see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
Theorems
A topological groupoid or Lie groupoid $C$ is called an étale groupoid if the source-map $s : Mor C \to Obj C$ is a local homeomorphism or local diffeomorphism, respectively, and hence exhibits the space of morphisms as an étale space over the space of objects.
In the situation of def. 1 it follows that all the other structure maps (target, identity, composition) are also local homeomorphisms, resp. local diffeomorphisms.
This means that an étale groupoid is equivalently an internal groupoid in the category whose objects are topological spaces/smooth manifolds and whose morphisms are local homeomorphisms/diffeomorphisms.
Definition 1 is not invariant under the general notion of equivalence of Lie groupoids, the equivalence between them regarded as smooth groupoids, specifically as differentiable stacks (“Morita equivalence”).
But it does make sense to take an étale smooth groupoid to be a smooth groupoid/differentiable stack which is equivalent, as such, to, hence is presented by an étale Lie groupoid as in def. 1. This notion has been called folitation groupoid in (Crainic-Moerdijk 00).
The following characterizes foliation groupoids
For a Lie groupoid $\mathcal{G}_\bullet$ the following are equivalent
$\mathcal{G}$ is a foliation groupoid, hence is equivalent, as a differentiable stack to an étale groupoid as in def. 1;
The Lie algebroid $(\mathfrak{g},\mathcal{G}_0)$ which corresponds to $\mathcal{G}$ under Lie differentiation has an injective anchor map;
hence the orbits of $\mathcal{G}$ form the leaves of a foliation, the foliation whose leaves are tangent to the vectors in the image of this anchor map;
All isotropy groups of $\mathcal{G}_\bullet$ are discrete groups.
This is (Crainic-Moerdijk 00, theorem 1).
In the literature one finds, roughly speaking, two different approaches to the study of étale groupoids. One approach is based on the construction of the convolution algebras associated to an étale groupoid, in the spirit of Connes’ noncommutative geometry, and involves the study of cyclic and Hochschild homology and cohomology of these algebras. The other approach uses methods of algebraic topology such as the construction of the classifying space of an étale groupoid and its (sheaf) cohomology groups.
For $X_\bullet$ an étale groupoid, there is a canonical morphism
to the Haefliger groupoid, example 3, of its manifold of objects. The kernel of this map is the ineffective part of $X_\bullet$. If the kernel vanishes, then $X$ is called an effective Lie groupoid.
(e.g. Carchedi 12, section 2.2)
The groupoid convolution algebra $C^\ast(\mathcal{G}_\bullet)$ of a Lie groupoid with its canonical atlas remembered has the structure of a Hopf algebroid. In (Mrčun 99, Kališnik-Mrčun 07) étale Lie groupoids are characterized dually by their Hopf algebroids (a refinement of Gelfand duality to noncommutative topology).
The 2-category of étale stacks with étale maps between them is equivalent to the 2-topos over the site of smooth manifolds with local diffeomorphisms between them
(Carchedi 12, theorem 3.4, corollary 3.3)
A smooth stack is an étale stack precisely if it is in the essential image of the left Kan extension along the non-full inclusion of sites
of smooth manifolds, with local diffeomorphisms on the left and all smooth functions on the right.
(Carchedi 12, theorem 3.5, corollary 3.4)
In particular:
A smooth stack is an effective étale stack precisely if under the prolongation of prop. 2 it is equivalent to the image of a sheaf (i.e. of a 0-truncated stack).
(Carchedi 12, corollary 4.1, corollary 4.2)
See at differential cohesion the section Etale objects.
Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions.
Every topological space may be regarded as an étale groupoid with only identity morphisms.
For $X$ a topological space and $\Gamma$ a discrete group with a continuous action $X \times \Gamma \to X$ on $X$, the action groupoid $X//\Gamma$ is étale.
The Haefliger groupoid $\Gamma^q$ has the Cartesian space $\mathbb{R}^q$ as its space of objects. A morphism $x \to y$ is a germ of a diffeomorphism $(\mathbb{R}^q,x) \to (\mathbb{R}^q, y)$.
This groupoid, and its geometric realization play a central role in foliation theory.
Every orbifold is an étale Lie groupoid.
A standard textbook account is section 5.5. of
The relation between étale groupoid and foliations is analyzed in detail in
See also at orbifold for basic and introductory literature.
Further discussion of étale groupoids and their properties includes
Marius Crainic, Ieke Moerdijk, A Homology Theory for Étale groupoids (journal)
David Carchedi, Sheaf Theory for Étale Geometric Stacks (arXiv:1011.6070)
David Carchedi, section 2.2, section 3 of Étale Stacks as Prolongations (arXiv:1212.2282)
The convolution-Hopf algebroids of étale Lie groupoids have been characterized in