# nLab jet groupoid

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

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.

• Arne Lorenz, Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method, Thesis (pdf) 2009