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)
In the context of generalized complex geometry one says for $X$ a manifold, $T X$ its tangent bundle and $T^* X$ the cotangent bundle that the fiberwise direct sum-bundle $T X \oplus T^* X$ is the generalized tangent bundle.
More generally, a vector bundle $E \to X$ that sits in an exact sequence $T^* X \to E \to T X$ is called a generalized tangent bundle, such as notably those underlying a Courant Lie 2-algebroid over $X$.
The ordinary tangent bundle is the canonical associated bundle to the general linear group-principal bundle classified by the morphism
to the smooth moduli stack of $GL(n)$.
Similarly there is a canonical morphism
to the moduli stack which is the delooping of the Narain group $O(n,n)$. This classifies the $O(n,n)$-principal bundle to which $T X \oplus T^* X$ is associated.
Where a reduction of the structure group of the tangent bundle along $\mathbf{B} O(n) \hookrightarrow \mathbf{B} GL(n)$ is equivalently a vielbein/orthogonal structure/Riemannian metric on $X$, so a reduction of the structure group of the generalized tangent bundle along $\mathbf{B} (O(n) \times O(n)) \to \mathbf{B}O(n,n)$ is a generalized vielbein, defining a type II geometry on $X$.
Other reductions yield other geometric notions, for instance:
reduction along $U(n,n) \to O(2n,2n)$ is a generalized complex structure;
further reduction along $SU(n,n) \to U(n,n) \to O(2n,2n)$ is a generalized Calabi-Yau manifold structure.