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 $BG$ and $BH$ for the corresponding delooping groupoids with a single object. Then the action groupoid $G//H$ is the homotopy fiber of the inclusion functor

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

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

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

that exhibits $G//H$ as the $G$-principal bundle over $BH$ which is classified by the cocycle $BH\to BG$.

Moreover, the decategorification of the action groupoid (its 0-truncation) is the ordinary quotient

$${\tau }_0(G//H)=G/H.$$\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

$$H\to G\to G//H\to BH\to BG.$$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 $H$ be a choice of cohesive structure. For instance choose

• $H=$ Disc∞Grpd for discrete higher Klein geometry (no actual geometric structure);

• $H=$ ETop∞Grpd for continuous higher Klein geometry (with topological structure);

• $H=$ Smooth∞Grpd for higher Klein geometry based on differential geometry;

• $H=$ SuperSmooth∞Grpd for the supergeometry version of higher Klein geometry

• and so on.

Definition

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

$$G//H\to BH\stackrel{i}{\to }BG$$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 $\mathrm{Stab}(f)\to G$ of $f$ in $X$.

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 $BH$ that is classified by the cocycle $i$.

  Continuing this fiber sequence further to the left yields the long fiber sequence

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

  This exhibits $G$ indeed as the fiber of $G//H\to BH$.

Examples

Higher super Poincaré Klein geometry

Let $H=$ SuperSmooth∞Grpd be the context for synthetic higher supergeometry.

Write ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{11}$ for the super L-∞ algebra called the supergravity Lie 6-algebra. This has a sub-super $L_{\mathrm{\infty }}$-algebra of the form

$$B(\mathrm{𝔰𝔬}(10,1)\oplus bℝ\oplus b^5ℝ)\hookrightarrow B{\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{11},$$\mathbf{B}(\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}) \hookrightarrow \mathbf{B}\mathfrak{sugra}_11 \,,

where

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

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

The quotient

$${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{11}/((\mathrm{𝔰𝔬}(10,1)\oplus bℝ\oplus b^5ℝ))$$\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

local model	global geometry
Klein geometry	Cartan geometry
Klein 2-geometry	Cartan 2-geometry
higher Klein geometry	higher Cartan geometry