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)
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In view of the congruence of the notions of homotopy type and type in homotopy type theory it makes sense to refer to an object in a cohesive (∞,1)-topos $\mathbf{H}$ such as as Smooth∞Grpd as a smooth homotopy type or smooth infinity-groupoid. Accordingly then an n-truncated object in $\mathbf{H}$ is a smooth $n$-type.
For instance a smooth 0-type is then an object in the sheaf topos $Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \simeq \mathbf{H}$ of smooth sets.