Haefliger groupoid in nLab
Context
Category theory
Topology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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fiber space, space attachment
Extra stuff, structure, properties
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Kolmogorov space, Hausdorff space, regular space, normal space
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sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
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open subspaces of compact Hausdorff spaces are locally compact
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compact spaces equivalently have converging subnet of every net
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continuous metric space valued function on compact metric space is uniformly continuous
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
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injective proper maps to locally compact spaces are equivalently the closed embeddings
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locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Contents
Definition
Variants and Generalizations
Properties
Classification of foliations
The Haefliger groupoid classifies foliations. See at Haefliger theorem.
Universal characterization
Consider in the following the union ℋ\mathcal{H} of Haefliger groupoids over all nn.
This implies (Carchedi 12, 3,2)
Sheaves and stacks on the Haefliger groupoid.
Consider in the following the union ℋ\mathcal{H} of Haefliger groupoids over all nn.
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(ℋ)≃St(SmthMfd et) St(\mathcal{H}) \simeq St(SmthMfd^{et})
References
Original articles:
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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)
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André Haefliger, Groupoïdes d’holonomie et espaces classiants , Astérisque 116 (1984), 70-97 (numdam:AST_1984__116__70_0)
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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:
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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)
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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)
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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)
Last revised on December 12, 2023 at 04:01:23. See the history of this page for a list of all contributions to it.