nLab Klein 2-geometry

Idea

Klein 2-geometry is an attempt to categorify Klein’s Erlangen program by replacing groups with 2-groups, which are like groups but categories instead of sets.

The idea of Klein’s original Erlangen program is to study geometries that have some group of symmetries, $G$. If this group acts transitively on some kind of geometrical figure - e.g. points, lines, etc. - the space of figures of this kind is $G/H$, where $H \subseteq G$ is the stabilizer of such a figure: that is, the subgroup of symmetries that preserve it.

To categorify this we’d like to replace $G$ with a 2-group, and replace $H$ with a “sub-2-group”. We then need to define the suitable analogue of the quotient $G/H$, and see in what sense $G$ acts transitively on this.

A suitable notion of sub-2-group is a faithful, but not necessarily full, functor into $G$. Quotients may be thought of as homogeneous orbifolds.

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

Discussion

Blog entries: I, II, III, IV, V, VI, VII, VIII, VIIIS, IX, X, XI, XII, and stabilizer.