nLab
Haefliger groupoid

Contents

Definition

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

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.

(Carchedi 12, theorem 3.3.)

This implies (Carchedi 12, 3,2)

Theorem

There is an equivalence

Θ:St(SmthMfd et)EtSt(SmthMfd) et \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.

(Carchedi 12, theorem 1.3)

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 Θ\Theta in theorem 1 of the sheaf on SmthMfd etSmthMfd^{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 nn.

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.

(Carchedi 12, theorem 3.1).

Proposition

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

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

(Carchedi 12, 3.2).

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

See also

Discussion of jet-restrictions of the Haefliger groupoid is in

  • Arne Lorenz, Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method, Thesis (pdf)

Revised on July 15, 2014 06:09:12 by Urs Schreiber (82.113.98.7)