derived smooth geometry
The idea of Klein’s original Erlangen program is to study geometries that have some group of symmetries, . If this group acts transitively on some kind of geometrical figure - e.g. points, lines, etc. - the space of figures of this kind is , where is the stabilizer of such a figure: that is, the subgroup of symmetries that preserve it.
To categorify this we’d like to replace with a 2-group, and replace with a “sub-2-group”. We then need to define the suitable analogue of the quotient , and see in what sense acts transitively on this.
|local model space||global geometry||differential cohomology||first order formulation of gravity|
|general||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|general||Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|