derived smooth geometry
A principal homogeneous -space is the total space of a -torsor over a point.
There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.
Under weak topological conditions (cf. Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces), every topological homogeneous space is isomorphic to a coset space for a closed subgroup (the stabilizer of a fixed point in ).