# nLab Haefliger groupoid

category theory

## Applications

#### Topology

topology

algebraic topology

# Contents

## Definition

###### 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 $(\mathbb{R}^n ,x) \to (\mathbb{R}^n ,y)$.

This is regarded as a topological or Lie étale groupoid via the canonical topology/smooth structure on $(\Gamma^n)_0 = \mathbb{R}^n$ and taking $s \colon (\Gamma^n)_1 \to (\Gamma^n)_0$ to be the étale space associated to the sheaf on $\mathbb{R}^n$ (with its canonical open cover Grothendieck topology) which is the sheafification of the presheaf that sends $U \subset \mathbb{R}^n$ to the set of all open embeddings of $U$ into $\mathbb{R}^n$.

The smooth stack represented by the smooth Haefliger gorupoid is also called the Haefliger stack.

## Variants and Generalizations

###### Remark

There is also the full smooth structure on the space of germs of diffeomorphisms. This gives a Lie groupoid whose underlying bare groupoid is the same as that of the Haefliger groupoid, but whose smooth structure is different.

###### Remark

More generally for a given integrable G-stucture there is a corresponding Haefliger groupoid, for instance for symplectic structures.

###### Remark

Instead of considering germs of local diffeomorphisms one may consider (just) order-$k$ jets of these. The resulting Lie groupoids are known as jet groupoids (see Lorenz 09)

## Properties

### 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