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 $X$).
Textbook accounts:
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
On homogeneous spaces with the same rational cohomology as a product of n-spheres:
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:
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