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\section*{jet bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{concrete}{Concrete}\dotfill \pageref*{concrete} \linebreak
\noindent\hyperlink{GeneralAbstractDefinition}{General abstract}\dotfill \pageref*{GeneralAbstractDefinition} \linebreak
\noindent\hyperlink{application}{Application}\dotfill \pageref*{application} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{concrete}{}\subsubsection*{{Concrete}}\label{concrete}

For

\begin{displaymath}
p \coloneqq E \to X
\end{displaymath}
a [[surjective submersion]] of [[smooth manifolds]] and $k \in \mathbb{N}$, the bundle

\begin{displaymath}
J^k P \to X
\end{displaymath}
of \textbf{order-$k$ jets of sections of $p$} is the bundle whose [[fiber]] over a point $x \in X$ is the space of equivalence classes of [[germ|germs]] of [[section|sections]] of $p$, where two germs are considered equivalent if their first $k$ partial [[derivative|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 $J^k(X,Y)$.

In order to pass to $k \to \infty$ to form the \emph{infinite jet bundle} $J^\infty P$ one forms the [[projective limit]] over the finite-order jet bundles,

\begin{displaymath}
J^\infty E \coloneqq \underset{\longleftarrow}{\lim}_k J^k E
  = 
  \underset{\longleftarrow}{\lim}
  \left(
    \cdots J^3 E \to J^2 E \to J^1 E \to E
  \right)
\end{displaymath}
but one has to decide in which category of [[infinite-dimensional manifolds]] to take this limit:

\begin{enumerate}%
\item one may form the limit formally, i.e. in [[pro-manifolds]]. This is what is implicit for instance in \hyperlink{Anderson}{Anderson, p.3-5};


\item one may form the limit in [[Fréchet manifolds]], this is farily explicit in (\hyperlink{Saunders89}{Saunders 89, chapter 7}). See at \emph{\href{Frechet+manifold#ProjectiveLimitsOfSmoothFiniteDimensionalManifolds}{Fr\'e{}chet manifold -- Projective limits of finite-dimensional manifolds}}. Beware that this is not equivalent to the pro-manifold structure (see the remark \href{Frechet+manifold#DifferenceBetweenProManifoldAndFrecherManifoldStructure}{here}). It makes sense to speak of \emph{[[locally pro-manifolds]]}.



\end{enumerate}
\hypertarget{GeneralAbstractDefinition}{}\subsubsection*{{General abstract}}\label{GeneralAbstractDefinition}

We discuss a [[general abstract]] definition of jet bundles.

Let $\mathbf{H}$ be an [[(∞,1)-topos]] equipped with [[differential cohesion]] with [[infinitesimal shape modality]] $\Im$ (or rather a tower $\Im_k$ of such, for each infinitesimal order $k \in \mathbb{N} \cup \{\infty\}$ ).

For $X \in \mathbf{H}$, write $\Im(X)$ for the corresponding [[de Rham space]] object.

Notice that we have the canonical morphism, the $X$-component of the [[unit of a monad|unit]] of the $\Im$-[[monad]]

\begin{displaymath}
i \colon X \to \Im(X)
\end{displaymath}
(``inclusion of constant paths into all infinitesimal paths'').

The corresponding [[base change]] [[geometric morphism]] is

\begin{displaymath}
(i^\ast \dashv i_\ast)
  \;\colon\;
  \mathbf{H}_{/X}
   \stackrel{\overset{i^*}{\longleftarrow}}{\underset{Jet := i_*}{\longrightarrow}}
  \mathbf{H}_{/\Im(X)}
\end{displaymath}
\begin{defn}
\label{GeneralAbstractDefinition}\hypertarget{GeneralAbstractDefinition}{}
The \emph{[[jet comonad]]} is the [[(∞,1)-comonad]]

\begin{displaymath}
i^\ast i_\ast
  \;\colon\;
  \mathbf{H}_{/X}
  \longrightarrow
  \mathbf{H}_{/X}
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Since [[base change]] gives even an [[adjoint triple]] $(i_! \dashv i^\ast \dashv i_\ast)$, there is a [[left adjoint]] $T_{inf} X \times_X (-)$ to the [[jet comonad]] of def. \ref{GeneralAbstractDefinition},

\begin{displaymath}
T_{inf}X \times_X (-) \;\dashv\; Jet
\end{displaymath}
where $T_{inf} X$ is the [[infinitesimal disk bundle]] of $X$, see at \emph{\href{differential+cohesion#RelationOfInfinitesimalDiskBundleToJetBundle}{differential cohesion -- infinitesimal disk bundle -- relation to jet bundles}}

\end{remark}
\begin{remark}
\label{LiteratureOnGeneralAbstractCharacterization}\hypertarget{LiteratureOnGeneralAbstractCharacterization}{}
In the context of [[differential geometry]] the fact that the jet bundle construction is a comonad was explicitly observed in (\hyperlink{Marvan86}{Marvan 86}, see also \hyperlink{Marvan93}{Marvan 93, section 1.1}, \hyperlink{Marvan89}{Marvan 89}). It is almost implicit in (\hyperlink{KrasilshchikVerbovetsky98}{Krasil'shchik-Verbovetsky 98, p. 13, p. 17}, \hyperlink{Krasilshchik99}{Krasilshchik 99, p. 25}).

In the context of [[synthetic differential geometry]] the fact that the jet bundle construction is [[right adjoint]] to the [[infinitesimal disk bundle]] construction is (\hyperlink{Kock80}{Kock 80, prop. 2.2}).

 In the context of [[algebraic geometry]] and of [[D-schemes]] as in (\hyperlink{BeilinsonDrinfeld}{BeilinsonDrinfeld, 2.3.2}, reviewed in \hyperlink{Paugam}{Paugam, section 2.3}), the base change comonad formulation inf def. \ref{GeneralAbstractDefinition} was noticed in (\hyperlink{Lurie}{Lurie, prop. 0.9}).

\end{remark}
In as in (\hyperlink{BeilinsonDrinfeld}{BeilinsonDrinfeld, 2.3.2}, reviewed in \hyperlink{Paugam}{Paugam, section 2.3}) jet bundles are expressed dually in terms of algebras in [[D-modules]]. We now indicate how the translation works.

\begin{remark}
\label{}\hypertarget{}{}
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]]

\begin{displaymath}
jet(E) \coloneqq \underset{X \to \Im X}{\prod} E
  \,.
\end{displaymath}
\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A [[quasicoherent (∞,1)-sheaf]] on $X$ is a morphism of [[(∞,2)-sheaves]]

\begin{displaymath}
X \to Mod
  \,.
\end{displaymath}
We write

\begin{displaymath}
QC(X) := Hom(X, Mod)
\end{displaymath}
for the [[stable (∞,1)-category]] of [[quasicoherent (∞,1)-sheaves]].

A \emph{[[D-module]]} on $X$ is a morphism of [[(∞,2)-sheaves]]

\begin{displaymath}
\Im (X) \to Mod
  \,.
\end{displaymath}
We write

\begin{displaymath}
DQC(X) := Hom(\Im (X), Mod)
\end{displaymath}
for the [[stable (∞,1)-category]] of D-modules.

\end{defn}
The \textbf{Jet algebra} functor is the [[left adjoint]] to the [[forgetful functor]] from [[associative algebra|commutative algebra]]s over $\mathcal{D}(X)$ to those over the [[structure sheaf]] $\mathcal{O}(X)$

\begin{displaymath}
(Jet \dashv F)
   :
   Alg_{\mathcal{D}(X)}
    \stackrel{\overset{Jet}{\leftarrow}}{\underset{F}{\to}}
   Alg_{\mathcal{O}(X)}
  \,.
\end{displaymath}
\hypertarget{application}{}\subsection*{{Application}}\label{application}

Typical [[Lagrangian]]s in [[quantum field theory]] are defined on jet bundles. Their [[variational calculus]] is governed by [[Euler-Lagrange equation]]s.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[jet prolongation]]


\item [[jet space]]


\item [[jet group]], [[jet groupoid]]


\item [[differential equation]]


\item [[evolutionary vector field]]


\item [[variational bicomplex]]

\begin{itemize}%
\item [[local Lagrangian]]


\item [[source form]], [[Lepage form]]



\end{itemize}

\item [[metric jet]]


\item [[h-principle]]


\item [[arithmetic jet space]]



\end{itemize}
[[!include infinitesimal and local - table]]

\begin{itemize}%
\item in [[homotopy theory]]/[[Goodwillie calculus]]: [[jet (∞,1)-category]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Exposition of [[variational calculus]] in terms of jet bundles and [[Lepage forms]] and aimed at examples from [[physics]] is in

\begin{itemize}%
\item [[Jana Musilová]], [[Stanislav Hronek]], \emph{The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories}, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) (\href{https://doi.org/10.1515/cm-2016-0012}{doi.org/10.1515/cm-2016-0012})

\end{itemize}
Textbook accounts and lecture notes include

\begin{itemize}%
\item [[Peter Michor]], \emph{Manifolds of differentiable mappings}, Shiva Publishing (1980) \href{http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf}{pdf}


\item [[David Saunders]], \emph{The geometry of jet bundles}, London Mathematical Society Lecture Note Series \textbf{142}, Cambridge Univ. Press 1989.


\item [[Joseph Krasil'shchik]] in collaboration with Barbara Prinari, \emph{Lectures on Linear Differential Operators over Commutative Algebras}, 1998 (\href{http://diffiety.ac.ru/preprint/99/01_99.pdf}{pdf})


\item Shihoko Ishii, \emph{Jet schemes, arc spaces and the Nash problem}, \href{http://arXiv.org/abs/0704.3327}{arXiv:math.AG/0704.3327}


\item G. Sardanashvily, \emph{Fibre bundles, jet manifolds and Lagrangian theory}, Lectures for theoreticians, \href{http://xxx.lanl.gov/abs/0908.1886}{arXiv:0908.1886}


\item [[Peter Olver]], \emph{Lectures on Lie groups and differential equation}, chapter 3, \emph{Jets and differential invariants}, 2012 (\href{http://www.math.umn.edu/~olver/sm_/j.pdf}{pdf})



\end{itemize}
Early accounts include

\begin{itemize}%
\item Hubert Goldschmidt, \emph{Integrability criteria for systems of nonlinear partial differential equations}, J. Differential Geom. Volume 1, Number 3-4 (1967), 269-307 (\href{http://projecteuclid.org/euclid.jdg/1214428094}{Euclid})

\end{itemize}
The algebra of smooth functions of just \emph{locally} finite order on the jet bundle (``[[locally pro-manifold]]'') was maybe first considered in

\begin{itemize}%
\item [[Floris Takens]], \emph{A global version of the inverse problem of the calculus of variations}, J. Differential Geom. Volume 14, Number 4 (1979), 543-562. (\href{https://projecteuclid.org/euclid.jdg/1214435235}{Euclid})

\end{itemize}
 Discussion of the [[Fréchet manifold]] structure on infinite jet bundles includes

\begin{itemize}%
\item [[David Saunders]], chapter 7 \emph{Infinite jet bundles} of \emph{The geometry of jet bundles}, London Mathematical Society Lecture Note Series \textbf{142}, Cambridge Univ. Press 1989.


\item M. Bauderon, \emph{Differential geometry and Lagrangian formalism in the calculus of variations}, in \emph{Differential Geometry, Calculus of Variations, and their Applications}, Lecture Notes in Pure and Applied Mathematics, 100, Marcel Dekker, Inc., N.Y., 1985, pp. 67-82.


\item [[C. T. J. Dodson]], George Galanis, Efstathios Vassiliou,, p. 109 and section 6.3 of \emph{Geometry in a Fr\'e{}chet Context: A Projective Limit Approach}, Cambridge University Press (2015)


\item [[Andrew Lewis]], \emph{The bundle of infinite jets} (2006) (\href{http://www.mast.queensu.ca/~andrew/notes/pdf/2006a.pdf}{pdf})



\end{itemize}
Discussion of finite-order jet bundles in tems of [[synthetic differential geometry]] is in

\begin{itemize}%
\item [[Anders Kock]], \emph{Formal manifolds and synthetic theory of jet bundles}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques (1980) Volume: 21, Issue: 3 (\href{http://www.numdam.org/item?id=CTGDC_1980__21_3_227_0}{Numdam})


\item [[Anders Kock]], section 2.7 of \emph{Synthetic geometry of manifolds}, Cambridge Tracts in Mathematics 180 (2010). (\href{http://home.imf.au.dk/kock/SGM-final.pdf}{pdf})



\end{itemize}
The [[jet comonad]] structure on the jet operation in the context of differential geometry is made explicit in

\begin{itemize}%
\item [[Michal Marvan]], \emph{A note on the category of partial differential equations}, in \emph{Differential geometry and its applications}, Proceedings of the Conference August 24-30, 1986, Brno ([[MarvanJetComonad.pdf:file]])

(notice that prop. 1.3 there is wrong, the correct version is in the thesis of the author)



\end{itemize}
with further developments in

\begin{itemize}%
\item [[Michal Marvan]] \emph{On the horizontal cohomology with general coefficients}, 1989 (\href{http://old.math.slu.cz/People/MichalMarvan/Annotations/horizontal.php}{web announcement}, \href{http://dml.cz/dmlcz/701469}{web archive})

\textbf{Abstract:} In the present paper the horizontal cohomology theory is interpreted as a special case of the Van Osdol bicohomology theory applied to what we call a ``jet comonad''. It follows that differential equations have well-defined cohomology groups with coefficients in linear differential equations.


\item [[Michal Marvan]], section 1.1 of \emph{On Zero-Curvature Representations of Partial Differential Equations}, (1993) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5631}{web})


\item [[Igor Khavkine]], [[Urs Schreiber]], \emph{[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]} (\href{https://arxiv.org/abs/1701.06238}{arXiv:1701.06238})



\end{itemize}
In the context of algebraic geometry, 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

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Notes on crystals and algebraic D-modules}, 2009 (\href{http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf}{pdf})

\end{itemize}
The explicit description in terms of formal duals of [[commutative monoids]] in [[D-module]]s is in

\begin{itemize}%
\item [[Alexander Beilinson]], [[Vladimir Drinfeld]], \emph{[[Chiral Algebras]]}

\end{itemize}
An exposition of this is in section 2.3 of

\begin{itemize}%
\item [[Frédéric Paugam]], \emph{Homotopical Poisson Reduction of gauge theories} (\href{http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf}{pdf})

\end{itemize}
A discussion of jet bundles with an eye towards discussion of the [[variational bicomplex]] on them is in chapter 1, section A of

\begin{itemize}%
\item [[Ian Anderson]], \emph{The variational bicomplex} ([[AndersonVariationalBicomplex.pdf:file]])

\end{itemize}
The [[de Rham complex]] and [[variational bicomplex]] of jet bundles is discussed in

\begin{itemize}%
\item G. Giachetta, L. Mangiarotti, [[Gennadi Sardanashvily]], \emph{Cohomology of the variational bicomplex on the infinite order jet space}, Journal of Mathematical Physics 42, 4272-4282 (2001) (\href{http://arxiv.org/abs/math/0006074}{arXiv:math/0006074})

\end{itemize}
where both versions (smooth functions being globally or locally of finite order) are discussed and compared.

Discussion of jet-restriction of the [[Haefliger groupoid]] is in

\begin{itemize}%
\item Arne Lorenz, \emph{Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method}, Thesis (\href{http://wwwb.math.rwth-aachen.de/~arne/Dissertation_Lorenz_Arne.pdf}{pdf})

\end{itemize}
Discussion of jet bundles in [[supergeometry]] includes

\begin{itemize}%
\item Arthemy V. Kiselev, Andrey O. Krutov, appendix of \emph{On the (non)removability of spectral parameters in $\mathbb{Z}_2$-graded zero-curvature representations and its applications} (\href{http://arxiv.org/abs/1301.7143}{arXiv:1301.7143})

\end{itemize}
See also

\begin{itemize}%
\item [[Joseph Krasil'shchik]], [[Alexander Verbovetsky]], \emph{Homological Methods in Equations of Mathematical Physics} (\href{http://arxiv.org/abs/math/9808130}{arXiv:math/9808130})

\end{itemize}
In the context of [[supermanifolds]], discussion is in

\begin{itemize}%
\item [[Gennadi Sardanashvily]], \emph{Graded infinite order jet manifolds}, \emph{Int. J. Geom. Methods Mod. Phys. v.4 (2007) 1335-1362} (\href{https://arxiv.org/abs/0708.2434}{arXiv:0708.2434})

\end{itemize}
[[!redirects jet bundles]]

[[!redirects jet]] [[!redirects jets]]

[[!redirects jet type]] [[!redirects jet types]]



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