# nLab jet bundle

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

A jet can be thought of as the infinitesimal germ of a section of some bundle or of a map between spaces. Jets are a coordinate free version of Taylor-polynomials and Taylor series.

## Definition

### Concrete

For $p : P \to X$ a surjective submersion of smooth manifolds and $k \in \mathbb{N}$, the bundle $J^k P \to X$ of $k$- order jets of sections of $p$ is the bundle whose fiber over a point $x \in X$ is the space of equivalence classes of germs of sections of $p$, where two germs are considered equivalent if their first $k$ partial derivatives at $x$ coincide.

In the case when $p$ is a trivial bundle $p:X\times Y \to X$ its sections are canonically in bijection with maps from $X$ to $Y$ and two sections have the same partial derivatives iff the partial derivatives of the corresponding maps from $X$ to $Y$ agree. So in this case the jet space $J^k P$ is the space of jets of maps from $X$ to $Y$ and commonly denoted with $J^k(X,Y)$.

### The infinite order jet bundle as the cofree $J^1$-coalgebra

Let $J^1$ be the functor on bundles over $X$ sending a bundle $p:P\to X$ to the bundle $J^1 p: J^1 P \to X$ of first order jets of sections of $p$. This functor is co-pointed: the natural projection $J^1 P \to P$ forgets first order information. A $J^1$-coalgebra is then by definition a bundle $p:P\to X$ with a bundle map $P\to J^1 P$, which is the same as a connection on $P$. With this, the infinite order jet bundle of sections of $p$ may be defined as the cofree $J^1$-coalgebra on $P$. By definition it comes with a bundle map $p_\infty: J^\infty P \to P$ (“forgetting all higher order information”) and a universal connection $C: J^\infty P \to J^1( J^\infty P)$ sometimes called the Cartan connection or infinite order contact structure on $J^\infty P$. Its universal property states that for any bundle $q:Q\to X$ with connection $K: Q \to J^1 Q$ and bundle map $\varphi: Q\to P$, there exists a unique prolongation of $\varphi$ denoted with $j^\infty \varphi: Q \to J^\infty P$ commuting with all given maps: $p_\infty \circ j^\infty \varphi =\varphi$ and $J^1(j^\infty \varphi)\circ K= C \circ j^\infty \varphi$.

In the particular case when $q:Q\to X$ is the identity $X\to X$ with its canonical connection, a morphisms of bundles $\varphi$ from $q$ to $p$ is the same as a section of $p$ and $j^\infty \varphi$ is traditionally called the infinite jet prolongation of the section $\varphi$.

When the the functor $J^1$ preserves sequential limits and the category of bundles over $X$ has equalizers, the cofree $J^1$-coalgebra can be constructed as a sequential limit:

$\cdots \to J^3 P \to J^2 P \to J^1 P \to J^0 P = P$

where $J^{n+1} P$ is recursively defined as the equalizer between $J^1 (J^n P ) \to J^n P \to J^1 ( J^{n-1} P)$ and $J^1 (J^n P) \to J^1 (J^{n-1} P)$, where the first arrow in $J^1 (J^n P ) \to J^n P \to J^1 ( J^{n-1} P)$ is the standard projection and the second one is the previously obtained equalizer (starting the induction with the identity $J^1 P \to J^1 P$), while the arrow $J^1(J^n P ) \to J^1( J^{n-1} P)$ is just the map obtained applying the functor $J^1$ to $J^n P \to J^{n-1} P$.

This sequantial limit construction should be a special case (and dual) to the one described in transfinite construction of free algebras.

(…)

### General abstract

We discuss a general abstract definition of jet bundles.

Let

• $\mathbf{H}$ be a cohesive (∞,1)-topos

• equipped with differential cohesion

$\mathbf{H} \stackrel{\hookrightarrow}{\stackrel{\overset{\Pi_{inf}}{\leftarrow}}{\stackrel{\overset{}{\longrightarrow}}{\underset{}{\leftarrow}}}} \mathbf{H}_{th}$

with infinitesimal shape modality $\Pi_{inf}$

• and equipped with an (∞,2)-sheaf

Mod $\colon \; \mathbf{H}^{op} \to$ Stab(∞,1)Cat

For $X \in \mathbf{H}$, write $\mathbf{\Pi}_{inf}(X)$ for the corresponding de Rham space object.

Notice that we have the canonical morphism

$i : X \to \mathbf{\Pi}_{inf}(X)$

(“inclusion of constant paths into all infinitesimal paths”).

###### Definition

Write

$Jet \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i^*}{\leftarrow}}{\underset{Jet := i_*}{\to}} \mathbf{H}_{/\mathbf{\Pi}_{inf}(X)}$

for the corresponding base change geometric morphism.

Its direct image may be called the jet bundle (∞,1)-functor . Or rather, if one regards the jet bundle again as a bundle over $X$, then the jet construction is the (∞,1)-comonad $i^\ast i_\ast$.

In the context of D-schemes this is (BeilinsonDrinfeld, 2.3.2). The abstract formulation as used here appears in (Lurie, prop. 0.9). See also (Paugam, section 2.3) for a review. There this is expressed dually in terms of algebras in D-modules. We indicate how the translation works

###### Remark

In terms of differential homotopy type theory this means that forming “jet types” of dependent types over $X$ is the dependent product operation along the unit of the infinitesimal shape modality

$jet(E) \coloneqq \underset{X \to \Pi_{inf}X}{\prod} E \,.$
###### Definition

A quasicoherent (∞,1)-sheaf on $X$ is a morphism of (∞,2)-sheaves

$X \to Mod \,.$

We write

$QC(X) := Hom(X, Mod)$

A D-module on $X$ is a morphism of (∞,2)-sheaves

$\mathbf{\Pi}_{inf}(X) \to Mod \,.$

We write

$DQC(X) := Hom(\mathbf{\Pi}_{inf}(X), Mod)$

for the stable (∞,1)-category of D-modules.

The Jet algebra functor is the left adjoint to the forgetful functor from commutative algebras over $\mathcal{D}(X)$ to those over the structure sheaf $\mathcal{O}(X)$

$(Jet \dashv F) : Alg_{\mathcal{D}(X)} \stackrel{\overset{Jet}{\leftarrow}}{\underset{F}{\to}} Alg_{\mathcal{O}(X)} \,.$

## Application

Typical Lagrangians in quantum field theory are defined on jet bundles. Their variational calculus is governed by Euler-Lagrange equations.

Examples of sequences of infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

## References

The abstract characterization of jet bundles as the direct images of base change along the de Rham space projection is noticed on p. 6 of

The explicit description in terms of formal duals of commutative monoids in D-modules is in

An exposition of this is in section 2.3 of

Standard textbook references include

• G. Sardanashvily, Fibre bundles, jet manifolds and Lagrangian theory, Lectures for theoreticians, arXiv:0908.1886

• Shihoko Ishii, Jet schemes, arc spaces and the Nash problem, arXiv:math.AG/0704.3327

• D. J. Saunders, The geometry of jet bundles, London Mathematical Society Lecture Note Series 142, Cambridge Univ. Press 1989.

A discussion of jet bundles with an eye towards discussion of the variational bicomplex on them is in chapter 1, section A of

• Ian Anderson, The variational bicomplex (pdf)

Discussion of jet-restriction of the Haefliger groupoid is in

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

Revised on April 23, 2014 07:10:26 by Michael Bächtold (147.88.200.8)