nLab
Klein 2-geometry

Context

Geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

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, GG. 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/HG/H, where HGH \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 GG with a 2-group, and replace HH with a “sub-2-group”. We then need to define the suitable analogue of the quotient G/HG/H, and see in what sense GG acts transitively on this.

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

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

Discussion

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

Revised on September 10, 2013 11:52:52 by Urs Schreiber (82.113.99.141)