higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Higher Klein geometry is the generalization of Klein geometry from traditional (differential) geometry to higher geometry:
where Klein geometry is about (Lie) groups and their quotients, higher Klein geometry is about (smooth) ∞-groups and their ∞-quotients.
The way that the generalization proceeds is clear after the following observation.
Let $G$ be a discrete group and $H \hookrightarrow G$ a subgroup. Write $\mathbf{B}G$ and $\mathbf{B}H$ for the corresponding delooping groupoids with a single object. Then the action groupoid $G//H$ is the homotopy fiber of the inclusion functor
in the (2,1)-category Grpd: we have a fiber sequence
that exhibits $G//H$ as the $G$-principal bundle over $\mathbf{B}H$ which is classified by the cocycle $\mathbf{B}H \to \mathbf{B}G$.
Moreover, the decategorification of the action groupoid (its 0-truncation) is the ordinary quotient
This should all be explained in detail at action groupoid.
The fact that a quotient is given by a homotopy fiber is a special case of the general theorem discussed at ∞-colimits with values in ∞Grpd
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 $G/H$ 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 $\mathbf{H}$ be a choice of cohesive structure. For instance choose
$\mathbf{H} =$ Disc∞Grpd for discrete higher Klein geometry (no actual geometric structure);
$\mathbf{H} =$ ETop∞Grpd for continuous higher Klein geometry (with topological structure);
$\mathbf{H} =$ Smooth∞Grpd for higher Klein geometry based on differential geometry;
$\mathbf{H} =$ SuperSmooth∞Grpd for the supergeometry version of higher Klein geometry
and so on.
An $\infty$-Klein geometry in $\mathbf{H}$ is a fiber sequence in $\mathbf{H}$
for $i$ any morphism between two connected objects, as indicated, hence $\Omega i : H \to G$ any morphism of ∞-group objects.
For $X$ an object equipped with a $G$-action and $f : Y \to X$ any morphism, the higher Klein geometry induced by “the shape $Y$ in $X$” is given by taking $i : H \to G$ be the stabilizer ∞-group $Stab(f) \to G$ of $f$ in $X$. See there at Examples – Stabilizers of shapes / Klein geometry.
By the discussion at looping and delooping, and using that a cohesive (∞,1)-topos has homotopy dimension 0 it follows that every connected object indeed is the delooping of an ∞-group object.
The above says that $G//H$ is the principal ∞-bundle over $\mathbf{B}H$ that is classified by the cocycle $i$.
Continuing this fiber sequence further to the left yields the long fiber sequence
This exhibits $G$ indeed as the fiber of $G//H \to \mathbf{B}H$.
Let $\mathbf{H} =$ SuperSmooth∞Grpd be the context for synthetic higher supergeometry.
Write $\mathfrak{sugra}_11$ for the super L-∞ algebra called the supergravity Lie 6-algebra. This has a sub-super $L_\infty$-algebra of the form
where
$\mathfrak{so}(d,1)$ is the special orthogonal Lie algebra;
$b^{n-1} \mathbb{R}$ is the line Lie n-algebra.
The quotient
is the super translation Lie algebra in 11-dimensions.
This higher Klein geometry is the local model for the higher Cartan geometry that describes 11-dimensional supergravity. See D'Auria-Fre formulation of supergravity for more on this.