higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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)
Derived differential geometry is higher differential geometry in an ambient (∞,1)-topos which is not 1-localic. This is the derived geometry corresponding to differential geometry. Typically this is specifically taken to be the derived geometry induced by the Lawvere theory for smooth algebras ($C^\infty$-rings):
See
Urs Schreiber, Seminar on derived differential geometry
Dominic Joyce, D-manifolds and d-orbifolds: a theory of derived differential geometry (pdf)
Dominic Joyce, Aarhus Masterclass on Derived Differential Geometry (videos)