higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
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 torsu?, then the coset $G/T$ play a central role in representation theory and cohomology, for instance in the splitting principle.
Under weak topological conditions (cf. Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces), 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 $X$).