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\begin{document}

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\section*{geometry of physics -- differentiation}

\begin{quote}%
This entry contains one chapter of the material at \emph{[[geometry of physics]]}.


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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{Differentiation}{\textbf{Differentiation}}\dotfill \pageref*{Differentiation} \linebreak
\noindent\hyperlink{DifferentiationLayerMod}{Model Layer}\dotfill \pageref*{DifferentiationLayerMod} \linebreak
\noindent\hyperlink{DifferentiationOfSmoothFunctionsAndDifferentialForms}{Differentiation of smooth functions and differential forms}\dotfill \pageref*{DifferentiationOfSmoothFunctionsAndDifferentialForms} \linebreak
\noindent\hyperlink{DifferentiationOnCoordinatePatches}{Differentiation on coordinate patches}\dotfill \pageref*{DifferentiationOnCoordinatePatches} \linebreak
\noindent\hyperlink{DifferentiationOnSmoothSpaces}{Differentiation on smooth spaces}\dotfill \pageref*{DifferentiationOnSmoothSpaces} \linebreak
\noindent\hyperlink{ElectromagneticFieldStrength}{Example: The electromagnetic field strength}\dotfill \pageref*{ElectromagneticFieldStrength} \linebreak
\noindent\hyperlink{VariationalCalculus}{Variational calculus}\dotfill \pageref*{VariationalCalculus} \linebreak
\noindent\hyperlink{discrete_points_of_a_smooth_space}{Discrete points of a smooth space}\dotfill \pageref*{discrete_points_of_a_smooth_space} \linebreak
\noindent\hyperlink{SmoothFunctionals}{Smooth functionals}\dotfill \pageref*{SmoothFunctionals} \linebreak
\noindent\hyperlink{FunctionalDerivative}{Functional derivative / variational derivative}\dotfill \pageref*{FunctionalDerivative} \linebreak
\noindent\hyperlink{EulerLagrangeEquations}{Euler-Lagrange equations}\dotfill \pageref*{EulerLagrangeEquations} \linebreak
\noindent\hyperlink{DGeometry}{$\mathcal{D}$-geometry}\dotfill \pageref*{DGeometry} \linebreak
\noindent\hyperlink{infinitesimal_smooth_loci}{Infinitesimal smooth loci}\dotfill \pageref*{infinitesimal_smooth_loci} \linebreak
\noindent\hyperlink{de_rham_space}{de Rham space}\dotfill \pageref*{de_rham_space} \linebreak
\noindent\hyperlink{jet_bundles}{Jet bundles}\dotfill \pageref*{jet_bundles} \linebreak
\noindent\hyperlink{DifferentiationLayerSem}{Semantic Layer}\dotfill \pageref*{DifferentiationLayerSem} \linebreak
\noindent\hyperlink{synthetic_differential_geometry}{Synthetic differential geometry}\dotfill \pageref*{synthetic_differential_geometry} \linebreak
\noindent\hyperlink{tangent_bundle}{Tangent bundle}\dotfill \pageref*{tangent_bundle} \linebreak
\noindent\hyperlink{differential_equations}{Differential equations}\dotfill \pageref*{differential_equations} \linebreak
\noindent\hyperlink{DifferentialCohesionOfToposOfSmoothSpaces}{Differential cohesion of the topos of smooth spaces}\dotfill \pageref*{DifferentialCohesionOfToposOfSmoothSpaces} \linebreak
\noindent\hyperlink{DifferentialCohesion}{Differential cohesion}\dotfill \pageref*{DifferentialCohesion} \linebreak
\noindent\hyperlink{DifferentiationSemLayerDeRhamSpace}{de Rham space}\dotfill \pageref*{DifferentiationSemLayerDeRhamSpace} \linebreak
\noindent\hyperlink{DifferentiationSemLayerJetBundle}{Jet bundle}\dotfill \pageref*{DifferentiationSemLayerJetBundle} \linebreak
\noindent\hyperlink{formally_tale__formally_unramified__formally_smooth}{Formally \'e{}tale / formally unramified / formally smooth}\dotfill \pageref*{formally_tale__formally_unramified__formally_smooth} \linebreak
\noindent\hyperlink{syntactic_layer}{Syntactic Layer}\dotfill \pageref*{syntactic_layer} \linebreak
\noindent\hyperlink{differential_homotopy_type_theory}{Differential homotopy type theory}\dotfill \pageref*{differential_homotopy_type_theory} \linebreak


\hypertarget{Differentiation}{}\subsection*{{\textbf{Differentiation}}}\label{Differentiation}

So far we have dealt with \emph{cohesive} structures for which there is a notion of \emph{smooth variation}, say of the position of a [[particle]] along a [[trajectory]] in [[spacetime]]. The idea of \emph{differentiation} is to measure the \emph{[[difference]]} between the position of two points on a cohesive trajectory in space as the difference between their [[worldline]] coordinates ``tends to 0'' without actually becoming 0. One also says that differentiation is forming ``[[infinitesimal space|infinitesimal]] differences'' of a cohesive process -- and we will make precise here what this means.

There are two stages to the theory of differentiation:

\begin{enumerate}%
\item We may think of differentiation as just a means to analyze more in detail the [[cohesive topos|cohesive structure]] already given, without adding new [[structure]], hence without a priori refining our notion of what a ``[[cohesion|cohesive]] [[trajectory]]'' is. Indeed, given any [[line object]] $\mathbb{R}$ in a [[cohesive ∞-topos]], there is canonically a homomorphism of cohesive spaces

\begin{displaymath}
\mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})
\end{displaymath}
from the line to the cohesive moduli space of closed [[differential 1-forms]], which is such that it sends a cohesive curve on the line to the differential form on this curve whose value at each point is the \emph{differential} of the curve, its rate of infinitesimal change at that point.

Below in \emph{\hyperlink{DifferentiationOfSmoothFunctionsAndDifferentialForms}{Differentiation of smooth functions and differetial forms}} we discuss this construction in the standard model of [[smooth infinity-groupoid|smooth cohesion]] for [[smooth spaces]], where it reproduces what traditionally is called the \emph{[[de Rham differential]]} $\mathbf{d}$.

Further below in \emph{\hyperlink{CohesiveDifferentiation}{Maurer-Cartan forms -- Cohesive differentiation}} we show how $\mathbf{d}$ comes out from just the abstract axioms of cohesionn.


\item We may think of differentiation as reflecting a refinement of smooth cohesion such that [[infinitesimal space|infinitesimal]] cohesive trajectories actually exist. Here, on top of having a \emph{measure} for how a cohesive trajectory changes infinitesimally at a given point, it makes sense to ask concretely if two points on a trajectory are infinitesimally close to each other. In this approach the very notion of cohesion is refined to include \emph{explicitly} a way to speak not just about a ``cohesive blob of points'', but to decide whether it is in fact just an ``infinitesimal cohesive blob of points'' -- an \emph{[[infinitesimally thickened point]]}.

[[differential geometry|Differential geometry]] with such an explicit notion of [[infinitesimal space|infinitesimals]] is known as \emph{[[synthetic differential geometry]]}: the [[axioms]] here allow one to \emph{synthesize} an [[infinitesimally thickened point]] and not just to reason about it \emph{as if} it existed.

In such a synthetic differential context then the differential $\mathbf{d}$ from above not just exists as a whole, but we can ``take it apart and re-synthesize it'' by realizing its value at each point literally as the ordinary [[difference]] between two \emph{infinitesimally close} points. Similarly, various other fundamental constructions in [[differential geometry]], such as that of [[tangent bundles]] and [[jet bundles]] have a usefully transparent axiomatic characterization in the presence of synthetic infinitesimals. ([[Sophus Lie]], one of the founding fathers of [[differential geometry]] \href{synthetic+differential+geometry#SophusLieQuote}{famously said} that he indeed found his theorems using such synthetic reasoning intuitively, and just did not publish them this way due to a lack of formalization of this language -- at his time. ) This we discuss in the Mod Layer in \emph{\hyperlink{DGeometry}{D-geometry}} below.



\end{enumerate}
In the \hyperlink{DifferentiationLayerSem}{Differentiation semantic layer} below we formalize differentiation, and these two aspects of it, by adding to the notion of \emph{[[cohesive topos]]} that of an \emph{[[infinitesimal cohesion|infinitesimal cohesive neighbourhood]]}.

Recalling that a [[cohesive topos]] is an abstract \emph{cohesive blob}, an infinitesimal cohesive neighbourhood is accordingly an infinitesimally thicked cohesive blob (which is itself again a cohesive blob):

\begin{displaymath}
\itexarray{
    CohesiveBlob &&\hookrightarrow&& InfinitesimallyThickenedCohesiveBlob
    \\
    & \searrow && \swarrow
    \\
    && Point
  }
  \;\;\;\;\;\;\;\;
  \itexarray{
    \mathbf{H} &&\stackrel{}{\hookrightarrow}&& \mathbf{H}_{th}
    \\
    & \searrow && \swarrow
    \\
    && DiscSpaces
  }
  \,.
\end{displaymath}
\hypertarget{DifferentiationLayerMod}{}\subsubsection*{{Model Layer}}\label{DifferentiationLayerMod}

We discuss

\begin{itemize}%
\item \hyperlink{DifferentiationOfSmoothFunctionsAndDifferentialForms}{Differentiation of smooth functions and differential forms}

\begin{itemize}%
\item first just \emph{\hyperlink{DifferentiationOnCoordinatePatches}{on coordinate patches}}


\item and then \emph{\hyperlink{DifferentiationOnSmoothSpaces}{on general smooth spaces}}.



\end{itemize}


\end{itemize}
By considering [[fiber products]] of smooth [[mapping spaces]] with [[discrete spaces]] of boundary configurations, we obtain from this the differentiation theory called

\begin{itemize}%
\item \emph{\hyperlink{VariationalCalculus}{Variational calculus}}

\begin{itemize}%
\item with the notion of \emph{\hyperlink{SmoothFunctionals}{Smooth functional}}


\item and \emph{\hyperlink{FunctionalDerivative}{Variational derivative}} of smooth functionals.


\item The central class of examples of this of interest in physics is the variation of [[action functionals]] that yields the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] in [[classical field theory]]. This we discuss in \emph{\hyperlink{EulerLagrangeEquations}{Euler-Lagrange equations}}.



\end{itemize}


\end{itemize}
\hypertarget{DifferentiationOfSmoothFunctionsAndDifferentialForms}{}\paragraph*{{Differentiation of smooth functions and differential forms}}\label{DifferentiationOfSmoothFunctionsAndDifferentialForms}

\hypertarget{DifferentiationOnCoordinatePatches}{}\paragraph*{{Differentiation on coordinate patches}}\label{DifferentiationOnCoordinatePatches}

By definition to [[smooth function]] $f \colon \mathbb{R} \to \mathbb{R}$ is associated its [[derivative]], a smooth function $f' \colon \mathbb{R} \to \mathbb{R}$. And more generally to a smooth function $f \colon \mathbb{R}^n \to \mathbb{R}$ are associated its [[partial derivatives]], smooth functions

\begin{displaymath}
\frac{\partial f}{\partial x^i} \colon \mathbb{R}^n \to \mathbb{R}
\end{displaymath}
for $1 \leq i \leq n$.

The [[de Rham differential]] collects all partial derivatives of a function into a single [[differential 1-form]]

\begin{defn}
\label{DeRhamDifferentialOn1Forms}\hypertarget{DeRhamDifferentialOn1Forms}{}
For $n \in \mathbb{N}$, The \textbf{[[de Rham differential]]} on [[smooth functions]] in $C^\infty(\mathbb{R}^n)$ is the [[function]]

\begin{displaymath}
\mathbf{d} \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)
\end{displaymath}
which takes $f \in C^\infty(\mathbb{R}^n)$ to

\begin{displaymath}
\mathbf{d}f \coloneqq \sum_{i = 1}^n \frac{\partial f}{\partial x^i} \mathbf{d} x^i
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For $x^i \colon \mathbb{R}^n \to \mathbb{R}$ one of the [[coordinate]] functions, the de Rham differential $\mathbf{d} x^i$ indeed coincides with the basis element of the same name according to def. \ref{Differential1FormsOnCartesianSpaces}, using that

\begin{displaymath}
\frac{\partial x^i}{\partial x^{j}} = 
  \left\{
    \itexarray{
      1 & | i = j
      \\
      0 & | otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
The de Rham differentials $\mathbf{d} \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$ for all $n \in \mathbb{N}$ are compatible with [[pullback of differential forms|pullback of differential 1-forms]], def. \ref{PullbackOfDifferential1FormsOnCartesianSpaces}, in that for each coordinate transformation $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^{k}$ the [[diagram]]

\begin{displaymath}
\itexarray{
     C^\infty(\mathbb{R}^k) 
      &\stackrel{\mathbf{d}}{\to}& 
    \Omega^1(\mathbb{R}^k)
     \\
     \downarrow^{\mathrlap{\phi^\ast}} && \downarrow^{\mathrlap{\phi^\ast}}
     \\
     C^\infty(\mathbb{R}^{\tilde k}) 
       &\stackrel{\mathbf{d}}{\to}&   
     \Omega^1(\mathbb{R}^{\tilde k})    
  }
\end{displaymath}
[[commuting diagram|commutes]].

\end{prop}
\begin{proof}
This is equivalently the statement of the \emph{[[chain rule]]} of [[differentiation]]: For any $f \in C^\infty(\mathbb{R}^k)$ we have on the one hand, by def. \ref{PullbackOfDifferential1FormsOnCartesianSpaces} and def. \ref{DeRhamDifferentialOn1Forms}

\begin{displaymath}
\begin{aligned}
    \mathbf{d} \phi^\ast f
    & = 
    \mathbf{d} (f \circ \phi)
    \\
    & =
    \sum_{j = 1}^{\tilde k} \frac{\partial (f \circ \phi)}{\partial x^j}\mathbf{d} x^j
  \end{aligned}
\end{displaymath}
and on the other hand, applying the definition in the other order,

\begin{displaymath}
\begin{aligned}
    \phi^* \mathbf{d}f
    & = 
    \phi^* \sum_{i = 1}^k \frac{\partial f}{\partial x^i} \mathbf{d}x^i
    \\
    & = \sum_{i = 1}^k \sum_{j = 1}^{\tilde k} 
     \left(\frac{\partial f}{\partial x^i}\circ \phi \right)
    \cdot 
    \left(\frac{\partial \phi^i}{\partial x^j}\right) \mathbf{d}x^j
  \end{aligned}
  \,.
\end{displaymath}
Both expressions agree precisely if for all $j$ we have

\begin{displaymath}
\frac{\partial (f \circ \phi)}{\partial x^j}
   =
  \sum_{i = 1}^k \left(\frac{\partial f}{\partial x^i}\circ \phi\right) \cdot \left(\frac{\partial \phi^i}{\partial x^j}\right)
  \;\;\;
  \in 
  C^\infty(\mathbb{R}^{\tilde k})
  \,.
\end{displaymath}
This is precisely the statement of the \emph{[[chain rule]]} for [[differentiation]].

\end{proof}
Notice that as smooth spaces $\mathbb{R} = \Omega^0 = C^\infty(-)$, by prop. \ref{SpaceOf0FormsIsRealLine}. Therefore the above says that

\begin{prop}
\label{}\hypertarget{}{}
The [[de Rham differential]], def. \ref{DeRhamDifferentialOn1Forms}, constitutes a homomorphism of smooth spaces, def. \ref{HomomorphismOfSmoothSpaces}

\begin{displaymath}
\mathbf{d} \colon \mathbb{R} \to \Omega^1
\end{displaymath}
from the [[real line]] to the universal smooth moduli space of differential 1-forms, def. \ref{SmoothModuliSpaceOfnForms}.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Below in \emph{\hyperlink{MaurerCartanFormOnLieGroup}{Maurer-Cartan form on a Lie group}} we discuss a more general abstract origin of $\mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl}$.

\end{remark}
We now extend the de Rham differential to differential forms of higher degree.

\begin{defn}
\label{DeRhamDifferentialInGeneralDegreeOverCartesianSpaces}\hypertarget{DeRhamDifferentialInGeneralDegreeOverCartesianSpaces}{}
For all $n \in \mathbb{N}$ let

\begin{displaymath}
\mathbf{d} 
    \colon 
  \Omega^\bullet(\mathbb{R}^n)
   \to 
  \Omega^\bullet(\mathbb{R}^n)
\end{displaymath}
be the unique extension of $\mathbf{d} \colon C^\infty(-) \to \Omega^1(-)$ to a degree-1 [[derivation]] with

\begin{displaymath}
\mathbf{d}\mathbf{d}x^i = 0
  \,.
\end{displaymath}
\end{defn}
(\ldots{})

\begin{prop}
\label{DeRhamDifferentialAsMorphismOfSmoothSpacesInEachDegree}\hypertarget{DeRhamDifferentialAsMorphismOfSmoothSpacesInEachDegree}{}
For each $n \in \mathbb{N}$ the de Rham differential of def. \ref{DeRhamDifferentialInGeneralDegreeOverCartesianSpaces} constitutes a homomorphism of smooth spaces

\begin{displaymath}
\mathbf{d} \colon \Omega^n \to \Omega^{n+1}
\end{displaymath}
form the universal smooth moduli space of differental $n$-forms to that of differential $n+1$-forms.

\end{prop}
\hypertarget{DifferentiationOnSmoothSpaces}{}\paragraph*{{Differentiation on smooth spaces}}\label{DifferentiationOnSmoothSpaces}

We now extend the notion of [[derivatives]] and [[de Rham differentials]] from [[smooth functions]] on [[Cartesian spaces]] to smooth functions on general [[smooth spaces]].

Recall from def. \ref{DifferentialnFormOnSmoothSpace} that the set of differential $n$-forms on a [[smooth space]] $X$ is $\Omega^n(X) \coloneqq Hom(X, \Omega^n)$.

\begin{defn}
\label{DeRhamDifferentialOverSmoothSpaces}\hypertarget{DeRhamDifferentialOverSmoothSpaces}{}
For $X \in Smooth0Type$ a smooth space and $n \in \mathbb{N}$, the \textbf{[[de Rham differential]]} on $n$-forms over $X$ is the [[function]]

\begin{displaymath}
\mathbf{d} \colon \Omega^n(X) \to \Omega^{n+1}(X)
\end{displaymath}
which is the postcomposition with the homomorphism of smooth spaces of prop. \ref{DeRhamDifferentialAsMorphismOfSmoothSpacesInEachDegree}:

\begin{displaymath}
Hom(X,\mathbf{d}) \colon Hom(X,\Omega^n) \to Hom(X,\Omega^{n+1})
  \,.
\end{displaymath}
\end{defn}
In particular the [[derivative]] of a smooth function $f \colon X \to \mathbb{R}$ is the [[composition|composite]]

\begin{displaymath}
\mathbf{d}f \colon X \stackrel{f}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1_{cl} \hookrightarrow \Omega^1
  \,.
\end{displaymath}
Below in \emph{\hyperlink{VariationIsDifferentiationOnSmoothSpaces}{Variation is differentiation on smooth spaces}} we find that this notion of differentiation of smooth functions on smooth spaces subsumes what traditionally is called \emph{[[variational calculus]]} of [[functionals]] on [[mapping spaces]].

\hypertarget{ElectromagneticFieldStrength}{}\paragraph*{{Example: The electromagnetic field strength}}\label{ElectromagneticFieldStrength}

for instance [[electromagnetic potential]]

\begin{displaymath}
A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3
\end{displaymath}
then the [[electromagnetic field strength]] is

\begin{displaymath}
F 
    \coloneqq
  \mathbf{d}A 
    = 
   E_1 \mathbf{d}x^1 \wedge \mathbf{d}t 
    + 
   E_2 \mathbf{d}x^2 \wedge \mathbf{d}t 
    + 
   E_3 \mathbf{d}x^3 \wedge \mathbf{d}t
    + 
    B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 
    + 
    B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 
    + 
    B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2
\end{displaymath}
with

\begin{displaymath}
E_i = \frac{\partial \phi}{\partial x^i} - \frac{\partial A_i}{\partial t}
\end{displaymath}
and

\begin{displaymath}
B_1 = \frac{\partial A_2}{\partial x^3} - \frac{\partial A_3}{\partial x^2}
\end{displaymath}
etc

This are the first 2 of 4 [[Maxwell equations]]: $\mathbf{d} F = 0$

(the other 2 are discussed below in \hyperlink{RiemannianGeometry}{Riemannian geometry})

for

\begin{displaymath}
A,A' : X \to \Omega^1(-)
\end{displaymath}
a [[gauge transformation]] $A \to A'$ is $\lambda : X \to \mathbb{R}$ with

\begin{displaymath}
A' = A + \mathbf{d} \lambda
\end{displaymath}
\hypertarget{VariationalCalculus}{}\paragraph*{{Variational calculus}}\label{VariationalCalculus}

Traditionally a \emph{[[functional]]} is a [[function]] which is sufficiently like a [[smooth function]], but defined not on a [[manifold]], but on a [[mapping space]] between manifolds. Also traditionally, a \emph{[[variational derivative]]} of such a functional is something aking to a [[derivative]], generalized to this context, and subject to the condition that all variations \emph{preserve some boundary conditions}.

We formulate this classical theory in the context of [[smooth spaces]]. Here a [[nonlinear functional|functional]] is simply a homomorphism of smooth spaces out of a smooth [[mapping space]], as in def. \ref{SmoothFunctionSpace}. We may impose \emph{respect for boundary conditions} by forming the [[fiber product]] of this mapping space with a \emph{discrete smooth space inclusion}, given in def. \ref{MapFromDiscretizationOfSmooth0Type} below. Then the \emph{variational derivative} is simply the ordinary derivative of def. \ref{DeRhamDifferentialOverSmoothSpaces}.

\hypertarget{discrete_points_of_a_smooth_space}{}\paragraph*{{Discrete points of a smooth space}}\label{discrete_points_of_a_smooth_space}

\begin{defn}
\label{DiscretizationOfSmooth0Type}\hypertarget{DiscretizationOfSmooth0Type}{}
For $X \in Smooth0Type$ a [[smooth space]], write

\begin{displaymath}
\Gamma X
  \coloneqq
  Hom(*,X)
  \in Set
\end{displaymath}
for its \emph{set of points}, the set of homomorphisms, def. \ref{HomomorphismOfSmoothSpaces}, from the point to $X$.

Write

\begin{displaymath}
\flat X \coloneqq Disc (\Gamma(X))
\end{displaymath}
for the discrete smooth space, def. \ref{DiscreteSmoothSpace}, on this set of points.

\end{defn}
\begin{defn}
\label{MapFromDiscretizationOfSmooth0Type}\hypertarget{MapFromDiscretizationOfSmooth0Type}{}
For every smooth space $X$ there is a canonical homomorphism of smooth spaces

\begin{displaymath}
\flat X \to X
  \,.
\end{displaymath}
This sends a plot $U \to \flat X$, which by definition of $Disc(-)$ is a point in $\Gamma X$, hence a homomorphism $x \colon * \to X$, to the plot $U \to * \stackrel{x}{\to} X$ of $X$.

\end{defn}
\hypertarget{SmoothFunctionals}{}\paragraph*{{Smooth functionals}}\label{SmoothFunctionals}

Let $X$ be a [[smooth manifold]]. Let $\Sigma$ be a [[smooth manifold|smooth]] [[manifold with boundary]] $\partial \Sigma \hookrightarrow \Sigma$.

Write

\begin{displaymath}
[\Sigma, X] \in Smooth0Type
\end{displaymath}
for the [[smooth space]] (a [[diffeological space]]) which is the [[mapping space]] from $\Sigma$ to $X$, hence such that for each $U \in$ [[CartSp]] we have

\begin{displaymath}
[\Sigma, X](U) = C^\infty(U \times \Sigma, X)
  \,.
\end{displaymath}
\begin{defn}
\label{MappingSpaceWithNonVaryingBoundary}\hypertarget{MappingSpaceWithNonVaryingBoundary}{}
Write

\begin{displaymath}
[\Sigma, X]_{\partial \Sigma}
  \coloneqq
  [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]
\end{displaymath}
for the [[pullback]] in smooth spaces

\begin{displaymath}
\itexarray{
    [\Sigma,X]_{\partial \Sigma}
    &\to&
    \flat [\partial \Sigma, X]
    \\
    \downarrow && \downarrow
    \\
    [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X]
  }
  \,,
\end{displaymath}
where

\begin{itemize}%
\item the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma, X]$ of configurations to the boundary;


\item the right vertical morphism is the homomorphism from def. \ref{MapFromDiscretizationOfSmooth0Type}.



\end{itemize}
\end{defn}
\begin{prop}
\label{PlotsOfMappingSpaceWithNonVaryingBoundary}\hypertarget{PlotsOfMappingSpaceWithNonVaryingBoundary}{}
The [[smooth space]] $[\Sigma, X]_{\partial \Sigma}$ is a [[diffeological space]] whose underlying set is $C^\infty(\Sigma,X)$ and whose $U$-plots for $U \in$ [[CartSp]] are smooth functions $\phi \colon U \times \Sigma \to X$ such that $\phi(-,s) \colon U \to X$ is the constant function for all $s \in \partial \Sigma \hookrightarrow \Sigma$.

\end{prop}
\begin{defn}
\label{SmoothFunctional}\hypertarget{SmoothFunctional}{}
A \textbf{[[nonlinear functional|functional]]} on the mapping space $[\Sigma, X]$ is a [[homomorphism]] of smooth spaces

\begin{displaymath}
S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{FunctionalDerivative}{}\paragraph*{{Functional derivative / variational derivative}}\label{FunctionalDerivative}

Write

\begin{displaymath}
\mathbf{d} \colon \mathbb{R} \to \Omega^1
\end{displaymath}
for the [[de Rham differential]] incarnated as a [[homomorphism]] of [[smooth spaces]] from the [[real line]] to the smooth [[moduli space]] of [[differential 1-forms]].

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{functional derivative}

\begin{displaymath}
\mathbf{d}S
  \in 
  \Omega^1([\Sigma,X]_{\partial \Sigma})
\end{displaymath}
of a functional $S$, def. \ref{SmoothFunctional}, is simply its [[de Rham differential]] as a homomorphism of smooth spaces, hence the composite

\begin{displaymath}
\mathbf{d} S \colon [
  \Sigma, X]_{\partial \Sigma} 
     \stackrel{S}{\to} 
  \mathbb{R}   
    \stackrel{\mathbf{d}}{\to}
  \Omega^1
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
This means that for each smooth parameter space $U \in$ [[CartSp]] and for each smooth $U$-parameterized collection

\begin{displaymath}
\phi \colon U \times \Sigma \to X
\end{displaymath}
of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,

\begin{displaymath}
S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})
\end{displaymath}
is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then

\begin{displaymath}
(\mathbf{d}S)_\phi \in \Omega^1(U)
\end{displaymath}
is the smooth [[differential 1-form]]

\begin{displaymath}
(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi))
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard [[interval]]. Let

\begin{displaymath}
L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))
\end{displaymath}
and consider the functional

\begin{displaymath}
S
  \colon
  ([0,1] \stackrel{\gamma}{\to} X)
  \mapsto
  \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t
  \,.
\end{displaymath}
Then for $U = \mathbb{R}$ and any smooth $U$-parameterized collection $\{\gamma_{u} \colon \Sigma \to X\}_{u \in U}$ the functional derivative takes the value

\begin{displaymath}
\begin{aligned}
   \mathbf{d}S_{\gamma_{(-)}}
   & =
   \left(
     \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt
   \right)
   \mathbf{d}u
   \\
    & = 
     \int_{0}^1 
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{d \gamma_u(t)}{d u}
      + 
       \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{\partial \dot \gamma_u(t)}{\partial u}
     \right) dt
     \mathbf{d} u
     \\
     & =
     \int_{0}^1 
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{d \gamma_u(t)}{d u}
      + 
       \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
       \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u}
     \right) dt
     \mathbf{d} u
     \\
     & =
     \int_{0}^1
     \left(
       \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t))
      - 
       \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t))
     \right)
     \frac{\partial \gamma_u(t)}{\partial u} dt
     \mathbf{d}u
  \end{aligned}
  \,.
\end{displaymath}
Here we used [[integration by parts]] and used that the boundary term vanishes

\begin{displaymath}
\begin{aligned}
    \int_{0}^1 \frac{\partial}{\partial t} 
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s))
      \frac{\partial \gamma_u(s)}{\partial u}
    \right)
    d s
    & = 
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1))
      \frac{\partial \gamma_u(1)}{\partial u}
    \right)
    - 
    \left(
      \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0))
      \frac{\partial \gamma_u(0)}{\partial u}
    \right)
    \\
    & = 0
  \end{aligned}
\end{displaymath}
since by prop. \ref{PlotsOfMappingSpaceWithNonVaryingBoundary} $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.

\end{example}
\hypertarget{EulerLagrangeEquations}{}\paragraph*{{Euler-Lagrange equations}}\label{EulerLagrangeEquations}

\begin{itemize}%
\item [[critical locus]]


\item [[Euler-Lagrange equation]]


\item [[equations of motion]]



\end{itemize}
\hypertarget{DGeometry}{}\paragraph*{{$\mathcal{D}$-geometry}}\label{DGeometry}

\begin{itemize}%
\item [[D-geometry]]

\end{itemize}
\hypertarget{infinitesimal_smooth_loci}{}\paragraph*{{Infinitesimal smooth loci}}\label{infinitesimal_smooth_loci}

\begin{defn}
\label{SmoothL}\hypertarget{SmoothL}{}
Write

\begin{displaymath}
SmoothLoci \coloneqq SmoothAlgebras^{op} \coloneqq Func^\times(CartSp,Set)
\end{displaymath}
for the category of [[spaces]] which are [[Isbell duality|formally dual]] to [[smooth algebras]]: the [[opposite category]] of that of [[smooth algebras]]. This is called the category of \textbf{[[smooth loci]]}.

\end{defn}
\begin{defn}
\label{InfinitesimalAlgebra}\hypertarget{InfinitesimalAlgebra}{}
A \textbf{[[Artin ring|smooth Artin algebra]]} (also called a ``Weil algebra'' in the [[synthetic differential geometry]]-literature) is a [[smooth algebra]] $A$ whose underlying $\mathbb{R}$-[[vector space]] is a [[direct sum]] of the form

\begin{displaymath}
A = \mathbb{R} \oplus V
  \,,
\end{displaymath}
where $V$ is of finite [[dimension]] and such that every element $v \in V \subset A$ is nilpotent, in that there is $n \in \mathbb{N}$ such that the $n$-fold product of $v$ with itself in $A$ vanishes: $v^n = 0$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The smallest smooth Artin algebra is the [[ring of dual numbers]], def. \ref{DualNumbers}, for which $V = \mathbb{R}$.

\end{example}
\begin{defn}
\label{InfinitesimallyThickenedPoints}\hypertarget{InfinitesimallyThickenedPoints}{}
Write

\begin{displaymath}
InfPoint \hookrightarrow SmoothLoci
\end{displaymath}
for the [[full subcategory]] of $SmoothLoci$, def. \ref{SmoothL}, of those that are duals of Artin algebras, def. \ref{InfinitesimalAlgebra}.

We call this the category of \textbf{[[infinitesimally thickened points]]}.

\end{defn}
\hypertarget{de_rham_space}{}\paragraph*{{de Rham space}}\label{de_rham_space}

\begin{itemize}%
\item [[de Rham space]]

\end{itemize}
\hypertarget{jet_bundles}{}\paragraph*{{Jet bundles}}\label{jet_bundles}

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

\end{itemize}
\hypertarget{DifferentiationLayerSem}{}\subsubsection*{{Semantic Layer}}\label{DifferentiationLayerSem}

\hypertarget{synthetic_differential_geometry}{}\paragraph*{{Synthetic differential geometry}}\label{synthetic_differential_geometry}

\begin{itemize}%
\item [[synthetic differential geometry]]

\end{itemize}
We have two [[full and faithful functors]]

\begin{displaymath}
CartSp \hookrightarrow SmoothLoci
\end{displaymath}
\begin{displaymath}
InfPoint \hookrightarrow SmoothLoci
 \,.
\end{displaymath}
\begin{defn}
\label{InfinitesimallyThickenedCartesianSpaces}\hypertarget{InfinitesimallyThickenedCartesianSpaces}{}
Write [[CartSp]]${}_{th} \hookrightarrow SmoothLocus$ for the [[full subcategory]] of that of [[smooth loci]], def. \ref{SmoothL}, on those of the form

\begin{displaymath}
\mathbf{U} = U \times D
\end{displaymath}
with $U \in CartSp \hookrightarrow SmoothLoci$ and $D \in InfPoint \hookrightarrow SmoothLoci$.

We may call this the category of \textbf{infinitesimally thickened Cartesian spaces} or or \textbf{[[formal smooth manifold|formal smooth Cartesian spaces]]}.

\end{defn}
The category $CartSp_{th}$ carries several [[coverages]] of interest. One is this:

\begin{defn}
\label{CahiersTopos}\hypertarget{CahiersTopos}{}
For $\mathbf{U} = U \times D \in CartSp_{th}$ say that a [[covering family]] is a set of morphisms in $CartSp_{th}$ of the form $\{ U_i \times D \stackrel{(\phi_i, id_D)}{\to} U \times D\}_i$ such that $\{ U_i \stackrel{\phi_i}{\to} U\}_i$ is a [[covering]] family in [[CartSp]].

The corresponding [[sheaf topos]] $Sh(CartSp_{th})$ is known as the \textbf{[[Cahiers topos]]}.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The [[Cahiers topos]] $Sh(CartSp_{th})$, def. \ref{CahiersTopos},\newline is a [[cohesive topos]].

\end{prop}
\begin{defn}
\label{}\hypertarget{}{}
We write

\begin{displaymath}
SynthDiff0Type \coloneqq Sh(CartSp_{th})
  \,.
\end{displaymath}
\end{defn}
\begin{itemize}%
\item [[synthetic differential infinity-groupoid]]

\end{itemize}
\hypertarget{tangent_bundle}{}\paragraph*{{Tangent bundle}}\label{tangent_bundle}

Write $D \in Sh(CartSp_{th})$ for the [[smooth locus]] formally dual to the [[ring of dual numbers]], def. \ref{}. Write

\begin{displaymath}
i \colon * \to D
\end{displaymath}
for the unique point inclusion.

\begin{defn}
\label{}\hypertarget{}{}
For $X \in SynthDiff0Type$, the [[internal hom]]

\begin{displaymath}
T X \coloneqq [D,X] \in SynthDiff0Type
\end{displaymath}
equipped with the morphism

\begin{displaymath}
[i,X] \colon [D,X] \to [*,X] \simeq X
\end{displaymath}
is the \textbf{[[tangent bundle]]} of $X$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For $X \in SynthDiff0Type$, a \textbf{[[vector field]]} $v$ on $X$ is a [[section]]

\begin{displaymath}
\itexarray{
     X &&\stackrel{v}{\to}&& T X
     \\
     & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{[i,D]}}
     \\
     && X
  }
  \,.
\end{displaymath}
The \textbf{smooth space of vector fields} is the [[internal hom]] in the [[slice topos]] over $X$

\begin{displaymath}
\mathbf{\Gamma}_X(T X ) \coloneqq [X,T X]_X
  \,.
\end{displaymath}
\end{defn}
\hypertarget{differential_equations}{}\paragraph*{{Differential equations}}\label{differential_equations}

\begin{itemize}%
\item [[equation]]


\item [[differential equation]]



\end{itemize}
\hypertarget{DifferentialCohesionOfToposOfSmoothSpaces}{}\paragraph*{{Differential cohesion of the topos of smooth spaces}}\label{DifferentialCohesionOfToposOfSmoothSpaces}

Recall the sites

\begin{itemize}%
\item [[CartSp]], def. \ref{TheDifferentiallyGoodOpenCoverCoverage};


\item [[infinitesimally thickened point|InfPoint]], def. \ref{InfinitesimallyThickenedPoints};


\item [[formal smooth manifold|CartSp]]${}_{th}$, def. \ref{InfinitesimallyThickenedCartesianSpaces}.



\end{itemize}
\begin{defn}
\label{TheMorphismsOfSitesForCartSpInfinitesimalNeighbourhood}\hypertarget{TheMorphismsOfSitesForCartSpInfinitesimalNeighbourhood}{}
Define [[functors]]

\begin{displaymath}
CartSp
   \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}}
  CartSp_{th}
   \stackrel{\iota}{\leftarrow}
  InfPoint
\end{displaymath}
by

\begin{itemize}%
\item $i \colon U \mapsto U \times *$;


\item $p \colon U \times D \mapsto U$


\item $\iota \colon D \mapsto * \times D$



\end{itemize}
\end{defn}
\begin{prop}
\label{DifferentialCohesiveStructureOfSmoothSpaces}\hypertarget{DifferentialCohesiveStructureOfSmoothSpaces}{}
All three functors in def. \ref{TheMorphismsOfSitesForCartSpInfinitesimalNeighbourhood} are [[morphisms of sites]]. The induced [[geometric morphism]] of [[sheaf toposes]] is of the form

\begin{displaymath}
Sh(CartSp)
   \stackrel{}{\stackrel{\overset{Lan_i}{\hookrightarrow}}{\stackrel{\overset{(-)\circ i}{\leftarrow}}{\stackrel{\overset{(-)\circ p}{\hookrightarrow}}{\underset{Ran_p}{\leftarrow}}}}}
  Sh(CartSp_{th})
   \stackrel{\overset{Lan_\iota}{\leftarrow}}{\underset{(-)\circ \iota}{\to}}
  Sh(InfPoint)
\end{displaymath}
where hence the morphism on the left is in particular an [[essential geometric morphism|essential]] [[geometric embedding]].

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The sequence of [[geometric morphisms]]

\begin{displaymath}
Sh(CartSp) \stackrel{p^*}{\to} Sh(CartSp_{th}) \stackrel{\iota^*}{\to} Sh(InfPoint)
\end{displaymath}
exhibit a [[homotopy cofiber sequence]] in the [[(2,1)-category]] [[Topos]].

\end{prop}
\begin{proof}
By the discussion at \emph{\href{%28infinity%2C1%29Topos#ExistenceOfLimitsAndColimits}{(∞,1)Topos -- Existence of limits and colimits}} the statement is equivalently that the [[inverse image]] functors

\begin{displaymath}
Sh(CartSp) 
    \stackrel{Lan_p}{\leftarrow}
  Sh(CartSp_{th})
    \stackrel{Lan_\iota}{\leftarrow}
  Sh(InfPoint)
\end{displaymath}
form a [[homotopy fiber sequence]] in [[(∞,1)Cat]]. Computing this in the [[model structure for quasi-categories]] after passing to [[nerves]], the morphism $N(Lan_p)$ is clearly an [[inner Kan fibration]] because of the [[subcategory]] inclusion $Sh(CartSp) \hookrightarrow Sh(CartSp_{th})$. So by the general discussion at [[homotopy pullback]] the homotopy fiber is given by the 1-categorical fiber of $N(Lan_p)$ in [[sSet]]. By the discussion at \emph{\href{http://ncatlab.org/nlab/show/Kan+extension#LeftKanOnRepresentables}{Left Kan extension - on representables}} $Lan_p$ acts as $p$ on [[representable functor|representables]]. The 1-categorical fiber of $N(p) \colon N(CartSp_{th}) \to N(CartSp)$ is evidently $N(InfPoint)$. Since $Lan_\iota$ is a [[left adjoint]] it preserves colimits and since ever sheaf is a colimit of representables, this is sufficient to imply the claim.

\end{proof}
\hypertarget{DifferentialCohesion}{}\paragraph*{{Differential cohesion}}\label{DifferentialCohesion}

We axiomatize the existence of infinitesimals by further [[modal logic|modalities]] on a [[cohesive topos]].

\begin{defn}
\label{DifferentialCohesiveTopos}\hypertarget{DifferentialCohesiveTopos}{}
Given a [[cohesive topos]] $\mathbf{H} = Cohesive0Type$ over a [[base topos]] [[Disc∞Grpd|Discrete0Type]], a structure of \textbf{[[differential cohesion]]} on $\mathbf{H}$ is an [[geometric embedding]] into a [[cohesive topos]] $\mathbf{H}_{th} = InfThickenedCohesive0Type$ with an extra [[left eadjoint]] that preserves the terminal object:

\begin{displaymath}
\itexarray{
  Cohesive0Type
   &&\stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}}&&
   InfThickenedCohesive0Type
   \\
   & {}_{\mathllap{\Gamma}}\searrow \nwarrow^{\mathrlap{Disc}} && {}^{\mathllap{\Gamma_{th}}}\swarrow \nearrow_{\mathrlap{Disc_{th}}}
   \\
   && Discrete0Type
  }
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Given [[differential cohesion]], def. \ref{DifferentialCohesiveTopos}, define the [[monad]]/[[comonad]] [[adjunction]]

\begin{displaymath}
(Red \dashv \Pi_{inf})
  \colon
  \mathbf{H}_{th}
  \stackrel{\overset{i_*}{\leftarrow}}{\underset{i^*}{\to}}
  \mathbf{H}
  \stackrel{\overset{i_!}{\leftarrow}}{\underset{i_*}{\to}}
  \mathbf{H}
  \,.
\end{displaymath}
We call $Red(X)$ the \textbf{[[reduced type]]} of $X$ and $\Pi_{inf}(X)$ the [[infinitesimal path ∞-groupoid]] of $X$.

For the $(i_* \dashv i^*)$-[[unit of an adjunction|unit]] we write

\begin{displaymath}
InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)
\end{displaymath}
and call it the \textbf{constant infinitesimal path inclusion} on $X$.

The $(i_* \dashv i^*)$-[[unit of an adjunction|counit]]

\begin{displaymath}
Red X \to X
\end{displaymath}
we call the \textbf{inclusion of the reduced part} of $X$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Given a [[geometric embedding]] of [[(∞,1)-topos|∞-toposes]]

\begin{displaymath}
CohesiveType \stackrel{i}{\hookrightarrow} InfThickenedCohesiveType
\end{displaymath}
exhibiting [[differential cohesion]], write

\begin{displaymath}
CohesiveType 
    \stackrel{i}{\hookrightarrow} 
  InfThickenedCohesiveType
   \to 
  InfinitesimalType
\end{displaymath}
for the corresponding [[homotopy cofiber sequence]] in [[(∞,1)-topos]]. The [[full sub-(∞,1)-category]] that is the [[kernel]] of the [[global section geometric morphism]] of $InfininitesimalType$ we call the [[(∞,1)-category]] of \textbf{synthetic [[∞-Lie algebras]]}

\begin{displaymath}
L_\infty Algebra \coloneqq ker(\Gamma) \hookrightarrow InfinitesimalType \stackrerl{\Gamma}{\to} DiscreteType
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For the moment see at \emph{\href{synthetic%20differential%20infinity-groupoid#LieDifferentiation}{Synthetic differential infinity-groupoid -- Lie differentiation}}.

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Setting

\begin{itemize}%
\item $\mathbf{H}_{th} \coloneqq Sh(CartSp_{th})$


\item $\mathbf{H} \coloneqq Sh(CartSp)$


\item $\mathbf{H}_{inf} \coloneqq Sh(InfPoint)$



\end{itemize}
makes prop. \ref{DifferentialCohesiveStructureOfSmoothSpaces} exhibit [[differential cohesion|differential cohesive structure]].

\end{prop}
\hypertarget{DifferentiationSemLayerDeRhamSpace}{}\paragraph*{{de Rham space}}\label{DifferentiationSemLayerDeRhamSpace}

\begin{defn}
\label{}\hypertarget{}{}
For $X \in \mathbf{H}_{th}$ we call $\Pi_{inf}(X) \in \mathbf{H}$ the \textbf{[[de Rham space]] object} of $X$.

\end{defn}
\hypertarget{DifferentiationSemLayerJetBundle}{}\paragraph*{{Jet bundle}}\label{DifferentiationSemLayerJetBundle}

\begin{defn}
\label{}\hypertarget{}{}
For $X \in \mathbf{H}$

\begin{displaymath}
Jet_X 
    \coloneqq 
  (InfinitesimalPathInclusion_X)_*
  \colon
  \mathbf{H}_{th}/X \to \mathbf{H}_{th}/\Pi_{inf}(X)
  \,.
\end{displaymath}
\end{defn}
For $(E \to X) \in \mathbf{H}_{th}/X$, $Jet_X(E)$ is the \textbf{[[jet bundle]]} of $E$.

\hypertarget{formally_tale__formally_unramified__formally_smooth}{}\paragraph*{{Formally \'e{}tale / formally unramified / formally smooth}}\label{formally_tale__formally_unramified__formally_smooth}

\begin{defn}
\label{FormallyEtaleMap}\hypertarget{FormallyEtaleMap}{}
A morphism $f \;\colon\; X \to Y$ in $\mathbf{H}$ is called a \textbf{[[formally étale morphism]]} if the naturality square of the $\mathbf{\Pi}_{inf}$-[[unit of a monad|unit]]

\begin{displaymath}
\itexarray{
    X &\stackrel{}{\to}& \mathbf{\Pi}_{inf}(X)
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}}
    \\
    Y &\to& \mathbf{\Pi}_{inf}(Y)
  }
\end{displaymath}
is an [[(∞,1)-pullback]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
If $X, Y \in$ [[SmoothMfd]] $\hookrightarrow$ $\mathbf{H} \stackrel{i_!}{\to} \mathbf{H}_{th}$ then for $f \colon X \to Y$ any morphism

\begin{enumerate}%
\item $f$ is [[formally étale morphism]] precisely if $f$ is a [[submersion]] of smooth manifolds;


\item $f$ is a [[formally unramified morphism]] precisely if it is an [[immersion]] of smooth manifolds;


\item $f$ is a [[formally smooth morphism]] precisely if it is a [[diffeomorphism]].



\end{enumerate}
\end{prop}
\hypertarget{syntactic_layer}{}\subsubsection*{{Syntactic Layer}}\label{syntactic_layer}

\hypertarget{differential_homotopy_type_theory}{}\paragraph*{{Differential homotopy type theory}}\label{differential_homotopy_type_theory}

\begin{itemize}%
\item [[differential homotopy type theory]]

\end{itemize}
(\ldots{})



\end{document}