# nLab Haefliger groupoid

category theory

## Applications

#### Topology

topology

algebraic topology

# Contents

## Definition

For $n \in \mathbb{N}$, the Haefliger groupoid $\Gamma^n$ is the groupoid whose set of objects is the Cartesian space $\mathbb{R}^n$ and for which a morphism $x \to y$ is a germ of a diffeomorphism $(\mathbb{R}^n ,x) \to (\mathbb{R}^n ,y)$.

## Properties

### Geometric structure

The Haefliger groupoid is naturally a topological groupoid. As such it is an étale groupoid.

### Classification of foliations

The Haefliger groupoid classifies foliations. See at Haefliger theorem.

### Universal characterization

Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.

###### Proposition

The Haefliger stack is a terminal object in the 2-category of étale stacks on the site of smooth manifolds with étale morphisms between them.

This implies (Carchedi 12, 3,2)

###### Theorem

There is an equivalence

$\Theta \colon St(SmthMfd^{et}) \simeq EtSt(SmthMfd)^{et}$

between stacks on the site of smooth manifolds with local diffeomorphisms between them and étale stacks with étale morphisms between them inside all smooth stacks.

###### Remark

This in turn implies for instance that the Haefliger groupoid for complex structures (Carchedi 12, p. 38) is simply the image under the equivalence $\Theta$ in theorem 1 of the sheaf on $SmthMfd^{et}$ which sends each smooth manifold to its set of complex structures. (…)

### Sheaves and stacks on the Haefliger groupoid.

Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.

###### Proposition

The category of sheaves over $\mathcal{H}$ is equivalently the category of sheaves on the site of smooth manifolds with local diffeomorphism between them.

###### Proposition

The 2-topos over the Haefliger stack is equivalent to the 2-topos over the site $SmthMfd^{et}$ of smooth manifolds with local diffeomorphisms between them:

$St(\mathcal{H}) \simeq St(SmthMfd^{et})$

## References

Original articles include

• André Haefliger, Groupoïdes d’holonomie et espaces classiants , Astérisque 116 (1984), 70-97

• Raoul Bott, Lectures on characteristic classes and foliations , Springer LNM 279, 1-94

A textbook account is in

Discussion in a broader context of étale stacks is in