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||global geometry|
|Klein geometry||Cartan geometry|
|Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry|