higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The Erlangen program (in German, erlanger Programm ) is a project, begun by Felix Klein at Erlangen in the 19th century, to study geometry from the perspective of group theory. The idea is to take the elementary building blocks of geometry to be not just Euclidean spaces but more generally homogeneous spaces. These have then also been called Klein geometries .
It was published in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen, translated by M. W. Haskell as A Comparative Review of Recent Researches in Geometry.
Elements of Klein geometry are recently being generalized fropm groups to groupoids and even categories or $\infty$-groupoids. See higher Klein geometry.
local model | global geometry |
---|---|
Klein geometry | Cartan geometry |
Klein 2-geometry | Cartan 2-geometry |
higher Klein geometry | higher Cartan geometry |
Vladimir Kisil, Erlangen Programme at Large: An Overview (arXiv:1106.1686)