CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
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.
More generally for a given integrable G-stucture there is a corresponding Haefliger groupoid, for instance for symplectic structures.
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)
The Haefliger groupoid classifies foliations. See at Haefliger theorem.
Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.
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)
There is an equivalence
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.
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. (…)
Consider in the following the union $\mathcal{H}$ of Haefliger groupoids over all $n$.
The category of sheaves over $\mathcal{H}$ is equivalently the category of sheaves on the site of smooth manifolds with local diffeomorphism between them.
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:
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
The geometric realization/shape modality for Haefliger-type groupoids is discussed in