higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
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
(AD-Renault 2000) Recalled for instance in (Sims & Williams 2012, p. 4).
The groupoid convolution algebra of an amenable topological groupoid is in the bootstrap category.
(Tu 99, prop. 10.7), recalled as (Uuye 11, example 3.6).
For an amenable Lie groupoid , the full groupoid convolution algebra and the reduced one are naturally isomorphic.
This is due to (AD-Renault 00), recalled for instance as (ANS 05, prop. 1.9)
C. Anantharaman-Delaroche, Jean Renault, Amenable Groupoids Monogr. Enseign. Math., vol. 36 Enseignement Math., Geneva (2000) (webpage, including extensive review)
Aidan Sims, Dana P. Williams, Amenability for Fell bundles over groupoids (arXiv:1201.0792)
Johannes Aastrup, Ryszard Nest, Elmar Schrohe, A Continuous Field of -algebras and the Tangent Groupoid for Manifolds with Boundary (arXiv:math/0507317)
Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, K-Theory 17 (1999), no. 3, 215–264. MR 1703305 (2000g:19004) (Portico, subscription needed)
Otgonbayar Uuye, A note on the Künneth theorem for nonnuclear C-algebras_ (arXiv:1111.7228)
Wikipedia: Amenable group
Last revised on February 10, 2025 at 04:20:56. See the history of this page for a list of all contributions to it.