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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Let $\pi : P \to X$ be a bundle in the category Diff of smooth manifolds.
The dg-algebra $\Omega^\bullet_{vert}(P)$ of vertical differential forms on $P$ is the quotient of the de Rham complex dg-algebra $\Omega^\bullet(P)$ of all forms on $P$, by the dg-ideal of horizontal differential forms, hence of all those forms that vanish when any one vector in their arguments is a vertical vector field in that it is in the kernel of the differential $d \pi : T P \to T X$.
For a trivial bundle $P = X \times F$ the underlying complex of $\Omega^\bullet_{vert}(P)$ is $\wedge^\bullet_{C^\infty(X \times F)} \Gamma(T^* F)$.
Last revised on September 19, 2023 at 05:17:14. See the history of this page for a list of all contributions to it.