∞-Lie theory

# Bisections of Lie groupoids

## 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 $\mathbf{H} =$ Smooth∞Grpd. Let $X \in \mathbf{H}$ 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 $\mathbf{X} \in \mathbf{H}_{/X}$

###### Definition

The smooth ∞-group of bisections of $\mathbf{X}$ is its automorphism ∞-group

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

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

$\array{ U &&\stackrel{\simeq}{\to}&& U \\ & \searrow &\swArrow_{\mathrlap{\eta}}& \swarrow \\ && X } \,.$

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

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

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 $\mathfrak{g} = Lie(\mathbf{BiSect}(X,U))$ for the Lie algebra of the group of bisections. Then $(C^\infty(X), \mathfrak{g})$ 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.

Revised on February 20, 2013 21:30:54 by Urs Schreiber (80.81.16.253)