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Klein 2-geometry

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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⊆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.

local modelglobal geometry
Klein geometryCartan geometry
Klein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometry

Discussion

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

Revised on August 31, 2011 17:29:14 by David Corfield (129.12.18.29)
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