# nLab higher Klein geometry

## Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

###### 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

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.

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $O(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz group $O(d-1,1)$Minkowski spacetime $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein 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 groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)$conformal parabolic subgroupMöbius space $S^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher 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

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