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.

local model spaceglobal geometrydifferential cohomologyfirst order formulation of gravity
generalKlein geometryCartan geometryCartan connection
examplesEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
Lorentzian supergeometysupergeometrysuperconnectionsupergravity
generalKlein 2-geometryCartan 2-geometry
higher 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)