derived smooth geometry
The way that the generalization proceeds is clear after the following observation.
This should all be explained in detail at action groupoid.
That fiber sequence continues to the left as
The above statement remains true verbatim if discrete groups are generalized to Lie groups – or other cohesive groups – if only we pass from the (2,1)-topos Grpd of discrete groupoids to the (2,1)-topos SmoothGrpd of smooth groupoids .
This follows with the discussion at smooth ∞-groupoid -- structures.
Since the quotient is what is called a Klein geometry and since by the above observations we have analogs of these quotients for higher cohesive groups, there is then an evident definition of higher Klein geometry :
Let be a choice of cohesive structure. For instance choose
Disc∞Grpd for discrete higher Klein geometry (no actual geometric structure);
and so on.
An -Klein geometry in is a fiber sequence in
For an object equipped with a -action and any morphism, the higher Klein geometry induced by “the shape in ” is given by taking be the stabilizer ∞-group of in . See there at Examples – Stabilizers of shapes / Klein geometry.
Continuing this fiber sequence further to the left yields the long fiber sequence
This exhibits indeed as the fiber of .
is the super translation Lie algebra in 11-dimensions.
|gauge group||stabilizer subgroup||local model space||local geometry||global geometry||differential cohomology||first order formulation of gravity|
|general||Lie group/algebraic group||subgroup (monomorphism)||quotient (“coset space”)||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean group||rotation group||Cartesian space||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Poincaré group||Lorentz group||Minkowski space||Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|super Poincaré group||spin group||super Minkowski spacetime||Lorentzian supergeometry||supergeometry||superconnection||supergravity|
|linear algebraic group||parabolic subgroup/Borel subgroup||flag variety||parabolic geometry|
|orthochronous Lorentz group||conformal geometry||conformal connection||conformal gravity|
|general||smooth 2-group||2-monomorphism||homotopy quotient||Klein 2-geometry||Cartan 2-geometry|
|cohesive ∞-group||∞-monomorphism (i.e. any homomorphism)||homotopy quotient of ∞-action||higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|
Such a formalization is offered in