higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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.
Blog entries: I, II, III, IV, V, VI, VII, VIII, VIIIS, IX, X, XI, XII, and stabilizer.