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)
Let $M$ be a differential manifold with differentiable left action of Lie group $G$, $G\times M\to M$ (respectively right action $M\times G\to G$). For example, the multiplication map of $G$ on itself. Then we define the left translations $L_g : m\mapsto g m$ (resp. right translations $R_g: m\mapsto m g$) for every $g\in G$, which are both diffeomorphisms of $M$.
A differential form on a Lie group $\omega \in \Omega^1(G)$ is called left invariant if for every $g \in G$ it is invariant under the pullback by the translation $L_g$
$(L_g)^* \omega = \omega$.
Analogously a form is right invariant if it is invariant under the pullback by right translations $R_g$. For a vector field $X$ one instead typically defines the invariance via the pushforward $(T L_g) X = (L_g)_* X$. Regarding that $L_g$ and $T_g$ are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.