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, . If this group acts transitively on some kind of geometrical figure - e.g. points, lines, etc. - the space of figures of this kind is , where is the stabilizer of such a figure: that is, the subgroup of symmetries that preserve it.
To categorify this we’d like to replace with a 2-group, and replace with a “sub-2-group”. We then need to define the suitable analogue of the quotient , and see in what sense acts transitively on this.
A suitable notion of sub-2-group is a faithful, but not necessarily full, functor into . Quotients may be thought of as homogeneous orbifolds.
Blog entries: I, II, III, IV, V, VI, VII, VIII, VIIIS, IX, X, XI, and stabilizer.