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)
For $f \colon X \to Y$ a smooth function between smooth manifold, and for $\omega \in \Omega^n(Y)$ a differential n-form, there is the pullback $n$ form $f^* \omega \in \Omega^n(X)$.
If differential forms are defined as linear duals to vectors then pullback is the dual operation to pushforward of a vector field?
Differential forms may be defined by Yoneda extension from differential forms on Cartesian spaces (see at geometry of physics -- differential forms).
For $X = \mathbb{R}^{\tilde k}$ and $Y = \mathbb{R}^k$ Cartesian spaces and $f \;\colon\; X \longrightarrow Y$ a smooth function between them, and on differential 1-forms
the pullback operation $f^\ast$ is given by
and hence
where
$f^* \omega_i$ is the pullback of functions defined by
the function
is the partial derivative of the $k$-th coordinate component of $f$ along the $j$the coordinate.
(compatiblity with the de Rham differential)
Pullback of differential forms commutes with the de Rham differential:
Hence it constitutes a chain map between the de Rham complexes
Under pullback differential forms form a presheaf on the catories CartSp and SmthMfd, in fact a sheaf with respect to the standard open cover-coverage.
A standard reference is
See also for instance section 2.7 of