} = g$ in any groupoid. A smooth involution is its own smooth inverse, hence a diffeomorphism. The identities $s \circ i = t$ and $t \circ i = s$ are the source/target clauses of the inversion axiom. $□$

Connections Master

A Lie group 03.03.01 is exactly a Lie groupoid with a single object: when $M$ is a point, $s = t$ is forced, the composable pairs are all of $G \times G$ , and the groupoid axioms collapse to the Lie-group axioms. The isotropy theorem of this unit then reduces to the tautology that the group is its own isotropy.
A smooth group action 03.03.02 produces the action groupoid $H ⋉ M$ , and the dictionary is exact: orbits of the groupoid are orbits of the action, isotropy groups are stabilizers, and transitivity of the groupoid is transitivity of the action. The groupoid viewpoint is what lets quotients $M / H$ that are badly behaved as spaces be replaced by the well-behaved groupoid $H ⋉ M$ .
A principal bundle 03.05.01 is the same data as a transitive Lie groupoid via the gauge-groupoid construction proved above: the structure group becomes the isotropy, and bundle reductions correspond to subgroupoids. This is the groupoid translation of the bundle theory used throughout gauge theory.
The fundamental groupoid 03.12.08 is the homotopy-theoretic Lie groupoid whose isotropy is $π_{1}$ and whose source fibres are universal covers; it is the prototype for thinking of paths-up-to-homotopy as invertible arrows and supplies the topological end of the example list.

Historical & philosophical context Master

The groupoid concept was isolated by Heinrich Brandt in 1927 in the arithmetic of quaternion ideals, and the differentiable version was introduced by Charles Ehresmann in 1959 as part of his program on differentiable categories and the structure of fibre bundles ^{[Ehresmann 1959]}. Ehresmann's groupoïdes différentiables arose directly from his work on connections and the holonomy of foliations; the gauge groupoid of a principal bundle is his construction. Jean Pradines, Ehresmann's student, developed the infinitesimal counterpart in a sequence of 1966–1968 notes, introducing what are now called Lie algebroids and stating the differentiation procedure from groupoids to algebroids ^{[Pradines 1966]}.

The theory was systematised by Kirill Mackenzie, whose 1987 lecture notes and 2005 monograph gave the field its standard definitions, the transitive-groupoid/principal-bundle correspondence, and the connection and cohomology theory ^{[Mackenzie 1987; Mackenzie Ch. I–II]}. The foliation-theoretic side — holonomy and monodromy groupoids — was advanced by Alan Connes in noncommutative geometry, where the groupoid $C^{*}$ -algebra replaces the often-pathological leaf space, and by Ieke Moerdijk and Janez Mrcun, whose 2003 text presents Lie groupoids as the organising language for foliations ^{[Moerdijk-Mrcun 2003]}. The integrability question — which Lie algebroids arise from Lie groupoids — was settled by Marius Crainic and Rui Loja Fernandes in 2003, identifying the monodromy obstruction that has no Lie-group analogue.

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}
}

@incollection{ehresmann1959,
  author    = {Ehresmann, Charles},
  title     = {Cat\'egories topologiques et cat\'egories diff\'erentiables},
  booktitle = {Colloque de G\'eom\'etrie Diff\'erentielle Globale (Bruxelles, 1958)},
  pages     = {137--150},
  publisher = {Centre Belge de Recherches Math\'ematiques},
  year      = {1959}
}

@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{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}
}

@article{brandt1927,
  author    = {Brandt, Heinrich},
  title     = {\"Uber eine Verallgemeinerung des Gruppenbegriffes},
  journal   = {Mathematische Annalen},
  volume    = {96},
  pages     = {360--366},
  year      = {1927}
}

Lie groupoid — a manifold of arrows over a manifold of objects, with smooth source/target submersions and smooth partial composition; the geometric home for relational symmetry and the global object whose linearisation is a Lie algebroid.

Prerequisites

03.02.02
03.02.03
03.03.01
03.03.02

Tier anchors

beginner: arrows between points that remember their endpoints — a symmetry that varies from point to point
intermediate: Moerdijk-Mrcun — Introduction to Foliations and Lie Groupoids Ch. 5
master: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. I–II

References

Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. I–II, Lie groupoids and their structure
Moerdijk-Mrcun — Introduction to Foliations and Lie Groupoids · Ch. 5, Lie groupoids
Mackenzie — Lie Groupoids and Lie Algebroids in Differential Geometry · LMS Lecture Notes 124 (1987), foundational definitions
Ehresmann — Catégories topologiques et catégories différentiables · Colloque de Géométrie Différentielle Globale (1959)

Estimated time

beginner: 18m
intermediate: 50m
master: 80m