higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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 (arXiv:1208.4948, web, pdf)
Dominic Joyce, Aarhus Masterclass on Derived Differential Geometry (videos)
Discussion via (differential) cohesive $(\infty,1)$-topos theory:
exposition:
Last revised on July 31, 2023 at 07:35:13. See the history of this page for a list of all contributions to it.