nLab
bisection of a Lie groupoid

Definition

In components

Definition

Let $(X_1\stackrel{(d_0,d_1)}{\to }{X}_0\times X_0)$ be a Lie groupoid.

A bisection of is a smooth function $\sigma :X_0\to X_1$ such that

1. $\sigma$ is a section of $d_1$;

2. $d_0\circ \sigma :X_0\to X_0$ is a diffeomorphism.

Bisections naturally form a group under pointwise composition in $X$, the group of bisections of the Lie groupoid.

Abstractly

Let $H=$ Smooth∞Grpd. Let $X\in H$ be equipped with an atlas, hence with an effective epimorphism $U\to X$ out of a 0-truncated object.

We may regard this atlas as an object in the slice (∞,1)-topos $X\in H_{/X}$

Definition

The smooth ∞-group of bisections of $X$ is its automorphism ∞-group

$$\mathrm{BiSect}(X,U)≔{\mathrm{Aut}}_{/X}(X,X).$$\mathbf{BiSect}(X,U) \coloneqq \mathbf{Aut}_{/X}(\mathbf{X}, \mathbf{X}) \,.

Remark

For $X$ a 1-groupoid as above and $U=X_0$, a bisection is precisely a smooth natural transformation of the form

$$\begin{array}{ccccc}U& & \stackrel{\simeq }{\to }& & U\\ & \searrow & {⇙}_{\eta }& \swarrow \\ & & X\end{array}.$$\array{ U &&\stackrel{\simeq}{\to}&& U \\ & \searrow &\swArrow_{\mathrlap{\eta}}& \swarrow \\ && X } \,.

Here the top morphism is a diffeomorphism $\varphi :X\to X$ and since the diagonal morphisms are identities onto the object manifold the component map of $\eta$ is

$$x\mapsto (x\stackrel{\eta (x)}{\to }\varphi (x)).$$x \mapsto (x \stackrel{\eta(x)}{\to} \phi(x)) \,.

This is precisely the bisection in the traditional sense of def. 1.

Properties

Relation to Lie-Rinehart algebras

For $U\to X$ a Lie groupoid with atlas as above, write $𝔤=\mathrm{Lie}(\mathrm{BiSect}(X,U))$ for the Lie algebra of the group of bisections. Then $(C^{\mathrm{\infty }}(X),𝔤)$ is the Lie-Rinehart algebra corresponding to the Lie algebroid of the Lie groupoid.

Relation to Atiyah groupoids

for the moment see at Atiyah groupoid and higher Atiyah groupoid.