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
A topological groupoid is an internal groupoid in the category Top.
So this is a groupoid with a topological space of objects and one of morphisms, and all structure maps (source, target, identity, composition, inverse) are continuous maps. Composition here refers to the map defined on the space of all composable morphisms.
A topological groupoid $C$ is called an open topological groupoid if the source map $s : Mor C \to Obj C$ is an open map.
It is called an étale groupoid if in addition $s$ is a local homeomorphism.
Every topos (Grothendieck topos) with enough points is the classifying topos of a topological groupoid. See there for more.
The notion of topological categories, hence of topological groupoids, goes back to
Their understanding as internal groupoids internal to TopologicalSpaces is often attributed to
but the simple notion of internalization and internal groupoids (Grothendieck 1960, 61) is hardly recognizable in this account.
Exposition:
Textbook account:
Many references on topological groupoids deal with them as models for topological stacks, see there for more.
On topological groupoids as a model for orbispaces:
André Haefliger, Groupoides d’holonomie et classifiants, Astérisque no. 116 (1984), p. 70-97 (numdam:AST_1984__116__70_0)
André Haefliger, Complexes of Groups and Orbihedra, in: E. Ghys, A. Haefliger, A Verjovsky (eds.), Proceedings of the Group Theory from a Geometrical Viewpoint, ICTP, Trieste, Italy , 26 March – 6 April 1990, World Scientific 1991 (doi:10.1142/1235)
On topological groupoids as presenting toposes with enough points:
On exponential objects among topological groupoids:
Last revised on November 16, 2022 at 15:28:03. See the history of this page for a list of all contributions to it.