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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A differential form $\omega \in \Omega^p(X)$ is called closed if the de Rham differential $d \colon \Omega^{p}(X) \to \Omega^{p+1}(X)$ sends it to zero: $d \omega = 0$, hence if it is in the kernel of the de Rham.
A differential form $\omega \in \Omega^{p+1}(X)$ is called exact if it is in the image of the de Rham differential: $\omega = d \alpha$, for some $\alpha \in \Omega^{p}(X)$.
The quotient of the vector space of closed differential forms by the exact differential forms of degree $p$ is the de Rham cohomology of $X$ in degree $p$.
Formalization of closed and co-exact differential forms in cohesive homotopy theory is discussed at differential cohomology hexagon.