topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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 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:
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
See also
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
The geometric realization/shape modality for Haefliger-type groupoids is discussed in
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