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 the space of jets of maps from $X$ to $Y$ and commonly denoted with $J^k(X,Y)$.
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:
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.
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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 infinitesimal and local structures
first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||
derivative | Taylor series | germ | smooth function | |||
tangent vector | jet | germ of curve | curve | |||
square-0 ring extension | nilpotent ring extension | ring extension | ||||
Lie algebra | formal group | local Lie group | Lie group | |||
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