nLab higher Klein geometry

Idea

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.

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

$$\mathbf{B}H \to \mathbf{B}G$$

in the (2,1)-category Grpd: we have a fiber sequence

$$G//H \to \mathbf{B}H \to \mathbf{B}G$$

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

$$\tau_0 (G//H) = G/H \,.$$

Proof

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

Remark

That fiber sequence continues to the left as

$$H \to G \to G//H \to \mathbf{B}H \to \mathbf{B}G \,.$$

Observation

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 :

Definition

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.

Definition

An $\infty$-Klein geometry in $\mathbf{H}$ is a fiber sequence in $\mathbf{H}$

$$G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G$$

for $i$ any morphism between two connected objects, as indicated, hence $\Omega i : H \to G$ any morphism of ∞-group objects.

Remark

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.

Remarks

• 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

  $$H \to G \to G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G$$

  This exhibits $G$ indeed as the fiber of $G//H \to \mathbf{B}H$.

Examples

Higher super Poincaré Klein geometry

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

$$\mathbf{B}(\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}) \hookrightarrow \mathbf{B}\mathfrak{sugra}_11 \,,$$

where

• $\mathfrak{so}(d,1)$ is the special orthogonal Lie algebra;

• $b^{n-1} \mathbb{R}$ is the line Lie n-algebra.

The quotient

$$\mathfrak{sugra}_11 / ((\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}))$$

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.

Related concepts

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

References

That there ought to be a systematic study of higher Klein geometry and higher Cartan geometry has been amplified by David Corfield since 2006.

Such a formalization is offered in

• differential cohomology in a cohesive topos

For more on this see at higher Cartan geometry and Higher Cartan Geometry.