A first idea of quantum field theory -- Fields

## Fields
 {#Fields}

In this chapter we discuss these topics:

* _[Field bundles](#FieldBundles)_

* _[Spaces of field histories](#NonFiniteDimensionalGeometry)_

* _[Infinitesimal geometry](#InfinitesimalGeometry)_

* _[Fermion fields and Supergeometry](#Supergeometry)_

$\,$

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 _[[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:

$$
  \Phi
    \;\colon\;
   \Sigma
     \longrightarrow
   F
  \,.
$$

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]]

$$
  E \;\coloneqq\; \Sigma \times F
$$

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

$$
  \array{
     E &\coloneqq& \Sigma \times F
     \\
     {}^{\mathllap{fb}}\downarrow & \swarrow_{\mathrlap{pr_1}}
     \\
     \Sigma
  }
  \,.
$$

This is then called the _[[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 _[[field fiber]]_. (See also at _[[fiber bundles in physics]]_.)

Given a [[field bundle]] $E \overset{fb}{\to}\Sigma$, then a _[[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 _[[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 _[[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]] [below](#Observables), because these are smooth functions on the [[space of field histories]].
In particular it is this smooth structure which allows to derive that _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:

$\,$

**aspects of the concept of [[field (physics)|fields]]**

| aspect | [[term]] | [[type]] |  description | def. |
|--|------------|-|---------|----|
| [[field bundle|field component]] | $\phi^a$, $\phi^a_{,\mu}$ |  $J^\infty_\Sigma(E) \to \mathbb{R}$ |[[coordinate function]] on [[jet bundle]] of [[field bundle]] | def. \ref{FieldsAndFieldBundles}, def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} |
| [[field history]] | $\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$ | $\Sigma \to J^\infty_\Sigma(E)$  | [[jet prolongation]] of [[section]] of [[field bundle]] | def. \ref{FieldsAndFieldBundles}, def. \ref{JetProlongation}  |
| [[observable|field observable]] | $\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x),   $ | $\Gamma_{\Sigma}(E) \to \mathbb{R}$  | [[derivative of a distribution|derivatives]] of [[delta-distribution|delta]]-[[functional]] on [[space of sections]] | def. \ref{Observable}, example \ref{PointEvaluationObservables} |
| 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} |

$\,$

**[[field bundles]]**
 {#FieldBundles}

+-- {: .num_defn #FieldsAndFieldBundles}
###### Definition
**([[field (physics)|fields]] and [[field histories]])**

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

$$
  \array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }
$$

called the _[[field bundle]]_,

Given a [[type]] of [[field|fields]] on $\Sigma$ this way, then a  _[[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

$$
  \Phi \;\colon\; \Sigma \longrightarrow E
$$

such that composed with the [[projection]] map it is the [[identity function]], i.e. such that

$$
  fb \circ \Phi = id
  \phantom{AAAAAAA}
  \array{
    && E
    \\
    & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    \Sigma & = & \Sigma
  }
  \,.
$$

The set of such [[sections]]/[[field histories]] is to be denoted

$$
 \label{SetOfFieldHistories}
  \Gamma_\Sigma(E)
  \;\coloneqq\;
  \left\{
    \array{
      && E
      \\
      & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}}
      \\
      \Sigma &=& \Sigma
    }
    \phantom{fb}
  \right\}
$$

=--

+-- {: .num_remark #FieldHistoriesAsHistoriesOfFieldConfigurations}
###### Remark
**([[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 _[[field configuration]]_ in space. As different spatial slices $\Sigma_p$ are chosen, one obtains such field configurations _at different times_. It is in this sense that the entirety of a section $\Phi \in \Gamma_\Sigma(E)$ is a _history_ of field configurations, hence a [[field history]] (def \ref{FieldsAndFieldBundles}).

=--

+-- {: .num_remark #PossibleFieldHistories}
###### Remark
**([[possible worlds|possible]] field histories)**

After we give the set $\Gamma_\Sigma(E)$ of field histories (eq:SetOfFieldHistories) [[differential geometry|differential geometric]] structure, below in example \ref{DiffeologicalSpaceOfFieldHistories} and example \ref{SupergeometricSpaceOfFieldHistories},
we call it the _[[space of field histories]]_. This should be read as space of _[[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

$$
  \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L}= 0}
    \overset{\phantom{AAA}}{\hookrightarrow}
  \Gamma_\Sigma(E)
$$

and called the _[[on-shell]] [[space of field histories]]_ (eq:InclusionOfOnShellSpaceOfFieldHistories).

=--

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:

+-- {: .num_example #TrivialVectorBundleAsAFieldBundle}
###### Example
**([[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 _[[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

$$
  ( (x^\mu), ( \phi^a ) )
  \,,
$$

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:

$$
  \Phi = ( \Phi^a )_{a = 1}^s
  \,.
$$

=--

+-- {: .num_example #RealScalarFieldBundle}
###### Example
**([[field bundle]] for [[real scalar field]])**

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

$$
  F \coloneqq \mathbb{R}
$$

is simply the [[real line]], then the corresponding trivial [[field bundle]] (def. \ref{FieldsAndFieldBundles})

$$
  \array{
    \Sigma \times \mathbb{R}
    \\
    {}^{\mathllap{pr_1}}\downarrow
    \\
    \Sigma
  }
$$

is the _[[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 _[[real scalar field]]_ on $\Sigma$. A configuration of this field is simply a [[smooth function]] on $\Sigma$ with values in the [[real numbers]]:

$$
  \label{SpaceOfFieldHistoriesOfRealScalarField}
  \Gamma_\Sigma(\Sigma \times \mathbb{R})
    \;\simeq\;
  C^\infty(\Sigma)
  \,.
$$

=--

+-- {: .num_example #Electromagnetism}
###### Example
**([[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]]

$$
  E \coloneqq T^\ast \Sigma
  \,.
$$

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]]

$$
  A \in \Gamma_\Sigma(T^\ast \Sigma) = \Omega^1(\Sigma)
$$

on [[spacetime]] (def. \ref{Differential1FormsOnCartesianSpaces}). Interpreted as a [[field history]] of the [[electromagnetic field]] on $\Sigma$, this is often called the _[[vector potential]]_.
Then the [[de Rham differential]] (def. \ref{deRhamDifferential}) of the [[vector potential]] is a [[differential 2-form]]

$$
  F \coloneqq d A
$$

known as the _[[Faraday tensor]]_. In the canonical coordinate basis 2-forms this may be expanded as

$$
  \label{TensorFaraday}
  F
  =
  \underoverset{i = 1}{p}{\sum}
  E_i d x^0 \wedge d x^i
  +
  \underset{1 \leq i \lt j \leq p}{\sum}
  B_{i j} d x^i \wedge d x^j
  \,.
$$

Here $(E_i)_{i = 1}^p$ are called the components of the _[[electric field]]_, while $(B_{i j})$ are called the
components of the _[[magnetic field]]_.

=--

+-- {: .num_example #YangMillsFieldOverMinkowski}
###### Example
**([[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

$$
  \label{LieAlgebraStructureConstants}
  [t_\alpha, t_\beta]
  \;=\;
  \gamma^\gamma{}_{\alpha \beta} t_\gamma
  \,.
$$

Over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}),
consider then the [[field bundle]] which is the [[cotangent bundle]] [[tensor product|tensored]] with the [[Lie algebra]] $\mathfrak{g}$

$$
  E \coloneqq T^\ast \Sigma \otimes \mathfrak{g}
  \,.
$$

This is the [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with induced [[field (physics)|field]] [[coordinates]]

$$
  ( a_\mu^\alpha )
  \,.
$$

A [[section]] of this bundle is a [[Lie algebra-valued differential 1-form]]

$$
  A \in \Gamma_\Sigma(T^\ast \Sigma \otimes \mathfrak{g}) = \Omega^1(\Sigma, \mathfrak{g})
  \,.
$$

with components

$$
  A^\ast(a_\mu^\alpha) = A^\alpha_\mu
  \,.
$$

This is called a [[field history]] for _[[Yang-Mills theory|Yang-Mills]] [[gauge theory]]_
(at least if $\mathfrak{g}$ is a _[[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]].

=--

For readers familiar with the concepts of _[[principal bundles]]_ and _[[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]].

+-- {: .num_example #YangMillsFieldInInstantonSector}
###### Example
**(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]] _in the [[instanton|topological sector]]_ $P$
is the [[tensor product of vector bundles]]

$$
  E \coloneqq \left(P \times^{ad}_G \mathfrak{g}\right) \otimes_\Sigma \left( T^\ast \Sigma \right)
$$

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

$$
  \nabla = \nabla_0 + A
  \,,
$$

where

$$
  A \in \Gamma_\Sigma(E)
$$

is a [[section]] of the above [[field bundle]], hence a [[Yang-Mills field|Yang-Mills]] [[field history]].

=--

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 _[[gauge fields]]_ (we consider this in more detail below in _[Gauge symmetries](#GaugeSymmetries)_). There are also _[[higher gauge fields]]_
with [[differential n-forms]] as [[field histories]]:

+-- {: .num_example #BField}
###### Example
**([[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

$$
  E \coloneqq \wedge^2_\Sigma T^\ast \Sigma
  \,.
$$

This is a [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with canonical [[field (physics)|field]] coordinates $(b_{\mu \nu})$ subject to

$$
  b_{\mu \nu} \;=\; - b_{\nu \mu}
  \,.
$$

A [[section]] of this bundle, hence a [[field history]], is a [[differential 2-form]] (def. \ref{DifferentialnForms})

$$
  B \in \Gamma_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) = \Omega^2(\Sigma)
$$

on [[spacetime]].

=--

$\,$

**[[space of field histories]]**
 {#NonFiniteDimensionalGeometry}

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 _[[space of sections]]_, to which
constructions of [[differential geometry]] apply (such as for the discussion of [[observables]] and [[states]] [below](#Observables) ). 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 _[[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 _smooth family_ of [[sections]]/[[field histories]], hence which [[functions]] of [[sets]]

$$
  \Phi_{(-)} \;\colon\; \mathbb{R}^n \longrightarrow \Gamma_\Sigma(E)
$$

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 _[[diffeological space]]_:

+-- {: .num_defn #DiffeologicalSpace}
###### Definition
**([[diffeological space]])**

A _[[diffeological space]]_ $X$ is

1. a [[set]] $X_s \in $ [[Set]];

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

   $$
     X(\mathbb{R}^n) \subset Hom_{Set}(\mathbb{R}^n_s, X_s) = \left\{ \mathbb{R}^n_s \to X_s  \right\}
   $$

   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 _smooth functions_ or _plots_ from $\mathbb{R}^n$ to $X$;

1. 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

   $$
     f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})
   $$

   to be thought of as the precomposition operation

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

such that

1. ([[constant functions]] are smooth)

   $$
     X(\mathbb{R}^0) = X_s
     \,,
   $$

1. ([[functor|functoriality]])

   1. 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 $X(\mathbb{R}^n)$;

   1. 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]]:

      $$
        f^\ast \circ g^\ast
        =
        (g \circ f)^\ast
      $$

      i.e.

      $$
        \array{
          && X(\mathbb{R}^{n_2})
          \\
          & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
          \\
          X(\mathbb{R}^{n_1})
          && \underset{ (g \circ f)^\ast }{\longleftarrow} &&
          X(\mathbb{R}^{n_3})
        }
      $$

1. ([[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

   $$
     X(\mathbb{R}^n)
       \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
     \underset{i \in I}{\prod} X(U_i)
   $$

   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]]:

   $$
     \phi_i\vert_{U_i \cap U_j} = \phi_j \vert_{U_i \cap U_j}
     \phantom{AAAAAA}
     \array{
       && U_i \cap U_j
       \\
       & \swarrow && \searrow
       \\
       U_i && && U_j
       \\
       & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
       \\
       && X
     }
   $$

Finally, given $X_1$ and $X_2$ two diffeological spaces, then a [[smooth function]] between them

$$
  f \;\colon\; X_1 \longrightarrow X_2
$$

is

* a [[function]] of the underlying sets

  $$
    f_s \;\colon\; (X_1)_s \longrightarrow (X_2)_s
  $$

such that

* 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 $X_2$:

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

(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}.)

=--

For more background on [[diffeological spaces]] see also _[[geometry of physics -- smooth sets]]_.

+-- {: .num_example #SmoothManifoldsAreDiffeologicalSpaces}
###### Example
**([[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 _[[Yoneda embedding]]_
over a _[[subcanonical site]]_.

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

=--

+-- {: .num_example #DiffeologicalSpaceOfFieldHistories}
###### Example
**([[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})

$$
  \label{SpaceOfFieldHistories}
  \Gamma_\Sigma(E) \in DiffeologicalSpaces
$$

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 $U$

$$
  \array{
    U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
    \\
    (u,x) &\mapsto& \Phi_u(x)
  }
$$

such that for each $u \in U$ we have $p \circ \Phi_{u}(-) = id_\Sigma$, i.e.

$$
  \array{
    && E
    \\
    & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
  }
  \,.
$$

=--

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.

+-- {: .num_example #FrechetManifoldsAreDiffeologicalSpaces}
###### Example
**([[Fréchet manifolds]] are [[diffeological spaces]])**

Consider the particular type of [[infinite-dimensional manifolds]] called _[[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]] ([this prop.](Fréchet+manifold#FFEmbeddingOfFrechetInDiffeological)):

$$
  FrechetManifolds \hookrightarrow DiffeologicalSpaces
  \,.
$$

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 _[[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 [this prop.](Fréchet+manifold#CompatibilityWithDiffeologicalMappingSpaces))

=--

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 _[[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):

+-- {: .num_defn #SmoothSet}
###### Definition
**([[smooth set]])**

A _[[smooth set]]_ $X$ is

1. for each $n \in \mathbb{N}$ a choice of [[set]]

   $$
     X(\mathbb{R}^n) \in Set
   $$

   to be called the set of _smooth functions_ or _plots_ from $\mathbb{R}^n$ to $X$;

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

   $$
     f^\ast \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1})
   $$

   to be thought of as the precomposition operation

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

such that

1. ([[functor|functoriality]])

   1. 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 $X(\mathbb{R}^n)$.

   1. 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]]:

      $$
        f^\ast \circ g^\ast
        =
        (g \circ f)^\ast
      $$

      i.e.

      $$
        \array{
          && X(\mathbb{R}^{n_2})
          \\
          & {}^{\mathllap{f^\ast}}\swarrow && \nwarrow^{\mathrlap{g^\ast}}
          \\
          X(\mathbb{R}^{n_1})
          && \underset{ (g \circ f)^\ast }{\longleftarrow} &&
          X(\mathbb{R}^{n_3})
        }
      $$

1. ([[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

   $$
     X(\mathbb{R}^n)
       \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
     \underset{i \in I}{\prod} X(U_i)
   $$

   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]]:

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

Finally, given $X_1$ and $X_2$ two [[smooth sets]], then a [[smooth function]] between them

$$
  f \;\colon\; X_1 \longrightarrow X_2
$$

is

* for each $n \in \mathbb{N}$ a [[function]]

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

such that

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

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

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

=--

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

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

+-- {: .num_example #SmoothSetsDiffeologicalSpaces}
###### Example
**([[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

$$
  CartesianSpaces
    \overset{\phantom{AAA}}{\hookrightarrow}
  DiffeologicalSpaces
    \overset{\phantom{AAA}}{\hookrightarrow}
  SmoothSets
  \,.
$$

=--

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 _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
_a posteriori_ concept of smooth functions from [[Cartesian spaces]] to [[smooth sets]]
coincides wth the _a priori_ concept, hence that we "may remove the quotation marks" in the above.
The following proposition says that this is indeed the case:

+-- {: .num_prop #CartSpYpnedaLemma}
###### Proposition
**(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]]

$$
  Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} X(\mathbb{R}^n)
$$

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 _[[bijection]]_.

This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the _would-be_ smooth functions
into $X$, end up being the _actual_ smooth functions into $X$.

=--

+-- {: .proof}
###### Proof

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

=--

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]]:

+-- {: .num_example #UniversalSmoothModuliSpaceOfDifferentialForms}
###### Example
**(universal [[smooth set|smooth]] [[moduli spaces]] of [[differential forms]])**

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

$$
  \mathbf{\Omega}^k \;\in\; SmoothSet
$$

defined as follows:

1. 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})

   $$
     \mathbf{\Omega}^k(\mathbb{R}^n) \;\coloneqq\; \Omega^k(\mathbb{R}^n)
   $$

1. 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}
   $$
     \array{
       \mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1})
       \\
       \downarrow^{\mathrlap{f}} && \uparrow^{\mathrlap{f^\ast}}
       \\
       \mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2})
     }
   $$

That this is [[functor|functorial]] is just the standard fact (eq: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]]:

$$
  \mathbf{\Omega}^0 \simeq \mathbb{R}^1
  \,.
$$

But for $k \geq 1$ we have that the set of plots on $\mathbb{R}^0 = \ast$ is a [[singleton]]

$$
  \mathbf{\Omega}^{k \geq 1}(\mathbb{R}^0) \simeq \{0\}
$$

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 _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 _[[modulating morphism]]_ $f_\omega \colon X \to \mathbf{\Omega}^k$:

$$
  \array{
    \{\omega\} &\overset{(f_\omega)^\ast}{\longleftarrow}& \{\omega_{univ}\}
    \\
    \\
    X &\underset{f_\omega}{\longrightarrow}& \mathbf{\Omega}^k
  }
$$

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

$$
  \mathbf{\Omega}^k \overset{\mathbf{d}}{\longrightarrow} \mathbf{\Omega}^{k+1}
$$

defined over $\mathbb{R}^n$ by the ordinary [[de Rham differential]] (def. \ref{deRhamDifferential})

$$
  \label{deRhamDifferentialUniversal}
  \Omega^k(\mathbb{R}^n) \overset{d}{\longrightarrow} \Omega^{k+1}(\mathbb{R}^n)
  \,.
$$

That this satisfies the compatibility with precomposition of plots

$$
  \array{
    \mathbb{R}^{n_1} && \Omega^k(\mathbb{R}^{n_1}) &\overset{d}{\longrightarrow}& \Omega^{k+1}(\mathbb{R}^{n_1})
    \\
    {}^{\mathllap{f}}\downarrow && \uparrow^{\mathrlap{f^\ast}} && \uparrow^{\mathrlap{f^\ast}}
    \\
    \mathbb{R}^{n_2} && \Omega^k(\mathbb{R}^{n_2}) &\underset{d}{\longrightarrow}& \Omega^k( \mathbb{R}^{n_2} )
  }
$$

is just the compatibility of [[pullback of differential forms]] with the [[de Rham differential]] of from prop. \ref{PullbackOfDifferentialForms}.

=--

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

+-- {: .num_defn #DifferentialFormsOnDiffeologicalSpaces}
###### Definition
**([[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]]

$$
  X \longrightarrow \mathbf{\Omega}^k
$$

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:

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

  $$
    \Phi^\ast(\omega) \in \Omega^\bullet(\mathbb{R}^n)
  $$

such that

* 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

  $$
    f^\ast\left(\Phi^\ast(\omega)\right)
    =
    ( f^\ast \Phi )^\ast(\omega)
  $$

  i.e.

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

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]]

$$
  X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^k
$$

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 (eq:deRhamDifferentialUniversal):

$$
  \label{FormsOnSmoothSetDeRhamDifferential}
  d \omega
    \;\colon\;
  X
    \overset{\omega}{\longrightarrow}
  \mathbf{\Omega}^k
    \overset{d}{\longrightarrow}
  \mathbf{\Omega}^{k+1}
  \,.
$$

Explicitly this means simply that for $\Phi \colon U \to X$ a plot, then

$$
  \Phi^\ast (d\omega)
  \;=\;
  d\left( \Phi^\ast \omega\right)
  \;\in\;
  \Omega^{k+1}(U)
  \,.
$$

=--

The usual operations on ordinary [[differential forms]] directly generalize plot-wise
to differential forms on [[diffeological spaces]] and more generally on [[smooth sets]]:

+-- {: .num_defn #ExteriorCalculusOnDiffeologicalSpaces}
###### Definition
**([[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

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

   $$
     d \omega \in \Omega^{n+1}(X)
   $$

   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:

   $$
     \Phi^\ast(d \omega) \coloneqq d \Phi^\ast(\omega)
     \,.
   $$

1. 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]]

   $$
     \omega_1 \wedge \omega_2 \;\in\; \Omega^{n_1 + n_2}(X)
   $$

   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:

   $$
     \Phi^\ast(\omega_1 \wedge \omega_2)
     \;\coloneqq\;
     \Phi^\ast(\omega_1) \wedge \Phi^\ast(\omega_2)
     \,.
   $$

=--

$\,$

**Infinitesimal geometry**
 {#InfinitesimalGeometry}

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 _[[infinitesimal neighbourhoods]]_
(example \ref{InfinitesimalNeighbourhood} below)
of these submanifolds. This appears notably in the construction of _[[phase spaces]]_ [below](#PhaseSpace).
Moreover, [[fermion fields]] such as the [[Dirac field]] (example \ref{DiracFieldBundle} below)
take values in _[[graded object|graded]]_ [[infinitesimal]] spaces, called _[[super spaces]]_
(discussed [below](#Supergeometry)). Therefore "infinitesimal geometry", sometimes called
_[[formal geometry]]_ (as in "[[formal scheme]]") or _[[synthetic differential geometry]]_
or _[[synthetic differential supergeometry]]_, is a central aspect of [[field theory]].

In order to mathematically grasp what _[[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

$$
  C^\infty(-) \;\colon\; SmoothManifolds \overset{\phantom{AAA}}{\hookrightarrow} (\mathbb{R} Algebras)^{op}
  \,.
$$

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 _defining_ a would-be [[space]], which would have $A$ as its [[algebra of functions]].

For example an _[[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:

+-- {: .num_defn #InfinitesimallyThickendSmoothManifold}
###### Definition
**([[infinitesimally thickened smooth manifold|infinitesimally thickened Cartesian space]])**

An _[[infinitesimally thickened point]]_

$$
  \mathbb{D} \coloneqq Spec(A)
$$

is represented by a [[commutative algebra]]
$A \in \mathbb{R}Alg$ which as a [[real vector space]] is a [[direct sum]]

$$
  A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V
$$

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

$$
  a^{n+1} = 0
  \,.
$$

More generally, an [[infinitesimally thickened manifold|infinitesimally thickened Cartesian space]]

$$
  \mathbb{R}^n \times \mathbb{D} \;\coloneqq\;  \mathbb{R}^n \times Spec(A)
$$

is represented by a [[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]] $n$ (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 _smooth function between two [[infinitesimally thickened manifolds|infinitesimally thickened Cartesian spaces]]_

$$
  \mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)
$$

is by definition [[formal dual|dually]] an $\mathbb{R}$-algebra [[homomorphism]] of the form

$$
  C^\infty(\mathbb{R}^{n_1}) \otimes A_1
    \overset{f^\ast}{\longleftarrow}
  C^\infty(\mathbb{R}^{n_2}) \otimes A_2
  \,.
$$

=--

+-- {: .num_example #InfinitesimalNeighbourhoodsInTheRealLine}
###### Example
**([[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 $\epsilon^2$:

$$
  (\mathbb{R}[ [\epsilon] ])/(\epsilon^2)
  \;\simeq_{\mathbb{R}}\;
  \mathbb{R} \oplus \epsilon \mathbb{R}
  \,.
$$

(This is sometimes called the _[[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

$$
   a + b \epsilon \in (\mathbb{R}[\epsilon])/(\epsilon^2)
$$

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 $\epsilon^2$:

$$
  \left(
    a_1 + b_1 \epsilon
  \right)
   \cdot
  \left(
    a_2 + b_2 \epsilon
  \right)
  \;=\;
  a_1 a_2 + ( a_1 b_2 + b_1 a_2 ) \epsilon
  \,.
$$

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]]

$$
  f \;\colon\; \mathbb{R} \to \mathbb{R}
$$

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:

$$
  \epsilon^2 = 0
  \,.
$$

This intuitive picture is now made precise by the concept of [[infinitesimally thickened points]] def. \ref{InfinitesimallyThickendSmoothManifold}, if we simply set

$$
  \mathbb{D}^1
    \;\coloneqq\;
  Spec\left(
    \mathbb{R}[ [\epsilon] ]/(\epsilon^2)
  \right)
$$

and observe that there is the [[monomorphism|inclusion]] of infinitesimally thickened Cartesian spaces

$$
  \mathbb{D}^1 \overset{\phantom{AA}i\phantom{AA} }{\hookrightarrow} \mathbb{R}^1
$$

which is dually given by the algebra homomorphism

$$
  \array{
    \mathbb{R} \oplus \epsilon \mathbb{R}
      &\overset{i^\ast}{\longleftarrow}&
    C^\infty(\mathbb{R}^1)
    \\
    f(0) + \frac{\partial f}{\partial x}(0)  &\longleftarrow& \{f\}
  }
$$

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

$$
  \begin{aligned}
    i^\ast(f \cdot g)
    & =
    (f \cdot g)(0) + \frac{\partial f \cdot g}{\partial x}(0) \epsilon
    \\
    & =
    f(0) \cdot g(0)
      +
    \left(
      \frac{\partial f}{\partial x}(0) \cdot g(0) +  f(0) \cdot \frac{\partial g}{\partial x}(0)
    \right) \epsilon
    \\
    & =
    \left(
      f(0) + \frac{\partial f}{\partial x}(0) \epsilon
    \right)
    \cdot
    \left(
      g(0) + \frac{\partial g}{\partial x}(0) \epsilon
    \right)
  \end{aligned}
$$

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:

$$
  \array{
    \mathbb{D}^1 &\overset{i}{\hookrightarrow}& \mathbb{R}^1
    \\
    & {}_{\mathllap{(\epsilon \mapsto f(0) + \frac{\partial f}{\partial x}(0) \epsilon) }}\searrow & \downarrow_{\mathrlap{f}}
    \\
    && \mathbb{R}^1
  }
$$

Similarly for each $k \in \mathbb{N}$ the algebra

$$
  (\mathbb{R}[ [ \epsilon ] ])/(\epsilon^{k+1})
$$

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

$$
  \mathbb{D}^1(k) \;\coloneqq\; Spec\left(  \left(\mathbb{R}[ [\epsilon] ]\right)/(\epsilon^{k+1}) \right)
  \,.
$$

This is now the [[subobject]] of the [[real line]]

$$
  \mathbb{D}^1(k) \overset{\phantom{AAA}}{\hookrightarrow} \mathbb{R}^1
$$

on those elements $\epsilon$ such that $\epsilon^{k+1} = 0$.

=--

([Kock 81](synthetic+differential+geometry#Kock81), [Kock 10](synthetic+differential+geometry#Kock10))

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

+-- {: .num_example #UniquePointOfInfinitesimalLine}
###### Example
**([[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 $\mathbb{D}^1(k)$:

$$
  \array{
    \mathbb{R}^n && \overset{\exists !}{\longrightarrow} &6 \mathbb{D}^1(k)
    \\
    & {}_{\mathllap{\exists !}}\searrow && \nearrow_{\mathrlap{\exists !}}
    \\
    && \mathbb{R}^0 = \ast
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

By definition such a morphism is [[formal duality|dually]] an algebra homomorphism

$$
  C^\infty(\mathbb{R}^n)
    \overset{f^\ast}{\longleftarrow}
  \left(
    \mathbb{R}[ [\epsilon] ])/(\epsilon^{k+1}
  \right)
    \simeq_{\mathbb{R}}
  \mathbb{R} \oplus \mathcal{O}(\epsilon)
$$

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:

$$
  f^\ast(1) = 1
  \,.
$$

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

$$
  \begin{aligned}
    \left(f^\ast(\epsilon)\right)^{k+1}
      & = f^\ast\left( \epsilon^{k+1}\right)
    \\
     & = f^\ast(0)
     \\ & = 0
  \end{aligned}
$$

But the only nilpotent element in $C^\infty(\mathbb{R}^n)$ is the zero element, and hence it follows that

$$
  f^\ast(\epsilon) = 0
  \,.
$$

Thus $f^\ast$ as above is uniquely fixed.

=--

+-- {: .num_example #SyntheticTangentVectorFields}
###### Example
**([[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

$$
  \array{
    \mathbb{R}^n \times \mathbb{D}^1
      && \overset{\tilde v}{\longrightarrow} &&
    \mathbb{R}^n
    \\
    & {}_{\mathllap{pr_1}}\searrow && \swarrow_{\mathrlap{id}}
    \\
    && \mathbb{R}^n
  }
$$

hence smoothly $X$-parameterized collections of morphisms

$$
  \tilde v_x \;\colon\; \mathbb{D}^1 \longrightarrow \mathbb{R}^n
$$

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}).

=--

+-- {: .proof}
###### Proof

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

$$
  (C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n))
  \longleftarrow
  C^\infty(\mathbb{R}^n)
$$

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

$$
  f + (\partial f) \epsilon
$$

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

$$
  (f_1 f_2 + (\partial (f_1 f_2))\epsilon )
  \;=\;
  (f_1 + (\partial f_1) \epsilon)
  \cdot
  (f_2 + (\partial f_2) \epsilon)
  \,.
$$

Multiplying this out and using that $\epsilon^2 = 0$, this is equivalent to

$$
  \partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2)
  \,.
$$

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]].

=--

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:

+-- {: .num_defn #FormalSmoothSet}
###### Definition
**([[formal smooth set]])**

A _[[formal smooth set]]_ $X$ is

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

   $$
     X(\mathbb{R}^n \times Spec(A)) \in Set
   $$

   to be called the set of _[[smooth functions]]_ or _plots_ from $\mathbb{R}^n \times Spec(A)$ to $X$;

1. 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

   $$
     f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))
   $$

   to be thought of as the precomposition operation

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

such that

1. ([[functor|functoriality]])

   1. 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 $X(\mathbb{R}^n \times Spec(A))$;

   1. 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]]:

      $$
        f^\ast \circ g^\ast = (g \circ f)^\ast
      $$

      i.e.

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

1. ([[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
   $$\{ U_i  \overset{f_i  }{\to} \mathbb{R}^n \}_{i \in I}$$
   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

   $$
     X(\mathbb{R}^n \times Spec(A))
       \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
     \underset{i \in I}{\prod} X(U_i \times Spec(A))
   $$

   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]]:

   $$
     \phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}
   $$

   i.e.

   $$
     \array{
       && (U_i \cap U_j) \times Spec(A)
       \\
       & \swarrow && \searrow
       \\
       U_i\times Spec(A) && && U_j \times Spec(A)
       \\
       & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
       \\
       && X
     }
   $$

Finally, given $X_1$ and $X_2$ two [[formal smooth sets]], then a [[smooth function]] between them

$$
  f \;\colon\; X_1 \longrightarrow X_2
$$

is

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

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

such that

* 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

  $$
    g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2))
    =
    f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1
  $$

  i.e.

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

=--

([Dubuc 79](Cahiers+topos#Dubuc79))

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

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

+-- {: .num_example #YonedaLemmaForFormalSmoothSets}
###### Example
**([[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:

$$
  X(\mathbb{R}^n \times \mathbb{D})
  \;\coloneqq\;
  Hom_{FormalCartSp}( \mathbb{R}^n \times \mathbb{D}, X )
  \,.
$$

(Stated more [[category theory|abstractly]], this is an instance of the _[[Yoneda embedding]]_ over a _[[subcanonical site]]_.)

=--

+-- {: .num_example #FormalSmoothSetsIncludedSmoothSet}
###### Example
**([[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]]

$$
  \mathbb{R}^n \times \mathbb{D} \longrightarrow \mathbb{R}^{k}
  \,,
  \phantom{AA}
  \mathbb{R}^k \longrightarrow X
$$

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

$$
  \array{
    && \mathbb{R}^n \times \mathbb{D}
    \\
    & \swarrow && \searrow
    \\
    \mathbb{R}^k && \overset{f}{\longrightarrow} && \mathbb{R}^{k'}
    \\
    \mathbb{R}^k && \underset{f}{\longrightarrow} && \mathbb{R}^{k'}
    \\
    & \searrow && \swarrow
    \\
    && X
  }
$$

Stated more [[category theory|abstractly]] this says that $X$ as a [[formal smooth set]]
is the _[[left Kan extension]]_ (see [this example](Kan+extension#CoendFormulaForPresheavesOfSets)) 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}).

=--

+-- {: .num_defn #ReductionAndInfinitesimalShape}
###### Definition
**([[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 _[[reduced object|reduction]]_

$$
 \Re\left(
   \mathbb{R}^n \times \mathbb{D}
 \right)
 \;\coloneqq\;
 \mathbb{R}^n
 \,.
$$

There is the canonical inclusion morphism

$$
  \Re\left(
    \mathbb{R}^n \times \mathbb{D}
  \right)
  =
  \mathbb{R}^n
  \overset{\phantom{AAAA}}{\hookrightarrow}
  \mathbb{R}^n \times \mathbb{D}
$$

which [[formal dual|dually]] corresponds to the [[homomorphism]] of [[commutative algebras]]

$$
  C^\infty(\mathbb{R}^n)
    \longleftarrow
  C^\infty(\mathbb{R}^n)
    \otimes_{\mathbb{R}}
  A
$$

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 _[[infinitesimal shape]]_ or _[[de Rham shape]]_ (also: _[[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:

$$
  (\Im X)( U ) \;\coloneqq\: X(\Re(U))
  \,.
$$

There is a canonical morphism of formal smooth set

$$
  \eta_X
    \;\colon\;
  X
    \longrightarrow
  \Im X
$$

which takes a plot

$$
  U = \mathbb{R}^n \times \mathbb{D} \overset{f}{\longrightarrow} X
$$

to the [[composition]]

$$
  \mathbb{R}^n \hookrightarrow \mathbb{R}^n \times \mathbb{D} \overset{f}{\hookrightarrow} X
$$

regarded as a plot of $\Im X$.

=--

+-- {: .num_example #MappingSpaceOutOfAnInfinitesimallyThickenedCartesianSpace}
###### Example
**([[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 _[[mapping space]]_

$$
  [X,Y] \;\in\; FormalSmoothSet
$$

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 $Y$:

$$
  [X,Y](U) \;\coloneqq\; Y(U \times X)
  \,.
$$

=--

+-- {: .num_example #TangentBundleSynthetic}
###### Example
**([[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

$$
  \mathbb{D}^1
    \hookrightarrow
  \mathbb{R}^1
$$

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

$$
  \array{
    T X &\simeq& [\mathbb{D}^1, X]
    \\
    {}^{\mathllap{tb}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{D}^1, X ]   }}
    \\
    X &\simeq& [\ast, X]
  }
$$

In particular this implies immediately that smooth [[sections]] (def. \ref{Sections}) of the tangent bundle

$$
  \array{
    && [\mathbb{D}^1, X] & \simeq T X
    \\
    & {}^{\mathllap{v}}\nearrow & \downarrow
    \\
    X &=& X
  }
$$

are equivalently morphisms of the form

$$
  \array{
    && X
    \\
    & {}^{\mathllap{\tilde v}}\nearrow & \downarrow^{\mathrlap{id}}
    \\
    X \times \mathbb{D}^1 &\underset{pr_1}{\longrightarrow}& X
  }
$$

which we had already identified with [[tangent vector fields]] (def. \ref{TangentVectorFields})
in example \ref{SyntheticTangentVectorFields}.

=--

+-- {: .proof}
###### Proof

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

=--

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:

+-- {: .num_example #InfinitesimalNeighbourhood}
###### Example
**([[infinitesimal neighbourhood]])**

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

$$
  \mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X
$$

such that their [[reduced object|reduction]] (def. \ref{ReductionAndInfinitesimalShape})

$$
  \mathbb{R}^n \hookrightarrow \mathbb{R}^n \times Spec(A) \overset{f}{\longrightarrow} X
$$

factors through a plot of $Y$.

=--

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]]
[below](#PhaseSpace).

+-- {: .num_defn #FieldHistoriesOnInfinitesimalNeighbourhoodOfSubmanifoldOfSpacetime}
###### Definition
**([[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 _set of field histories restricted to $S$_, to be denoted

$$
  \label{SpaceOfFieldHistoriesInHigherCodimension}
  \Gamma_{S}(E) \coloneqq \Gamma_{N_\Sigma(S)}( E\vert_{N_\Sigma S} ) \in \mathbf{H}
$$

is the set of section of $E$ restricted to the [[infinitesimal neighbourhood]] $N_\Sigma(S)$ (example \ref{InfinitesimalNeighbourhood}).

=--

$\,$

We close the discussion of [[synthetic differential geometry|infinitesimal differential geometry]] by explaining how we may recover the concept of _[[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.

+-- {: .num_defn #FormalSmoothSetLocalDiffeomorphism}
###### Definition
**([[local diffeomorphism]] of [[formal smooth sets]])**

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

$$
  f \;\colon\; X \overset{et}{\longrightarrow} Y
  \,,
$$

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

$$
  X(\mathbb{R}^n \times \mathbb{D})
  \;\simeq\;
  Y(\mathbb{R}^n \times \mathbb{D})
    \underset{Y(\mathbb{R}^n)}{\times^f}
  X(\mathbb{R}^n)
$$

i. e.

$$
  \array{
     && X(\mathbb{R}^n \times \mathbb{D})
     \\
     & \swarrow && \searrow
     \\
     Y(\mathbb{R}^n \times \mathbb{D}) && \text{(pb)} && X(\mathbb{R}^n)
     \\
     & \searrow && \swarrow
     \\
     && Y(\mathbb{R}^n )
  }
$$

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]]

$$
  \array{
    X &\overset{\eta_X}{\longrightarrow}& \Im X
    \\
    {}^{\mathllap{f}}\downarrow &\text{(pb)}& \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\underset{\eta_Y}{\longrightarrow}& \Im Y
  }
$$

=--

([Khavkine-Schreiber 17, def. 3.1](local+diffeomorphism#KhavkineSchreiber17))

+-- {: .num_example #AbstractLocalDiffeomorphismsOfCartesianSpaces}
###### Example
**([[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}.

=--

([Khavkine-Schreiber 17, prop. 3.2](geometry+of+physics+--+manifolds+and+orbifolds#KhavkineSchreiber17))

+-- {: .num_defn #SmoothManifoldInsideDiffeologicalSpaces}
###### Definition/Proposition
**([[smooth manifolds]])**

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

* a [[diffeological space]] (def. \ref{DiffeologicalSpace})

such that

1. 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

   1. every point $x \in X_s$ is in the [[image]] of at least one of the $\phi_i$;

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

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

=--

([Khavkine-Schreiber 17, example 3.4](geometry+of+physics+--+manifolds+and+orbifolds#KhavkineSchreiber17))

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

$\,$

**[[fermion fields]] and [[supergeometry]]**
 {#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 _[[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]] [above](#InfinitesimalGeometry), where now the [[infinitesimal]]
elements in the [[algebra of functions]] may be equipped with a [[graded object|grading]]: one speaks of _[[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 _[[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 _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 _[[geometry of physics -- supersymmetry]]_.

+-- {: .num_defn #SupercommutativeSuperalgebra}
###### Definition
**([[supercommutative superalgebra]])**

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

$$
  A \simeq_{\mathbb{R}} A_{even} \oplus A_{odd}
  \,,
$$

such that the product in the algebra respects the multiplication in the [[cyclic group|cyclic]] [[group of order 2]] $\mathbb{Z}/2 = \{even, odd\}$:

$$
  \left.
  \array{
    A_{even} \cdot A_{even}
    \\
    A_{odd} \cdot A_{odd}
  }
  \right\}
  \subset A_{even}
  \phantom{AAAA}
  \left.
  \array{
    A_{odd} \cdot A_{even}
    \\
    A_{even} \cdot A_{odd}
  }
  \right\}
  \subset A_{odd}
  \,.
$$

This is called a _[[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

$$
  a_i \in A_{{\vert a_i\vert}} \subset A
$$

we have

$$
  a_1 \cdot a_2 = (-1)^{{\vert a_1 \vert \vert a_2 \vert}} a_2 \cdot a_1
  \,.
$$

A _[[homomorphism]] of [[superalgebras]]_

$$
  f \;\colon\; A \longrightarrow A'
$$

is a [[homomorphism]] of [[associative algebras]] over the [[real numbers]] such that the $\mathbb{Z}/2$-[[graded object|grading]]
is respected in that

$$
  f(A_{even}) \subset A'_{even}
  \phantom{AAAAA}
  f(A_{odd}) \subset A'_{odd}
  \,.
$$

=--

For more details on superalgebra see at _[[geometry of physics -- superalgebra]]_.

+-- {: .num_example #GrassmannAlgebra}
###### Example
**(basic examples of [[supercommutative superalgebras]])**

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

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

1. 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

   $$
     \theta^i \cdot \theta^j = - \theta^j \cdot \theta^i
     \,.
   $$

   In particular this implies that all the $\theta^i$ are [[infinitesimal]] (def. \ref{InfinitesimallyThickendSmoothManifold}):

   $$
     \theta^i \cdot \theta^i = 0
     \,.
   $$

1. For $A_1$ and $A_2$ two [[supercommutative superalgebras]], there is their _[[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

   $$
     C^\infty(X) \otimes_{\mathbb{R}} \wedge^\bullet((\mathbb{R}^s)\ast)
   $$

   whose elements may uniquely be expanded in the form

   $$
      f + f_i \theta^i + f_{i j} \theta^i \theta^j + f_{i j k} \theta^i \theta^j \theta^k + \cdots + f_{i_1 \cdots i_s} \theta^{i_1} \cdots \theta^{i_s}
     \,,
   $$

   where the $f_{i_1 \cdots i_k} \in C^\infty(X)$ are smooth functions on $X$ which are skew-symmetric in their indices.

=--

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

+-- {: .num_defn #SuperCartesianSpace}
###### Definition
**([[super Cartesian space]])**

A _[[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]]

$$
  A \simeq_{\mathbb{R}} \langle 1 \rangle \oplus V
$$

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

$$
  a^{n+1} = 0
  \,.
$$

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

$$
  \label{StandardSuperpoints}
  \mathbb{R}^{0 \vert s}
    \;\coloneqq\;
  Spec\left(  \wedge^\bullet (\mathbb{R}^s)^\ast \right)
$$

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

$$
  \begin{aligned}
    \mathbb{R}^{b\vert s}
      & \coloneqq
    \mathbb{R}^b \times \mathbb{R}^{0 \vert s}
    \\
    & =
    \mathbb{R}^b \times Spec( \wedge^\bullet(\mathbb{R}^s)^\ast )
    \\
     & =
     Spec\left(  C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet (\mathbb{R}^s)^\ast  \right)
  \end{aligned}
$$

for the corresponding super Cartesian spaces whose algebra of functions is as in example \ref{GrassmannAlgebra}.

We say that a _smooth function_ between two [[super Cartesian spaces]]

$$
  \mathbb{R}^{n_1} \times Spec(A_1) \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \times Spec(A_2)
$$

is by definition [[formal dual|dually]] a [[homomorphism]] of [[supercommutative superalgebras]] (def. \ref{SupercommutativeSuperalgebra}) of the form

$$
  C^\infty(\mathbb{R}^{n_1}) \otimes A_1
    \overset{f^\ast}{\longleftarrow}
  C^\infty(\mathbb{R}^{n_2}) \otimes A_2
  \,.
$$

=--

+-- {: .num_example #SuperpointInducedByFiniteDimensionalVectorSpace}
###### Example
**([[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

$$
  V_{odd} \simeq \mathbb{R}^{0 \vert dim(V)}
  \,.
$$

=--

All the [[differential geometry]] over [[Cartesian space]] that we discussed [above](#Geometry)
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 [above](#DifferentialFormsAndCartanCalculus).

+-- {: .num_defn #DifferentialFormOnSuperCartesianSpaces}
###### Definition
**([[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

$$
  C^\infty(\mathbb{R}^{b\vert s})
  \;=\;
  C^\infty(\mathbb{R}^b) \otimes_{\mathbb{R}} \wedge^\bullet \mathbb{R}^s
$$

with canonical even-graded [[coordinate functions]] $(x^i)_{i = 1^b}$ and odd-graded coordinate functions $(\theta^j)_{j = 1}^s$.

Then the _[[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]]

$$
  \Omega^\bullet(\mathbb{R}^{b|s})
  \;\coloneqq\;
  C^\infty(\mathbb{R}^{b|s})
    \otimes_{\mathbb{R}}
   \wedge^\bullet \langle
      d x^1, \cdots, d x^b,
      \;
      d \theta^1, \cdots, d\theta^s
   \rangle
$$

which is generated over $C^\infty(\mathbb{R}^{b\vert s})$ from new generators

$$
  \underset{
    \text{deg} = (1,even)
  }{\underbrace{ d x^i }}
  \phantom{AAAAA}
  \underset{
    \text{deg} = (1,odd)
  }{
  \underbrace{
    d \theta^j
  }
  }
$$

whose [[differential]] is defined in degree-0 by

$$
  d f
    \;\coloneqq\;
  \frac{\partial f}{\partial x^i} d x^i
    +
  \frac{\partial f}{\partial \theta^j} d \theta^j
$$

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 $\sigma$:

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

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

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

$$
  \alpha \wedge \beta
  \;
  =
  \;
  (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta}
  \;
  \beta \wedge \alpha
  \,.
$$

Hence there are _two_ contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

1. there is a "cohomological sign" which for commuting an $n_1$-forms past an $n_2$-form is $(-1)^{n_1 n_2}$;

1. 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 $(-1)^{\sigma_1 \sigma_2}$.

For example:

$$
  x^{a_1} (dx^{a_2}) \;=\; + (dx^{a_2}) x^{a_1}
$$

$$
  \theta^\alpha (dx^a) \;=\; + (dx^a) \theta^\alpha
$$

$$
  \theta^{\alpha_1} (d\theta^{\alpha_2})
   \;=\;
  - (d\theta^{\alpha_2}) \theta^{\alpha_1}
$$

$$
  dx^{a_1}
  \wedge
  d x^{a_2}
  \;=\;
  -
  d x^{a_2}
  \wedge
  d x^{a_1}
$$

$$
  dx^a
  \wedge
  d \theta^{\alpha}
  \;=\;
  -
  d\theta^{\alpha}
  \wedge
  d x^a
$$

$$
  d\theta^{\alpha_1}
  \wedge
  d \theta^{\alpha_2}
  \;=\;
  +
  d\theta^{\alpha_2}
  \wedge
  d \theta^{\alpha_1}
$$

=--

(e.g. [Castellani-D'Auria-Fré 91 (II.2.106) and (II.2.109)](signs+in+supergeometry#CastellaniDAuriaFre91), [Deligne-Freed 99, section 6](signs+in+supergeometry#DeligneFreed99))

Beware that there is also _another_ sign rule for [[super differential forms]]
used in the literature. See at _[[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:

+-- {: .num_defn #SuperFormalSmoothSet}
###### Definition
**([[super formal smooth set|super smooth set]])**

A _[[super formal smooth set|super smooth set]]_ $X$ is

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

   $$
     X(\mathbb{R}^n \times Spec(A)) \in Set
   $$

   to be called the set of _[[smooth functions]]_ or _plots_ from $\mathbb{R}^n \times Spec(A)$ to $X$;

1. 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

   $$
     f^\ast \;\colon\; X(\mathbb{R}^{n_2} \times Spec(A_2)) \longrightarrow X(\mathbb{R}^{n_1} \times Spec(A_1))
   $$

   to be thought of as the precomposition operation

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

such that

1. ([[functor|functoriality]])

   1. 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 $X(\mathbb{R}^n \times Spec(A))$.

   1. 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]]:

      $$
        f^\ast \circ g^\ast
        =
        (g \circ f)^\ast
      $$

      i.e.

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

1. ([[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
   $$\{ U_i  \overset{f_i  }{\to} \mathbb{R}^n \}_{i \in I}$$
   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

   $$
     X(\mathbb{R}^n \times Spec(A))
       \overset{( (f_i)^\ast )_{i \in I}  }{\hookrightarrow}
     \underset{i \in I}{\prod} X(U_i \times Spec(A))
   $$

   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]]:

   $$
     \phi_i\vert_{((U_i \cap U_j) \times Spec(A)} = \phi_j \vert_{(U_i \cap U_j)\times Spec(A)}
   $$

   i.e.

   $$
     \array{
       && (U_i \cap U_j) \times Spec(A)
       \\
       & \swarrow && \searrow
       \\
       U_i\times Spec(A) && && U_j \times Spec(A)
       \\
       & {}_{\mathrlap{\Phi_i}}\searrow && \swarrow_{\mathrlap{\Phi_j}}
       \\
       && X
     }
   $$

Finally, given $X_1$ and $X_2$ two [[super formal smooth sets]], then a [[smooth function]] between them

$$
  f \;\colon\; X_1 \longrightarrow X_2
$$

is

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

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

such that

* 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

  $$
    g^\ast_2 \circ f_\ast(\mathbb{R}^{n_2} \times Spec(A_2))
    =
    f_\ast(\mathbb{R}^{n_1} \times Spec(A_1)) \circ g^\ast_1
  $$

  i.e.

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

=--

([Yetter 88](synthetic+differential+supergeometry#Yetter88))

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

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

+-- {: .num_example #SuperSmoothSetSuperCartesianSpaces}
###### Example
**([[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 _[[Yoneda embedding]]_
over a _[[subcanonical site]]_.

=--

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

+-- {: .num_prop #SuperCartSpYpnedaLemma}
###### Proposition
**(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]]

$$
  Hom_{SmoothSet}(\mathbb{R}^n , X) \overset{\simeq}{\longrightarrow} X(\mathbb{R}^n)
$$

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 _[[bijection]]_.

This says that the plots of $X$, which initially bootstrap $X$ into being as declaring the _would-be_ smooth functions
into $X$, end up being the _actual_ smooth functions into $X$.

=--

+-- {: .proof}
###### Proof

This is the statement of the _[[Yoneda lemma]]_ over the [[site]] of [[super Cartesian spaces]].

=--

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}:

+-- {: .num_defn #SuperSmoothManifolds}
###### Definition/Proposition
**([[supermanifolds]])**

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

* a [[super smooth set]] (def. \ref{SuperFormalSmoothSet})

such that

1. 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

   1. 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 $\mathbb{R}^{b\vert s}_i$;

   1. 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]];

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

=--

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

+-- {: .num_example #ModuliOfSDifferentialForms}
###### Example
**(universal [[smooth set|smooth]] [[moduli spaces]] of [[super differential forms]])**

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

$$
  \mathbf{\Omega}^n \;\in\; SuperSmoothSet
$$

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 $n$

$$
  \mathbf{\Omega}^n(U) \;\coloneqq\; \Omega^n(U)
$$

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

$$
  \label{SuperUniversalDeRhamDifferential}
  d \;\colon\; \mathbf{\Omega}^n \longrightarrow \mathbf{\Omega}^{n+1}
  \,.
$$

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

$$
  \Omega^n(X)
   \;\coloneqq\;
  Hom_{SuperSmoothSet}(X,\mathbf{\Omega}^n)
$$

and we define the [[de Rham differential]] on these to be given by postcomposition with (eq:SuperUniversalDeRhamDifferential).

=--

$\,$

+-- {: .num_defn #FermionicBosonicFields}
###### Definition
**([[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

* the even-graded [[sections]] are called the _[[bosonic field|bosonic]]_ [[field histories]];

* the odd-graded [[sections]] are called the _[[fermionic field|fermionic]]_ [[field histories]].

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
_[[bosonic field|bosonic]]_ field coordinates, and the $\psi^A$ are called the _[[fermionic field|fermionic]]_ field coordinates.

=--

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}:

+-- {: .num_example #SupergeometricSpaceOfFieldHistories}
###### Example
**([[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 _[[space of sections]]_, hence the _[[space of field histories]]_, is the [[super formal smooth set]] (def. \ref{SuperFormalSmoothSet})

$$
  \Gamma_\Sigma(E) \in SuperSmoothSet
$$

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})

$$
  \array{
    U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E
  }
$$

which make the following [[commuting diagram|diagram commute]]:

$$
  \array{
    && E
    \\
    & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}}
    \\
    U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma
  }
  \,.
$$

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})

$$
  \array{
    C^\infty(U \times \Sigma) &\overset{\left(\Phi_{(-)}(-)\right)^\ast}{\longleftarrow}& C^\infty(E)
  }
$$

which make the following [[commuting diagram|diagram commute]]:

$$
  \array{
    && C^\infty(E)
    \\
    & {}^{\mathllap{\left( \Phi_{(-)}(-) \right)^\ast }}\nearrow & \uparrow^{\mathrlap{fb^\ast}}
    \\
    C^\infty(U \times \Sigma) &\underset{pr_2^\ast}{\longleftarrow}& C^\infty(\Sigma)
  }
  \,.
$$

=--

We will focus on discussing the [[supergeometry|supergeometric]] [[space of field histories]] (example \ref{SupergeometricSpaceOfFieldHistories}) of the _[[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.

+-- {: .num_defn #MappingSpaceOutOfASuperCartesianSpace}
###### Definition
**([[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 _[[mapping space]]_

$$
  [X,Y] \;\in\; SuperSmoothSet
$$

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 $Y$:

$$
  [X,Y](U) \;\coloneqq\; Y(U \times X)
  \,.
$$

=--

In direct generalization of the [[synthetic differential geometry|synthetic]] [[tangent bundle]]
construction (example \ref{TangentBundleSynthetic}) to supergeometry we have

+-- {: .num_defn #TangentBundleOdd}
###### Definition
**([[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]] (eq:StandardSuperpoints) then the [[supergeometry|supergeometry]]-[[mapping space]]

$$
  \array{
    T_{odd} X & \coloneqq& [\mathbb{R}^{0\vert 1}, X]
    \\
    {}^{\mathllap{tb_{odd}}}\downarrow && \downarrow^{\mathrlap{ [ \ast \to \mathbb{R}^{0 \vert 1}, X ] }}
    \\
    X & = & X
  }
$$

is called the _[[odd tangent bundle]]_ of $X$.

=--

+-- {: .num_example #SuperpointsMapping}
###### Example
**([[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]] $V_{odd} \times V$

$$
  [\mathbb{R}^{0\vert 1}, V_{odd}]
  \;\simeq\;
  V_{odd} \times V
  \,.
$$

By def. \ref{TangentBundleOdd} this says that $V_{odd} \times V$ is the "[[odd tangent bundle]]" of $V_{odd}$.

=--

+-- {: .proof}
###### Proof

Let $U$ be any [[super Cartesian space]]. Then by definition we have the
following sequence of [[natural bijections]] of sets of plots

$$
  \begin{aligned}
    \left[
      \mathbb{R}^{0\vert 1}, V_{odd}
    \right](U)
    & =
    Hom_{SuperSmoothSet}( \mathbb{R}^{0\vert 1} \times U, V_{odd} )
    \\
    &
    \simeq
    Hom_{\mathbb{R}sAlg}( \wedge^\bullet(V^\ast)\,,\, C^\infty(U)[\theta]/(\theta^2) )
    \\
    &
    \simeq
    Hom_{\mathbb{R}Vect}( V^\ast \,,\, (C^\infty(U)_{odd} \oplus C^\infty(U)_{even}\langle \theta\rangle  )
    \\
    & \simeq
    Hom_{\mathbb{R}Vect}( V^\ast\,,\, C^\infty(U)_{odd} ) \,\times\, Hom_{\mathbb{R}Vect}( V^\ast, C^\infty(U)_{even} )
    \\
    & \simeq
    V_{odd}(U) \times V(U)
    \\
    & \simeq
    (V_{odd} \times V)(U)
  \end{aligned}
$$

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]].

=--

With this [[supergeometry]] in hand we finally turn to defining the [[Dirac field]] species:

+-- {: .num_example #DiracFieldBundle}
###### Example
**([[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

$$
  S \;=\;
  \left\{
    \array{
      \mathbb{R}^2 \oplus \mathbb{R}^2 & \vert & p + 1 = 2+1
      \\
      \mathbb{C}^2 \oplus \mathbb{C}^2  &\vert& p + 1 = 3 + 1
      \\
      \mathbb{H}^2 \oplus \mathbb{H}^2 &\vert& p + 1 = 5 + 1
      \\
      \mathbb{O}^2  \oplus \mathbb{O}^2 &\vert& p + 1 = 9 + 1
    }
  \right.
$$

With

$$
  S_{odd} \simeq \mathbb{R}^{0 \vert dim(S)}
$$

the corresponding [[superpoint]] (example \ref{SuperpointInducedByFiniteDimensionalVectorSpace}),
then the [[field bundle]] for the _[[Dirac field]]_ on $\Sigma$ is

$$
  E \;\coloneqq\; \Sigma \times S_{odd} \overset{pr_1}{\to} \Sigma
  \,,
$$

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]] $S_{odd}$:

$$
  \left(
   \psi^A
  \right)_{A = 1}^4
   \;=\;
  \left(
     (\chi_a), (\xi^{\dagger \dot a})
  \right)_{a,\dot a = 1,2}
  \,.
$$

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:

$$
  \label{DiracFieldCoordinatesAnticommute}
  \psi^\alpha  \psi^{\beta} = - \psi^{\beta} \psi^\alpha
  \,.
$$

<img src="https://ncatlab.org/nlab/files/DermisekAnticommutingSpinorFieldCoordinates.jpg" width="400">

> snippet grabbed from ([Dermisek 09](Dirac+field#DermisekI8))

=--

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)

+-- {: .num_defn #DiracSpaceOfFieldHistories}
###### Proposition
**([[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]]

$$
  \Gamma_\Sigma(\Sigma \times S_{odd})
  \;\in\;
  SuperSmoothSet
$$

from example \ref{SupergeometricSpaceOfFieldHistories} has the following properties:

1. 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

   $$
     \Psi_{(-)}\;\colon\;\mathbb{R}^n \overset{ 0 }{\longrightarrow} \Gamma_\Sigma(\Sigma \times S_{odd})
   $$

   and this corresponds to the [[zero section]], hence to the trivial [[Dirac field]]

   $$
     \Psi^A_{(-)} = 0
     \,.
   $$

1. {#OddParameterizedFieldHistories} 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

   $$
     \Phi_{(-)} \;\colon\; \mathbb{R}^{n\vert 1} \longrightarrow \Gamma_\Sigma(\Sigma \times S_{odd})
   $$

   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:

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

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}):

$$
  \Gamma_\Sigma(\Sigma \times S_{odd})
    \;\simeq\;
  \left(
    \Gamma_\Sigma(E\times S_{even})
  \right)_{odd}
  \,.
$$

=--

+-- {: .proof}
###### Proof

In the first case, the plot is a morphism of [[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}) of the form

$$
  \mathbb{R}^n \times \mathbb{R}^{p,1} \longrightarrow S_{odd}
  \,.
$$

By definitions this is [[formal duality|dually]] homomorphism of real [[supercommutative superalgebras]]

$$
  C^\infty(\mathbb{R}^n \times \mathbb{R}^{p,1})
    \longleftarrow
  \wedge^\bullet S^\ast
$$

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

$$
  \mathbb{R}^{n\vert 1}
    \overset{\Phi_{(-)}}{\longrightarrow}
  S_{odd}
$$

is by def. \ref{TangentBundleOdd} [[natural bijection|naturally identified]] with a morphism of the form

$$
  \mathbb{R}^n
    \overset{\Psi_{(-)}}{\longrightarrow}
  [\mathbb{R}^{0 \vert 1}, S_{odd}]
    \simeq
  S_{odd} \times S_{even}
  \,,
$$

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

$$
  \left(
    \array{
      \mathbb{R}^n &\overset{}{\longrightarrow}& S_{odd}\,,
      \\
      \mathbb{R}^n &\overset{}{\longrightarrow}& S_{even}
    }
  \right)
  \,.
$$

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.

=--

+-- {: .num_remark #DiracFieldSupergeometric}
###### Remark
**([[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]]:

1. On the one hand a [[field history]] $\Psi$ of the [[Dirac field]] is _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}.

1. 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]].

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 $\theta$.

=--

$\,$

This concludes our discussion of the concept of _[[field (physics)|fields]]_ itself.
In the [following chapter](#FieldVariations) we consider the [[variational calculus]] of fields.