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\section*{A first idea of quantum field theory -- Fields}

\hypertarget{Fields}{}\subsection*{{Fields}}\label{Fields}

In this chapter we discuss these topics:

\begin{itemize}%
\item \emph{\hyperlink{FieldBundles}{Field bundles}}


\item \emph{\hyperlink{NonFiniteDimensionalGeometry}{Spaces of field histories}}


\item \emph{\hyperlink{InfinitesimalGeometry}{Infinitesimal geometry}}


\item \emph{\hyperlink{Supergeometry}{Fermion fields and Supergeometry}}



\end{itemize}
$\,$

A [[field history]] on a given [[spacetime]] $\Sigma$ (a history of spatial [[field configurations]], see remark \ref{FieldHistoriesAsHistoriesOfFieldConfigurations} below) is a [[quantity]] assigned to each point of spacetime (each [[event]]), such that this assignment varies smoothly with spacetime points. For instance an \emph{[[electromagnetic field]] [[field history|history]]} (example \ref{Electromagnetism} below) is at each point of spacetime a collection of [[vectors]] that encode the direction in which a [[charged particle]] passing through that point would feel a [[force]] (the ``[[Lorentz force]]'', see example \ref{Electromagnetism} below).

This is readily formalized (def. \ref{FieldsAndFieldBundles} below): If $F$ denotes the [[smooth manifold]] of ``values'' that the given kind of field may take at any spacetime point, then a field history $\Phi$ is modeled as a [[smooth function]] from spacetime to this space of values:

\begin{displaymath}
\Phi
    \;\colon\;
   \Sigma
     \longrightarrow
   F
  \,.
\end{displaymath}
It will be useful to unify [[spacetime]] and the space of [[field (physics)|field]] values (the [[field fiber]]) into a single manifold, the [[Cartesian product]]

\begin{displaymath}
E \;\coloneqq\; \Sigma \times F
\end{displaymath}
and to think of this equipped with the [[projection]] map onto the first factor as a [[fiber bundle]] of spaces of field values over spacetime

\begin{displaymath}
\itexarray{
     E &\coloneqq& \Sigma \times F
     \\
     {}^{\mathllap{fb}}\downarrow & \swarrow_{\mathrlap{pr_1}}
     \\
     \Sigma
  }
  \,.
\end{displaymath}
This is then called the \emph{[[field bundle]]}, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the [[fiber]] of this [[fiber bundle]] (def. \ref{FiberBundle}), it is sometimes also called the \emph{[[field fiber]]}. (See also at \emph{[[fiber bundles in physics]]}.)

Given a [[field bundle]] $E \overset{fb}{\to}\Sigma$, then a \emph{[[field history]]} is a [[section]] of that bundle (def. \ref{Sections}). The discussion of [[field theory]] concerns the [[space of field histories|space of all possible field histories]], hence the [[space of sections]] of the [[field bundle]] (example \ref{DiffeologicalSpaceOfFieldHistories} below). This is a very ``large'' [[generalized smooth space]], called a \emph{[[diffeological space]]} (def. \ref{DiffeologicalSpace} below).

Or rather, in the presence of [[fermion fields]] such as the [[Dirac field]] (example \ref{DiracFieldBundle} below), the [[Pauli exclusion principle]] demands that the [[field bundle]] is a [[supermanifold|super-manifold]], and that the fermionic [[space of field histories]] (example \ref{DiracSpaceOfFieldHistories} below) is a [[supergeometry|super-geometric]] [[generalized smooth space]]: a \emph{[[super smooth set]]} (def. \ref{SuperFormalSmoothSet} below).

This smooth structure on the [[space of field histories]] will be crucial when we discuss [[observables]] of a [[field theory]] \hyperlink{Observables}{below}, because these are smooth functions on the [[space of field histories]]. In particular it is this smooth structure which allows to derive that \emph{linear} observables of a [[free field theory]] are given by [[distributions]] (prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions}) below. Among these are the point evaluation observables ([[delta distributions]]) which are traditionally denoted by the same symbol as the [[field histories]] themselves.

Hence there are these aspects of the concept of ``[[field (physics)|field]]'' in [[physics]], which are closely related, but crucially different:

$\,$

\textbf{aspects of the concept of [[field (physics)|fields]]}

\newline | averaging of [[observable|field observable]] | $\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$ | $\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$ | [[operator-valued distribution|observable-valued distribution]] | def. \ref{RegularLinearFieldObservables} | | [[algebra of quantum observables]] | $\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$ | $\mathbb{C}Alg$ | [[non-commutative algebra]] [[structure]] on [[observable|field observables]] | def. \ref{WickAlgebraOfFreeQuantumField}, def. \ref{GeneratingFunctionsForCorrelationFunctions} |

$\,$

\textbf{[[field bundles]]}

\begin{defn}
\label{FieldsAndFieldBundles}\hypertarget{FieldsAndFieldBundles}{}
\textbf{([[field (physics)|fields]] and [[field histories]])}

Given a [[spacetime]] $\Sigma$, then a \emph{[[type]] of [[field|fields]]} on $\Sigma$ is a [[smooth set|smooth]] [[fiber bundle]] (def. \ref{FiberBundle})

\begin{displaymath}
\itexarray{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }
\end{displaymath}
called the \emph{[[field bundle]]},

Given a [[type]] of [[field|fields]] on $\Sigma$ this way, then a \emph{[[field history]]} of that type on $\Sigma$ is a [[term]] of that [[type]], hence is a smooth [[section]] (def. \ref{Sections}) of this [[bundle]], namely a [[smooth function]] of the form

\begin{displaymath}
\Phi \;\colon\; \Sigma \longrightarrow E
\end{displaymath}
such that composed with the [[projection]] map it is the [[identity function]], i.e. such that

\begin{displaymath}
fb \circ \Phi = id
  \phantom{AAAAAAA}
  \itexarray{
    && E
    \\
    & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    \Sigma & = & \Sigma
  }
  \,.
\end{displaymath}
The set of such [[sections]]/[[field histories]] is to be denoted

\begin{equation}
\Gamma_\Sigma(E)
  \;\coloneqq\;
  \left\{
    \itexarray{
      && E
      \\
      & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
      \\
      \Sigma &=& \Sigma
    }
    \phantom{fb}
  \right\}
\label{SetOfFieldHistories}\end{equation}
\end{defn}
\begin{remark}
\label{FieldHistoriesAsHistoriesOfFieldConfigurations}\hypertarget{FieldHistoriesAsHistoriesOfFieldConfigurations}{}
\textbf{([[field histories]] are histories of spatial [[field configurations]])}

Given a [[section]] $\Phi \in \Gamma_\Sigma(E)$ of the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and given a [[spacelike]] (def. \ref{SpacelikeTimelikeLightlike}) [[submanifold]] $\Sigma_p \hookrightarrow \Sigma$ (def. \ref{SmoothManifoldInsideDiffeologicalSpaces}) of [[spacetime]] in [[codimension]] 1, then the [[restriction]] $\Phi\vert_{\Sigma_p}$ of $\Phi$ to $\Sigma_p$ may be thought of as a \emph{[[field configuration]]} in space. As different spatial slices $\Sigma_p$ are chosen, one obtains such field configurations \emph{at different times}. It is in this sense that the entirety of a section $\Phi \in \Gamma_\Sigma(E)$ is a \emph{history} of field configurations, hence a [[field history]] (def \ref{FieldsAndFieldBundles}).

\end{remark}
\begin{remark}
\label{PossibleFieldHistories}\hypertarget{PossibleFieldHistories}{}
\textbf{([[possible worlds|possible]] field histories)}

After we give the set $\Gamma_\Sigma(E)$ of field histories \eqref{SetOfFieldHistories} [[differential geometry|differential geometric]] structure, below in example \ref{DiffeologicalSpaceOfFieldHistories} and example \ref{SupergeometricSpaceOfFieldHistories}, we call it the \emph{[[space of field histories]]}. This should be read as space of \emph{[[possibility|possible]]} field histories; containing all field histories that qualify as being of the [[type]] specified by the [[field bundle]] $E$.

After we obtain [[equations of motion]] below in def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}, these serve as the ``laws of nature'' that field histories should obey, and they define the subspace of those field histories that do solve the equations of motion; this will be denoted

\begin{displaymath}
\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}= 0}
    \overset{\phantom{AAA}}{\hookrightarrow}
  \Gamma_\Sigma(E)
\end{displaymath}
and called the \emph{[[on-shell]] [[space of field histories]]} \eqref{InclusionOfOnShellSpaceOfFieldHistories}.

\end{remark}
For the time being, not to get distracted from the basic idea of [[quantum field theory]], we will focus on the following simple special case of field bundles:

\begin{example}
\label{TrivialVectorBundleAsAFieldBundle}\hypertarget{TrivialVectorBundleAsAFieldBundle}{}
\textbf{([[trivial vector bundle]] as a [[field bundle]])}

In applications the [[field fiber]] $F = V$ is often a [[finite dimensional vector space]]. In this case the [[trivial bundle|trivial]] [[field bundle]] with [[fiber]] $F$ is of course a \emph{[[trivial vector bundle|trivial]] [[vector bundle]]} (def. \ref{VectorBundle}).

Choosing any [[linear basis]] $(\phi^a)_{a = 1}^s$ of the field fiber, then over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) we have canonical [[coordinates]] on the total space of the field bundle

\begin{displaymath}
( (x^\mu), ( \phi^a ) )
  \,,
\end{displaymath}
where the index $\mu$ ranges from $0$ to $p$, while the index $a$ ranges from 1 to $s$.

If this trivial vector bundle is regarded as a [[field bundle]] according to def. \ref{FieldsAndFieldBundles}, then a field history $\Phi$ is equivalently an $s$-[[tuple]] of [[real number|real]]-valued [[smooth functions]] $\Phi^a \colon \Sigma \to \mathbb{R}$ on spacetime:

\begin{displaymath}
\Phi = ( \Phi^a )_{a = 1}^s
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{RealScalarFieldBundle}\hypertarget{RealScalarFieldBundle}{}
\textbf{([[field bundle]] for [[real scalar field]])}

If $\Sigma$ is a [[spacetime]] and if the [[field fiber]]

\begin{displaymath}
F \coloneqq \mathbb{R}
\end{displaymath}
is simply the [[real line]], then the corresponding trivial [[field bundle]] (def. \ref{FieldsAndFieldBundles})

\begin{displaymath}
\itexarray{
    \Sigma \times \mathbb{R}
    \\
    {}^{\mathllap{pr_1}}\downarrow
    \\
    \Sigma
  }
\end{displaymath}
is the \emph{[[trivial fiber bundle|trivial]] [[real line bundle]]} (a special case of example \ref{TrivialVectorBundleAsAFieldBundle}) and the corresponding [[field (physics)|field]] type (def. \ref{FieldsAndFieldBundles}) is called the \emph{[[real scalar field]]} on $\Sigma$. A configuration of this field is simply a [[smooth function]] on $\Sigma$ with values in the [[real numbers]]:

\begin{equation}
\Gamma_\Sigma(\Sigma \times \mathbb{R})
    \;\simeq\;
  C^\infty(\Sigma)
  \,.
\label{SpaceOfFieldHistoriesOfRealScalarField}\end{equation}
\end{example}
\begin{example}
\label{Electromagnetism}\hypertarget{Electromagnetism}{}
\textbf{([[field bundle]] for [[electromagnetic field]])}

On [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) be given by the [[cotangent bundle]]

\begin{displaymath}
E \coloneqq T^\ast \Sigma
  \,.
\end{displaymath}
This is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with canonical [[field (physics)|field]] coordinates $(a_\mu)$.

A [[section]] of this bundle, hence a [[field history]], is a [[differential 1-form]]

\begin{displaymath}
A \in \Gamma_\Sigma(T^\ast \Sigma) = \Omega^1(\Sigma)
\end{displaymath}
on [[spacetime]] (def. \ref{Differential1FormsOnCartesianSpaces}). Interpreted as a [[field history]] of the [[electromagnetic field]] on $\Sigma$, this is often called the \emph{[[vector potential]]}. Then the [[de Rham differential]] (def. \ref{deRhamDifferential}) of the [[vector potential]] is a [[differential 2-form]]

\begin{displaymath}
F \coloneqq d A
\end{displaymath}
known as the \emph{[[Faraday tensor]]}. In the canonical coordinate basis 2-forms this may be expanded as

\begin{equation}
F
  =
  \underoverset{i = 1}{p}{\sum}
  E_i d x^0 \wedge d x^i
  +
  \underset{1 \leq i \lt j \leq p}{\sum}
  B_{i j} d x^i \wedge d x^j
  \,.
\label{TensorFaraday}\end{equation}
Here $(E_i)_{i = 1}^p$ are called the components of the \emph{[[electric field]]}, while $(B_{i j})$ are called the components of the \emph{[[magnetic field]]}.

\end{example}
\begin{example}
\label{YangMillsFieldOverMinkowski}\hypertarget{YangMillsFieldOverMinkowski}{}
\textbf{([[field bundle]] for [[Yang-Mills field]] over [[Minkowski spacetime]])}

Let $\mathfrak{g}$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]] with [[linear basis]] $(t_\alpha)$, in terms of which the [[Lie bracket]] is given by

\begin{equation}
[t_\alpha, t_\beta]
  \;=\;
  \gamma^\gamma{}_{\alpha \beta} t_\gamma
  \,.
\label{LieAlgebraStructureConstants}\end{equation}
Over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), consider then the [[field bundle]] which is the [[cotangent bundle]] [[tensor product|tensored]] with the [[Lie algebra]] $\mathfrak{g}$

\begin{displaymath}
E \coloneqq T^\ast \Sigma \otimes \mathfrak{g}
  \,.
\end{displaymath}
This is the [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with induced [[field (physics)|field]] [[coordinates]]

\begin{displaymath}
( a_\mu^\alpha )
  \,.
\end{displaymath}
A [[section]] of this bundle is a [[Lie algebra-valued differential 1-form]]

\begin{displaymath}
A \in \Gamma_\Sigma(T^\ast \Sigma \otimes \mathfrak{g}) = \Omega^1(\Sigma, \mathfrak{g})
  \,.
\end{displaymath}
with components

\begin{displaymath}
A^\ast(a_\mu^\alpha) = A^\alpha_\mu
  \,.
\end{displaymath}
This is called a [[field history]] for \emph{[[Yang-Mills theory|Yang-Mills]] [[gauge theory]]} (at least if $\mathfrak{g}$ is a \emph{[[semisimple Lie algebra]]}, see example \ref{YangMillsLagrangian} below).

For $\mathfrak{g} = \mathbb{R}$ is the [[line Lie algebra]], this reduces to the case of the [[electromagnetic field]] (example \ref{Electromagnetism}).

For $\mathfrak{g} = \mathfrak{su}(3)$ this is a [[field (physics)|field]] history for the [[gauge field]] of the [[strong nuclear force]] in [[quantum chromodynamics]].

\end{example}
For readers familiar with the concepts of \emph{[[principal bundles]]} and \emph{[[connections on a bundle]]} we include the following example \ref{YangMillsFieldInInstantonSector} which generalizes the [[Yang-Mills field]] over [[Minkowski spacetime]] from example \ref{YangMillsFieldOverMinkowski} to the situation over general [[spacetimes]].

\begin{example}
\label{YangMillsFieldInInstantonSector}\hypertarget{YangMillsFieldInInstantonSector}{}
\textbf{(general [[Yang-Mills field]] in fixed [[instanton|topological sector]])}

Let $\Sigma$ be any [[spacetime]] [[manifold]] and let $G$ be a [[compact Lie group]] with [[Lie algebra]] denoted $\mathfrak{g}$. Let $P \overset{is}{\to} \Sigma$ be a $G$-[[principal bundle]] and $\nabla_0$ a chosen [[connection on a bundle|connection]] on it, to be called the [[background field|background]] $G$-[[Yang-Mills theory|Yang-Mills]] field.

Then the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) for $G$-[[Yang-Mills theory]] \emph{in the [[instanton|topological sector]]} $P$ is the [[tensor product of vector bundles]]

\begin{displaymath}
E \coloneqq \left(P \times^{ad}_G \mathfrak{g}\right) \otimes_\Sigma \left( T^\ast \Sigma \right)
\end{displaymath}
of the [[adjoint bundle]] of $P$ and the [[cotangent bundle]] of $\Sigma$.

With the choice of $\nabla_0$, every (other) connection $\nabla$ on $P$ uniquely decomposes as

\begin{displaymath}
\nabla = \nabla_0 + A
  \,,
\end{displaymath}
where

\begin{displaymath}
A \in \Gamma_\Sigma(E)
\end{displaymath}
is a [[section]] of the above [[field bundle]], hence a [[Yang-Mills field|Yang-Mills]] [[field history]].

\end{example}
The [[electromagnetic field]] (def. \ref{Electromagnetism}) and the [[Yang-Mills field]] (def. \ref{YangMillsFieldOverMinkowski}, def. \ref{YangMillsFieldInInstantonSector}) with [[differential 1-forms]] as [[field histories]] are the basic examples of \emph{[[gauge fields]]} (we consider this in more detail below in \emph{\hyperlink{GaugeSymmetries}{Gauge symmetries}}). There are also \emph{[[higher gauge fields]]} with [[differential n-forms]] as [[field histories]]:

\begin{example}
\label{BField}\hypertarget{BField}{}
\textbf{([[field bundle]] for [[B-field]])}

On [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let the [[field bundle]] (def. \ref{FieldsAndFieldBundles}) be given by the skew-symmetrized [[tensor product of vector bundles]] of the [[cotangent bundle]] with itself

\begin{displaymath}
E \coloneqq \wedge^2_\Sigma T^\ast \Sigma
  \,.
\end{displaymath}
This is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with canonical [[field (physics)|field]] coordinates $(b_{\mu \nu})$ subject to

\begin{displaymath}
b_{\mu \nu} \;=\; - b_{\nu \mu}
  \,.
\end{displaymath}
A [[section]] of this bundle, hence a [[field history]], is a [[differential 2-form]] (def. \ref{DifferentialnForms})

\begin{displaymath}
B \in \Gamma_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) = \Omega^2(\Sigma)
\end{displaymath}
on [[spacetime]].

\end{example}
$\,$

\textbf{[[space of field histories]]}

Given any [[field bundle]], we will eventually need to regard the set of all [[field histories]] $\Gamma_\Sigma(E)$ as a ``[[smooth set]]'' itself, a smooth \emph{[[space of sections]]}, to which constructions of [[differential geometry]] apply (such as for the discussion of [[observables]] and [[states]] \hyperlink{Observables}{below} ). Notably we need to be talking about [[differential forms]] on $\Gamma_\Sigma(E)$.

However, a [[space of sections]] $\Gamma_\Sigma(E)$ does not in general carry the structure of a [[smooth manifold]]; and it carries the correct smooth structure of an [[infinite dimensional manifold]] only if $\Sigma$ is a [[compact space]] (see at \emph{[[manifold structure of mapping spaces]]}). Even if it does carry [[infinite dimensional manifold]] structure, inspection shows that this is more [[structure]] than actually needed for the discussion of [[field theory]]. Namely it turns out below that all we need to know is what counts as a \emph{smooth family} of [[sections]]/[[field histories]], hence which [[functions]] of [[sets]]

\begin{displaymath}
\Phi_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \Gamma_\Sigma(E)
\end{displaymath}
from any [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) into $\Gamma_\Sigma(E)$ count as [[smooth functions]], subject to some basic consistency condition on this choice.

This [[structure]] on $\Gamma_\Sigma(E)$ is called the structure of a \emph{[[diffeological space]]}:

\begin{defn}
\label{DiffeologicalSpace}\hypertarget{DiffeologicalSpace}{}
\textbf{([[diffeological space]])}

A \emph{[[diffeological space]]} $X$ is

\begin{enumerate}%
\item a [[set]] $X_s \in$ [[Set]];


\item for each $n \in \mathbb{N}$ a choice of [[subset]]

\begin{displaymath}
X(\mathbb{R}^n) \subset Hom_{Set}(\mathbb{R}^n_s, X_s) = \left\{ \mathbb{R}^n_s \to X_s  \right\}
\end{displaymath}
of the [[function set|set of functions]] from the underlying set $\mathbb{R}^n_s$ of $\mathbb{R}^n$ to $X_s$, to be called the \emph{smooth functions} or \emph{plots} from $\mathbb{R}^n$ to $X$;


\item for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$ between [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) a choice of function

\begin{displaymath}
f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})
\end{displaymath}
to be thought of as the precomposition operation

\begin{displaymath}
\left(
    \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
  \right)
  \;\overset{f^\ast}{\mapsto}\;
  \left(
    \mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\Phi}{\to} X
  \right)
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item ([[constant functions]] are smooth)

\begin{displaymath}
X(\mathbb{R}^0) = X_s
  \,,
\end{displaymath}

\item ([[functor|functoriality]])

\begin{enumerate}%
\item If $id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n$ is the [[identity function]] on $\mathbb{R}^n$, then $\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)$ is the identity function on the set of plots $X(\mathbb{R}^n)$;


\item If $\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}$ are two [[composition|composable]] [[smooth functions]] between [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), then pullback of plots along them consecutively equals the pullback along the [[composition]]:

\begin{displaymath}
f^\ast \circ g^\ast
  =
  (g \circ f)^\ast
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
 && X(\mathbb{R}^{n_2})
 \\
 & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
 \\
 X(\mathbb{R}^{n_1})
 && \underset{ (g \circ f)^\ast }{\longleftarrow} &&
 X(\mathbb{R}^{n_3})
  }
\end{displaymath}


\end{enumerate}

\item ([[sheaf|gluing]])

If $\{ U_i \overset{f_i}{\to} \mathbb{R}^n\}_{i \in I}$ is a [[differentiably good open cover]] of a [[Cartesian space]] (def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts $\mathbb{R}^n$-plots of $X$ to a set of $U_i$-plots

\begin{displaymath}
X(\mathbb{R}^n)
    \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
  \underset{i \in I}{\prod} X(U_i)
\end{displaymath}
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are ``[[matching families]]'' in that they agree on [[intersections]]:

\begin{displaymath}
\phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j}
  \phantom{AAAAAA}
  \itexarray{
    && U_i \cap U_j
    \\
    & \swarrow && \searrow
    \\
    U_i && && U_j
    \\
    & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
    \\
    && X
  }
\end{displaymath}


\end{enumerate}
Finally, given $X_1$ and $X_2$ two diffeological spaces, then a [[smooth function]] between them

\begin{displaymath}
f \;\colon\; X_1 \longrightarrow X_2
\end{displaymath}
is

\begin{itemize}%
\item a [[function]] of the underlying sets

\begin{displaymath}
f_s \;\colon\; (X_1)_s \longrightarrow (X_2)_s
\end{displaymath}


\end{itemize}
such that

\begin{itemize}%
\item for $\Phi \in X(\mathbb{R}^n)$ a plot of $X_1$, then the [[composition]] $f_s \circ \Phi_s$ is a plot $f_\ast(\Phi)$ of $X_2$:

\begin{displaymath}
\itexarray{
    && \mathbb{R}^n
    \\
    & {}^{\mathllap{\Phi}}\swarrow && \searrow^{\mathrlap{f_\ast(\Phi)}}
    \\
    X_1 && \underset{f}{\longrightarrow} && X_2
  }
  \,.
\end{displaymath}


\end{itemize}
(Stated more [[category theory|abstractly]], this says simply that [[diffeological spaces]] are the [[concrete sheaves]] on the [[site]] of [[Cartesian spaces]] from def. \ref{DifferentiablyGoodOpenCover}.)

\end{defn}
For more background on [[diffeological spaces]] see also \emph{[[geometry of physics -- smooth sets]]}.

\begin{example}
\label{SmoothManifoldsAreDiffeologicalSpaces}\hypertarget{SmoothManifoldsAreDiffeologicalSpaces}{}
\textbf{([[Cartesian spaces]] are [[diffeological spaces]])}

Let $X$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) Then it becomes a [[diffeological space]] (def. \ref{DiffeologicalSpace}) by declaring its plots $\Phi \in X(\mathbb{R}^n)$ to the ordinary [[smooth functions]] $\Phi \colon \mathbb{R}^n \to X$.

Under this identification, a function $f \;\colon\; (X_1)_s \to (X_2)_s$ between the underlying sets of two [[Cartesian spaces]] is a [[smooth function]] in the ordinary sense precisely if it is a smooth function in the sense of [[diffeological spaces]].

Stated more [[category theory|abstractly]], this statement is an example of the \emph{[[Yoneda embedding]]} over a \emph{[[subcanonical site]]}.

More generally, the same construction makes every [[smooth manifold]] a [[smooth set]].

\end{example}
\begin{example}
\label{DiffeologicalSpaceOfFieldHistories}\hypertarget{DiffeologicalSpaceOfFieldHistories}{}
\textbf{([[diffeological space|diffeological]] [[space of field histories]])}

Let $E \overset{fb}{\to} \Sigma$ be a smooth [[field bundle]] (def. \ref{FieldsAndFieldBundles}). Then the set $\Gamma_\Sigma(E)$ of [[field histories]]/[[sections]] (def. \ref{FieldsAndFieldBundles}) becomes a [[diffeological space]] (def. \ref{DiffeologicalSpace})

\begin{equation}
\Gamma_\Sigma(E) \in DiffeologicalSpaces
\label{SpaceOfFieldHistories}\end{equation}
by declaring that a smooth family $\Phi_{(-)}$ of field histories, parameterized over any [[Cartesian space]] $U$ is a smooth function out of the [[Cartesian product]] manifold of $\Sigma$ with $U$

\begin{displaymath}
\itexarray{
    U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
    \\
    (u,x) &\mapsto& \Phi_u(x)
  }
\end{displaymath}
such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.

\begin{displaymath}
\itexarray{
    && E
    \\
    & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
  }
  \,.
\end{displaymath}
\end{example}
The following example \ref{FrechetManifoldsAreDiffeologicalSpaces} is included only for readers who wonder how [[infinite-dimensional manifolds]] fit in. Since we will never actually use [[infinite-dimensional manifold]]-structure, this example is may be ignored.

\begin{example}
\label{FrechetManifoldsAreDiffeologicalSpaces}\hypertarget{FrechetManifoldsAreDiffeologicalSpaces}{}
\textbf{([[Fréchet manifolds]] are [[diffeological spaces]])}

Consider the particular type of [[infinite-dimensional manifolds]] called \emph{[[Fréchet manifolds]]}. Since ordinary [[smooth manifolds]] $U$ are an example, for $X$ a [[Fréchet manifold]] there is a concept of [[smooth functions]] $U \to X$. Hence we may give $X$ the structure of a [[diffeological space]] (def. \ref{DiffeologicalSpace}) by declaring the plots over $U$ to be these smooth functions $U \to X$, with the evident postcomposition action.

It turns out that then that for $X$ and $Y$ two [[Fréchet manifolds]], there is a [[natural bijection]] between the [[smooth functions]] $X \to Y$ between them regarded as [[Fréchet manifolds]] and regarded as [[diffeological spaces]. Hence it does not matter which of the two perspectives we take (unless of course a [[diffeological space]] more general than a [[Fréchet manifolds]] enters the picture, at which point the second definition generalizes, whereas the first does not).

Stated more [[category theory|abstractly]], this means that [[Fréchet manifolds]] form a [[full subcategory]] of that of [[diffeological spaces]] (\href{Fr&#233;chet+manifold#FFEmbeddingOfFrechetInDiffeological}{this prop.}):

\begin{displaymath}
FrechetManifolds \hookrightarrow DiffeologicalSpaces
  \,.
\end{displaymath}
If $\Sigma$ is a [[compact space|compact]] [[smooth manifold]] and $E \simeq \Sigma \times F \to \Sigma$ is a [[trivial fiber bundle]] with [[fiber]] $F$ a [[smooth manifold]], then the set of [[sections]] $\Gamma_\Sigma(E)$ carries a standard structure of a [[Fréchet manifold]] (see at \emph{[[manifold structure of mapping spaces]]}). Under the above inclusion of [[Fréchet manifolds]] into [[diffeological spaces]], this [[smooth structure]] agrees with that from example \ref{DiffeologicalSpaceOfFieldHistories} (see \href{Fr&#233;chet+manifold#CompatibilityWithDiffeologicalMappingSpaces}{this prop.})

\end{example}
Once the step from [[smooth manifolds]] to [[diffeological spaces]] (def. \ref{DiffeologicalSpace}) is made, characterizing the [[smooth structure]] of the space entirely by how we may probe it by mapping smooth Cartesian spaces into it, it becomes clear that the underlying set $X_s$ of a diffeological space $X$ is not actually crucial to support the concept: The space is already entirely defined [[structuralism|structurally]] by the system of smooth plots it has, and the underlying set $X_s$ is recovered from these as the set of plots from the point $\mathbb{R}^0$.

This is crucial for [[field theory]]: the [[spaces of field histories]] of [[fermionic fields]] (def. \ref{FermionicBosonicFields} below) such as the \emph{[[Dirac field]]} (example \ref{DiracSpaceOfFieldHistories} below) do not have underlying sets of points the way [[diffeological spaces]] have. Informally, the reason is that a point is a [[bosonic]] object, while and the nature of [[fermionic fields]] is [[antimodal type|the opposite of]] bosonic.

But we may just as well drop the mentioning of the underlying set $X_s$ in the definition of [[generalized smooth spaces]]. By simply stripping this requirement off of def. \ref{DiffeologicalSpace} we obtain the following more general and more useful definition (still ``bosonic'', though, the [[supergeometry|supergeometric]] version is def. \ref{SuperFormalSmoothSet} below):

\begin{defn}
\label{SmoothSet}\hypertarget{SmoothSet}{}
\textbf{([[smooth set]])}

A \emph{[[smooth set]]} $X$ is

\begin{enumerate}%
\item for each $n \in \mathbb{N}$ a choice of [[set]]

\begin{displaymath}
X(\mathbb{R}^n) \in Set
\end{displaymath}
to be called the set of \emph{smooth functions} or \emph{plots} from $\mathbb{R}^n$ to $X$;


\item for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$ between [[Cartesian spaces]] a choice of function

\begin{displaymath}
f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})
\end{displaymath}
to be thought of as the precomposition operation

\begin{displaymath}
\left(
    \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
  \right)
  \;\overset{f^\ast}{\mapsto}\;
  \left(
    \mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{\Phi}{\to} X
  \right)
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item ([[functor|functoriality]])

\begin{enumerate}%
\item If $id_{\mathbb{R}^n} \;\colon\; \mathbb{R}^n \to \mathbb{R}^n$ is the [[identity function]] on $\mathbb{R}^n$, then $\left(id_{\mathbb{R}^n}\right)^\ast \;\colon\; X(\mathbb{R}^n) \to X(\mathbb{R}^n)$ is the [[identity function]] on the set of plots $X(\mathbb{R}^n)$.


\item If $\mathbb{R}^{n_1} \overset{f}{\to} \mathbb{R}^{n_2} \overset{g}{\to} \mathbb{R}^{n_3}$ are two [[composition|composable]] [[smooth functions]] between [[Cartesian spaces]], then consecutive pullback of plots along them equals the pullback along the [[composition]]:

\begin{displaymath}
f^\ast \circ g^\ast
  =
  (g \circ f)^\ast
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
 && X(\mathbb{R}^{n_2})
 \\
 & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
 \\
 X(\mathbb{R}^{n_1})
 && \underset{ (g \circ f)^\ast }{\longleftarrow} &&
 X(\mathbb{R}^{n_3})
  }
\end{displaymath}


\end{enumerate}

\item ([[sheaf|gluing]])

If $\{ U_i \overset{f_i}{\to} \mathbb{R}^n\}_{i \in I}$ is a [[differentiably good open cover]] of a [[Cartesian space]] (def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts $\mathbb{R}^n$-plots of $X$ to a set of $U_i$-plots

\begin{displaymath}
X(\mathbb{R}^n)
    \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
  \underset{i \in I}{\prod} X(U_i)
\end{displaymath}
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are ``[[matching families]]'' in that they agree on [[intersections]]:

\begin{displaymath}
\phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j}
  \phantom{AAAA}
  \text{i.e.}
  \phantom{AAAA}
  \itexarray{
    && U_i \cap U_j
    \\
    & \swarrow && \searrow
    \\
    U_i && && U_j
    \\
    & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
    \\
    && X
  }
\end{displaymath}


\end{enumerate}
Finally, given $X_1$ and $X_2$ two [[smooth sets]], then a [[smooth function]] between them

\begin{displaymath}
f \;\colon\; X_1 \longrightarrow X_2
\end{displaymath}
is

\begin{itemize}%
\item for each $n \in \mathbb{N}$ a [[function]]

\begin{displaymath}
f_\ast(\mathbb{R}^n)
  \;\colon\;
  X_1(\mathbb{R}^n) \longrightarrow X_2(\mathbb{R}^n)
\end{displaymath}


\end{itemize}
such that

\begin{itemize}%
\item for each [[smooth function]] $g \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between [[Cartesian spaces]] we have

\begin{displaymath}
g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2})
  =
  f_\ast(\mathbb{R}^{n_1}) \circ g^\ast_1
  \phantom{AAAAA}
  \text{i.e.}
  \phantom{AAAAA}
  \text{i.e.}
  \phantom{AAAAA}
  \itexarray{
    X_1(\mathbb{R}^{n_2})
      &\overset{f_\ast(\mathbb{R}^{n_2})}{\longrightarrow}&
    X_2(\mathbb{R}^{n_2})
    \\
    \mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2}
    \\
    X_1(\mathbb{R}^{n_1})
      &\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}&
    X_2(\mathbb{R}^{n_1})
  }
\end{displaymath}


\end{itemize}
Stated more [[category theory|abstractly]], this simply says that [[smooth sets]] are the \emph{[[sheaves]] on the [[site]] of [[Cartesian spaces]] from def. \ref{DifferentiablyGoodOpenCover}.}

\end{defn}
Basing [[differential geometry]] on [[smooth sets]] is an instance of the general approach to [[geometry]] called \emph{[[functorial geometry]]} or \emph{[[topos theory]]}. For more background on this see at \emph{[[geometry of physics -- smooth sets]]}.

First we verify that the concept of smooth sets is a consistent generalization:

\begin{example}
\label{SmoothSetsDiffeologicalSpaces}\hypertarget{SmoothSetsDiffeologicalSpaces}{}
\textbf{([[diffeological spaces]] are [[smooth sets]])}

Every [[diffeological space]] $X$ (def. \ref{DiffeologicalSpace}) is a [[smooth set]] (def. \ref{SmoothSet}) simply by [[forgetful functor|forgetting]] its underlying set of points and remembering only its sets of plot.

In particular therefore each [[Cartesian space]] $\mathbb{R}^n$ is canonically a [[smooth set]] by example \ref{SmoothManifoldsAreDiffeologicalSpaces}.

Moreover, given any two [[diffeological spaces]], then the [[morphisms]] $f \colon X \to Y$ between them, regarded as diffeological spaces, are [[natural bijection|the same]] as the morphisms as [[smooth sets]].

Stated more [[category theory|abstractly]], this means that we have [[full subcategory]] inclusions

\begin{displaymath}
CartesianSpaces
    \overset{\phantom{AAA}}{\hookrightarrow}
  DiffeologicalSpaces
    \overset{\phantom{AAA}}{\hookrightarrow}
  SmoothSets
  \,.
\end{displaymath}
\end{example}
Recall, for the next proposition \ref{CartSpYpnedaLemma}, that in the definition \ref{SmoothSet} of a [[smooth set]] $X$ the sets $X(\mathbb{R}^n)$ are abstract sets which are \emph{to be thought of} as would-be smooth functions ``$\mathbb{R}^n \to X$''. Inside def. \ref{SmoothSet} this only makes sense in quotation marks, since inside that definition the smooth set $X$ is only being defined, so that inside that definition there is not yet an actual concept of smooth functions of the form ``$\mathbb{R}^n \to X$''.

But now that the definition of [[smooth sets]] and of [[morphisms]] between them has been stated, and seeing that [[Cartesian space]] $\mathbb{R}^n$ are examples of [[smooth sets]], by example \ref{SmoothSetsDiffeologicalSpaces}, there is now an actual concept of smooth functions $\mathbb{R}^n \to X$, namely as smooth sets. For the concept of smooth sets to be consistent, it ought to be true that this \emph{a posteriori} concept of smooth functions from [[Cartesian spaces]] to [[smooth sets]] coincides wth the \emph{a priori} concept, hence that we ``may remove the quotation marks'' in the above. The following proposition says that this is indeed the case:

\begin{prop}
\label{CartSpYpnedaLemma}\hypertarget{CartSpYpnedaLemma}{}
\textbf{(plots of a [[smooth set]] really are the [[smooth functions]] into the smooth set)}

Let $X$ be a [[smooth set]] (def. \ref{SmoothSet}). For $n \in \mathbb{R}$, there is a [[natural transformation|natural]] [[function]]

\begin{displaymath}
Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} X(\mathbb{R}^n)
\end{displaymath}
from the set of homomorphisms of smooth sets from $\mathbb{R}^n$ (regarded as a smooth set via example \ref{SmoothSetsDiffeologicalSpaces}) to $X$, to the set of plots of $X$ over $\mathbb{R}^n$, given by evaluating on the [[identity function|identity]] plot $id_{\mathbb{R}^n}$.

This function is a \emph{[[bijection]]}.

This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the \emph{would-be} smooth functions into $X$, end up being the \emph{actual} smooth functions into $X$.

\end{prop}
\begin{proof}
This elementary but profound fact is called the \emph{[[Yoneda lemma]]}, here in its incarnation over the [[site]] of [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).

\end{proof}
A key class of examples of [[smooth sets]] (def. \ref{SmoothSet}) that are not [[diffeological spaces]] (def. \ref{DiffeologicalSpace}) are universal smooth [[moduli spaces]] of [[differential forms]]:

\begin{example}
\label{UniversalSmoothModuliSpaceOfDifferentialForms}\hypertarget{UniversalSmoothModuliSpaceOfDifferentialForms}{}
\textbf{(universal [[smooth set|smooth]] [[moduli spaces]] of [[differential forms]])}

For $k \in \mathbb{N}$ there is a [[smooth set]] (def. \ref{SmoothSet})

\begin{displaymath}
\mathbf{\Omega}^k \;\in\; SmoothSet
\end{displaymath}
defined as follows:

\begin{enumerate}%
\item for $n \in \mathbb{N}$ the set of plots from $\mathbb{R}^n$ to $\mathbf{\Omega}^k$ is the set of smooth [[differential forms|differential k-forms]] on $\mathbb{R}^n$ (def. \ref{DifferentialnForms})

\begin{displaymath}
\mathbf{\Omega}^k(\mathbb{R}^n) \;\coloneqq\; \Omega^k(\mathbb{R}^n)
\end{displaymath}

\item for $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[smooth function]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) the operation of pullback of plots along $f$ is just the [[pullback of differential forms]] $f^\ast$ from prop. \ref{PullbackOfDifferentialForms}

\begin{displaymath}
\itexarray{
    \mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1})
    \\
    \downarrow^{\mathrlap{f}} && \uparrow^{\mathrlap{f^\ast}}
    \\
    \mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2})
  }
\end{displaymath}


\end{enumerate}
That this is [[functor|functorial]] is just the standard fact \eqref{PullbackOfDiffereentialFormsCompatibleWithComposition} from prop. \ref{PullbackOfDifferentialForms}.

For $k = 1$ the smooth set $\mathbf{\Omega}^0$ actually is a [[diffeological space]], in fact under the identification of example \ref{SmoothSetsDiffeologicalSpaces} this is just the [[real line]]:

\begin{displaymath}
\mathbf{\Omega}^0 \simeq \mathbb{R}^1
  \,.
\end{displaymath}
But for $k \geq 1$ we have that the set of plots on $\mathbb{R}^0 = \ast$ is a [[singleton]]

\begin{displaymath}
\mathbf{\Omega}^{k \geq 1}(\mathbb{R}^0) \simeq \{0\}
\end{displaymath}
consisting just of the zero differential form. The only diffeological space with this property is $\mathbb{R}^0 = \ast$ itself. But $\mathbf{\Omega}^{k \geq 1}$ is far from being that trivial: even though its would-be underlying set is a single point, for all $n \geq k$ it admits an infinite set of plots. Therefore the smooth sets $\mathbf{\Omega}^k$ for $k \geq$ are not diffeological spaces.

That the [[smooth set]] $\mathbf{\Omega}^k$ indeed deserves to be addressed as the \emph{universal [[moduli space]] of [[differential n-forms|differential k-forms]]} follows from prop. \ref{CartSpYpnedaLemma}: The universal moduli space of $k$-forms ought to carry a universal differential $k$-forms $\omega_{univ} \in \Omega^k(\mathbf{\Omega}^k)$ such that every differential $k$-form $\omega$ on any $\mathbb{R}^n$ arises as the [[pullback of differential forms]] of this universal one along some \emph{[[modulating morphism]]} $f_\omega \colon X \to \mathbf{\Omega}^k$:

\begin{displaymath}
\itexarray{
    \{\omega\} &\overset{(f_\omega)^\ast}{\longleftarrow}& \{\omega_{univ}\}
    \\
    \\
    X &\underset{f_\omega}{\longrightarrow}& \mathbf{\Omega}^k
  }
\end{displaymath}
But with prop. \ref{CartSpYpnedaLemma} this is precisely what the definition of the plots of $\mathbf{\Omega}^k$ says.

Similarly, all the usual operations on differential form now have their universal archetype on the universal [[moduli spaces]] of differential forms

In particular, for $k \in \mathbb{N}$ there is a canonical [[morphism]] of [[smooth sets]] of the form

\begin{displaymath}
\mathbf{\Omega}^k \overset{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{k+1}
\end{displaymath}
defined over $\mathbb{R}^n$ by the ordinary [[de Rham differential]] (def. \ref{deRhamDifferential})

\begin{equation}
\Omega^k(\mathbb{R}^n) \overset{d}{\longrightarrow} \Omega^{k+1}(\mathbb{R}^n)
  \,.
\label{deRhamDifferentialUniversal}\end{equation}
That this satisfies the compatibility with precomposition of plots

\begin{displaymath}
\itexarray{
    \mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1}) &\overset{d}{\longrightarrow}& \Omega^{k+1}(\mathbb{R}^{n_1})
    \\
    {}^{\mathllap{f}}\downarrow && \uparrow^{\mathrlap{f^\ast}} && \uparrow^{\mathrlap{f^\ast}}
    \\
    \mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2}) &\underset{d}{\longrightarrow}& \Omega^k( \mathbb{R}^{n_2} )
  }
\end{displaymath}
is just the compatibility of [[pullback of differential forms]] with the [[de Rham differential]] of from prop. \ref{PullbackOfDifferentialForms}.

\end{example}
The upshot is that we now have a good definition of [[differential forms]] on any [[diffeological space]] and more generally on any [[smooth set]]:

\begin{defn}
\label{DifferentialFormsOnDiffeologicalSpaces}\hypertarget{DifferentialFormsOnDiffeologicalSpaces}{}
\textbf{([[differential forms]] on [[smooth sets]])}

Let $X$ be a [[diffeological space]] (def. \ref{DiffeologicalSpace}) or more generally a [[smooth set]] (def. \ref{SmoothSet}) then a [[differential form|differential k-form]] $\omega$ on $X$ is equivalently a [[morphism]] of [[smooth sets]]

\begin{displaymath}
X \longrightarrow \mathbf{\Omega}^k
\end{displaymath}
from $X$ to the universal [[smooth set|smooth]] [[moduli space]] of differential froms from example \ref{UniversalSmoothModuliSpaceOfDifferentialForms}.

Concretely, by unwinding the definitions of $\mathbf{\Omega}^k$ and of [[morphisms]] of smooth sets, this means that such a differential form is:

\begin{itemize}%
\item for each $n \in \mathbb{N}$ and each plot $\mathbb{R}^n \overset{\Phi}{\to} X$ an ordinary [[differential form]]

\begin{displaymath}
\Phi^\ast(\omega) \in \Omega^\bullet(\mathbb{R}^n)
\end{displaymath}


\end{itemize}
such that

\begin{itemize}%
\item for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ between [[Cartesian spaces]] the ordinary [[pullback of differential forms]] along $f$ is compatible with these choices, in that for every plot $\mathbb{R}^{n_2} \overset{\Phi}{\to} X$ we have

\begin{displaymath}
f^\ast\left(\Phi^\ast(\omega)\right)
  =
  ( f^\ast \Phi )^\ast(\omega)
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
    \mathbb{R}^{n_1} && \overset{f}{\longrightarrow} && \mathbb{R}^{n_2}
    \\
    & {}_{\mathllap{f^\ast \Phi}}\searrow && \swarrow_{\mathrlap{\Phi}}
    \\
    && X
  }
  \phantom{AAAA}
  \itexarray{
    \Omega^\bullet( \mathbb{R}^{n_1} ) && \overset{f^\ast}{\longleftarrow} && \Omega^\bullet(\mathbb{R}^{n_2})
    \\
    & {}_{\mathllap{(f^\ast \Phi)^\ast}}\nwarrow && \nearrow_{\mathrlap{\Phi^\ast}}
    \\
    && \Omega^\bullet(X)
  }
  \,.
\end{displaymath}


\end{itemize}
We write $\Omega^\bullet(X)$ for the set of differential forms on the smooth set $X$ defined this way.

Moreover, given a [[differential form|differential k-form]]

\begin{displaymath}
X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^k
\end{displaymath}
on a [[smooth set]] $X$ this way, then its [[de Rham differential]] $d \omega \in \Omega^{k+1}(X)$ is given by the [[composition|composite]] of [[morphisms]] of [[smooth sets]] with the universal de Rham differential from \eqref{deRhamDifferentialUniversal}:

\begin{equation}
d \omega
    \;\colon\;
  X
    \overset{\omega}{\longrightarrow}
  \mathbf{\Omega}^k
    \overset{d}{\longrightarrow}
  \mathbf{\Omega}^{k+1}
  \,.
\label{FormsOnSmoothSetDeRhamDifferential}\end{equation}
Explicitly this means simply that for $\Phi \colon U \to X$ a plot, then

\begin{displaymath}
\Phi^\ast (d\omega)
  \;=\;
  d\left( \Phi^\ast \omega\right)
  \;\in\;
  \Omega^{k+1}(U)
  \,.
\end{displaymath}
\end{defn}
The usual operations on ordinary [[differential forms]] directly generalize plot-wise to differential forms on [[diffeological spaces]] and more generally on [[smooth sets]]:

\begin{defn}
\label{ExteriorCalculusOnDiffeologicalSpaces}\hypertarget{ExteriorCalculusOnDiffeologicalSpaces}{}
\textbf{([[exterior differential]] and [[exterior product]] on [[smooth sets]])}

Let $X$ be a [[diffeological space]] (def. \ref{DiffeologicalSpace}) or more generally a [[smooth set]] (def. \ref{SmoothSet}). Then

\begin{enumerate}%
\item For $\omega \in \Omega^n(X)$ a [[differential form]] on $X$ (def. \ref{DifferentialFormsOnDiffeologicalSpaces}) its [[exterior differential]]

\begin{displaymath}
d \omega \in \Omega^{n+1}(X)
\end{displaymath}
is defined on any plot $\mathbb{R}^n \overset{\Phi}{\to} X$ as the ordinary [[exterior differential]] of the pullback of $\omega$ along that plot:

\begin{displaymath}
\Phi^\ast(d \omega) \coloneqq d \Phi^\ast(\omega)
  \,.
\end{displaymath}

\item For $\omega_1 \in \Omega^{n_1}$ and $\omega_2 \in \Omega^{n_2}(X)$ two differential forms on $X$ (def. \ref{DifferentialFormsOnDiffeologicalSpaces}) then their [[exterior product]]

\begin{displaymath}
\omega_1 \wedge \omega_2 \;\in\; \Omega^{n_1 + n_2}(X)
\end{displaymath}
is the differential form defined on any plot $\mathbb{R}^n \overset{\Phi}{\to} X$ as the ordinary exterior product of the pullback of th differential forms $\omega_1$ and $\omega_2$ to this plot:

\begin{displaymath}
\Phi^\ast(\omega_1 \wedge \omega_2)
  \;\coloneqq\;
  \Phi^\ast(\omega_1) \wedge \Phi^\ast(\omega_2)
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
$\,$

\textbf{Infinitesimal geometry}

It is crucial in [[field theory]] that we consider [[field histories]] not only over all of [[spacetime]], but also restricted to [[submanifolds]] of spacetime. Or rather, what is actually of interest are the restrictions of the field histories to the \emph{[[infinitesimal neighbourhoods]]} (example \ref{InfinitesimalNeighbourhood} below) of these submanifolds. This appears notably in the construction of \emph{[[phase spaces]]} \hyperlink{PhaseSpace}{below}. Moreover, [[fermion fields]] such as the [[Dirac field]] (example \ref{DiracFieldBundle} below) take values in \emph{[[graded object|graded]]} [[infinitesimal]] spaces, called \emph{[[super spaces]]} (discussed \hyperlink{Supergeometry}{below}). Therefore ``infinitesimal geometry'', sometimes called \emph{[[formal geometry]]} (as in ``[[formal scheme]]'') or \emph{[[synthetic differential geometry]]} or \emph{[[synthetic differential supergeometry]]}, is a central aspect of [[field theory]].

In order to mathematically grasp what \emph{[[infinitesimal neighbourhoods]]} are, we appeal to the first magic algebraic property of differential geometry from prop. \ref{AlgebraicFactsOfDifferentialGeometry}, which says that we may recognize [[smooth manifolds]] $X$ [[formal dual|dually]] in terms of their [[commutative algebras]] $C^\infty(X)$ of [[smooth functions]] on them

\begin{displaymath}
C^\infty(-) \;\colon\; SmoothManifolds \overset{\phantom{AAA}}{\hookrightarrow} (\mathbb{R} Algebras)^{op}
  \,.
\end{displaymath}
But since there are of course more [[associative algebras|algebras]] $A \in \mathbb{R}Algebras$ than arise this way from smooth manifolds, we may turn this around and try to regard any algebra $A$ as \emph{defining} a would-be [[space]], which would have $A$ as its [[algebra of functions]].

For example an \emph{[[infinitesimally thickened point]]} should be a space which is ``so small'' that every smooth function $f$ on it which vanishes at the origin takes values so tiny that some finite power of them is not just even more tiny, but actually vanishes:

\begin{defn}
\label{InfinitesimallyThickendSmoothManifold}\hypertarget{InfinitesimallyThickendSmoothManifold}{}
\textbf{([[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]])}

An \emph{[[infinitesimally thickened point]]}

\begin{displaymath}
\mathbb{D} \coloneqq Spec(A)
\end{displaymath}
is represented by a [[commutative algebra]] $A \in \mathbb{R}Alg$ which as a [[real vector space]] is a [[direct sum]]

\begin{displaymath}
A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V
\end{displaymath}
of the 1-dimensional space $\langle 1 \rangle = \mathbb{R}$ of multiples of 1 with a [[finite dimensional vector space]] $V$ that is a [[nilpotent ideal]] in that for each element $a \in V$ there exists a [[natural number]] $n \in \mathbb{N}$ such that

\begin{displaymath}
a^{n+1} = 0
  \,.
\end{displaymath}
More generally, an [[infinitesimally thickened manifold|infinitesimally thickened Cartesian space]]

\begin{displaymath}
\mathbb{R}^n \times \mathbb{D} \;\coloneqq\;  \mathbb{R}^n \times Spec(A)
\end{displaymath}
is represented by a [[commutative algebra]]

\begin{displaymath}
C^\infty(\mathbb{R}^n) \otimes A \;\in\; \mathbb{R} Alg
\end{displaymath}
which is the [[tensor product of algebras]] of the algebra of smooth functions $C^\infty(\mathbb{R}^n)$ on an actual [[Cartesian space]] of some [[dimension]] $n$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), with an algebra of functions $A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V$ of an infinitesimally thickened point, as above.

We say that a \emph{smooth function between two [[infinitesimally thickened manifolds|infinitesimally thickened Cartesian spaces]]}

\begin{displaymath}
\mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)
\end{displaymath}
is by definition [[formal dual|dually]] an $\mathbb{R}$-algebra [[homomorphism]] of the form

\begin{displaymath}
C^\infty(\mathbb{R}^{n_1}) \otimes A_1
    \overset{f^\ast}{\longleftarrow}
  C^\infty(\mathbb{R}^{n_2}) \otimes A_2
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{InfinitesimalNeighbourhoodsInTheRealLine}\hypertarget{InfinitesimalNeighbourhoodsInTheRealLine}{}
\textbf{([[infinitesimal neighbourhoods]] in the [[real line]] )}

Consider the [[quotient ring|quotient algebra]] of the [[formal power series algebra]] $\mathbb{R}[ [\epsilon] ]$ in a single parameter $\epsilon$ by the ideal generated by $\epsilon^2$:

\begin{displaymath}
(\mathbb{R}[ [\epsilon] ])/(\epsilon^2)
  \;\simeq_{\mathbb{R}}\;
  \mathbb{R} \oplus \epsilon \mathbb{R}
  \,.
\end{displaymath}
(This is sometimes called the \emph{[[algebra of dual numbers]]}, for no good reason.) The underlying [[real vector space]] of this algebra is, as show, the [[direct sum]] of the multiples of 1 with the multiples of $\epsilon$. A general element in this algebra is of the form

\begin{displaymath}
a + b \epsilon \in (\mathbb{R}[\epsilon])/(\epsilon^2)
\end{displaymath}
where $a,b \in \mathbb{R}$ are [[real numbers]]. The product in this algebra is given by ``multiplying out'' as usual, and discarding all terms proportional to $\epsilon^2$:

\begin{displaymath}
\left(
    a_1 + b_1 \epsilon
  \right)
   \cdot
  \left(
    a_2 + b_2 \epsilon
  \right)
  \;=\;
  a_1 a_2 + ( a_1 b_2 + b_1 a_2 ) \epsilon
  \,.
\end{displaymath}
We may think of an element $a + b \epsilon$ as the truncation to first order of a [[Taylor series]] at the origin of a [[smooth function]] on the [[real line]]

\begin{displaymath}
f \;\colon\; \mathbb{R} \to \mathbb{R}
\end{displaymath}
where $a = f(0)$ is the value of the function at the origin, and where $b = \frac{\partial f}{\partial x}(0)$ is its first [[derivative]] at the origin.

Therefore this algebra behaves like the algebra of smooth function on an [[infinitesimal neighbourhood]] $\mathbb{D}^1$ of $0 \in \mathbb{R}$ which is so tiny that its [[generalized element|elements]] $\epsilon \in \mathbb{D}^1 \hookrightarrow \mathbb{R}$ become, upon squaring them, not just tinier, but actually zero:

\begin{displaymath}
\epsilon^2 = 0
  \,.
\end{displaymath}
This intuitive picture is now made precise by the concept of [[infinitesimally thickened points]] def. \ref{InfinitesimallyThickendSmoothManifold}, if we simply set

\begin{displaymath}
\mathbb{D}^1
    \;\coloneqq\;
  Spec\left(
    \mathbb{R}[ [\epsilon] ]/(\epsilon^2)
  \right)
\end{displaymath}
and observe that there is the [[monomorphism|inclusion]] of infinitesimally thickened Cartesian spaces

\begin{displaymath}
\mathbb{D}^1 \overset{\phantom{AA}i\phantom{AA} }{\hookrightarrow} \mathbb{R}^1
\end{displaymath}
which is dually given by the algebra homomorphism

\begin{displaymath}
\itexarray{
    \mathbb{R} \oplus \epsilon \mathbb{R}
      &\overset{i^\ast}{\longleftarrow}&
    C^\infty(\mathbb{R}^1)
    \\
    f(0) + \frac{\partial f}{\partial x}(0)  &\longleftarrow& \{f\}
  }
\end{displaymath}
which sends a [[smooth function]] to its value $f(0)$ at zero plus $\epsilon$ times its [[derivative]] at zero. Observe that this is indeed a [[homomorphism]] of algebras due to the [[product law]] of [[differentiation]], which says that

\begin{displaymath}
\begin{aligned}
    i^\ast(f \cdot g)
    & =
    (f \cdot g)(0) + \frac{\partial f \cdot g}{\partial x}(0) \epsilon
    \\
    & =
    f(0) \cdot g(0)
      +
    \left(
      \frac{\partial f}{\partial x}(0) \cdot g(0) +  f(0) \cdot \frac{\partial g}{\partial x}(0)
    \right) \epsilon
    \\
    & =
    \left(
      f(0) + \frac{\partial f}{\partial x}(0) \epsilon
    \right)
    \cdot
    \left(
      g(0) + \frac{\partial g}{\partial x}(0) \epsilon
    \right)
  \end{aligned}
\end{displaymath}
Hence we see that restricting a smooth function to the infinitesimal neighbourhood of a point is equivalent to restricting attention to its [[Taylor series]] to the given order at that point:

\begin{displaymath}
\itexarray{
    \mathbb{D}^1 &\overset{i}{\hookrightarrow}& \mathbb{R}^1
    \\
    & {}_{\mathllap{(\epsilon \mapsto f(0) + \frac{\partial f}{\partial x}(0) \epsilon) }}\searrow & \downarrow_{\mathrlap{f}}
    \\
    && \mathbb{R}^1
  }
\end{displaymath}
Similarly for each $k \in \mathbb{N}$ the algebra

\begin{displaymath}
(\mathbb{R}[ [ \epsilon ] ])/(\epsilon^{k+1})
\end{displaymath}
may be thought of as the algebra of [[Taylor series]] at the origin of $\mathbb{R}$ of [[smooth functions]] $\mathbb{R} \to \mathbb{R}$, where all terms of order higher than $k$ are discarded. The corresponding [[infinitesimally thickened point]] is often denoted

\begin{displaymath}
\mathbb{D}^1(k) \;\coloneqq\; Spec\left(  \left(\mathbb{R}[ [\epsilon] ]\right)/(\epsilon^{k+1}) \right)
  \,.
\end{displaymath}
This is now the [[subobject]] of the [[real line]]

\begin{displaymath}
\mathbb{D}^1(k) \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{R}^1
\end{displaymath}
on those elements $\epsilon$ such that $\epsilon^{k+1} = 0$.

\end{example}
(\href{synthetic+differential+geometry#Kock81}{Kock 81}, \href{synthetic+differential+geometry#Kock10}{Kock 10})

The following example \ref{UniquePointOfInfinitesimalLine} shows that infinitesimal thickening is invisible for ordinary spaces when mapping \emph{out} of these. In contrast example \ref{SyntheticTangentVectorFields} further below shows that the morphisms \emph{into} an ordinary space out of an infinitesimal space are interesting: these are [[tangent vectors]] and their higher order infinitesimal analogs.

\begin{example}
\label{UniquePointOfInfinitesimalLine}\hypertarget{UniquePointOfInfinitesimalLine}{}
\textbf{([[infinitesimal]] line $\mathbb{D}^1$ has unique [[global point]])}

For $\mathbb{R}^n$ any ordinary [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) and $D^1(k) \hookrightarrow \mathbb{R}^1$ the order-$k$ [[infinitesimal neighbourhood]] of the origin in the [[real line]] from example \ref{InfinitesimalNeighbourhoodsInTheRealLine}, there is exactly only one possible morphism of [[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] from $\mathbb{R}^n$ to $\mathbb{D}^1(k)$:

\begin{displaymath}
\itexarray{
    \mathbb{R}^n && \overset{\exists !}{\longrightarrow} &6 \mathbb{D}^1(k)
    \\
    & {}_{\mathllap{\exists !}}\searrow && \nearrow_{\mathrlap{\exists !}}
    \\
    && \mathbb{R}^0 = \ast
  }
  \,.
\end{displaymath}
\end{example}
\begin{proof}
By definition such a morphism is [[formal duality|dually]] an algebra homomorphism

\begin{displaymath}
C^\infty(\mathbb{R}^n)
    \overset{f^\ast}{\longleftarrow}
  \left(
    \mathbb{R}[ [\epsilon] ])/(\epsilon^{k+1}
  \right)
    \simeq_{\mathbb{R}}
  \mathbb{R} \oplus \mathcal{O}(\epsilon)
\end{displaymath}
from the higher order ``[[algebra of dual numbers]]'' to the [[algebra of functions|algebra of]] [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}).

Now this being an $\mathbb{R}$-algebra homomorphism, its action on the multiples $c \in \mathbb{R}$ of the identity is fixed:

\begin{displaymath}
f^\ast(1) = 1
  \,.
\end{displaymath}
All the remaining elements are proportional to $\epsilon$, and hence are nilpotent. However, by the [[homomorphism]] property of an algebra homomorphism it follows that it must send nilpotent elements $\epsilon$ to nilpotent elements $f(\epsilon)$, because

\begin{displaymath}
\begin{aligned}
    \left(f^\ast(\epsilon)\right)^{k+1}
      & = f^\ast\left( \epsilon^{k+1}\right)
    \\
     & = f^\ast(0)
     \\ & = 0
  \end{aligned}
\end{displaymath}
But the only nilpotent element in $C^\infty(\mathbb{R}^n)$ is the zero element, and hence it follows that

\begin{displaymath}
f^\ast(\epsilon) = 0
  \,.
\end{displaymath}
Thus $f^\ast$ as above is uniquely fixed.

\end{proof}
\begin{example}
\label{SyntheticTangentVectorFields}\hypertarget{SyntheticTangentVectorFields}{}
\textbf{([[synthetic differential geometry|synthetic]] [[tangent vector fields]])}

Let $\mathbb{R}^n$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), regarded as an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) and consider $\mathbb{D}^1 \coloneqq Spec( (\mathbb{R}[ [\epsilon] ])/(\epsilon^2) )$ the first order infinitesimal line from example \ref{InfinitesimalNeighbourhoodsInTheRealLine}.

Then homomorphisms of [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] of the form

\begin{displaymath}
\itexarray{
    \mathbb{R}^n \times \mathbb{D}^1
      && \overset{\tilde v}{\longrightarrow} &&
    \mathbb{R}^n
    \\
    & {}_{\mathllap{pr_1}}\searrow && \swarrow_{\mathrlap{id}}
    \\
    && \mathbb{R}^n
  }
\end{displaymath}
hence smoothly $X$-parameterized collections of morphisms

\begin{displaymath}
\tilde v_x \;\colon\; \mathbb{D}^1 \longrightarrow \mathbb{R}^n
\end{displaymath}
which send the unique base point $\Re(\mathbb{D}^1) = \ast$ (example \ref{UniquePointOfInfinitesimalLine}) to $x \in \mathbb{R}^n$, are in [[natural bijection]] with [[tangent vector fields]] $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ (example \ref{TangentVectorFields}).

\end{example}
\begin{proof}
By definition, the morphisms in question are [[formal duality|dually]] $\mathbb{R}$-[[associative algebra|algebra]] [[homomorphisms]] of the form

\begin{displaymath}
(C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n))
  \longleftarrow
  C^\infty(\mathbb{R}^n)
\end{displaymath}
which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to

\begin{displaymath}
f + (\partial f) \epsilon
\end{displaymath}
for some smooth function $(\partial f) \in C^\infty(\mathbb{R}^n)$. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all $f_1,f_2 \in C^\infty(\mathbb{R}^n)$ we have

\begin{displaymath}
(f_1 f_2 + (\partial (f_1 f_2))\epsilon )
  \;=\;
  (f_1 + (\partial f_1) \epsilon)
  \cdot
  (f_2 + (\partial f_2) \epsilon)
  \,.
\end{displaymath}
Multiplying this out and using that $\epsilon^2 = 0$, this is equivalent to

\begin{displaymath}
\partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2)
  \,.
\end{displaymath}
This in turn means equivalently that $\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ is a [[derivation]].

With this the statement follows with the third magic algebraic property of smooth functions (prop. \ref{AlgebraicFactsOfDifferentialGeometry}): [[derivations of smooth functions are vector fields]].

\end{proof}
We need to consider infinitesimally thickened spaces more general than the thickenings of just Cartesian spaces in def. \ref{InfinitesimallyThickendSmoothManifold}. But just as [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) serve as the local test geometries to induce the general concept of [[diffeological spaces]] and [[smooth sets]] (def. \ref{SmoothSet}), so using infinitesimally thickened Cartesian spaces as test geometries immediately induces the corresponding generalization of smooth sets with infinitesimals:

\begin{defn}
\label{FormalSmoothSet}\hypertarget{FormalSmoothSet}{}
\textbf{([[formal smooth set]])}

A \emph{[[formal smooth set]]} $X$ is

\begin{enumerate}%
\item for each [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] $\mathbb{R}^n \times Spec(A)$ (def. \ref{InfinitesimallyThickendSmoothManifold}) a [[set]]

\begin{displaymath}
X(\mathbb{R}^n \times Spec(A)) \in Set
\end{displaymath}
to be called the set of \emph{[[smooth functions]]} or \emph{plots} from $\mathbb{R}^n \times Spec(A)$ to $X$;


\item for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \times Spec(A_1) \longrightarrow \mathbb{R}^{n_2} \times Spec(A_2)$ between [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] a choice of function

\begin{displaymath}
f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))
\end{displaymath}
to be thought of as the precomposition operation

\begin{displaymath}
\left(
    \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
  \right)
  \;\overset{f^\ast}{\mapsto}\;
  \left(
    \mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{\Phi}{\to} X
  \right)
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item ([[functor|functoriality]])

\begin{enumerate}%
\item If $id_{\mathbb{R}^n \times Spec(A)} \;\colon\; \mathbb{R}^n \times Spec(A) \to \mathbb{R}^n \times Spec(A)$ is the [[identity function]] on $\mathbb{R}^n \times Spec(A)$, then $\left(id_{\mathbb{R}^n \times Spec(A)}\right)^\ast \;\colon\; X(\mathbb{R}^n \times Spec(A)) \to X(\mathbb{R}^n \times Spec(A))$ is the [[identity function]] on the set of plots $X(\mathbb{R}^n \times Spec(A))$;


\item If $\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{g}{\to} \mathbb{R}^{n_3} \times Spec(A_3)$ are two [[composition|composable]] [[smooth functions]] between infinitesimally thickened Cartesian spaces, then pullback of plots along them consecutively equals the pullback along the [[composition]]:

\begin{displaymath}
f^\ast \circ g^\ast = (g \circ f)^\ast
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
 && X(\mathbb{R}^{n_2} \times Spec(A_2))
 \\
 & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
 \\
 X(\mathbb{R}^{n_1} \times Spec(A_1))
 && \underset{ (g \circ f)^\ast }{\longleftarrow} &&
 X(\mathbb{R}^{n_3} \times Spec(A_3))
  }
\end{displaymath}


\end{enumerate}

\item ([[sheaf|gluing]])

If $\{ U_i \times Spec(A) \overset{f_i \times id_{Spec(A)}}{\to} \mathbb{R}^n \times Spec(A)\}_{i \in I}$ is such that

\begin{displaymath}
\{ U_i  \overset{f_i  }{\to} \mathbb{R}^n \}_{i \in I}
\end{displaymath}
in a [[differentiably good open cover]] (def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts $\mathbb{R}^n \times Spec(A)$-plots of $X$ to a set of $U_i \times Spec(A)$-plots

\begin{displaymath}
X(\mathbb{R}^n \times Spec(A))
    \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
  \underset{i \in I}{\prod} X(U_i \times Spec(A))
\end{displaymath}
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are ``[[matching families]]'' in that they agree on [[intersections]]:

\begin{displaymath}
\phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
    && (U_i \cap U_j) \times Spec(A)
    \\
    & \swarrow && \searrow
    \\
    U_i\times Spec(A) && && U_j \times Spec(A)
    \\
    & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
    \\
    && X
  }
\end{displaymath}


\end{enumerate}
Finally, given $X_1$ and $X_2$ two [[formal smooth sets]], then a [[smooth function]] between them

\begin{displaymath}
f \;\colon\; X_1 \longrightarrow X_2
\end{displaymath}
is

\begin{itemize}%
\item for each [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] $\mathbb{R}^n \times Spec(A)$ (def. \ref{InfinitesimallyThickendSmoothManifold}) a function

\begin{displaymath}
f_\ast(\mathbb{R}^n \times Spec(A))
  \;\colon\;
  X_1(\mathbb{R}^n \times Spec(A)) \longrightarrow X_2(\mathbb{R}^n \times Spec(A))
\end{displaymath}


\end{itemize}
such that

\begin{itemize}%
\item for each [[smooth function]] $g \colon \mathbb{R}^{n_1} \times Spec(A_1) \to \mathbb{R}^{n_2} \times Spec(A_2)$ between infinitesimally thickened Cartesian spaces we have

\begin{displaymath}
g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2))
  =
  f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
    X_1(\mathbb{R}^{n_2} \times Spec(A_2))
      &\overset{f_\ast(\mathbb{R}^{n_2}\times Spec(A_2) )}{\longrightarrow}&
    X_2(\mathbb{R}^{n_2} \times Spec(A_2))
    \\
    \mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2}
    \\
    X_1(\mathbb{R}^{n_1} \times Spec(A_1))
      &\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}&
    X_2(\mathbb{R}^{n_1} \times Spec(A_1))
  }
\end{displaymath}


\end{itemize}
\end{defn}
(\href{Cahiers+topos#Dubuc79}{Dubuc 79})

Basing [[synthetic differential geometry|infinitesimal geometry]] on [[formal smooth sets]] is an instance of the general approach to [[geometry]] called \emph{[[functorial geometry]]} or \emph{[[topos theory]]}. For more background on this see at \emph{[[geometry of physics -- manifolds and orbifolds]]}.

We have the evident generalization of example \ref{SmoothManifoldsAreDiffeologicalSpaces} to smooth geometry with [[infinitesimals]]:

\begin{example}
\label{YonedaLemmaForFormalSmoothSets}\hypertarget{YonedaLemmaForFormalSmoothSets}{}
\textbf{([[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] are [[formal smooth sets]])}

For $X$ an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}), it becomes a [[formal smooth set]] according to def. \ref{FormalSmoothSet} by taking its plots out of some $\mathbb{R}^n \times \mathbb{D}$ to be the homomorphism of infinitesimally thickened Cartesian spaces:

\begin{displaymath}
X(\mathbb{R}^n \times \mathbb{D})
  \;\coloneqq\;
  Hom_{FormalCartSp}( \mathbb{R}^n \times \mathbb{D}, X )
  \,.
\end{displaymath}
(Stated more [[category theory|abstractly]], this is an instance of the \emph{[[Yoneda embedding]]} over a \emph{[[subcanonical site]]}.)

\end{example}
\begin{example}
\label{FormalSmoothSetsIncludedSmoothSet}\hypertarget{FormalSmoothSetsIncludedSmoothSet}{}
\textbf{([[smooth sets]] are [[formal smooth sets]])}

Let $X$ be a [[smooth set]] (def. \ref{SmoothSet}). Then $X$ becomes a [[formal smooth set]] (def. \ref{FormalSmoothSet}) by declaring the set of plots $X(\mathbb{R}^n \times \mathbb{D})$ over an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) to be [[equivalence classes]] of [[pairs]]

\begin{displaymath}
\mathbb{R}^n \times \mathbb{D} \longrightarrow \mathbb{R}^{k}
  \,,
  \phantom{AA}
  \mathbb{R}^k \longrightarrow X
\end{displaymath}
of a [[morphism]] of infinitesimally thickened Cartesian spaces and of a plot of $X$, as shown, subject to the [[equivalence relation]] which identifies two such pairs if there exists a smooth function $f \colon \mathbb{R}^k \to \mathbb{R}^{k'}$ such that

\begin{displaymath}
\itexarray{
    && \mathbb{R}^n \times \mathbb{D}
    \\
    & \swarrow && \searrow
    \\
    \mathbb{R}^k && \overset{f}{\longrightarrow} && \mathbb{R}^{k'}
    \\
    \mathbb{R}^k && \underset{f}{\longrightarrow} && \mathbb{R}^{k'}
    \\
    & \searrow && \swarrow
    \\
    && X
  }
\end{displaymath}
Stated more [[category theory|abstractly]] this says that $X$ as a [[formal smooth set]] is the \emph{[[left Kan extension]]} (see \href{Kan+extension#CoendFormulaForPresheavesOfSets}{this example}) of $X$ as a [[smooth set]] along the [[functor]] that [[full subcategory|includes]] [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) into [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] (def. \ref{InfinitesimallyThickendSmoothManifold}).

\end{example}
\begin{defn}
\label{ReductionAndInfinitesimalShape}\hypertarget{ReductionAndInfinitesimalShape}{}
\textbf{([[reduction modality|reduction]] and [[infinitesimal shape]])}

For $\mathbb{R}^n \times \mathbb{D}$ an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) we say that the underlying ordinary [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is its \emph{[[reduced object|reduction]]}

\begin{displaymath}
\Re\left(
   \mathbb{R}^n \times \mathbb{D}
 \right)
 \;\coloneqq\;
 \mathbb{R}^n
 \,.
\end{displaymath}
There is the canonical inclusion morphism

\begin{displaymath}
\Re\left(
    \mathbb{R}^n \times \mathbb{D}
  \right)
  =
  \mathbb{R}^n
  \overset{\phantom{AAAA}}{\hookrightarrow}
  \mathbb{R}^n \times \mathbb{D}
\end{displaymath}
which [[formal dual|dually]] corresponds to the [[homomorphism]] of [[commutative algebras]]

\begin{displaymath}
C^\infty(\mathbb{R}^n)
    \longleftarrow
  C^\infty(\mathbb{R}^n)
    \otimes_{\mathbb{R}}
  A
\end{displaymath}
which is the identity on all smooth functions $f \in C^\infty(\mathbb{R}^n)$ and is zero on all elements $a \in V \subset A$ in the nilpotent ideal of $A$ (as in example \ref{UniquePointOfInfinitesimalLine}).

Given any [[formal smooth set]] $X$, we say that its \emph{[[infinitesimal shape]]} or \emph{[[de Rham shape]]} (also: \emph{[[de Rham stack]]}) is the [[formal smooth set]] $\Im X$ (def. \ref{FormalSmoothSet}) defined to have as plots the [[reduction modality|reductions]] of the plots of $X$, according to the above:

\begin{displaymath}
(\Im X)( U ) \;\coloneqq\: X(\Re(U))
  \,.
\end{displaymath}
There is a canonical morphism of formal smooth set

\begin{displaymath}
\eta_X
    \;\colon\;
  X
    \longrightarrow
  \Im X
\end{displaymath}
which takes a plot

\begin{displaymath}
U = \mathbb{R}^n \times \mathbb{D} \overset{f}{\longrightarrow} X
\end{displaymath}
to the [[composition]]

\begin{displaymath}
\mathbb{R}^n \hookrightarrow \mathbb{R}^n \times \mathbb{D} \overset{f}{\hookrightarrow} X
\end{displaymath}
regarded as a plot of $\Im X$.

\end{defn}
\begin{example}
\label{MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}\hypertarget{MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}{}
\textbf{([[mapping space]] out of an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]])}

Let $X$ be an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) and let $Y$ be a [[formal smooth set]] (def. \ref{FormalSmoothSet}). Then the \emph{[[mapping space]]}

\begin{displaymath}
[X,Y] \;\in\; FormalSmoothSet
\end{displaymath}
of smooth functions from $X$ to $Y$ is the [[formal smooth set]] whose $U$-plots are the morphisms of [[formal smooth sets]] from the [[Cartesian product]] of [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] $U \times X$ to $Y$, hence the $U \times X$-plots of $Y$:

\begin{displaymath}
[X,Y](U) \;\coloneqq\; Y(U \times X)
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{TangentBundleSynthetic}\hypertarget{TangentBundleSynthetic}{}
\textbf{([[synthetic differential geometry|synthetic]] [[tangent bundle]])}

Let $X \coloneqq \mathbb{R}^n$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) regarded as an [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (\ref{InfinitesimallyThickendSmoothManifold}) and thus regarded as a [[formal smooth set]] (def. \ref{FormalSmoothSet}) by example \ref{YonedaLemmaForFormalSmoothSets}. Consider the infinitesimal line

\begin{displaymath}
\mathbb{D}^1
    \hookrightarrow
  \mathbb{R}^1
\end{displaymath}
from example \ref{InfinitesimalNeighbourhoodsInTheRealLine}. Then the [[mapping space]] $[\mathbb{D}^1, X]$ (example \ref{MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}) is the total space of the [[tangent bundle]] $T X$ (example \ref{TangentVectorFields}). Moreover, under restriction along the [[reduced object|reduction]] $\ast \longrightarrow \mathbb{D}^1$, this is the full [[tangent bundle]] [[projection]], in that there is a [[natural isomorphism]] of [[formal smooth sets]] of the form

\begin{displaymath}
\itexarray{
    T X &\simeq& [\mathbb{D}^1, X]
    \\
    {}^{\mathllap{tb}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{D}^1, X ]   }}
    \\
    X &\simeq& [\ast, X]
  }
\end{displaymath}
In particular this implies immediately that smooth [[sections]] (def. \ref{Sections}) of the tangent bundle

\begin{displaymath}
\itexarray{
    && [\mathbb{D}^1, X] & \simeq T X
    \\
    & {}^{\mathllap{v}}\nearrow & \downarrow
    \\
    X &=& X
  }
\end{displaymath}
are equivalently morphisms of the form

\begin{displaymath}
\itexarray{
    && X
    \\
    & {}^{\mathllap{\tilde v}}\nearrow & \downarrow^{\mathrlap{id}}
    \\
    X \times \mathbb{D}^1 &\underset{pr_1}{\longrightarrow}& X
  }
\end{displaymath}
which we had already identified with [[tangent vector fields]] (def. \ref{TangentVectorFields}) in example \ref{SyntheticTangentVectorFields}.

\end{example}
\begin{proof}
This follows by an analogous argument as in example \ref{SyntheticTangentVectorFields}, using the [[Hadamard lemma]].

\end{proof}
While in [[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] (def. \ref{InfinitesimallyThickendSmoothManifold}) only [[infinitesimals]] to any [[finite number|finite]] order may exist, in [[formal smooth sets]] (def. \ref{FormalSmoothSet}) we may find infinitesimals to any arbitrary finite order:

\begin{example}
\label{InfinitesimalNeighbourhood}\hypertarget{InfinitesimalNeighbourhood}{}
\textbf{([[infinitesimal neighbourhood]])}

Let $X$ be a [[formal smooth sets]] (def. \ref{FormalSmoothSet}) $Y \hookrightarrow X$ a sub-formal smooth set. Then the \emph{[[infinitesimal neighbourhood]]} to arbitrary infinitesimal order of $Y$ in $X$ is the [[formal smooth set]] $N_X Y$ whose plots are those plots of $X$

\begin{displaymath}
\mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X
\end{displaymath}
such that their [[reduced object|reduction]] (def. \ref{ReductionAndInfinitesimalShape})

\begin{displaymath}
\mathbb{R}^n \hookrightarrow \mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X
\end{displaymath}
factors through a plot of $Y$.

\end{example}
This allows to grasp the restriction of [[field histories]] to the [[infinitesimal neighbourhood]] of a [[submanifold]] of [[spacetime]], which will be crucial for the discussion of [[phase spaces]] \hyperlink{PhaseSpace}{below}.

\begin{defn}
\label{FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}\hypertarget{FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}{}
\textbf{([[field histories]] on [[infinitesimal neighbourhood]] of [[submanifold]] of [[spacetime]])}

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and let $S \hookrightarrow \Sigma$ be a [[submanifold]] of [[spacetime]].

We write $N_\Sigma(S) \hookrightarrow \Sigma$ for its [[infinitesimal neighbourhood]] in $\Sigma$ (def. \ref{InfinitesimalNeighbourhood}).

Then the \emph{set of field histories restricted to $S$}, to be denoted

\begin{equation}
\Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}
\label{SpaceOfFieldHistoriesInHigherCodimension}\end{equation}
is the set of section of $E$ restricted to the [[infinitesimal neighbourhood]] $N_\Sigma(S)$ (example \ref{InfinitesimalNeighbourhood}).

\end{defn}
$\,$

We close the discussion of [[synthetic differential geometry|infinitesimal differential geometry]] by explaining how we may recover the concept of \emph{[[smooth manifolds]]} inside the more general [[formal smooth sets]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below). The key point is that the presence of [[infinitesimals]] in the theory allows an intrinsic definition of [[local diffeomorphisms]]/[[formally étale morphism]] (def. \ref{FormalSmoothSetLocalDiffeomorphism} and example \ref{AbstractLocalDiffeomorphismsOfCartesianSpaces} below). It is noteworthy that the only role this concept plays in the development of [[field theory]] below is that [[smooth manifolds]] admit [[triangulations]] by smooth [[singular simplices]] (def. \ref{SingularSimplicesInCartesianSpaces}) so that the concept of [[fiber integration|fiber]] [[integration of differential forms]] is well defined over manifolds.

\begin{defn}
\label{FormalSmoothSetLocalDiffeomorphism}\hypertarget{FormalSmoothSetLocalDiffeomorphism}{}
\textbf{([[local diffeomorphism]] of [[formal smooth sets]])}

Let $X,Y$ be [[formal smooth sets]] (def. \ref{FormalSmoothSet}). Then a [[morphism]] between them is called a \emph{[[local diffeomorphism]]} or \emph{[[formally étale morphism]]}, denoted

\begin{displaymath}
f \;\colon\; X \overset{et}{\longrightarrow} Y
  \,,
\end{displaymath}
if $f$ if for each [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]] (def. \ref{InfinitesimallyThickendSmoothManifold}) $\mathbb{R}^n \times \mathbb{D}$ we have a natural identification between the $\mathbb{R}^n \times \mathbb{D}$-plots of $X$ with those $\mathbb{R}^n n\times \mathbb{D}$-plots of $Y$ whose [[reduction modality|reduction]] (def. \ref{ReductionAndInfinitesimalShape}) comes from an $\mathbb{R}^n$-plot of $X$, hence if we have a [[natural transformation|natural]] [[fiber product]] of [[sets]] of plots

\begin{displaymath}
X(\mathbb{R}^n \times \mathbb{D})
  \;\simeq\;
  Y(\mathbb{R}^n \times \mathbb{D})
    \underset{Y(\mathbb{R}^n)}{\times^f}
  X(\mathbb{R}^n)
\end{displaymath}
i. e.

\begin{displaymath}
\itexarray{
     && X(\mathbb{R}^n \times \mathbb{D})
     \\
     & \swarrow && \searrow
     \\
     Y(\mathbb{R}^n \times \mathbb{D}) && \text{(pb)} && X(\mathbb{R}^n)
     \\
     & \searrow && \swarrow
     \\
     && Y(\mathbb{R}^n )
  }
\end{displaymath}
for all [[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian spaces]] $\mathbb{R}^n \times \mathbb{D}$.

Stated more [[category theory|abstractly]], this means that the [[naturality square]] of the [[unit of a monad|unit]] of the [[infinitesimal shape]] $\Im$ (def. \ref{ReductionAndInfinitesimalShape}) is a [[pullback square]]

\begin{displaymath}
\itexarray{
    X &\overset{\eta_X}{\longrightarrow}& \Im X
    \\
    {}^{\mathllap{f}}\downarrow &\text{(pb)}& \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\underset{\eta_Y}{\longrightarrow}& \Im Y
  }
\end{displaymath}
\end{defn}
(\href{local+diffeomorphism#KhavkineSchreiber17}{Khavkine-Schreiber 17, def. 3.1})

\begin{example}
\label{AbstractLocalDiffeomorphismsOfCartesianSpaces}\hypertarget{AbstractLocalDiffeomorphismsOfCartesianSpaces}{}
\textbf{([[local diffeomorphism]] between [[Cartesian spaces]] from the general definition)}

For $X,Y \in CartSp$ two ordinary [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), regarded as [[formal smooth sets]] by example \ref{YonedaLemmaForFormalSmoothSets} then a [[morphism]] $f \colon X \to Y$ between them is a [[local diffeomorphism]] in the general sense of def. \ref{FormalSmoothSetLocalDiffeomorphism} precisely if it is so in the ordinary sense of def. \ref{LocalDiffeomorphismBetweenCartesianSpaces}.

\end{example}
(\href{geometry+of+physics+--+manifolds+and+orbifolds#KhavkineSchreiber17}{Khavkine-Schreiber 17, prop. 3.2})

\begin{defn}
\label{SmoothManifoldInsideDiffeologicalSpaces}\hypertarget{SmoothManifoldInsideDiffeologicalSpaces}{}
\textbf{([[smooth manifolds]])}

A \emph{[[smooth manifold]]} $X$ of [[dimension]] $n \in \mathbb{N}$ is

\begin{itemize}%
\item a [[diffeological space]] (def. \ref{DiffeologicalSpace})

\end{itemize}
such that

\begin{enumerate}%
\item there exists an [[indexed set]] $\{ \mathbb{R}^n \overset{\phi_i}{\to} X\}_{i \in I}$ of morphisms of [[formal smooth sets]] (def. \ref{FormalSmoothSet}) from [[Cartesian spaces]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) (regarded as [[formal smooth sets]] via example \ref{SmoothManifoldsAreDiffeologicalSpaces}, example \ref{SmoothSetsDiffeologicalSpaces} and example \ref{FormalSmoothSetsIncludedSmoothSet}) into $X$, (regarded as a [[formal smooth set]] via example \ref{SmoothSetsDiffeologicalSpaces} and example \ref{FormalSmoothSetsIncludedSmoothSet}) such that

\begin{enumerate}%
\item every point $x \in X_s$ is in the [[image]] of at least one of the $\phi_i$;


\item every $\phi_i$ is a [[local diffeomorphism]] according to def. \ref{FormalSmoothSetLocalDiffeomorphism};



\end{enumerate}

\item the [[final topology]] induced by the set of plots of $X$ makes $X_s$ a [[paracompact Hausdorff space]].



\end{enumerate}
\end{defn}
(\href{geometry+of+physics+--+manifolds+and+orbifolds#KhavkineSchreiber17}{Khavkine-Schreiber 17, example 3.4})

For more on [[smooth manifolds]] from the perspective of [[formal smooth sets]] see at \emph{[[geometry of physics -- manifolds and orbifolds]]}.

$\,$

\textbf{[[fermion fields]] and [[supergeometry]]}

Field theories of interest crucially involve [[fermionic fields]] (def. \ref{FermionicBosonicFields} below), such as the [[Dirac field]] (example \ref{DiracFieldBundle} below), which are subject to the ``[[Pauli exclusion principle]]'', a key reason for the [[stability of matter]]. Mathematically this principle means that these [[field (physics)|fields]] have [[field bundles]] whose [[field fiber]] is not an ordinary [[manifold]], but an odd-graded \emph{[[supermanifold]]} (more on this in remark \ref{LagrangianDensityOfDiracFieldSupergeometricNature} and remark \ref{SupergeometricNatureOfDiracEquation} below).

This ``[[supergeometry]]'' is an immediate generalization of the [[synthetic differential geometry|infinitesimal geometry]] \hyperlink{InfinitesimalGeometry}{above}, where now the [[infinitesimal]] elements in the [[algebra of functions]] may be equipped with a [[graded object|grading]]: one speaks of \emph{[[superalgebra]]}.

The ``super''-terminology for something as down-to-earth as the mathematical principle behind the [[stability of matter]] may seem unfortunate. For better or worse, this terminology has become standard since the middle of the 20th century. But the concept that today is called \emph{[[supercommutative superalgebra]]} was in fact first considered by [[Ausdehnungslehre|Grassmann 1844]] who got it right (``[[Ausdehnungslehre]]'') but apparently was too far ahead of his time and remained unappreciated.

Beware that considering [[supergeometry]] does \emph{not} necessarily involve considering ``[[supersymmetry]]''. Supergeometry is the geometry of [[fermion fields]] (def. \ref{FermionicBosonicFields} below), and that all [[matter]] fields in the [[observable universe]] are fermionic has been [[experiment|experimentally]] established since the [[Stern-Gerlach experiment]] in 1922. Supersymmetry, on the other hand, is a hypothetical extension of [[spacetime]]-[[symmetry]] within the context of [[supergeometry]]. Here we do not discuss supersymmetry; for details see instead at \emph{[[geometry of physics -- supersymmetry]]}.

\begin{defn}
\label{SupercommutativeSuperalgebra}\hypertarget{SupercommutativeSuperalgebra}{}
\textbf{([[supercommutative superalgebra]])}

A \emph{[[real number|real]] $\mathbb{Z}/2$-[[graded algebra]]} or \emph{[[superalgebra]]} is an [[associative algebra]] $A$ over the [[real numbers]] together with a [[direct sum]] decomposition of its underlying [[real vector space]]

\begin{displaymath}
A \simeq_{\mathbb{R}} A_{even} \oplus A_{odd}
  \,,
\end{displaymath}
such that the product in the algebra respects the multiplication in the [[cyclic group|cyclic]] [[group of order 2]] $\mathbb{Z}/2 = \{even, odd\}$:

\begin{displaymath}
\left.
  \itexarray{
    A_{even} \cdot A_{even}
    \\
    A_{odd} \cdot A_{odd}
  }
  \right\}
  \subset A_{even}
  \phantom{AAAA}
  \left.
  \itexarray{
    A_{odd} \cdot A_{even}
    \\
    A_{even} \cdot A_{odd}
  }
  \right\}
  \subset A_{odd}
  \,.
\end{displaymath}
This is called a \emph{[[supercommutative superalgebra]]} if for all elements $a_1, a_2 \in A$ which are of homogeneous degree ${\vert a_i \vert} \in \mathbb{Z}/2 = \{even, odd\}$ in that

\begin{displaymath}
a_i \in A_{{\vert a_i\vert}} \subset A
\end{displaymath}
we have

\begin{displaymath}
a_1 \cdot a_2 = (-1)^{{\vert a_1 \vert \vert a_2 \vert}} a_2 \cdot a_1
  \,.
\end{displaymath}
A \emph{[[homomorphism]] of [[superalgebras]]}

\begin{displaymath}
f \;\colon\; A \longrightarrow A'
\end{displaymath}
is a [[homomorphism]] of [[associative algebras]] over the [[real numbers]] such that the $\mathbb{Z}/2$-[[graded object|grading]] is respected in that

\begin{displaymath}
f(A_{even}) \subset A'_{even}
  \phantom{AAAAA}
  f(A_{odd}) \subset A'_{odd}
  \,.
\end{displaymath}
\end{defn}
For more details on superalgebra see at \emph{[[geometry of physics -- superalgebra]]}.

\begin{example}
\label{GrassmannAlgebra}\hypertarget{GrassmannAlgebra}{}
\textbf{(basic examples of [[supercommutative superalgebras]])}

Basic examples of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra}) include the following:

\begin{enumerate}%
\item Every [[commutative algebra]] $A$ becomes a [[supercommutative superalgebra]] by declaring it to be all in even degree: $A = A_{even}$.


\item For $V$ a [[finite dimensional vector space|finite dimensional]] [[real vector space]], then the [[Grassmann algebra]] $A \coloneqq \wedge^\bullet_{\mathbb{R}} V^\ast$ is a supercommutative superalgebra with $A_{even} \coloneqq \wedge^{even} V^\ast$ and $A_{odd} \coloneqq \wedge^{odd} V^\ast$.

More explicitly, if $V = \mathbb{R}^s$ is a [[Cartesian space]] with canonical dual [[coordinates]] $(\theta^i)_{i = 1}^s$ then the [[Grassmann algebra]] $\wedge^\bullet (\mathbb{R}^s)^\ast$ is the real algebra which is [[generators and relations|generated]] from the $\theta^i$ regarded in odd degree and hence subject to the relation

\begin{displaymath}
\theta^i \cdot \theta^j = - \theta^j \cdot \theta^i
  \,.
\end{displaymath}
In particular this implies that all the $\theta^i$ are [[infinitesimal]] (def. \ref{InfinitesimallyThickendSmoothManifold}):

\begin{displaymath}
\theta^i \cdot \theta^i = 0
  \,.
\end{displaymath}

\item For $A_1$ and $A_2$ two [[supercommutative superalgebras]], there is their \emph{[[tensor product of algebras|tensor product]]} supercommutative superalgebra $A_1 \otimes_{\mathbb{R}} A_2$. For example for $X$ a [[smooth manifold]] with ordinary algebra of smooth functions $C^\infty(X)$ regarded as a supercommutative superalgebra by the first example above, the tensor product with a [[Grassmann algebra]] (second example above) is the supercommutative superalgebta

\begin{displaymath}
C^\infty(X) \otimes_{\mathbb{R}} \wedge^\bullet((\mathbb{R}^s)\ast)
\end{displaymath}
whose elements may uniquely be expanded in the form

\begin{displaymath}
f + f_i \theta^i + f_{i j} \theta^i \theta^j + f_{i j k} \theta^i \theta^j \theta^k + \cdots + f_{i_1 \cdots i_s} \theta^{i_1} \cdots \theta^{i_s}
  \,,
\end{displaymath}
where the $f_{i_1 \cdots i_k} \in C^\infty(X)$ are smooth functions on $X$ which are skew-symmetric in their indices.



\end{enumerate}
\end{example}
The following is the direct super-algebraic analog of the definition of [[infinitesimally thickened smooth manifolds|infinitesimally thickened Cartesian spaces]] (def. \ref{InfinitesimallyThickendSmoothManifold}):

\begin{defn}
\label{SuperCartesianSpace}\hypertarget{SuperCartesianSpace}{}
\textbf{([[super Cartesian space]])}

A \emph{[[superpoint]]} $Spec(A)$ is represented by a [[super-commutative superalgebra]] $A$ (def. \ref{SupercommutativeSuperalgebra}) which as a $\mathbb{Z}/2$-[[graded vector space]] ([[super vector space]]) is a [[direct sum]]

\begin{displaymath}
A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V
\end{displaymath}
of the 1-dimensional even vector space $\langle 1 \rangle = \mathbb{R}$ of multiples of 1, with a [[finite dimensional vector space|finite dimensional]] [[super vector space]] $V$ that is a [[nilpotent ideal]] in $A$ in that for each element $a \in V$ there exists a [[natural number]] $n \in \mathbb{N}$ such that

\begin{displaymath}
a^{n+1} = 0
  \,.
\end{displaymath}
More generally, a [[super Cartesian space]] $\mathbb{R}^n \times Spec(A)$ is represented by a [[super-commutative algebra]] $C^\infty(\mathbb{R}^n) \otimes A \in \mathbb{R} Alg$ which is the [[tensor product of algebras]] of the algebra of smooth functions $C^\infty(\mathbb{R}^n)$ on an actual [[Cartesian space]] of some [[dimension]] $n$, with an algebra of functions $A \simeq_{\mathbb{R}} \langle 1\rangle \oplus V$ of a [[superpoint]] (example \ref{GrassmannAlgebra}).

Specifically, for $s \in \mathbb{N}$, there is the superpoint

\begin{equation}
\mathbb{R}^{0 \vert s}
    \;\coloneqq\;
  Spec\left(  \wedge^\bullet (\mathbb{R}^s)^\ast \right)
\label{StandardSuperpoints}\end{equation}
whose [[algebra of functions]] is by definition the [[Grassmann algebra]] on $s$ generators $(\theta^i)_{i = 1}^s$ in odd degree (example \ref{GrassmannAlgebra}).

We write

\begin{displaymath}
\begin{aligned}
    \mathbb{R}^{b\vert s}
      & \coloneqq
    \mathbb{R}^b \times \mathbb{R}^{0 \vert s}
    \\
    & =
    \mathbb{R}^b \times Spec( \wedge^\bullet(\mathbb{R}^s)^\ast )
    \\
     & =
     Spec\left(  C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet (\mathbb{R}^s)^\ast  \right)
  \end{aligned}
\end{displaymath}
for the corresponding super Cartesian spaces whose algebra of functions is as in example \ref{GrassmannAlgebra}.

We say that a \emph{smooth function} between two [[super Cartesian spaces]]

\begin{displaymath}
\mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)
\end{displaymath}
is by definition [[formal dual|dually]] a [[homomorphism]] of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra}) of the form

\begin{displaymath}
C^\infty(\mathbb{R}^{n_1}) \otimes A_1
    \overset{f^\ast}{\longleftarrow}
  C^\infty(\mathbb{R}^{n_2}) \otimes A_2
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{SuperpointInducedByFiniteDimensionalVectorSpace}\hypertarget{SuperpointInducedByFiniteDimensionalVectorSpace}{}
\textbf{([[superpoint]] induced by a [[finite dimensional vector space]])}

Let $V$ be a [[finite dimensional vector space|finite dimensional]] [[real vector space]]. With $V^\ast$ denoting its [[dual vector space]] write $\wedge^\bullet V^\ast$ for the [[Grassmann algebra]] that it generates. This being a [[supercommutative algebra]], it defines a [[superpoint]] (def. \ref{SuperCartesianSpace}).

We denote this superpoint by

\begin{displaymath}
V_{odd} \simeq \mathbb{R}^{0 \vert dim(V)}
  \,.
\end{displaymath}
\end{example}
All the [[differential geometry]] over [[Cartesian space]] that we discussed \hyperlink{Geometry}{above} generalizes immediately to [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}) if we stricly adhere to the [[signs in supergeometry|super sign rule]] which says that whenever two odd-graded elements swap places, a minus sign is picked up. In particular we have the following generalization of the [[de Rham complex]] on [[Cartesian spaces]] discussed \hyperlink{DifferentialFormsAndCartanCalculus}{above}.

\begin{defn}
\label{DifferentialFormOnSuperCartesianSpaces}\hypertarget{DifferentialFormOnSuperCartesianSpaces}{}
\textbf{([[super differential forms]] on [[super Cartesian spaces]])}

For $\mathbb{R}^{b\vert s}$ a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}), hence the [[formal dual]] of the [[supercommutative superalgebra]] of the form

\begin{displaymath}
C^\infty(\mathbb{R}^{b\vert s})
  \;=\;
  C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet \mathbb{R}^s
\end{displaymath}
with canonical even-graded [[coordinate functions]] $(x^i)_{i = 1^b}$ and odd-graded coordinate functions $(\theta^j)_{j = 1}^s$.

Then the \emph{[[de Rham complex]] of [[super differential forms]] on $\mathbb{R}^{b\vert s}$} is, in super-generalization of def. \ref{DifferentialnForms}, the $\mathbb{Z} \times (\mathbb{Z}/2)$-[[graded commutative algebra]]

\begin{displaymath}
\Omega^\bullet(\mathbb{R}^{b|s})
  \;\coloneqq\;
  C^\infty(\mathbb{R}^{b|s})
    \otimes_{\mathbb{R}}
   \wedge^\bullet \langle
      d x^1, \cdots, d x^b,
      \;
      d \theta^1, \cdots, d\theta^s
   \rangle
\end{displaymath}
which is generated over $C^\infty(\mathbb{R}^{b\vert s})$ from new generators

\begin{displaymath}
\underset{
    \text{deg} = (1,even)
  }{\underbrace{ d x^i }}
  \phantom{AAAAA}
  \underset{
    \text{deg} = (1,odd)
  }{
  \underbrace{
    d \theta^j
  }
  }
\end{displaymath}
whose [[differential]] is defined in degree-0 by

\begin{displaymath}
d f
    \;\coloneqq\;
  \frac{\partial f}{\partial x^i} d x^i
    +
  \frac{\partial f}{\partial \theta^j} d \theta^j
\end{displaymath}
and extended from there as a bigraded [[derivation]] of bi-degree $(1,even)$, in super-generalization of def. \ref{deRhamDifferential}.

Accordingly, the operation of contraction with [[tangent vector fields]] (def. \ref{ContractionOfFormsWithVectorFields}) has bi-degree $(-1,\sigma)$ if the tangent vector has super-degree $\sigma$:

\begin{tabular}{l|l}
generator&bi-degree\\
\hline 
$\phantom{AA} x^a$&(0,even)\\
$\phantom{AA} \theta^\alpha$&(0,odd)\\
$\phantom{AA} dx^a$&(1,even)\\
$\phantom{AA} d\theta^\alpha$&(1,odd)\\
\end{tabular}

\begin{tabular}{l|l}
derivation&bi-degree\\
\hline 
$\phantom{AA} d$&(1,even)\\
$\phantom{AA}\iota_{\partial x^a}$&(-1, even)\\
$\phantom{AA}\iota_{\partial \theta^\alpha}$&(-1,odd)\\
\end{tabular}

This means that if $\alpha \in \Omega^\bullet(\mathbb{R}^{b\vert s})$ is in bidegree $(n_\alpha, \sigma_\alpha)$, and $\beta \in \Omega^\bullet(\mathbb{R}^{b \vert \sigma})$ is in bidegree $(n_\beta, \sigma_\beta)$, then

\begin{displaymath}
\alpha \wedge \beta
  \;
  =
  \;
  (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta}
  \;
  \beta \wedge \alpha
  \,.
\end{displaymath}
Hence there are \emph{two} contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

\begin{enumerate}%
\item there is a ``cohomological sign'' which for commuting an $n_1$-forms past an $n_2$-form is $(-1)^{n_1 n_2}$;


\item in addition there is a ``super-grading'' sign which for commuting a $\sigma_1$-graded coordinate function past a $\sigma_2$-graded coordinate function (possibly under the de Rham differential) is $(-1)^{\sigma_1 \sigma_2}$.



\end{enumerate}
For example:

\begin{displaymath}
x^{a_1} (dx^{a_2}) \;=\; + (dx^{a_2}) x^{a_1}
\end{displaymath}
\begin{displaymath}
\theta^\alpha (dx^a) \;=\; + (dx^a) \theta^\alpha
\end{displaymath}
\begin{displaymath}
\theta^{\alpha_1} (d\theta^{\alpha_2})
   \;=\;
  - (d\theta^{\alpha_2}) \theta^{\alpha_1}
\end{displaymath}
\begin{displaymath}
dx^{a_1}
  \wedge
  d x^{a_2}
  \;=\;
  -
  d x^{a_2}
  \wedge
  d x^{a_1}
\end{displaymath}
\begin{displaymath}
dx^a
  \wedge
  d \theta^{\alpha}
  \;=\;
  -
  d\theta^{\alpha}
  \wedge
  d x^a
\end{displaymath}
\begin{displaymath}
d\theta^{\alpha_1}
  \wedge
  d \theta^{\alpha_2}
  \;=\;
  +
  d\theta^{\alpha_2}
  \wedge
  d \theta^{\alpha_1}
\end{displaymath}
\end{defn}
(e.g. \href{signs+in+supergeometry#CastellaniDAuriaFre91}{Castellani-D'Auria-Fr\'e{} 91 (II.2.106) and (II.2.109)}, \href{signs+in+supergeometry#DeligneFreed99}{Deligne-Freed 99, section 6})

Beware that there is also \emph{another} sign rule for [[super differential forms]] used in the literature. See at \emph{[[signs in supergeometry]]} for further discussion.

$\,$

It is clear now by direct analogy with the definition of [[formal smooth sets]] (def. \ref{FormalSmoothSet}) what the corresponding [[supergeometry|supergeometric]] generalization is. For definiteness we spell it out yet once more:

\begin{defn}
\label{SuperFormalSmoothSet}\hypertarget{SuperFormalSmoothSet}{}
\textbf{([[super formal smooth set|super smooth set]])}

A \emph{[[super formal smooth set|super smooth set]]} $X$ is

\begin{enumerate}%
\item for each [[super Cartesian space]] $\mathbb{R}^n \times Spec(A)$ (def. \ref{SuperCartesianSpace}) a [[set]]

\begin{displaymath}
X(\mathbb{R}^n \times Spec(A)) \in Set
\end{displaymath}
to be called the set of \emph{[[smooth functions]]} or \emph{plots} from $\mathbb{R}^n \times Spec(A)$ to $X$;


\item for each [[smooth function]] $f \;\colon\; \mathbb{R}^{n_1} \times Spec(A_1) \longrightarrow \mathbb{R}^{n_2} \times Spec(A_2)$ between [[super Cartesian spaces]] a choice of function

\begin{displaymath}
f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))
\end{displaymath}
to be thought of as the precomposition operation

\begin{displaymath}
\left(
    \mathbb{R}^{n_2} \overset{\Phi}{\longrightarrow} X
  \right)
  \;\overset{f^\ast}{\mapsto}\;
  \left(
    \mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{\Phi}{\to} X
  \right)
\end{displaymath}


\end{enumerate}
such that

\begin{enumerate}%
\item ([[functor|functoriality]])

\begin{enumerate}%
\item If $id_{\mathbb{R}^n \times Spec(A)} \;\colon\; \mathbb{R}^n \times Spec(A) \to \mathbb{R}^n \times Spec(A)$ is the [[identity function]] on $\mathbb{R}^n \times Spec(A)$, then $\left(id_{\mathbb{R}^n \times Spec(A)}\right)^\ast \;\colon\; X(\mathbb{R}^n \times Spec(A)) \to X(\mathbb{R}^n \times Spec(A))$ is the [[identity function]] on the set of plots $X(\mathbb{R}^n \times Spec(A))$.


\item If $\mathbb{R}^{n_1}\times Spec(A_1) \overset{f}{\to} \mathbb{R}^{n_2} \times Spec(A_2) \overset{g}{\to} \mathbb{R}^{n_3} \times Spec(A_3)$ are two [[composition|composable]] [[smooth functions]] between infinitesimally thickened Cartesian spaces, then pullback of plots along them consecutively equals the pullback along the [[composition]]:

\begin{displaymath}
f^\ast \circ g^\ast
  =
  (g \circ f)^\ast
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
 && X(\mathbb{R}^{n_2} \times Spec(A_2))
 \\
 & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
 \\
 X(\mathbb{R}^{n_1} \times Spec(A_1))
 && \underset{ (g \circ f)^\ast }{\longleftarrow} &&
 X(\mathbb{R}^{n_3} \times Spec(A_3))
  }
\end{displaymath}


\end{enumerate}

\item ([[sheaf|gluing]])

If $\{ U_i \times Spec(A) \overset{f_i \times id_{Spec(A)}}{\to} \mathbb{R}^n \times Spec(A)\}_{i \in I}$ is such that

\begin{displaymath}
\{ U_i  \overset{f_i  }{\to} \mathbb{R}^n \}_{i \in I}
\end{displaymath}
is a [[differentiably good open cover]] (def. \ref{DifferentiablyGoodOpenCover}) then the function which restricts $\mathbb{R}^n \times Spec(A)$-plots of $X$ to a set of $U_i \times Spec(A)$-plots

\begin{displaymath}
X(\mathbb{R}^n \times Spec(A))
    \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
  \underset{i \in I}{\prod} X(U_i \times Spec(A))
\end{displaymath}
is a [[bijection]] onto the set of those [[tuples]] $(\Phi_i \in X(U_i))_{i \in I}$ of plots, which are ``[[matching families]]'' in that they agree on [[intersections]]:

\begin{displaymath}
\phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
    && (U_i \cap U_j) \times Spec(A)
    \\
    & \swarrow && \searrow
    \\
    U_i\times Spec(A) && && U_j \times Spec(A)
    \\
    & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
    \\
    && X
  }
\end{displaymath}


\end{enumerate}
Finally, given $X_1$ and $X_2$ two [[super formal smooth sets]], then a [[smooth function]] between them

\begin{displaymath}
f \;\colon\; X_1 \longrightarrow X_2
\end{displaymath}
is

\begin{itemize}%
\item for each [[super Cartesian space]] $\mathbb{R}^n \times Spec(A)$ a function

\begin{displaymath}
f_\ast(\mathbb{R}^n \times Spec(A))
  \;\colon\;
  X_1(\mathbb{R}^n \times Spec(A)) \longrightarrow X_2(\mathbb{R}^n \times Spec(A))
\end{displaymath}


\end{itemize}
such that

\begin{itemize}%
\item for each [[smooth function]] $g \colon \mathbb{R}^{n_1} \times Spec(A_1) \to \mathbb{R}^{n_2} \times Spec(A_2)$ between super Cartesian spaces we have

\begin{displaymath}
g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2))
  =
  f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1
\end{displaymath}
i.e.

\begin{displaymath}
\itexarray{
    X_1(\mathbb{R}^{n_2} \times Spec(A_2))
      &\overset{f_\ast(\mathbb{R}^{n_2}\times Spec(A_2) )}{\longrightarrow}&
    X_2(\mathbb{R}^{n_2} \times Spec(A_2))
    \\
    \mathllap{g_1^\ast}\downarrow && \downarrow\mathrlap{g^\ast_2}
    \\
    X_1(\mathbb{R}^{n_1} \times Spec(A_1))
      &\underset{f_\ast(\mathbb{R}^{n_1})}{\longrightarrow}&
    X_2(\mathbb{R}^{n_1} \times Spec(A_1))
  }
\end{displaymath}


\end{itemize}
\end{defn}
(\href{synthetic+differential+supergeometry#Yetter88}{Yetter 88})

Basing [[supergeometry]] on [[super formal smooth sets]] is an instance of the general approach to [[geometry]] called \emph{[[functorial geometry]]} or \emph{[[topos theory]]}. For more background on this see at \emph{[[geometry of physics -- supergeometry]]}.

In direct generalization of example \ref{SmoothManifoldsAreDiffeologicalSpaces} we have:

\begin{example}
\label{SuperSmoothSetSuperCartesianSpaces}\hypertarget{SuperSmoothSetSuperCartesianSpaces}{}
\textbf{([[super Cartesian spaces]] are [[super formal smooth sets|super smooth sets]])}

Let $X$ be a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) Then it becomes a [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet}) by declaring its plots $\Phi \in X(\mathbb{R}^n \times \mathbb{D})$ to the algebra homomorphisms $C^\infty(\mathbb{R}^n \times \mathbb{D}) \leftarrow C^\infty(\mathbb{R}^{b\vert s})$.

Under this identification, morphisms between [[super Cartesian spaces]] are in [[natural bijection]] with their morphisms regarded as [[super formal smooth set|super smooth sets]].

Stated more [[category theory|abstractly]], this statement is an example of the \emph{[[Yoneda embedding]]} over a \emph{[[subcanonical site]]}.

\end{example}
Similarly, in direct generalization of prop. \ref{CartSpYpnedaLemma} we have:

\begin{prop}
\label{SuperCartSpYpnedaLemma}\hypertarget{SuperCartSpYpnedaLemma}{}
\textbf{(plots of a [[super formal smooth set|super smooth set]] really are the [[smooth functions]] into the smooth smooth set)}

Let $X$ be a [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet}). For $\mathbb{R}^n \times \mathbb{D}$ any [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) there is a [[natural transformation|natural]] [[function]]

\begin{displaymath}
Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\simeq}{\longrightarrow} X(\mathbb{R}^n)
\end{displaymath}
from the set of homomorphisms of super smooth sets from $\mathbb{R}^n \times \mathbb{D}$ (regarded as a super smooth set via example \ref{SuperSmoothSetSuperCartesianSpaces}) to $X$, to the set of plots of $X$ over $\mathbb{R}^n \times \mathbb{D}$, given by evaluating on the [[identity function|identity]] plot $id_{\mathbb{R}^n \times \mathbb{D}}$.

This function is a \emph{[[bijection]]}.

This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the \emph{would-be} smooth functions into $X$, end up being the \emph{actual} smooth functions into $X$.

\end{prop}
\begin{proof}
This is the statement of the \emph{[[Yoneda lemma]]} over the [[site]] of [[super Cartesian spaces]].

\end{proof}
We do not need to consider here [[supermanifolds]] more general than the [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}). But for those readers familiar with the concept we include the following direct analog of the characterization of [[smooth manifolds]] according to def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces}:

\begin{defn}
\label{SuperSmoothManifolds}\hypertarget{SuperSmoothManifolds}{}
\textbf{([[supermanifolds]])}

A \emph{[[supermanifold]]} $X$ of [[dimension]] super-dimension $(b,s) \in \mathbb{N} \times \mathbb{N}$ is

\begin{itemize}%
\item a [[super smooth set]] (def. \ref{SuperFormalSmoothSet})

\end{itemize}
such that

\begin{enumerate}%
\item there exists an [[indexed set]] $\{ \mathbb{R}^{b\vert s} \overset{\phi_i}{\to} X\}_{i \in I}$ of morphisms of [[super formal smooth sets|super smooth sets]] (def. \ref{SuperFormalSmoothSet}) from [[super Cartesian spaces]] $\mathbb{R}^{b\vert s}$ (def. \ref{SuperCartesianSpace}) (regarded as [[super formal smooth set|super smooth sets]] via example \ref{SuperSmoothSetSuperCartesianSpaces} into $X$, such that

\begin{enumerate}%
\item for every plot $\mathbb{R}^n \times \mathbb{D} \to X$ there is a [[differentiably good open cover]] (def. \ref{DifferentiablyGoodOpenCover}) restricted to which the plot factors through the $\mathbb{R}^{b\vert s}_i$;


\item every $\phi_i$ is a [[local diffeomorphism]] according to def. \ref{FormalSmoothSetLocalDiffeomorphism}, now with respect not just to [[infinitesimally thickened points]], but with respect to [[superpoints]];



\end{enumerate}

\item the [[bosonic modality|bosonic]] part of $X$ is a [[smooth manifold]] according to def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces}.



\end{enumerate}
\end{defn}
Finally we have the evident generalization of the smooth moduli space $\mathbf{\Omega}^\bullet$ of [[differential forms]] from example \ref{UniversalSmoothModuliSpaceOfDifferentialForms} to [[supergeometry]]

\begin{example}
\label{ModuliOfSDifferentialForms}\hypertarget{ModuliOfSDifferentialForms}{}
\textbf{(universal [[smooth set|smooth]] [[moduli spaces]] of [[super differential forms]])}

For $n \in \mathbf{M}$ write

\begin{displaymath}
\mathbf{\Omega}^n \;\in\; SuperSmoothSet
\end{displaymath}
for the [[super smooth set]] (def. \ref{SuperSmoothSetSuperCartesianSpaces}) whose set of plots on a [[super Cartesian space]] $U \in SuperCartSp$ (def. \ref{SuperCartesianSpace}) is the set of [[super differential forms]] (def. \ref{DifferentialFormOnSuperCartesianSpaces}) of cohomolgical degree $n$

\begin{displaymath}
\mathbf{\Omega}^n(U) \;\coloneqq\; \Omega^n(U)
\end{displaymath}
and whose maps of plots is given by [[pullback of differential forms|pullback]] of super differential forms.

The [[de Rham differential]] on [[super differential forms]] applied plot-wise yields a morpism of super smooth sets

\begin{equation}
d \;\colon\; \mathbf{\Omega}^n \longrightarrow \mathbf{\Omega}^{n+1}
  \,.
\label{SuperUniversalDeRhamDifferential}\end{equation}
As before in def. \ref{DifferentialFormsOnDiffeologicalSpaces} we then define for any [[super smooth set]] $X \in SuperSmoothSet$ its set of differential $n$-forms to be

\begin{displaymath}
\Omega^n(X)
   \;\coloneqq\;
  Hom_{SuperSmoothSet}(X,\mathbf{\Omega}^n)
\end{displaymath}
and we define the [[de Rham differential]] on these to be given by postcomposition with \eqref{SuperUniversalDeRhamDifferential}.

\end{example}
$\,$

\begin{defn}
\label{FermionicBosonicFields}\hypertarget{FermionicBosonicFields}{}
\textbf{([[bosonic fields]] and [[fermionic fields]])}

For $\Sigma$ a [[spacetime]], such as [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) if a [[fiber bundle]] $E \overset{fb}{\longrightarrow} \Sigma$ with total space a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) (or more generally a [[supermanifold]], def./prop. \ref{SuperSmoothManifolds}) is regarded as a super-[[field bundle]] (def. \ref{FieldsAndFieldBundles}), then

\begin{itemize}%
\item the even-graded [[sections]] are called the \emph{[[bosonic field|bosonic]]} [[field histories]];


\item the odd-graded [[sections]] are called the \emph{[[fermionic field|fermionic]]} [[field histories]].



\end{itemize}
In components, if $E = \Sigma \times F$ is a [[trivial bundle]] with [[fiber]] a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) with even-graded [[coordinates]] $(\phi^a)$ and odd-graded [[coordinates]] $(\psi^A)$, then the $\phi^a$ are called the \emph{[[bosonic field|bosonic]]} field coordinates, and the $\psi^A$ are called the \emph{[[fermionic field|fermionic]]} field coordinates.

\end{defn}
What is crucial for the discussion of [[field theory]] is the following immediate [[supergeometry|supergeometric]] analog of the smooth structure on the [[space of field histories]] from example \ref{DiffeologicalSpaceOfFieldHistories}:

\begin{example}
\label{SupergeometricSpaceOfFieldHistories}\hypertarget{SupergeometricSpaceOfFieldHistories}{}
\textbf{([[super smooth set|supergeometric]] [[space of field histories]])}

Let $E \overset{fb}{\to} \Sigma$ be a super-[[field bundle]] (def. \ref{FieldsAndFieldBundles}, def. \ref{FermionicBosonicFields}).

Then the \emph{[[space of sections]]}, hence the \emph{[[space of field histories]]}, is the [[super formal smooth set]] (def. \ref{SuperFormalSmoothSet})

\begin{displaymath}
\Gamma_\Sigma(E) \in SuperSmoothSet
\end{displaymath}
whose plots $\Phi_{(-)}$ for a given [[Cartesian space]] $\mathbb{R}^n$ and [[superpoint]] $\mathbb{D}$ (def. \ref{SuperCartesianSpace}) with the [[Cartesian products]] $U \coloneqq \mathbb{R}^n \times \mathbb{D}$ and $U \times \Sigma$ regarded as [[super formal smooth set|super smooth sets]] according to example \ref{SuperSmoothSetSuperCartesianSpaces} are defined to be the [[morphisms]] of [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet})

\begin{displaymath}
\itexarray{
    U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
  }
\end{displaymath}
which make the following [[commuting diagram|diagram commute]]:

\begin{displaymath}
\itexarray{
    && E
    \\
    & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
  }
  \,.
\end{displaymath}
Explicitly, if $\Sigma$ is a [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and $E = \Sigma \times F$ a [[trivial bundle|trivial]] [[field bundle]] with [[field fiber]] a [[super vector space]] (example \ref{TrivialVectorBundleAsAFieldBundle}, example \ref{FermionicBosonicFields}) this means [[formal duality|dually]] that a plot $\Phi_{(-)}$ of the super smooth set of field histories is a [[homomorphism]] of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra})

\begin{displaymath}
\itexarray{
    C^\infty(U \times \Sigma) &\overset{\left(\Phi_{(-)}(-)\right)^\ast}{\longleftarrow}& C^\infty(E)
  }
\end{displaymath}
which make the following [[commuting diagram|diagram commute]]:

\begin{displaymath}
\itexarray{
    && C^\infty(E)
    \\
    & {}^{\mathllap{\left( \Phi_{(-)}(-) \right)^\ast }}\nearrow & \uparrow^{\mathrlap{fb^\ast}}
    \\
    C^\infty(U \times \Sigma) &\underset{pr_2^\ast}{\longleftarrow}& C^\infty(\Sigma)
  }
  \,.
\end{displaymath}
\end{example}
We will focus on discussing the [[supergeometry|supergeometric]] [[space of field histories]] (example \ref{SupergeometricSpaceOfFieldHistories}) of the \emph{[[Dirac field]]} (def. \ref{DiracFieldBundle} below). This we consider below in example \ref{DiracFieldBundle}; but first we discuss now some relevant basics of general [[supergeometry]].

Example \ref{SupergeometricSpaceOfFieldHistories} is really a special case of a general relative [[mapping space]]-construction as in example \ref{MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}. This immediately generalizes also to the [[supergeometry|supergeometric]] context.

\begin{defn}
\label{MappingSpaceOutOfASuperCartesianSpace}\hypertarget{MappingSpaceOutOfASuperCartesianSpace}{}
\textbf{([[supergeometry|super]]-[[mapping space]] out of a [[super Cartesian space]])}

Let $X$ be a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) and let $Y$ be a [[super formal smooth sets|super smooth set]] (def. \ref{SuperFormalSmoothSet}). Then the \emph{[[mapping space]]}

\begin{displaymath}
[X,Y] \;\in\; SuperSmoothSet
\end{displaymath}
of super smooth functions from $X$ to $Y$ is the [[super formal smooth set]] whose $U$-plots are the morphisms of [[super formal smooth set|super smooth set]] from the [[Cartesian product]] of [[super Cartesian space]] $U \times X$ to $Y$, hence the $U \times X$-plots of $Y$:

\begin{displaymath}
[X,Y](U) \;\coloneqq\; Y(U \times X)
  \,.
\end{displaymath}
\end{defn}
In direct generalization of the [[synthetic differential geometry|synthetic]] [[tangent bundle]] construction (example \ref{TangentBundleSynthetic}) to supergeometry we have

\begin{defn}
\label{TangentBundleOdd}\hypertarget{TangentBundleOdd}{}
\textbf{([[odd tangent bundle]])}

Let $X$ be a [[super formal smooth set|super smooth set]] (def. \ref{SuperFormalSmoothSet}) and $\mathbb{R}^{0 \vert 1}$ the [[superpoint]] \eqref{StandardSuperpoints} then the [[supergeometry|supergeometry]]-[[mapping space]]

\begin{displaymath}
\itexarray{
    T_{odd} X & \coloneqq& [\mathbb{R}^{0\vert 1}, X]
    \\
    {}^{\mathllap{tb_{odd}}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{R}^{0 \vert 1}, X ] }}
    \\
    X & = & X
  }
\end{displaymath}
is called the \emph{[[odd tangent bundle]]} of $X$.

\end{defn}
\begin{example}
\label{SuperpointsMapping}\hypertarget{SuperpointsMapping}{}
\textbf{([[mapping space]] of [[superpoints]])}

Let $V$ be a [[finite dimensional vector space|finite dimensional]] [[real vector space]] and consider its corresponding [[superpoint]] $V_{odd}$ from exampe \ref{SuperpointInducedByFiniteDimensionalVectorSpace}. Then the [[mapping space]] (def. \ref{MappingSpaceOutOfASuperCartesianSpace}) out of the [[superpoint]] $\mathbb{R}^{0\vert 1}$ (def. \ref{SuperCartesianSpace}) into $V_{odd}$ is the [[Cartesian product]] $V_{odd} \times V$

\begin{displaymath}
[\mathbb{R}^{0\vert 1}, V_{odd}]
  \;\simeq\;
  V_{odd} \times V
  \,.
\end{displaymath}
By def. \ref{TangentBundleOdd} this says that $V_{odd} \times V$ is the ``[[odd tangent bundle]]'' of $V_{odd}$.

\end{example}
\begin{proof}
Let $U$ be any [[super Cartesian space]]. Then by definition we have the following sequence of [[natural bijections]] of sets of plots

\begin{displaymath}
\begin{aligned}
    \left[
      \mathbb{R}^{0\vert 1}, V_{odd}
    \right](U)
    & =
    Hom_{SuperSmoothSet}( \mathbb{R}^{0\vert 1} \times U, V_{odd} )
    \\
    &
    \simeq
    Hom_{\mathbb{R}sAlg}( \wedge^\bullet(V^\ast)\,,\, C^\infty(U)[\theta]/(\theta^2) )
    \\
    &
    \simeq
    Hom_{\mathbb{R}Vect}( V^\ast \,,\, (C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta\rangle  )
    \\
    & \simeq
    Hom_{\mathbb{R}Vect}( V^\ast\,,\, C^\infty(U)_{odd} ) \,\times\, Hom_{\mathbb{R}Vect}( V^\ast, C^\infty(U)_{even} )
    \\
    & \simeq
    V_{odd}(U) \times V(U)
    \\
    & \simeq
    (V_{odd} \times V)(U)
  \end{aligned}
\end{displaymath}
Here in the third line we used that the [[Grassmann algebra]] $\wedge^\bullet V^\ast$ is [[free construction|free]] on its generators in $V^\ast$, meaning that a homomorphism of [[supercommutative superalgebras]] out of the Grassmann algebra is uniquely fixed by the underlying degree-preserving [[linear function]] on these generators. Since in a [[Grassmann algebra]] all the generators are in odd degree, this is equivalently a linear map from $V^\ast$ to the odd-graded [[real vector space]] underlying $C^\infty(U)[\theta](\theta^2)$, which is the [[direct sum]] $C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta \rangle$.

Then in the fourth line we used that [[finite coproduct|finite]] [[direct sums]] are [[Cartesian products]], so that linear maps into a direct sum are [[pairs]] of linear maps into the direct summands.

That all these [[bijections]] are [[natural bijection|natural]] means that they are compatible with morphisms $U \to U'$ and therefore this says that $[\mathbb{R}^{0\vert 1}, V_{odd}]$ and $V_{odd} \times V$ are the same as seen by super-smooth plots, hence that they are [[isomorphism|isomorphic]] as [[super formal smooth set|super smooth sets]].

\end{proof}
With this [[supergeometry]] in hand we finally turn to defining the [[Dirac field]] species:

\begin{example}
\label{DiracFieldBundle}\hypertarget{DiracFieldBundle}{}
\textbf{([[field bundle]] for [[Dirac field]])}

For $\Sigma$ being [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}), of dimension $2+1$, $3+1$, $5+1$ or $9+1$, let $S$ be the [[spin representation]] from prop. \ref{SpinorRepsByNormedDivisionAlgebra}, whose underlying [[real vector space]] is

\begin{displaymath}
S \;=\;
  \left\{
    \itexarray{
      \mathbb{R}^2 \oplus \mathbb{R}^2 & \vert & p + 1 = 2+1
      \\
      \mathbb{C}^2 \oplus \mathbb{C}^2  &\vert& p + 1 = 3 + 1
      \\
      \mathbb{H}^2 \oplus \mathbb{H}^2 &\vert& p + 1 = 5 + 1
      \\
      \mathbb{O}^2  \oplus \mathbb{O}^2 &\vert& p + 1 = 9 + 1
    }
  \right.
\end{displaymath}
With

\begin{displaymath}
S_{odd} \simeq \mathbb{R}^{0 \vert dim(S)}
\end{displaymath}
the corresponding [[superpoint]] (example \ref{SuperpointInducedByFiniteDimensionalVectorSpace}), then the [[field bundle]] for the \emph{[[Dirac field]]} on $\Sigma$ is

\begin{displaymath}
E \;\coloneqq\; \Sigma \times S_{odd} \overset{pr_1}{\to} \Sigma
  \,,
\end{displaymath}
hence the [[field fiber]] is the [[superpoint]] $S_{odd}$. This is the corresponding [[spinor bundle]] on [[Minkowski spacetime]], with fiber in odd super-degree.

The traditional two-component [[spinor]] basis from remark \ref{TwoComponentSpinorNotation} provides [[fermionic field]] coordinates (def. \ref{FermionicBosonicFields}) on the [[field fiber]] $S_{odd}$:

\begin{displaymath}
\left(
   \psi^A
  \right)_{A = 1}^4
   \;=\;
  \left(
     (\chi_a), (\xi^{\dagger \dot a})
  \right)_{a,\dot a = 1,2}
  \,.
\end{displaymath}
Notice that these are $\mathbb{K}$-valued odd functions: For instance if $\mathbb{K} = \mathbb{C}$ then each $\chi_a$ in turn has two components, a [[real part]] and an [[imaginary part]].

A key point with the [[field bundle]] of the [[Dirac field]] (example \ref{DiracFieldBundle}) is that the field fiber coordinates $(\psi^A)$ or $\left((\chi_a), (\xi^{\dagger \dot a})\right)$ are now odd-graded elements in the function algebra on the field fiber, which is the [[Grassmann algebra]] $C^\infty(S_{odd}) =  \wedge^\bullet(S^\ast)$. Therefore they anti-commute with each other:

\begin{equation}
\psi^\alpha  \psi^{\beta} = - \psi^{\beta} \psi^\alpha
  \,.
\label{DiracFieldCoordinatesAnticommute}\end{equation}


\begin{quote}%
snippet grabbed from (\href{Dirac+field#DermisekI8}{Dermisek 09})


\end{quote}
\end{example}
We analyze the special nature of the [[supergeometry|supergeometry]] [[space of field histories]] of the [[Dirac field]] a little (prop. \ref{DiracSpaceOfFieldHistories}) below and conclude by highlighting the crucial role of [[supergeometry]] (remark \ref{DiracFieldSupergeometric} below)

\begin{defn}
\label{DiracSpaceOfFieldHistories}\hypertarget{DiracSpaceOfFieldHistories}{}
\textbf{([[space of field histories]] of the [[Dirac field]])}

Let $E = \Sigma \times S_{odd} \overset{pr_1}{\to}  \Sigma$ be the super-[[field bundle]] (def. \ref{FermionicBosonicFields}) for the [[Dirac field]] over [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1}$ from example \ref{DiracFieldBundle}.

Then the corresponding [[supergeometry|supergeometric]] [[space of field histories]]

\begin{displaymath}
\Gamma_\Sigma(\Sigma \times S_{odd})
  \;\in\;
  SuperSmoothSet
\end{displaymath}
from example \ref{SupergeometricSpaceOfFieldHistories} has the following properties:

\begin{enumerate}%
\item For $U = \mathbb{R}^n$ an ordinary [[Cartesian space]] (with no super-geometric thickening, def. \ref{SuperCartesianSpace}) there is only a single $U$-parameterized collection of [[field histories]], hence a single plot

\begin{displaymath}
\Psi_{(-)}\;\colon\;\mathbb{R}^n \overset{ 0 }{\longrightarrow} \Gamma_\Sigma(\Sigma \times S_{odd})
\end{displaymath}
and this corresponds to the [[zero section]], hence to the trivial [[Dirac field]]

\begin{displaymath}
\Psi^A_{(-)} = 0
  \,.
\end{displaymath}

\item For $U = \mathbb{R}^{n \vert 1}$ a [[super Cartesian space]] (\ref{SuperCartesianSpace}) with a single super-odd dimension, then $U$-parameterized collections of field histories

\begin{displaymath}
\Phi_{(-)} \;\colon\; \mathbb{R}^{n\vert 1} \longrightarrow \Gamma_\Sigma(\Sigma \times S_{odd})
\end{displaymath}
are in [[natural bijection]] with plots of sections of the [[bosonic field|bosonic]]-field bundle with field fiber $S_{even} = S$ the [[spin representation]] regarded as an ordinary vector space:

\begin{displaymath}
\Psi_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \Gamma_\Sigma(\Sigma \times S_{even})
\,.
\end{displaymath}


\end{enumerate}
Moreover, these two kinds of plots determine the fermionic field space completely: It is in fact [[isomorphism|isomorphic]], as a [[super vector space]], to the bosonic field space shifted to odd degree (as in example \ref{SuperpointInducedByFiniteDimensionalVectorSpace}):

\begin{displaymath}
\Gamma_\Sigma(\Sigma \times S_{odd})
    \;\simeq\;
  \left(
    \Gamma_\Sigma(E\times S_{even})
  \right)_{odd}
  \,.
\end{displaymath}
\end{defn}
\begin{proof}
In the first case, the plot is a morphism of [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}) of the form

\begin{displaymath}
\mathbb{R}^n \times \mathbb{R}^{p,1} \longrightarrow S_{odd}
  \,.
\end{displaymath}
By definitions this is [[formal duality|dually]] homomorphism of real [[supercommutative superalgebras]]

\begin{displaymath}
C^\infty(\mathbb{R}^n \times \mathbb{R}^{p,1}) 
    \longleftarrow 
  \wedge^\bullet S^\ast
\end{displaymath}
from the [[Grassmann algebra]] on the [[dual vector space]] of the [[spin representation]] $S$ to the ordinary algebras of [[smooth functions]] on $\mathbb{R}^n \times \mathbb{R}^{p,1}$. But the latter has no elements in odd degree, and hence all the Grassmann generators need to be send to zero.

For the second case, notice that a morphism of the form

\begin{displaymath}
\mathbb{R}^{n\vert 1}
    \overset{\Phi_{(-)}}{\longrightarrow}
  S_{odd}
\end{displaymath}
is by def. \ref{TangentBundleOdd} [[natural bijection|naturally identified]] with a morphism of the form

\begin{displaymath}
\mathbb{R}^n 
    \overset{\Psi_{(-)}}{\longrightarrow} 
  [\mathbb{R}^{0 \vert 1}, S_{odd}] 
    \simeq 
  S_{odd} \times S_{even}
  \,,
\end{displaymath}
where the identification on the right is from example \ref{SuperpointsMapping}.

By the [[universal property|nature]] of [[Cartesian products]] these morphisms in turn are [[natural bijection|naturally identified]] with [[pairs]] of morphisms of the form

\begin{displaymath}
\left(
    \itexarray{
      \mathbb{R}^n &\overset{}{\longrightarrow}& S_{odd}\,,
      \\
      \mathbb{R}^n &\overset{}{\longrightarrow}& S_{even}
    }
  \right)
  \,.
\end{displaymath}
Now, as in the first point above, here the first component is uniquely fixed to be the [[zero morphism]] $\mathbb{R}^n \overset{0}{\to} S_{odd}$; and hence only the second component is free to choose. This is precisely the claim to be shown.

\end{proof}
\begin{remark}
\label{DiracFieldSupergeometric}\hypertarget{DiracFieldSupergeometric}{}
\textbf{([[supergeometry|supergeometric]] nature of the [[Dirac field]])}

Proposition \ref{DiracSpaceOfFieldHistories} how two basic facts about the [[Dirac field]], which may superficially seem to be in tension with each other, are properly unified by [[supergeometry]]:

\begin{enumerate}%
\item On the one hand a [[field history]] $\Psi$ of the [[Dirac field]] is \emph{not} an ordinary section of an ordinary [[vector bundle]]. In particular its component functions $\psi^A$ anti-commute with each other, which is not the case for ordinary functions, and this is crucial for the [[Lagrangian density]] of the Dirac field to be well defined, we come to this below in example \ref{LagrangianDensityForDiracField}.


\item On the other hand a [[field history]] of the [[Dirac field]] is supposed to be a [[spinor]], hence a [[section]] of a [[spinor bundle]], which is an ordinary [[vector bundle]].



\end{enumerate}
Therefore prop. \ref{DiracSpaceOfFieldHistories} serves to shows how, even though a Dirac field is not defined to be an ordinary section of an ordinary vector bundle, it is nevertheless encoded by such an ordinary section: One says that this ordinary section is a ``[[superfield]]-component'' of the Dirac field, the one linear in a Grassmann variable $\theta$.

\end{remark}
$\,$

This concludes our discussion of the concept of \emph{[[field (physics)|fields]]} itself. In the \hyperlink{FieldVariations}{following chapter} we consider the [[variational calculus]] of fields.



\end{document}