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)
Given a smooth manifold $X$, and given $k \in \mathbb{N}\cup \{\infty\}$, there is a Lie groupoid whose objects are the points of $X$ and whose morphisms between two such points $x\to y$ are order-$k$ jets of local diffeomorphisms taking $x$ to $y$.
The automorphism groups of objects in these groupoids are jet groups.
Hence these Lie groupoids are often called jet groupoids, e.g. (Lorenz 09). If one passes from jets to germs of local diffeomorphisms then one arrives essentially at the Haefliger groupoid of $X$ (except that this has a more discrete smooth structure on its set of morphisms).
In a context of synthetic differential geometry or differential cohesion there is the bundle $T_{inf}^k X\to X$ of order-$k$ infinitesimal neighbourhoods in $X$. In terms of this the jet groupoid is the Atiyah groupoid of $T_{inf}^k X$, the groupoid whose morphisms between objects $x$ and $y$ are isomorphism of the fibers of this bundle over these points.
Last revised on January 4, 2015 at 23:47:15. See the history of this page for a list of all contributions to it.