synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Cartan geometry is geometry of spaces that are locally (infinitesimally, tangentially) like coset spaces $G/H$, i.e. like Klein geometries. Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling without sliding’ the ‘model geometry’ $G/H$ along it. Hence Cartan geometry may be thought of as the globalization of the program of Klein geometry initiated in the Erlangen program.
Cartan geometry subsumes many types of geometry, such as notably Riemannian geometry, conformal geometry, parabolic geometry and many more. As a Cartan geometry is defined by principal connection data (hence by cocycles in nonabelian differential cohomology) this means that it serves to express all these kinds of geometries in connection data. This is used notably in the first order formulation of gravity, which was the motivating example in the original text (Cartan 32).
A Cartan geometry is a space equipped with a Cartan connection. See there for more.
The original article is
Textbook accounts are in
R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlangen program Springer (1997)
Andreas Cap, Jan Slovák, chapter 1 of Parabolic Geometries I – Background and General Theory, AMS 2009
For more see at Cartan connection – References.
Discussion in homotopy type theory is in
See also
wikipedia: Cartan connection
The blog discussion of Derek Wise, MacDowell-Mansouri gravity and Cartan geometry.