higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
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derived smooth geometry
Given a topological group or algebraic group or Lie group, etc., $G$, a homogeneous $G$-space is a topological space or scheme, or smooth manifold etc. with transitive $G$-action.
A principal homogeneous $G$-space is the total space of a $G$-torsor over a point.
There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.
A special case of homogeneous spaces are coset spaces arising from the quotient $G/H$ of a group $G$ by a subgroup. For the case of Lie groups this is also called Klein geometry.
Specifically for $G$ a compact Lie group and $T\hookrightarrow G$ a maximal torus, then the coset $G/T$ play a central role in representation theory and cohomology, for instance in the splitting principle.
In analysis and number theory, certain functions on certain coset spaces play a role as automorphic forms (e.g. modular forms). See there for more.
Under weak topological conditions (cf. Helgason), every topological homogeneous space $M$ is isomorphic to a coset space $G/H$ for a closed subgroup $H\subset G$ (the stabilizer of a fixed point in $X$).
Glen Bredon, Section I.4 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)
Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces
The following article has categorical analysis of relation between the total space of a principal bundle and of the corresponding quotient space both for the classical case and for noncommutative generalizations:
Last revised on April 8, 2021 at 01:56:40. See the history of this page for a list of all contributions to it.