nLab Haefliger groupoid

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 groupoid 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.

(Carchedi 12, theorem 3.3.)

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.

(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 $\Theta$ in theorem 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.

(Carchedi 12, theorem 3.1).

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:

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

(Carchedi 12, 3.2).

References

Original articles:

• André Haefliger, Homotopy and integrability, In: N.H. Kuiper (ed.) Manifolds — Amsterdam 1970. Lecture Notes in Mathematics, vol 197. Springer 1971 (doi:10.1007/BFb0068615)

• André Haefliger, Groupoïdes d’holonomie et espaces classiants , Astérisque 116 (1984), 70-97 (numdam:AST_1984__116__70_0)

• Raoul Bott, Lectures on characteristic classes and foliations, Springer LNM 279, 1-94 (doi:10.1007/BFb0058509)

A textbook account is in

• Ieke Moerdijk, Janez Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics 91, 2003. x+173 pp. ISBN: 0-521-83197-0

See also

• Wikipedia, Haefliger structure

Discussion in a broader context of étale stacks and étale ∞-stacks:

• David Carchedi, section 2.2, section 3 of: Étale Stacks as Prolongations, Advances in Mathematics Volume 352, 20 August 2019, Pages 56-132 (arXiv:1212.2282)

• David Carchedi, p. 104-105 of: Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi, Memoirs of the American Mathematical Society 2020; 120 (arXiv:1312.2204, ISBN:978-1-4704-5810-2)

• David Carchedi, On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces, Advances of Mathematics, Volume 294, 2016, Pages 756-818 (arXiv:1504.02394)

Discussion of jet groupoids includes

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

The geometric realization/shape modality for Haefliger-type groupoids is discussed in

• David Carchedi, On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces (arXiv:1504.02394)