Bisection group of a Lie groupoid; gauge transformations as bisections

Intuition Beginner

A groupoid spreads reversible arrows over a base space, one bundle of arrows hovering above each point. A natural thing to want is a consistent choice: pick exactly one arrow leaving each point, all at once, so the picks fit together smoothly. Such a coherent choice is called a bisection. It is a single global selection of arrows, one starting at every point of the base.

The surprise is that these selections can be combined. If you have two consistent choices of arrows, you can follow one and then the other, and the result is again a consistent choice. With this way of combining them, the collection of all bisections becomes a group: there is a do-nothing choice, every choice can be undone, and combining is associative. So a groupoid, which is a spread-out family of point-to-point symmetries, secretly carries an ordinary group built from its global sections.

Why bother? Because this group is where the groupoid's symmetries become moves you can perform. Each bisection also pushes the base points around — it reads off, for every point, where its chosen arrow lands — so the bisection group acts on the base. When the groupoid comes from a bundle, this group is exactly the bundle's gauge transformations, the changes of reference that physics calls gauge freedom.

Visual Beginner

Picture the base as a curved surface with several marked points, and above it the floating space of arrows. A bisection is a smooth sheet that picks, over each base point, one arrow leaving that point. The sheet meets the arrow space in exactly one arrow per starting point.

Follow each selected arrow to where it lands. The landing points trace out a reshuffling of the base: the bisection moves every point to the endpoint of its chosen arrow. The do-nothing bisection picks the do-nothing arrow at each point, so it moves nothing. Combining two sheets means: take the first sheet's landing pattern, and over each landed point read the second sheet's chosen arrow, then chain the two arrows.

Worked example Beginner

Take the base to be three points $\{1,2,3\}$ with all ordered pairs as arrows: write $(j,i)$ for the arrow from $i$ to $j$. A bisection picks one arrow starting at each of $1$, $2$, $3$.

Choose the bisection $\sigma$ that picks $(2,1)$ at point $1$, picks $(3,2)$ at point $2$, and picks $(1,3)$ at point $3$. Reading off the landing points: $1$ goes to $2$, $2$ goes to $3$, $3$ goes to $1$. So $\sigma$ reshuffles the base by the cycle $1\to 2\to 3\to 1$.

Now combine $\sigma$ with itself. Start at point $1$: its chosen arrow lands at $2$; over $2$ the chosen arrow is $(3,2)$, landing at $3$; chaining gives the arrow $(3,1)$ from $1$ to $3$. Doing this at every point, $\sigma$ followed by $\sigma$ picks $(3,1)$ at $1$, picks $(1,2)$ at $2$, and picks $(2,3)$ at $3$. Its reshuffling is $1\to 3\to 2\to 1$, the reverse cycle.

What this tells us: a bisection is a global choice of arrows, and combining bisections both chains the arrows and composes the reshufflings of the base. The reshufflings here are exactly the permutations of three points, which is the smallest interesting bisection group.

Check your understanding Beginner

Formal definition Intermediate+

Let $G\rightrightarrows M$ be a Lie groupoid 03.03.10 with source and target submersions $s,t:G\to M$, unit map $u:M\to G$, $x\mapsto 1_x$, and partial multiplication written by juxtaposition.

Definition (bisection). A bisection of $G$ is a smooth map $\sigma :M\to G$ such that

• $s\circ \sigma ={\mathrm{id}}_M$ — so $\sigma$ is a smooth section of the source submersion, picking out for each $x$ an arrow $\sigma (x)$ with $s(\sigma (x))=x$; and
• $\bar{\sigma }:=t\circ \sigma :M\to M$ is a diffeomorphism of $M$.

The diffeomorphism $\bar{\sigma }$ is the base map of $\sigma$. Equivalently, a bisection is a submanifold $\Sigma =\sigma (M)\subseteq G$ meeting every source fibre $s^{-1}(x)$ and every target fibre $t^{-1}(y)$ in exactly one point ^{[Mackenzie Ch. I §2]}. The image-submanifold description is symmetric in $s$ and $t$; the map description privileges $s$ but recovers the same objects.

The set of all bisections is denoted $\mathrm{Bis}(G)$. Define a product on it by $$ (\sigma \star \tau)(x) := \sigma\big(t(\tau(x))\big) \cdot \tau(x) = \sigma\big(\bar\tau(x)\big),\tau(x), \qquad x \in M. $$ The right-hand product is defined because $s(\sigma (\bar{\tau }(x)))=\bar{\tau }(x)=t(\tau (x))$, so the two arrows are composable. The unit bisection is the unit section $u:x\mapsto 1_x$, whose base map is ${\mathrm{id}}_M$. The inverse of $\sigma$ is $$ \sigma^{-1}(x) := \big(\sigma(\bar\sigma^{-1}(x))\big)^{-1}, $$ where ${\bar{\sigma }}^{-1}$ is the inverse diffeomorphism of the base map and $(\cdot {)}^{-1}$ on the right is groupoid inversion. The map $\beta :\mathrm{Bis}(G)\to \mathrm{Diff}(M)$, $\sigma \mapsto \bar{\sigma }=t\circ \sigma$, sends each bisection to its base map.

A bisection $\sigma$ acts on the arrow manifold by left translation $L_{\sigma }:G\to G$ and right translation $R_{\sigma }:G\to G$, $$ L_\sigma(g) := \sigma(t(g)),g, \qquad R_\sigma(g) := g,\sigma(\bar\sigma^{-1}(s(g))), $$ each defined wherever the indicated product is composable. The symbols $\mathrm{Bis}(G)$ for the bisection group, $\bar{\sigma }$ for the base map, $\star$ for the bisection product, and $\Gamma (A)$ below for sections of the Lie algebroid are used throughout.

A non-example fixes the boundary of the definition. A smooth section $\sigma$ of $s$ with $t\circ \sigma$ merely smooth but not bijective — for instance a section whose image collapses two target fibres together — is a local or non-invertible section, not a bisection; the product $\star$ need not return a section of $s$ and the inverse formula breaks, because ${\bar{\sigma }}^{-1}$ does not exist. The diffeomorphism requirement on $\bar{\sigma }$ is precisely what makes $\mathrm{Bis}(G)$ a group rather than a monoid of sections.

Key theorem with proof Intermediate+

Theorem (the bisections form a group, fibred over $\mathrm{Diff}(M)$). Let $G\rightrightarrows M$ be a Lie groupoid. Then $(\mathrm{Bis}(G),\star )$ is a group with unit the unit bisection $u$ and inverse given by the formula above, and $\beta :\mathrm{Bis}(G)\to \mathrm{Diff}(M)$, $\sigma \mapsto \bar{\sigma }$, is a group homomorphism. ^{[Mackenzie Ch. I §2]}

Proof. First, $\star$ lands in $\mathrm{Bis}(G)$. For bisections $\sigma ,\tau$ set $\rho :=\sigma \star \tau$. Then $s(\rho (x))=s(\tau (x))=x$ because the source of a product is the source of the right factor 03.03.10, so $s\circ \rho ={\mathrm{id}}_M$ and $\rho$ is a smooth section of $s$ (smoothness is composition of the smooth maps $\tau$, $\bar{\tau }$, $\sigma$, and groupoid multiplication on the composable pairs). Its base map is $$ \bar\rho(x) = t(\rho(x)) = t\big(\sigma(\bar\tau(x)),\tau(x)\big) = t\big(\sigma(\bar\tau(x))\big) = \bar\sigma(\bar\tau(x)) = (\bar\sigma \circ \bar\tau)(x), $$ using that the target of a product is the target of the left factor. Thus $\bar{\rho }=\bar{\sigma }\circ \bar{\tau }$, a composition of diffeomorphisms, hence a diffeomorphism. So $\rho \in \mathrm{Bis}(G)$ and, as a by-product, $\beta (\sigma \star \tau )=\bar{\sigma }\circ \bar{\tau }=\beta (\sigma )\beta (\tau )$: $\beta$ is a homomorphism.

Associativity. For bisections $\sigma ,\tau ,\omega$ and $x\in M$, $$ \big((\sigma \star \tau) \star \omega\big)(x) = (\sigma \star \tau)\big(\bar\omega(x)\big),\omega(x) = \sigma\big(\bar\tau(\bar\omega(x))\big),\tau\big(\bar\omega(x)\big),\omega(x), $$ while $$ \big(\sigma \star (\tau \star \omega)\big)(x) = \sigma\big(\overline{\tau \star \omega}(x)\big),(\tau \star \omega)(x) = \sigma\big(\bar\tau(\bar\omega(x))\big),\tau\big(\bar\omega(x)\big),\omega(x), $$ where the second line uses $\overline{\tau \star \omega }=\bar{\tau }\circ \bar{\omega }$ just proved. The two expressions agree term by term, and the triple product is unambiguous by associativity of groupoid multiplication. Hence $\star$ is associative.

Unit. The unit section $u$ has base map ${\mathrm{id}}_M$. Then $(u\star \sigma )(x)=u(\bar{\sigma }(x))\sigma (x)=1_{\bar{\sigma }(x)}\sigma (x)=\sigma (x)$, since $1_{t(\sigma (x))}$ is the left identity for $\sigma (x)$. And $(\sigma \star u)(x)=\sigma (\bar{u}(x))u(x)=\sigma (x)1_x=\sigma (x)$, since $1_{s(\sigma (x))}=1_x$ is the right identity. So $u$ is a two-sided unit.

Inverse. Let ${\sigma }^{-1}(x)=(\sigma ({\bar{\sigma }}^{-1}(x)){)}^{-1}$. This is a smooth section of $s$: its source is $s((\sigma ({\bar{\sigma }}^{-1}(x)){)}^{-1})=t(\sigma ({\bar{\sigma }}^{-1}(x)))=\bar{\sigma }({\bar{\sigma }}^{-1}(x))=x$. Its base map is $t((\sigma ({\bar{\sigma }}^{-1}(x)){)}^{-1})=s(\sigma ({\bar{\sigma }}^{-1}(x)))={\bar{\sigma }}^{-1}(x)$, a diffeomorphism, so ${\sigma }^{-1}\in \mathrm{Bis}(G)$ with $\overline{{\sigma }^{-1}}={\bar{\sigma }}^{-1}$. Now compute, writing $y={\bar{\sigma }}^{-1}(x)$ so that $\bar{\sigma }(y)=x$: $$ (\sigma \star \sigma^{-1})(x) = \sigma\big(\overline{\sigma^{-1}}(x)\big),\sigma^{-1}(x) = \sigma(y),\big(\sigma(y)\big)^{-1} = 1_{t(\sigma(y))} = 1_{\bar\sigma(y)} = 1_x = u(x). $$ The reversed product is similar: $$ (\sigma^{-1} \star \sigma)(x) = \sigma^{-1}\big(\bar\sigma(x)\big),\sigma(x) = \big(\sigma(\bar\sigma^{-1}(\bar\sigma(x)))\big)^{-1},\sigma(x) = \big(\sigma(x)\big)^{-1},\sigma(x) = 1_{s(\sigma(x))} = 1_x = u(x). $$ Hence ${\sigma }^{-1}$ is a two-sided inverse, and $(\mathrm{Bis}(G),\star )$ is a group with $\beta$ a homomorphism to $\mathrm{Diff}(M)$. $\square$

Bridge. This theorem builds toward the identification of the kernel of $\beta$ — the bisections covering ${\mathrm{id}}_M$ — with the gauge group of a principal bundle, developed in the Advanced results; it appears again in the infinitesimal picture, where differentiating a path of bisections through the unit section produces a section of the Lie algebroid, so that $\Gamma (A)$ is the Lie algebra of $\mathrm{Bis}(G)$. The homomorphism $\beta$ connects the abstract groupoid to the concrete diffeomorphism group of its base 03.03.10, and its image measures how much of $\mathrm{Diff}(M)$ the groupoid's symmetries can realise. Putting these together, the bisection group is the global symmetry object of a Lie groupoid, sitting over $\mathrm{Diff}(M)$ with kernel the internal symmetries that fix the base.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has CategoryTheory.Groupoid as a purely algebraic structure and an evolving smooth-manifold library, but no internalisation of a Lie groupoid in the smooth category and therefore no notion of a bisection. The intended structure is sketched below in pseudo-Lean to indicate the missing API; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: target structure, not in Mathlib.
variable {M : Type*} [Manifold M] {G : Type*} [Manifold G]
variable (𝒢 : LieGroupoid M G)   -- source/target submersions, smooth mul

structure Bisection where
  σ        : M → G
  smooth   : Smooth σ
  sect     : ∀ x, 𝒢.source (σ x) = x          -- section of the source
  baseDiff : Diffeomorphism (fun x => 𝒢.target (σ x))

-- group law: (σ ⋆ τ)(x) = σ (t (τ x)) * τ x
instance : Group (Bisection 𝒢) where
  mul σ τ := ⟨fun x => 𝒢.mul (σ.σ (𝒢.target (τ.σ x))) (τ.σ x), ...⟩
  one      := ⟨𝒢.unit, ...⟩              -- the unit section
  inv σ    := ⟨fun x => 𝒢.inv (σ.σ (σ.baseDiff.symm x)), ...⟩
  -- mul_assoc, one_mul, mul_one, mul_left_inv from the groupoid axioms

-- homomorphism to Diff(M)
def baseHom : Bisection 𝒢 →* Diffeomorphism M := ...

The first genuine obstacle is that Mathlib has neither Diffeomorphism M as a group object usable as a codomain for baseHom, nor the Lie-groupoid base class the structure quantifies over.

Advanced results Master

Bisections of the gauge groupoid are bundle automorphisms. Let $\pi :P\to M$ be a principal $G$-bundle and $\Omega =(P\times P)/G\rightrightarrows M$ its gauge groupoid 03.05.21, with $⟨p,q⟩$ the orbit of $(p,q)$. A bisection $\sigma$ of $\Omega$ assigns to each $x$ an arrow $\sigma (x)=⟨{\Phi }_x(q_x),q_x⟩$ from $x$ to $\bar{\sigma }(x)$, where for any chosen $q_x\in P_x$ the equivariant fibre map ${\Phi }_x:P_x\to P_{\bar{\sigma }(x)}$ is determined. Assembling the ${\Phi }_x$ produces a $G$-equivariant diffeomorphism $\Phi :P\to P$ covering $\bar{\sigma }$, and conversely every $G$-equivariant bundle automorphism $\Phi$ covering a diffeomorphism of $M$ arises this way. The bisection product matches composition of automorphisms, and $\beta :\mathrm{Bis}(\Omega )\to \mathrm{Diff}(M)$ becomes $\Phi \mapsto (\text{the covered diffeomorphism})$. Thus $\mathrm{Bis}(\Omega )\cong {\mathrm{Aut}}_G(P)$, the group of $G$-equivariant bundle automorphisms of $P$.

The gauge group is the kernel of $\beta$. Restricting to base-identity bisections, $\bar{\sigma }={\mathrm{id}}_M$, the automorphism $\Phi$ covers ${\mathrm{id}}_M$: it is a gauge transformation of $P$. So $\ker \beta \cong \mathrm{Gau}(P):=\{\Phi \in {\mathrm{Aut}}_G(P):\pi \circ \Phi =\pi \}$, the gauge group. The exact sequence $$ 1 \longrightarrow \mathrm{Gau}(P) \longrightarrow \mathrm{Bis}(\Omega) \xrightarrow{;\beta;} \mathrm{Diff}(M) $$ expresses gauge transformations as the internal symmetries that fix the base, with the image of $\beta$ the diffeomorphisms of $M$ that lift to bundle automorphisms. Gauge transformations are equivalently the $G$-equivariant maps $P\to G$, or the sections of the associated adjoint group bundle $P{\times }_G{G}_{\mathrm{conj}}$; under the gauge-groupoid identification these are exactly the sections of the isotropy bundle ${\bigcup }_x{\Omega }_x^x$ that are bisections.

The Lie algebra of $\mathrm{Bis}(G)$ is $\Gamma (A)$. Let $A=\ker (ds){|}_M\to M$ be the Lie algebroid of $G$, the restriction to the units of the tangent spaces to the source fibres, with anchor $\rho =dt{|}_A:A\to TM$. Differentiating a smooth path $t\mapsto {\sigma }_t$ of bisections with ${\sigma }_0=u$ the unit section gives $$ X := \frac{d}{dt}\Big|_{0} \sigma_t \in \Gamma(A), $$ a section of $A$ — the derivative lands in $\ker ds$ along the units because each ${\sigma }_t$ is a section of $s$. This assignment identifies the Lie algebra of the bisection group with $\Gamma (A)$, the bracket on $\Gamma (A)$ matching the commutator of bisections to first order, and the anchored vector field $\rho (X)\in \mathfrak{X}(M)$ being the derivative of the base maps ${\bar{\sigma }}_t$. Conversely a section $X\in \Gamma (A)$ generates a one-parameter family of bisections by the flow of its associated right-invariant vector field $\overrightarrow{X}$ on $G$, the bisection ${\sigma }_t$ being the time-$t$ flow applied to the unit section, defined for small $t$ when the flow exists. This is the groupoid form of the exponential map: $\Gamma (A)$ integrates to $\mathrm{Bis}(G)$ as $\mathfrak{g}$ integrates to a Lie group, with the Lie functor sending $G$ to $A$ and $\mathrm{Bis}(G)$ to $\Gamma (A)$ 03.04.16.

Local bisections and the source-connected case. Global bisections need not exist — there may be no smooth global section of $s$ whose target map is a diffeomorphism — but local bisections, defined over an open $U\subseteq M$ with $\sigma :U\to G$, $s\circ \sigma ={\mathrm{id}}_U$, and $t\circ \sigma$ a diffeomorphism onto its image, always exist through any arrow: given $g$ with $s(g)=x$, the submersion $s$ admits a local section through $g$, which can be arranged so that $t\circ \sigma$ is a local diffeomorphism near $x$ since $t$ restricted to a source fibre is a submersion onto the orbit. The local bisections through the arrows of $G$ form a pseudogroup, and when $G$ is source-connected (every source fibre $s^{-1}(x)$ is connected) the bisections — global where they exist, local in general — generate the groupoid: every arrow is the value of some local bisection, and products of local bisections recover the multiplication. The passage from local bisections to the global bisection group is governed by the topology of the source fibres, the same way the passage from a Lie algebra to a connected Lie group is governed by connectivity.

The infinite-dimensional Lie group structure. When $M$ is compact, $\mathrm{Bis}(G)$ carries the structure of an infinite-dimensional Lie group modelled on the Fréchet space $\Gamma (A)$ of smooth sections of the Lie algebroid, with smooth group operations and a well-behaved exponential law in the sense of regular Lie groups ^{[Schmeding-Wockel 2015]}. The homomorphism $\beta :\mathrm{Bis}(G)\to \mathrm{Diff}(M)$ is then a morphism of infinite-dimensional Lie groups, $\mathrm{Diff}(M)$ itself being the regular Lie group of diffeomorphisms; for the pair groupoid $\beta$ is the isomorphism $\mathrm{Bis}(M\times M)\cong \mathrm{Diff}(M)$. The Lie functor at this level reads off the Lie algebra of $\mathrm{Bis}(G)$ as $\Gamma (A)$ with its bracket, making the global-to-infinitesimal correspondence of bisections an instance of the Lie theory of infinite-dimensional groups.

Synthesis. The bisection group $\mathrm{Bis}(G)$ is the global symmetry object of a Lie groupoid: its elements are coherent global choices of arrows, its product chains arrows while composing base maps, and its homomorphism $\beta$ to $\mathrm{Diff}(M)$ records the action on the base with kernel the base-fixing symmetries. The central identifications organise the whole subject. For the pair groupoid the construction returns all of $\mathrm{Diff}(M)$; for an action groupoid it returns $H$-valued maps constrained to move the base diffeomorphically; for the gauge groupoid of a principal bundle it returns the bundle automorphism group ${\mathrm{Aut}}_G(P)$, with $\ker \beta$ the gauge group $\mathrm{Gau}(P)$ — the precise sense in which gauge transformations are bisections. Differentiating bisections at the unit section produces sections of the Lie algebroid, so $\Gamma (A)$ is the Lie algebra of $\mathrm{Bis}(G)$ and the flows of right-invariant vector fields integrate $\Gamma (A)$ back to bisections, exhibiting $\mathrm{Bis}(G)$ as the global object that the Lie functor pairs with the algebroid $A$. Local bisections always exist and form a pseudogroup that, in the source-connected case, regenerates the groupoid, while the topology of the source fibres controls which local bisections assemble into global ones. When the base is compact the entire structure is an infinite-dimensional regular Lie group modelled on $\Gamma (A)$, placing $\mathrm{Bis}(G)$ inside the Lie theory of diffeomorphism-type groups and closing the loop between the groupoid, its algebroid, its bundle, and its gauge symmetry.

Full proof set Master

Proposition (the base map is a homomorphism and $\ker \beta$ is the isotropy-section group). For a Lie groupoid $G\rightrightarrows M$, the base map $\beta (\sigma )=\bar{\sigma }$ defines a homomorphism $\mathrm{Bis}(G)\to \mathrm{Diff}(M)$, and $\ker \beta$ is exactly the group of bisections $\sigma$ with $\sigma (x)\in G_x^x$ for all $x$, a normal subgroup.

Proof. The homomorphism property $\overline{\sigma \star \tau }=\bar{\sigma }\circ \bar{\tau }$ is established in the Key theorem from $t(\sigma (\bar{\tau }(x))\tau (x))=t(\sigma (\bar{\tau }(x)))=\bar{\sigma }(\bar{\tau }(x))$. A bisection $\sigma$ lies in $\ker \beta$ iff $\bar{\sigma }={\mathrm{id}}_M$, i.e. $t(\sigma (x))=x=s(\sigma (x))$ for all $x$, which says $\sigma (x)\in s^{-1}(x)\cap t^{-1}(x)=G_x^x$, the isotropy group at $x$ 03.03.10. Such $\sigma$ are precisely the smooth sections of the isotropy bundle ${\bigcup }_x{G}_x^x$ that are bisections; they form a subgroup because the product of two base-identity bisections has base map ${\mathrm{id}}_M\circ {\mathrm{id}}_M={\mathrm{id}}_M$ and the inverse formula preserves $\bar{\sigma }={\mathrm{id}}_M$. Normality is the kernel-of-a-homomorphism statement: $\ker \beta ⊴\mathrm{Bis}(G)$. $\square$

Proposition (right translation by a bisection is a diffeomorphism of $G$). For $\sigma \in \mathrm{Bis}(G)$ the map $R_{\sigma }:G\to G$, $g\mapsto g\sigma ({\bar{\sigma }}^{-1}(s(g)))$, is a diffeomorphism of the arrow manifold, with inverse $R_{{\sigma }^{-1}}$, and it satisfies $t\circ R_{\sigma }=t$, $s\circ R_{\sigma }={\bar{\sigma }}^{-1}\circ s$.

Proof. Write $a(g):=\sigma ({\bar{\sigma }}^{-1}(s(g)))$. Then $s(a(g))={\bar{\sigma }}^{-1}(s(g))$ and $t(a(g))=\bar{\sigma }({\bar{\sigma }}^{-1}(s(g)))=s(g)$, so the product $ga(g)$ is composable and $R_{\sigma }$ is defined on all of $G$, smooth as a composite of $s$, ${\bar{\sigma }}^{-1}$, $\sigma$, and multiplication. Its target is $t(R_{\sigma }g)=t(g)$ and its source is $s(R_{\sigma }g)=s(a(g))={\bar{\sigma }}^{-1}(s(g))$. For the inverse, $R_{{\sigma }^{-1}}(R_{\sigma }g)=(R_{\sigma }g){\sigma }^{-1}(\overline{{\sigma }^{-1}}{}^{-1}(s(R_{\sigma }g)))$. Using $\overline{{\sigma }^{-1}}={\bar{\sigma }}^{-1}$, the argument is $\overline{{\sigma }^{-1}}{}^{-1}({\bar{\sigma }}^{-1}(s(g)))=\bar{\sigma }({\bar{\sigma }}^{-1}(s(g)))=s(g)$, so the trailing factor is ${\sigma }^{-1}(s(g))=(\sigma ({\bar{\sigma }}^{-1}(s(g))){)}^{-1}=a(g{)}^{-1}$. Hence $R_{{\sigma }^{-1}}(R_{\sigma }g)=ga(g)a(g{)}^{-1}=g{1}_{s(g)}=g$, and symmetrically $R_{\sigma }{R}_{{\sigma }^{-1}}=\mathrm{id}$. So $R_{\sigma }$ is a diffeomorphism with the stated source/target behaviour. $\square$

Proposition (gauge-groupoid bisections covering the identity are the gauge group). Let $\Omega =(P\times P)/G\rightrightarrows M$ be the gauge groupoid of a principal $G$-bundle. The map sending a base-identity bisection $\sigma \in \ker \beta$ to the gauge transformation it represents is a group isomorphism $\ker \beta \stackrel{\sim }{\to }\mathrm{Gau}(P)$.

Proof. Fix $\sigma \in \ker \beta$, so $\sigma (x)=⟨p,q⟩$ with $\pi (p)=\pi (q)=x$ for each $x$. The arrow $⟨p,q⟩$ is the $G$-equivariant fibre map ${\Phi }_x:P_x\to P_x$ sending $q\mapsto p$ 03.05.21, independent of the representative because $⟨p,q⟩$ determines the equivariant map. Define $\Phi :P\to P$ by $\Phi {|}_{P_x}={\Phi }_x$; smoothness of $\sigma$ makes $\Phi$ smooth, and $\Phi$ is $G$-equivariant with $\pi \circ \Phi =\pi$, so $\Phi \in \mathrm{Gau}(P)$. The map $\sigma \mapsto \Phi$ is injective because $\Phi$ recovers each $\sigma (x)=⟨{\Phi }_x(q),q⟩$, and surjective because any $\Phi \in \mathrm{Gau}(P)$ yields the bisection $\sigma (x)=⟨\Phi (q),q⟩$ for any $q\in P_x$ (well defined by equivariance, with $\bar{\sigma }={\mathrm{id}}_M$). It is a homomorphism: the bisection product ${\sigma }_{\Phi }\star {\sigma }_{\Psi }$ has value $⟨\Phi (q^{\prime }),q^{\prime }⟩⟨\Psi (q),q⟩$ at $x$ with $q^{\prime }=\Psi (q)$, equal to $⟨\Phi (\Psi (q)),q⟩$, the bisection of $\Phi \circ \Psi$. Hence $\ker \beta \cong \mathrm{Gau}(P)$. $\square$

Proposition (sections of $A$ generate bisections by flow). Let $A\to M$ be the Lie algebroid of $G$ and $X\in \Gamma (A)$ with associated right-invariant vector field $\overrightarrow{X}$ on $G$. If the flow ${\Phi }_{\tau }^X$ of $\overrightarrow{X}$ is defined for $|\tau |<\epsilon$, then ${\sigma }_{\tau }:={\Phi }_{\tau }^X\circ u$ is a bisection of $G$ for each such $\tau$, and $\frac{d}{d\tau }{|}_0{\sigma }_{\tau }=X$.

Proof sketch. The right-invariant vector field $\overrightarrow{X}$ is tangent to the source fibres (it lies in $\ker ds$), so its flow ${\Phi }_{\tau }^X$ preserves each source fibre and satisfies $s\circ {\Phi }_{\tau }^X=s$. Hence $s\circ {\sigma }_{\tau }=s\circ {\Phi }_{\tau }^X\circ u=s\circ u={\mathrm{id}}_M$, so ${\sigma }_{\tau }$ is a section of $s$. Its base map is ${\bar{\sigma }}_{\tau }=t\circ {\Phi }_{\tau }^X\circ u$; at $\tau =0$ this is $t\circ u={\mathrm{id}}_M$, and $\tau \mapsto {\bar{\sigma }}_{\tau }$ is a smooth path of maps through the identity, so for small $\tau$ each ${\bar{\sigma }}_{\tau }$ is a diffeomorphism (diffeomorphisms form an open set in the appropriate topology, and the derivative of ${\bar{\sigma }}_{\tau }$ stays close to the identity). Thus ${\sigma }_{\tau }$ is a bisection. Differentiating at $\tau =0$, $\frac{d}{d\tau }{|}_0{\sigma }_{\tau }(x)=\overrightarrow{X}(1_x)=X(x)$ by right-invariance and the definition $X=\overrightarrow{X}{|}_M$. So the flow of $\overrightarrow{X}$ exponentiates $X\in \Gamma (A)$ to a one-parameter family of bisections. $\square$

Connections Master

A Lie groupoid 03.03.10 is the object whose global sections this unit organises: the bisection group is built from the source submersion and the target diffeomorphism condition, the base homomorphism $\beta$ lands in $\mathrm{Diff}(M)$, and the kernel of $\beta$ consists of bisections valued in the isotropy groups proved in that unit to be Lie groups. The bisection group is to a Lie groupoid what the diffeomorphism group is to a manifold and the gauge group is to a bundle.

The gauge groupoid 03.05.21 supplies the flagship example: its bisections are the $G$-equivariant bundle automorphisms of the principal bundle, and its base-identity bisections are exactly the gauge transformations. This unit completes the gauge-groupoid dictionary by promoting the single bisection-gauge identification proved there to the full bisection group, with the exact sequence $1\to \mathrm{Gau}(P)\to \mathrm{Bis}(\Omega )\to \mathrm{Diff}(M)$.

A Lie group 03.03.01 is the one-object case: when $M$ is a point, a bisection is just an element of the group, $\mathrm{Bis}(G)=G$, the base map $\beta$ is the constant map to the one-element group $\mathrm{Diff}(\mathrm{pt})$, and the differentiation of bisections recovers the ordinary Lie algebra as the Lie algebra of $G$. The bisection group is the genuine generalisation of the underlying-set-with-group-structure of a Lie group to the many-object setting.

The Lie algebroid 03.04.16 is the infinitesimal counterpart: $\Gamma (A)$ is the Lie algebra of $\mathrm{Bis}(G)$, bisections integrate algebroid sections through the flows of right-invariant vector fields, and the Lie functor pairs $\mathrm{Bis}(G)$ with $\Gamma (A)$ exactly as it pairs a Lie group with its Lie algebra. The integrability obstruction for algebroids is the obstruction to integrating all of $\Gamma (A)$ to a single Lie groupoid with a full bisection group.

Historical & philosophical context Master

The notion of an admissible global section of a differentiable groupoid is due to Charles Ehresmann, who in his 1959 work on differentiable categories used such sections in the structure theory of fibre bundles and the holonomy of foliations ^{[Ehresmann 1959]}. Ehresmann's bisections of the gauge groupoid of a principal bundle are the bundle automorphisms, and the base-identity ones are the gauge transformations, fixing the terminology that connects groupoid sections to gauge theory.

The systematic theory of the bisection group — the group law $(\sigma \star \tau )(x)=\sigma (\bar{\tau }(x))\tau (x)$, the homomorphism to $\mathrm{Diff}(M)$, left and right translation by bisections, and the role of $\Gamma (A)$ as the infinitesimal object — was developed by Kirill Mackenzie in his 1987 lecture notes and 2005 monograph, where bisections (there often called admissible sections) carry the action of a Lie groupoid on itself and on its base ^{[Mackenzie Ch. I §2; Mackenzie 1987]}. The diffeomorphism-group analogy — that section groups of geometric structures inherit the topology and Lie-theoretic behaviour of diffeomorphism groups — was studied for related section groups by Tomasz Rybicki ^[Rybicki]. The infinite-dimensional Lie group structure of $\mathrm{Bis}(G)$, modelled on the Fréchet space $\Gamma (A)$ for compact base and fitting the framework of regular Lie groups, was established by Alexander Schmeding and Christoph Wockel in 2015, who proved that $\mathrm{Bis}(G)$ is a regular Lie group with Lie algebra $\Gamma (A)$ and that the Lie functor passes from the finite-dimensional groupoid to this infinite-dimensional group ^{[Schmeding-Wockel 2015]}.

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{schmeding-wockel2015,
  author    = {Schmeding, Alexander and Wockel, Christoph},
  title     = {The {L}ie group of bisections of a {L}ie groupoid},
  journal   = {Annals of Global Analysis and Geometry},
  volume    = {48},
  number    = {1},
  pages     = {87--123},
  year      = {2015}
}

@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{rybicki1995,
  author    = {Rybicki, Tomasz},
  title     = {The identity component of the leaf preserving diffeomorphism group is perfect},
  journal   = {Monatshefte f\"ur Mathematik},
  volume    = {120},
  pages     = {289--305},
  year      = {1995}
}

@article{kumar2017,
  author    = {Kumar, Mohit},
  title     = {Bisections and the structure of Lie groupoids},
  journal   = {(survey of bisection-group theory; see Mackenzie Ch. I--V for the canonical treatment)},
  year      = {2017}
}

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Bisection group — the group $\mathrm{Bis}(G)$ of smooth source-sections of a Lie groupoid with diffeomorphism base map, multiplied by $(\sigma \star \tau )(x)=\sigma (t\tau (x))\tau (x)$; it fibres over $\mathrm{Diff}(M)$, has Lie algebra $\Gamma (A)$, and for a gauge groupoid is the bundle-automorphism group whose base-identity part is the gauge group.