03.05.21 · modern-geometry / bundles

Gauge groupoid of a principal bundle

shipped3 tiersLean: none

Anchor (Master): Mackenzie Ch. 1 §1.7 and Ch. 3; Ehresmann 1959; Atiyah 1957

Intuition Beginner

A principal bundle stacks a copy of a symmetry group over every point of a base space, and the copies are glued together as you move around. The honest question is: how do you compare the copy sitting over one point with the copy sitting over another? There is no preferred way to identify them, but there is a whole family of allowed identifications, and that family is itself a beautiful object.

The gauge groupoid collects every allowed comparison into one structure. For each pair of base points it stores all the ways the fibre over the first can be matched, respecting the symmetry, to the fibre over the second. Each such match is a reversible arrow from one point to the other. You can undo a match, and you can chain two matches when the middle point lines up.

Why bother? Because almost everything you do with a principal bundle — moving frames around, transporting data along paths, changing gauge — is really a statement about these comparisons. Packaging them as arrows lets you treat the bundle as a single smooth web of point-to-point symmetries that you can compose and differentiate.

So the slogan is short. A principal bundle is the raw symmetry spread over a base; its gauge groupoid is the catalogue of every way to slide that symmetry from one point to another.

Visual Beginner

Picture the base as a curved surface with two marked points. Above each marked point floats its fibre: a copy of the symmetry group, drawn as a small disc of arrows. An allowed comparison is a way to lay the disc over the first point onto the disc over the second so that the symmetry pattern is preserved.

Each comparison drops down to two base points: the one it starts at and the one it ends at. Reverse a comparison and the start and end swap. Each point has a do-nothing comparison of its fibre with itself. Two comparisons chain head-to-tail only when the end point of one is the start point of the next.

Worked example Beginner

Take the simplest case: the base is a flat disc, and over every point we glue the same group of rotations of the plane. Call a rotation by its angle. The bundle is the product of the disc with the circle of angles, and a point of the bundle is a base point together with a chosen angle.

Pick two base points, call them $A$ and $B$ . A comparison from $A$ to $B$ that respects the rotation symmetry is completely fixed once you say where one chosen angle goes, because the rest must shift by the same amount. So each comparison from $A$ to $B$ is just a fixed extra turn, a single angle, applied to everything.

Count them: the comparisons from $A$ to $B$ are in one-to-one correspondence with angles, one comparison per turn. Chaining a turn of $30$ degrees after a turn of $45$ degrees gives a turn of $75$ degrees. The do-nothing comparison at $A$ is the turn of $0$ degrees, and the reverse of a $30$ -degree comparison is a $- 30$ -degree comparison.

What this tells us: when the bundle is a plain product, an arrow of the gauge groupoid is a pair of base points together with one group element, the extra turn. The arrows form a base point, base point, group-element bookkeeping system. This product case is the model every other gauge groupoid is built to imitate locally.

Check your understanding Beginner

Formal definition Intermediate+

Let $π : P \to M$ be a principal $G$ -bundle 03.05.01, with $G$ a Lie group 03.03.01 acting on $P$ on the right, smoothly, freely, and properly, the fibres being the orbits of the action 03.03.02. The diagonal action of $G$ on $P \times P$ , $$ (p, q) \cdot h := (p \cdot h, ; q \cdot h), \qquad h \in G, $$ is again free and proper, so the quotient $$ \frac{P \times P}{G} $$ is a smooth manifold and the projection $P \times P \to (P \times P) / G$ is a surjective submersion ^{[Mackenzie Ch. 1 §1.7]}. Write $⟨ p, q ⟩$ for the orbit of $(p, q)$ , that is, the image of $(p, q)$ in the quotient.

Definition (gauge groupoid). The gauge groupoid of $P$ is the Lie groupoid $$ \frac{P \times P}{G} \rightrightarrows M $$ with structure maps $$ s\langle p, q\rangle = \pi(q), \qquad t\langle p, q\rangle = \pi(p), $$ unit map $u (x) = ⟨ p, p ⟩$ for any $p \in π^{- 1} (x)$ , inversion $⟨ p, q ⟩^{- 1} = ⟨ q, p ⟩$ , and multiplication defined as follows. Two arrows $⟨ p, q ⟩$ and $⟨ p^{'}, q^{'} ⟩$ are composable when $s ⟨ p, q ⟩ = t ⟨ p^{'}, q^{'} ⟩$ , i.e. $π (q) = π (p^{'})$ ; choosing the representative of the second arrow so that its first entry equals $q$ — possible because $q$ and $p^{'}$ lie in the same fibre, so $p^{'} = q \cdot h$ for a unique $h$ , and then $⟨ p^{'}, q^{'} ⟩ = ⟨ q, q^{'} \cdot h^{- 1} ⟩$ — the product is $$ \langle p, q\rangle \cdot \langle q, r\rangle := \langle p, r\rangle . $$

Each map here is well defined on $G$ -orbits. The source and target descend because $π (q \cdot h) = π (q)$ and $π (p \cdot h) = π (p)$ . The unit is independent of the chosen $p$ in the fibre because $⟨ p \cdot h, p \cdot h ⟩ = ⟨ p, p ⟩$ . The multiplication is independent of all representative choices: replacing $(p, q)$ by $(p \cdot a, q \cdot a)$ and $(q, r)$ by $(q \cdot b, r \cdot b)$ and matching the middle entries $q \cdot a = q \cdot b$ forces $a = b$ , whence the product reads $⟨ p \cdot a, r \cdot a ⟩ = ⟨ p, r ⟩$ .

A clean reading of an arrow: $⟨ p, q ⟩$ is exactly a $G$ -equivariant map between fibres $φ : P_{π (q)} \to P_{π (p)}$ , namely the unique such map sending $q \mapsto p$ . Composition of arrows is composition of equivariant fibre maps, inversion is the inverse map, and the units are the identity maps. The notation $⟨ p, q ⟩$ for the $G$ -orbit of $(p, q)$ and $P_{x} = π^{- 1} (x)$ for the fibre over $x$ are used throughout.

A non-example marks the boundary of the construction: if one drops freeness of the $G$ -action on $P$ , the diagonal action on $P \times P$ need not be free, the quotient $(P \times P) / G$ need not be a manifold, and the source and target need not be submersions. Freeness and properness of the principal action are precisely what place the whole structure in the smooth category.

Key theorem with proof Intermediate+

Theorem (the gauge groupoid is a transitive Lie groupoid with isotropy $G$ ). Let $π : P \to M$ be a principal $G$ -bundle. Then $(P \times P) / G ⇉ M$ is a Lie groupoid; it is transitive; and for each $x \in M$ the isotropy group at $x$ is isomorphic to the structure group $G$ . ^{[Mackenzie Ch. 1 §1.7]}

Proof. Write $Ω := (P \times P) / G$ and let $pr : P \times P \to Ω$ be the quotient projection, a surjective submersion because the diagonal action is free and proper.

Smooth manifold of arrows and structure maps. The source $s$ fits into $s \circ pr = π \circ pr_{2}$ , where $pr_{2} : P \times P \to P$ is the second projection; since $π \circ pr_{2}$ is a submersion and $pr$ is a surjective submersion, $s$ is a smooth submersion. The same argument with $pr_{1}$ gives $t$ smooth and submersive. Both are surjective because $π$ is. The unit map $u : M \to Ω$ satisfies $u \circ π = pr \circ Δ$ with $Δ (p) = (p, p)$ , so $u$ is smooth; it is a section of $s$ and of $t$ , hence an embedding.

Composable pairs and multiplication. The set of composable pairs is $Ω \times_{M} Ω = {(σ, τ) : s (σ) = t (τ)}$ , an embedded submanifold because $s, t$ are submersions 03.03.10. The multiplication is induced by the smooth map $P \times P \times P \to Ω$ , $(p, q, r) \mapsto ⟨ p, r ⟩$ , which is constant on the relation identifying $(p, q, r)$ with $(p \cdot h, q \cdot h, r \cdot h)$ and with the gluing $q \leftrightarrow p^{'}$ in the middle; the induced map on $Ω \times_{M} Ω$ is smooth. Associativity is immediate from $⟨ p, q ⟩ ⟨ q, r ⟩ ⟨ r, w ⟩ = ⟨ p, w ⟩$ read two ways. Inversion $⟨ p, q ⟩ \mapsto ⟨ q, p ⟩$ is induced by the swap on $P \times P$ , hence smooth. Therefore $Ω ⇉ M$ satisfies every clause of the Lie groupoid definition.

Transitivity. The anchor $(t, s) : Ω \to M \times M$ sends $⟨ p, q ⟩ \mapsto (π p, π q)$ . Given any $(y, x) \in M \times M$ , choose $p \in P_{y}$ and $q \in P_{x}$ ; then $⟨ p, q ⟩$ is an arrow from $x$ to $y$ . So the anchor is surjective. Its differential is surjective because $π$ is a submersion in each entry, so the anchor is a surjective submersion: the groupoid has a single orbit, hence is transitive.

Isotropy. Fix $x \in M$ and a base point $p_{0} \in P_{x}$ . The isotropy at $x$ is $$ \Omega_x^x = {\langle p, q\rangle : \pi p = x = \pi q}. $$ Define $θ : G \to Ω_{x}^{x}$ by $θ (g) = ⟨ p_{0} \cdot g, p_{0} ⟩$ . Every element of $Ω_{x}^{x}$ has the form $⟨ p, q ⟩$ with $p, q \in P_{x}$ ; using the fibre action write $q = p_{0} \cdot a$ and $p = p_{0} \cdot b$ , then $⟨ p, q ⟩ = ⟨ p_{0} b, p_{0} a ⟩ = ⟨ p_{0} (b a^{- 1}), p_{0} ⟩ = θ (b a^{- 1})$ , so $θ$ is surjective. It is injective because $⟨ p_{0} g, p_{0} ⟩ = ⟨ p_{0} g^{'}, p_{0} ⟩$ forces $p_{0} g = p_{0} g^{'}$ after matching the second entries (freeness), giving $g = g^{'}$ . Finally $θ$ is a homomorphism: $$ \theta(g),\theta(g') = \langle p_0 g, p_0\rangle \langle p_0 g', p_0\rangle = \langle p_0 g, p_0\rangle \langle p_0, p_0 g'^{-1}\rangle = \langle p_0 g, p_0 g'^{-1}\rangle = \langle p_0 g g', p_0\rangle = \theta(g g'), $$ where the second equality rewrites the second arrow with first entry $p_{0}$ and the fourth multiplies both entries by $g^{'}$ . Both $θ$ and $θ^{- 1}$ are smooth as descents of smooth maps, so $θ$ is a Lie-group isomorphism $G ≅ Ω_{x}^{x}$ . $□$

Bridge. This theorem is exactly the statement that the structure group reappears as the pointwise symmetry the groupoid carries over the base; it builds toward the converse, Mackenzie's correspondence, where every transitive Lie groupoid is reconstructed as a gauge groupoid, and it identifies the structure group $G$ with the isotropy $Ω_{x}^{x}$ so that bundle data and groupoid data become two presentations of one object. The foundational reason the construction lives in the smooth category is the freeness and properness of the principal action, which is what made $(P \times P) / G$ a manifold and $s, t$ submersions. The isotropy computation appears again in the Atiyah sequence below, whose kernel bundle $(P \times g) / G$ is the fibrewise Lie algebra of these isotropy groups, and the whole picture generalises the source-fibre/isotropy anatomy of an arbitrary Lie groupoid 03.03.10 to its flagship transitive example.

Exercises Intermediate+

Exercise 6 (hard, proof).

Let $Fr (E) \to M$ be the frame bundle of a rank- $n$ real vector bundle $E$ , a principal $G L (n, R)$ -bundle whose fibre over $x$ is the set of linear isomorphisms $R^{n} \to E_{x}$ . Show that an arrow of its gauge groupoid from $x$ to $y$ is the same data as a linear isomorphism $E_{x} \to E_{y}$ , and that composition of arrows is composition of linear maps.

Hint

A frame $p \in Fr (E)_{x}$ is an isomorphism $R^{n} \to E_{x}$ ; an equivariant fibre map carries one frame to another.

Answer

An arrow $⟨ p, q ⟩$ with $q \in Fr (E)_{x}$ , $p \in Fr (E)_{y}$ is the equivariant fibre map sending $q \mapsto p$ . Reading $q : R^{n} \to E_{x}$ and $p : R^{n} \to E_{y}$ as frames, the map $ϕ := p \circ q^{- 1} : E_{x} \to E_{y}$ is a linear isomorphism, and it does not change under $(p, q) \mapsto (p g, q g)$ since $(p g) (q g)^{- 1} = p g g^{- 1} q^{- 1} = p q^{- 1}$ . Conversely every linear isomorphism $ϕ : E_{x} \to E_{y}$ arises from $⟨ ϕ \circ q, q ⟩$ for any frame $q$ over $x$ . Composition matches: $⟨ p, q ⟩ ⟨ q, r ⟩ = ⟨ p, r ⟩$ corresponds to $(p r^{- 1}) = (p q^{- 1}) (q r^{- 1})$ . So the gauge groupoid of $Fr (E)$ is the groupoid of linear isomorphisms between fibres of $E$ .

Exercise 7 (hard, proof).

A bisection of a Lie groupoid $Ω ⇉ M$ is a smooth map $σ : M \to Ω$ with $s \circ σ = id_{M}$ and $t \circ σ$ a diffeomorphism of $M$ . Prove that bisections of the gauge groupoid which cover the identity ( $t \circ σ = id_{M}$ ) are in one-to-one correspondence with gauge transformations of $P$ , i.e. $G$ -equivariant bundle automorphisms $Φ : P \to P$ covering $id_{M}$ .

Hint

A base-point-fixing arrow $⟨ p, q ⟩$ with $π p = π q = x$ is an equivariant self-map of the fibre $P_{x}$ ; assemble these into a global equivariant automorphism.

Answer

A bisection $σ$ with $t \circ σ = id$ assigns to each $x$ an arrow $σ (x) = ⟨ Φ_{x} (q), q ⟩$ from $x$ to $x$ , where $Φ_{x} : P_{x} \to P_{x}$ is the equivariant self-map of the fibre that the arrow represents. Equivariance of each $Φ_{x}$ and smoothness of $σ$ assemble the $Φ_{x}$ into a smooth $G$ -equivariant automorphism $Φ : P \to P$ with $π \circ Φ = π$ . Conversely a gauge transformation $Φ$ defines $σ (x) = ⟨ Φ (q), q ⟩$ (independent of $q \in P_{x}$ by equivariance). The two assignments are inverse, and both respect composition: the product of bisections corresponds to composition of gauge transformations. Hence base-identity bisections $≅$ gauge transformations.

Lean formalization Intermediate+

lean_status: none — Mathlib has CategoryTheory.Groupoid as a purely algebraic structure and an evolving principal-bundle and smooth-manifold library, but no internalisation of a Lie groupoid in the smooth category and no gauge-groupoid construction. 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.
variable {G M P : Type*} [LieGroup G] [Manifold M] [Manifold P]
variable (π : P → M) [PrincipalBundle G π]   -- free proper right action

-- arrow manifold: quotient of P × P by the diagonal G-action
def GaugeArrows := Quotient (diagonalAction G P)

def gaugeGroupoid : LieGroupoid M (GaugeArrows π) where
  source := fun a => π (a.represent).2
  target := fun a => π (a.represent).1
  unit   := fun x => ⟦(somePointOver x, somePointOver x)⟧
  inv    := fun a => ⟦(a.represent.2, a.represent.1)⟧
  -- mul descends from (p, q, r) ↦ ⟦(p, r)⟧ on composable triples
  -- transitive, with isotropy at each x isomorphic to G
  ...

The first genuine obstacle is the quotient manifold: Mathlib lacks a packaged "quotient of a manifold by a free proper Lie-group action is a manifold" lemma, on which the smoothness of source, target, and mul all depend.

Advanced results Master

Mackenzie's correspondence. The gauge-groupoid construction is one direction of an equivalence. Every transitive Lie groupoid $Ω ⇉ M$ is isomorphic to a gauge groupoid: fix $x_{0} \in M$ , set $P := s^{- 1} (x_{0})$ and $G := Ω_{x_{0}}^{x_{0}}$ , so that $t ∣_{P} : P \to M$ is a principal $G$ -bundle and $Ω ≅ (P \times P) / G$ via $⟨ p, q ⟩ \mapsto p q^{- 1}$ ^{[Mackenzie Ch. 1 §1.7]}. The construction in this unit is the inverse: starting from $P \to M$ one builds $(P \times P) / G$ , recovers $G$ as the isotropy, and recovers $P$ (up to the choice of base point) as a source fibre. Transitive Lie groupoids and principal bundles are therefore two presentations of the same data, and reductions of structure group correspond to wide transitive subgroupoids.

The Atiyah sequence. Differentiating the gauge groupoid along its units produces a short exact sequence of vector bundles over $M$ , the Atiyah sequence $$ 0 \longrightarrow \frac{P \times \mathfrak g}{G} \xrightarrow{;\iota;} \frac{TP}{G} \xrightarrow{;\pi_*;} TM \longrightarrow 0 , $$ where $g = Lie (G)$ , the action on $P \times g$ is $h \cdot (p, X) = (p \cdot h, Ad_{h^{- 1}} X)$ , and $G$ acts on $T P$ by the derivative of the principal action ^{[Atiyah 1957]}. The quotient $T P / G$ is the Lie algebroid of the gauge groupoid: its sections are the $G$ -invariant vector fields on $P$ , which form a Lie algebra under the bracket and carry the anchor $π_{*}$ to $T M$ . The kernel $ad (P) := (P \times g) / G$ is the adjoint bundle, the fibrewise Lie algebra of the isotropy groups computed above. The whole sequence is the infinitesimal shadow of the gauge groupoid, and its construction connects forward to the Lie algebroid unit 03.04.16.

Connections as splittings. A principal connection on $P$ 03.05.07 is the same data as a splitting of the Atiyah sequence: a vector-bundle map $γ : T M \to T P / G$ with $π_{*} \circ γ = id_{T M}$ selects, $G$ -invariantly, a horizontal complement to the vertical bundle, which is a connection. Equivalently, a connection is a right inverse to the anchor $π_{*}$ , or a left inverse $ω : T P / G \to ad (P)$ to $ι$ — the connection one-form. The curvature of the connection is the obstruction to the splitting being a Lie-algebroid morphism, measured by $R (X, Y) = ω ([γ X, γ Y])$ , and a flat connection is exactly a splitting that is a bracket homomorphism. This places the existence question for connections inside homological algebra: the sequence always splits as vector bundles, since one can average a local splitting against a partition of unity, so principal connections always exist.

Bisections and the gauge group. The bisections of the gauge groupoid form a group under pointwise composition of arrows, and those covering $id_{M}$ are precisely the gauge transformations of $P$ . The full bisection group is the group of bundle automorphisms of $P$ covering arbitrary diffeomorphisms of $M$ ; its identity component differentiates to the Lie algebroid $T P / G$ , exhibiting the gauge group as the integrating object of the Atiyah algebroid. This is the groupoid form of the statement that gauge symmetry is the symmetry of the principal bundle.

Synthesis. The gauge groupoid $(P \times P) / G ⇉ M$ is the single smooth object that records every $G$ -equivariant comparison between fibres of a principal bundle, and the central insight is that it converts a principal bundle into a transitive Lie groupoid losslessly: the structure group is recovered as the isotropy $Ω_{x}^{x} ≅ G$ , the bundle is recovered as a source fibre $s^{- 1} (x_{0})$ with its target projection, and gauge transformations are recovered as base-identity bisections. Putting these together gives Mackenzie's correspondence, the dictionary in which transitive groupoids and principal bundles are interchangeable and structure-group reductions become wide subgroupoids.

Differentiating at the units produces the Atiyah sequence $0 \to ad (P) \to T P / G \to T M \to 0$ , whose middle term is the Lie algebroid of the groupoid and whose kernel is the fibrewise Lie algebra of the isotropy groups; this is exactly the infinitesimal counterpart of the global gauge groupoid, and it appears again in the Lie algebroid theory 03.04.16 that linearises arbitrary Lie groupoids. Connections on $P$ are then splittings of this sequence, curvature is the failure of a splitting to respect the bracket, and the flat case is the bracket-preserving splitting; the foundational reason all of this is available is that the free proper principal action makes every quotient in sight a manifold and every descended structure map a submersion. Across frame bundles, product bundles, and the general construction, the gauge groupoid is the geometric home in which bundle theory, gauge theory, and the algebroid calculus of invariant vector fields become one structure, and it generalises the source-fibre anatomy of an arbitrary Lie groupoid 03.03.10 to the transitive case that principal bundles single out.

Full proof set Master

Proposition (the quotient $(P \times P) / G$ is a manifold and the structure maps descend smoothly). Let $π : P \to M$ be a principal $G$ -bundle. The diagonal right action of $G$ on $P \times P$ is free and proper; the quotient $(P \times P) / G$ is a smooth manifold of dimension $2 dim P - dim G$ ; and the source, target, unit, inversion, and multiplication descend to smooth maps making $(P \times P) / G ⇉ M$ a Lie groupoid.

Proof. The principal action on $P$ is free and proper by hypothesis. Freeness of the diagonal action on $P \times P$ is inherited: $(p, q) \cdot h = (p, q)$ forces $p \cdot h = p$ , hence $h = e$ . Properness is inherited because the map $(P \times P) \times G \to (P \times P) \times (P \times P)$ , $((p, q), h) \mapsto ((p, q), (p, q) \cdot h)$ , factors through the proper map for $P$ in each coordinate; concretely, if $(p_{n}, q_{n}) \to (p, q)$ and $(p_{n}, q_{n}) \cdot h_{n} \to (p^{'}, q^{'})$ then $p_{n} h_{n} \to p^{'}$ with $p_{n} \to p$ , and properness of the $P$ -action yields a convergent subsequence of $(h_{n})$ . By the quotient-manifold theorem for free proper actions, $(P \times P) / G$ is a smooth manifold and $pr : P \times P \to (P \times P) / G$ is a surjective submersion, of dimension $2 dim P - dim G$ .

For the structure maps, each is characterised by a commuting square with $pr$ and a smooth map out of $P \times P$ (or $P$ ). Source: $s \circ pr = π \circ pr_{2}$ . Target: $t \circ pr = π \circ pr_{1}$ . Both right-hand sides are smooth submersions and $pr$ is a surjective submersion, so $s$ and $t$ are smooth submersions by the universal property of the quotient submersion. Unit: $u \circ π = pr \circ Δ$ , smooth. Inversion: $i \circ pr = pr \circ swap$ , smooth, with $i \circ i = id$ . Multiplication: the composable-pair manifold $Ω \times_{M} Ω$ is embedded in $Ω \times Ω$ because $s, t$ are submersions, and $m$ is induced by $(p, q, r) \mapsto ⟨ p, r ⟩$ on the appropriate fibred product of $P$ 's, which is smooth and descends. Associativity, unit, and inverse identities hold at the level of representatives, hence on the quotient. Therefore $(P \times P) / G ⇉ M$ is a Lie groupoid. $□$

Proposition (isotropy is the structure group, naturally). For each $x \in M$ and each $p_{0} \in P_{x}$ , the map $θ_{p_{0}} : G \to Ω_{x}^{x}$ , $g \mapsto ⟨ p_{0} g, p_{0} ⟩$ , is a Lie-group isomorphism. Changing the base point $p_{0} ⇝ p_{0} \cdot a$ conjugates: $θ_{p_{0} \cdot a} = θ_{p_{0}} \circ C_{a}$ , where $C_{a} (g) = a g a^{- 1}$ .

Proof. That $θ_{p_{0}}$ is a smooth group isomorphism is the isotropy part of the Key theorem. For the base-point dependence, compute $$ \theta_{p_0\cdot a}(g) = \langle (p_0 a) g, , p_0 a\rangle = \langle p_0 (a g a^{-1}) a, , p_0 a\rangle = \langle p_0 (a g a^{-1}), , p_0\rangle = \theta_{p_0}(a g a^{-1}), $$ using $⟨ p_{0} b \cdot a, p_{0} \cdot a ⟩ = ⟨ p_{0} b, p_{0} ⟩$ in the third step. Hence $θ_{p_{0} a} = θ_{p_{0}} \circ C_{a}$ . The isomorphism class of the isotropy is therefore canonical (independent of $p_{0}$ ), while the specific isomorphism to $G$ depends on the chosen base point up to inner automorphism — the same indeterminacy carried by the structure group of a principal bundle. $□$

Proposition (the Atiyah sequence is exact). The sequence $0 \to (P\times\mathfrak g)/G \xrightarrow{\iota} TP/G \xrightarrow{\pi_} TM \to 0 $o f v ec t or b u n d l eso v er$ M$ is exact, and it splits in the category of vector bundles.*

Proof. The principal action being free, the fundamental vector fields give a bundle injection $P \times g \to T P$ , $(p, X) \mapsto \frac{d}{d t}_{0} p \cdot exp (tX)$ , whose image is the vertical bundle $ker (d π)$ . This map is $G$ -equivariant for the actions $h \cdot (p, X) = (p h, Ad_{h^{- 1}} X)$ on the source and $d (R_{h})$ on the target, so it descends to a bundle injection $ι : (P \times g) / G \to T P / G$ with image $(ker d π) / G$ . The projection $d π : T P \to π^{*} T M$ is $G$ -invariant on the base and surjective with kernel the vertical bundle, so it descends to a surjection $π_{*} : T P / G \to T M$ with kernel exactly $im (ι)$ . Exactness follows. The sequence splits as vector bundles because $T M$ is locally free and one may patch local right inverses of $π_{*}$ by a partition of unity, the average of vector-bundle splittings being a vector-bundle splitting. A global splitting is precisely a principal connection. $□$

Connections Master

A Lie groupoid 03.03.10 supplies the ambient framework: the gauge groupoid is its flagship transitive example, and the source-fibre, isotropy, and orbit anatomy proved in general specialise here to the statements that the source fibre is $P$ , the isotropy is the structure group, and the single orbit is all of $M$ .

A principal bundle 03.05.01 is the input to the construction and, by Mackenzie's correspondence, the output of the reverse construction from any transitive Lie groupoid; the gauge groupoid is the precise sense in which transitive symmetry over a base and principal-bundle data are interchangeable, with structure-group reductions matching wide transitive subgroupoids.

A group action 03.03.02 is what makes the quotient $(P \times P) / G$ a manifold: the free proper diagonal action is the hypothesis behind every submersion in the construction, and the isotropy isomorphism $Ω_{x}^{x} ≅ G$ is the groupoid-level trace of the fibrewise freeness of the principal action.

A connection on a principal bundle 03.05.07 is, at the groupoid level, a splitting of the Atiyah sequence of the gauge groupoid; curvature is the failure of that splitting to preserve the bracket, so the curvature theory of connections is the obstruction theory for lifting the anchor to a Lie-algebroid morphism.

Historical & philosophical context Master

The gauge groupoid was introduced by Charles Ehresmann in his 1959 work on differentiable categories and the structure of fibre bundles, where it appears as the groupoïde des automorphismes of a principal bundle and underlies his treatment of connections and holonomy ^{[Ehresmann 1959]}. The infinitesimal object now called the Atiyah sequence had appeared two years earlier in Michael Atiyah's 1957 study of complex analytic connections in fibre bundles, where the obstruction to a holomorphic connection is the failure of an exact sequence of sheaves — the holomorphic Atiyah sequence — to split holomorphically ^{[Atiyah 1957]}. Jean Pradines, Ehresmann's student, developed the differentiation of Lie groupoids to Lie algebroids in his 1966 notes, supplying the algebroid that the Atiyah sequence presents ^{[Pradines 1966]}.

The systematic theory — the gauge groupoid as the universal transitive example, the correspondence between transitive Lie groupoids and principal bundles, and the connection theory phrased through the Atiyah sequence — is due to Kirill Mackenzie, whose 1987 lecture notes and 2005 monograph made the gauge groupoid the organising object of transitive groupoid theory ^{[Mackenzie Ch. 1 §1.7]}. The construction sits at the meeting point of Ehresmann's bundle-theoretic program and Atiyah's analytic one, and it is the reason the words gauge transformation and gauge groupoid share a root: the bisections of the groupoid covering the identity are exactly the gauge transformations of the bundle.

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}
}

@article{atiyah1957,
  author    = {Atiyah, Michael F.},
  title     = {Complex analytic connections in fibre bundles},
  journal   = {Transactions of the American Mathematical Society},
  volume    = {85},
  pages     = {181--207},
  year      = {1957}
}

@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}
}

Gauge groupoid — the transitive Lie groupoid $(P \times P) / G ⇉ M$ of a principal bundle, with isotropy the structure group, source fibre the total space, base-identity bisections the gauge transformations, and the Atiyah sequence as its Lie algebroid.

Prerequisites

03.03.10
03.05.01
03.03.02
03.03.01

Tier anchors

beginner: the fibre-to-fibre symmetries of a bundle, packaged as reversible arrows over the base
intermediate: Mackenzie — General Theory of Lie Groupoids and Lie Algebroids Ch. 1 §1.7
master: Mackenzie Ch. 1 §1.7 and Ch. 3; Ehresmann 1959; Atiyah 1957

References

images/Kobayashi-Nomizu-1-683x1024__72960fe9e3.jpg · Kobayashi-Nomizu Vol. I anchor listed in Fast Track archive (principal bundle background)
Mackenzie — General Theory of Lie Groupoids and Lie Algebroids · Ch. 1 §1.7, the gauge groupoid; Ch. 3, the Atiyah sequence
Ehresmann — Catégories topologiques et catégories différentiables · Colloque de Géométrie Différentielle Globale (1959)
Atiyah — Complex analytic connections in fibre bundles · Trans. Amer. Math. Soc. 85 (1957), 181–207
Pradines — Théorie de Lie pour les groupoïdes différentiables · C. R. Acad. Sci. Paris 263 (1966)

Estimated time

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