nLab
bisection of a Lie groupoid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Bisections of Lie groupoids

Definition

In components

Definition

Let (X 1(d 0,d 1)X 0×X 0)(X_1 \stackrel{(d_0, d_1)}{\to} X_0 \times X_0) be a Lie groupoid.

A bisection of is a smooth function σ:X 0X 1\sigma : X_0 \to X_1 such that

  1. σ\sigma is a section of d 1d_1;

  2. d 0σ:X 0X 0d_0 \circ \sigma : X_0 \to X_0 is a diffeomorphism.

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

Abstractly

Let H=\mathbf{H} = Smooth∞Grpd. Let XHX \in \mathbf{H} be equipped with an atlas, hence with an effective epimorphism UXU \to X out of a 0-truncated object.

We may regard this atlas as an object in the slice (∞,1)-topos XH /X\mathbf{X} \in \mathbf{H}_{/X}

Definition

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

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

For XX a 1-groupoid as above and U=X 0U = X_0, a bisection is precisely a smooth natural transformation of the form

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

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

x(xη(x)ϕ(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 UXU \to X a Lie groupoid with atlas as above, write 𝔤=Lie(BiSect(X,U))\mathfrak{g} = Lie(\mathbf{BiSect}(X,U)) for the Lie algebra of the group of bisections. Then (C (X),𝔤)(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)