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.
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 called the space of jets of maps from $X$ to $Y$ and commonly denoted with $J^k(X,Y)$.
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We discuss a general abstract definition of jet bundles.
Let
$\mathbf{H}$ be a cohesive (∞,1)-topos
equipped with differential cohesion
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
(“inclusion of constant paths into all infinitesimal paths”).
Write
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
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
A quasicoherent (∞,1)-sheaf on $X$ is a morphism of (∞,2)-sheaves
We write
for the stable (∞,1)-category of quasicoherent (∞,1)-sheaves.
A D-module on $X$ is a morphism of (∞,2)-sheaves
We write
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 $\mathcal{O}(X)$
Typical Lagrangians in quantum field theory are defined on jet bundles. Their variational calculus is governed by Euler-Lagrange equations.
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | open neighbourhood | |||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
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
Discussion of jet-restriction of the Haefliger groupoid is in