nLab
Haefliger groupoid

Contents

Definition

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).

This is regarded as a topological or Lie étale groupoid via the canonical topology/smooth structure on (Γ n) 0= n(\Gamma^n)_0 = \mathbb{R}^n and taking s:(Γ n) 1(Γ n) 0s \colon (\Gamma^n)_1 \to (\Gamma^n)_0 to be the étale space associated to the sheaf on n\mathbb{R}^n (with its canonical open cover Grothendieck topology) which is the sheafification of the presheaf that sends U nU \subset \mathbb{R}^n to the set of all open embeddings of UU into n\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-kk 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 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 groupoids includes

  • Arne Lorenz, Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method, Thesis (pdf) 2009

Revised on December 29, 2014 17:30:23 by Urs Schreiber (127.0.0.1)