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

\hypertarget{Geometry}{}\subsection*{{Geometry}}\label{Geometry}

The [[geometry of physics]] is \emph{[[differential geometry]]}. This is the flavor of [[geometry]] which is modeled on [[Cartesian spaces]] $\mathbb{R}^n$ with [[smooth functions]] between them. Here we briefly review the basics of [[differential geometry]] on [[Cartesian spaces]].

In principle the only \textbf{background} assumed of the reader here is

\begin{enumerate}%
\item usual \emph{[[structural set theory|naive set theory]]} (e.g. [[Sets for Mathematics|Lawvere-Rosebrugh 03]]);


\item the concept of the \emph{[[continuum]]}: the [[real line]] $\mathbb{R}$, the [[plane]] $\mathbb{R}^2$, etc.


\item the concepts of \emph{[[differentiation]]} and \emph{[[integration]]} of functions on such [[Cartesian spaces]];



\end{enumerate}
hence essentially the content of multi-variable [[differential calculus]].

We now discuss:

\begin{itemize}%
\item \emph{\hyperlink{SmoothFunctions}{Abstract coordinate systems}}


\item \emph{\hyperlink{BundlesAndSections}{Fiber bundles}}


\item \emph{\hyperlink{SyntheticDifferentialGeometry}{Synthetic differential geometry}}


\item \emph{\hyperlink{DifferentialFormsAndCartanCalculus}{Differential forms}}



\end{itemize}
As we uncover [[Lagrangian field theory]] further below, we discover ever more general concepts of ``[[space]]'' in differential geometry, such as \emph{[[smooth manifolds]]}, \emph{[[diffeological spaces]]}, \emph{[[infinitesimal neighbourhoods]]}, \emph{[[supermanifolds]]}, \emph{[[Lie algebroids]]} and \emph{[[super L-∞ algebra|super]] [[Lie ∞-algebroids]]}. We introduce these incrementally as we go along:

\textbf{more general [[spaces]] in [[differential geometry]] introduced further below}

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l}
&&&&&&&&&&&[[higher differential geometry]]\\
\hline 
\textbf{[[differential geometry]]}&[[smooth manifolds]]  (def. \ref{SmoothManifoldInsideDiffeologicalSpaces})&$\hookrightarrow$&[[diffeological spaces]]  (def. \ref{DiffeologicalSpace})&$\hookrightarrow$&[[smooth sets]]  (def. \ref{SmoothSet})&$\hookrightarrow$&[[formal smooth sets]]  (def. \ref{FormalSmoothSet})&$\hookrightarrow$&[[super formal smooth sets]]  (def. \ref{SuperFormalSmoothSet})&$\hookrightarrow$&[[super formal smooth ∞-groupoids]]  (not needed in fully [[perturbative QFT]])\\
\textbf{[[infinitesimal]] [[formal geometry}&geometry]],  [[Lie theory]]**&&&&&&&[[infinitesimally thickened points]]  (def. \ref{InfinitesimallyThickendSmoothManifold})&&[[superpoints]]  (def. \ref{SuperCartesianSpace})&\\
&&&&&&&&&&&\textbf{[[higher Lie theory]]}\\
\textbf{needed in [[QFT]] for:}&[[spacetime]] (def. \ref{MinkowskiSpacetime})&&[[space of field histories]]  (def. \ref{DiffeologicalSpaceOfFieldHistories})&&&&[[Cauchy surface]] (def. \ref{CauchySurface}),  [[perturbation theory]] (def. \ref{LocalObservablesOnInfinitesimalNeighbourhood})&&[[Dirac field]] (expl. \ref{DiracFieldBundle}), [[Pauli exclusion principle]]&&[[infinitesimal gauge symmetry]]/[[BRST complex]]  (expl. \ref{LocalOffShellBRSTComplex})\\
\end{tabular}

$\,$

\textbf{Abstract coordinate systems}

What characterizes [[differential geometry]] is that it models [[geometry]] on \emph{the [[continuum]]}, namely the [[real line]] $\mathbb{R}$, together with its [[Cartesian products]] $\mathbb{R}^n$, regarded with its canonical [[smooth structure]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem} below). We may think of these \emph{[[Cartesian spaces]]} $\mathbb{R}^n$ as the ``abstract [[coordinate systems]]'' and of the [[smooth functions]] between them as the ``abstract [[coordinate transformations]]''.

We will eventually consider \hyperlink{FieldBundles}{below} much more general ``[[smooth spaces]]'' $X$ than just the [[Cartesian spaces]] $\mathbb{R}^n$; but all of them are going to be understood by ``laying out abstract coordinate systems'' inside them, in the general sense of having smooth functions $f \colon \mathbb{R}^n \to X$ mapping a Cartesian space smoothly into them. All structure on [[generalized smooth spaces]] $X$ is thereby reduced to \emph{compatible systems} of structures on just [[Cartesian spaces]], one for each smooth ``probe'' $f\colon \mathbb{R}^n \to X$. This is called ``[[functorial geometry]]''.

Notice that the popular concept of a \emph{[[smooth manifold]]} (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below) is essentially that of a [[smooth space]] which \emph{locally looks just like} a [[Cartesian space]], in that there exist sufficiently many $f \colon \mathbb{R}^n \to X$ which are ([[open map|open]]) \emph{[[isomorphisms]]} onto their [[images]]. Historically it was a long process to arrive at the insight that it is wrong to \emph{fix} such local coordinate identifications $f$, or to have any structure depend on such a choice. But it is useful to go one step further:

In [[functorial geometry]] we do not even focus attention on those $f \colon \mathbb{R}^n \to X$ that are isomorphisms onto their image, but consider \emph{all} ``probes'' of $X$ by ``abstract coordinate systems''. This makes [[differential geometry]] both simpler as well as more powerful. The analogous insight for [[algebraic geometry]] is due to \href{functorial+geometry#Grothendieck65}{Grothendieck 65}; it was transported to [[differential geometry]] by \href{synthetic+differential+geometry#Lawvere67}{Lawvere 67}.

This allows to combine the best of two superficially disjoint worlds: On the one hand we may reduce all constructions and computations to [[coordinates]], the way traditionally done in the [[physics]] literature; on the other hand we have full conceptual control over the coordinate-free generalized spaces analyzed thereby. What makes this work is that all [[coordinate]]-constructions are \emph{[[functorial geometry|functorially]]} considered over all abstract coordinate systems.

$\,$

\begin{defn}
\label{CartesianSpacesAndSmoothFunctionsBetweenThem}\hypertarget{CartesianSpacesAndSmoothFunctionsBetweenThem}{}
\textbf{([[Cartesian spaces]] and [[smooth functions]] between them)}

For $n \in \mathbb{N}$ we say that the set $\mathbb{R}^n$ of [[n-tuples]] of [[real numbers]] is a \emph{[[Cartesian space]]}. This comes with the canonical [[coordinate functions]]

\begin{displaymath}
x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}
\end{displaymath}
which send an [[n-tuple]] of real numbers to the $k$th element in the tuple, for $k \in \{1, \cdots, n\}$.

For

\begin{displaymath}
f \;\colon\; \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n'}
\end{displaymath}
any [[function]] between [[Cartesian spaces]], we may ask whether its [[partial derivative]] along the $k$th coordinate exists, denoted

\begin{displaymath}
\frac{\partial f}{\partial x^k}
  \;\colon\;
  \mathbb{R}^{n}
    \longrightarrow
  \mathbb{R}^{n'}
  \,.
\end{displaymath}
If this exists, we may in turn ask that the [[partial derivative]] of the partial derivative exists

\begin{displaymath}
\frac{\partial^2 f}{\partial x^{k_1} \partial x^{k_2}}
  \coloneqq
  \frac{\partial}{\partial x^{k_2}} \frac{\partial f}{\partial x^{k_1}}
\end{displaymath}
and so on.

A general higher [[partial derivative]] obtained this way is, if it exists, indexed by an [[n-tuple]] of [[natural numbers]] $\alpha \in \mathbb{N}^n$ and denoted

\begin{equation}
\partial_\alpha
    \;\coloneqq\;
  \frac{
    \partial^{\vert \alpha \vert} f
  }{
    \partial^{\alpha_1} x^1
    \partial^{\alpha_2} x^2
      \cdots
    \partial^{\alpha_n} x^n
  }
  \,,
\label{PartialDerivativeWithManyIndices}\end{equation}
where ${\vert \alpha\vert} \coloneqq \underoverset{n}{i = 1}{\sum} \alpha_i$ is the total \emph{order} of the partial derivative.

If all partial derivative to all orders $\alpha \in \mathbb{N}^n$ of a [[function]] $f \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ exist, then $f$ is called a \emph{[[smooth function]]}.

\end{defn}
Of course the [[composition]] $g \circ f$ of two smooth functions is again a [[smooth function]].

\begin{displaymath}
\itexarray{
    && \mathbb{R}^{n_2}
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    \mathbb{R}^{n_1}
      && \underset{g \circ f}{\longrightarrow} &&
    \mathbb{R}^{n_3}
  }
  \,.
\end{displaymath}
The inclined reader may notice that this means that [[Cartesian spaces]] with [[smooth functions]] between them constitute a \emph{[[category]]} (``[[CartSp]]''); but the reader not so inclined may ignore this.

For the following it is useful to think of each [[Cartesian space]] as an \emph{abstract [[coordinate system]]}. We will be dealing with various [[generalized smooth spaces]] (see the table \hyperlink{NotionsOfGeometry}{below}), but they will all be characterized by a prescription for how to smoothly map abstract coordinate systems into them.

\begin{example}
\label{CoordinateFunctionsAreSmoothFunctions}\hypertarget{CoordinateFunctionsAreSmoothFunctions}{}
\textbf{([[coordinate functions]] are [[smooth functions]])}

Given a [[Cartesian space]] $\mathbb{R}^n$, then all its [[coordinate functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})

\begin{displaymath}
x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}
\end{displaymath}
are [[smooth functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).

For

\begin{displaymath}
f \colon \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
\end{displaymath}
any [[smooth function]] and $a \in \{1, 2, \cdots, n_2\}$ write

\begin{displaymath}
f^a \coloneqq x^a \circ f
  \;\colon\;
  \mathbb{R}^{n_1}
    \overset{f}{\longrightarrow}
  \mathbb{R}^{n_2}
    \overset{x^a}{\longrightarrow}
  \mathbb{R}
\end{displaymath}
. for its [[composition]] with this [[coordinate function]].

\end{example}
\begin{example}
\label{AlgebraOfSmoothFunctionsOnCartesianSpaces}\hypertarget{AlgebraOfSmoothFunctionsOnCartesianSpaces}{}
\textbf{([[algebra of functions|algebra of]] [[smooth functions]] on [[Cartesian spaces]])}

For each $n \in \mathbb{N}$, the set

\begin{displaymath}
C^\infty(\mathbb{R}^n)
   \;\coloneqq\;
  Hom_{CartSp}(\mathbb{R}^n, \mathbb{R})
\end{displaymath}
of [[real number]]-valued [[smooth functions]] $f \colon \mathbb{R}^n \to \mathbb{R}$ on the $n$-dimensional [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) becomes a [[commutative algebra|commutative]] [[associative algebra]] over the [[ring]] of [[real numbers]] by pointwise addition and multiplication in $\mathbb{R}$: for $f,g \in C^\infty(\mathbb{R}^n)$ and $x \in \mathbb{R}^n$

\begin{enumerate}%
\item $(f + g)(x) \coloneqq f(x) + g(x)$


\item $(f \cdot g)(x) \coloneqq f(x) \cdot g(x)$.



\end{enumerate}
The inclusion

\begin{displaymath}
\mathbb{R} \overset{const}{\hookrightarrow} C^\infty(\mathbb{R}^n)
\end{displaymath}
is given by the [[constant functions]].

We call this the \emph{[[real numbers|real]] [[algebra of functions|algebra of]] [[smooth functions]]} on $\mathbb{R}^n$:

\begin{displaymath}
C^\infty(\mathbb{R}^n)
  \;\in\;
  \mathbb{R} Alg
  \,.
\end{displaymath}
If

\begin{displaymath}
f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
\end{displaymath}
is any [[smooth function]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) then [[composition|pre-composition]] with $f$ (``[[pullback of functions]]'')

\begin{displaymath}
\itexarray{
    C^\infty(\mathbb{R}^{n_2})
      &\overset{f^\ast}{\longrightarrow}&
    C^\infty(\mathbb{R}^{n_1})
    \\
    g &\mapsto& f^\ast g \coloneqq g \circ f
  }
\end{displaymath}
is an [[associative algebra|algebra]] [[homomorphism]]. Moreover, this is clearly compatible with [[composition]] in that

\begin{displaymath}
f_1^\ast(f_2^\ast g) = (f_2 \circ f_1)^\ast g
  \,.
\end{displaymath}
Stated more [[category theory|abstractly]], this means that assigning [[algebra of functions|algebras]] of [[smooth functions]] is a [[functor]]

\begin{displaymath}
C^\infty(-)
  \;\colon\;
  CartSp \longrightarrow \mathbb{R} Alg^{op}
\end{displaymath}
from the [[category]] [[CartSp]] of [[Cartesian spaces]] and [[smooth functions]] between them (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), to the [[opposite category|opposite]] of the category $\mathbb{R}$[[Alg]] of $\mathbb{R}$-[[associative algebra|algebras]].

\end{example}
\begin{defn}
\label{LocalDiffeomorphismBetweenCartesianSpaces}\hypertarget{LocalDiffeomorphismBetweenCartesianSpaces}{}
\textbf{([[local diffeomorphisms]] and [[open embeddings]] of [[Cartesian spaces]])}

A [[smooth function]] $f \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ from one [[Cartesian space]] to itself (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is called a \emph{[[local diffeomorphism]]}, denoted

\begin{displaymath}
f \;\colon\; \mathbb{R}^{n} \overset{et}{\longrightarrow} \mathbb{R}^n
\end{displaymath}
if the [[determinant]] of the [[matrix]] of [[partial derivatives]] (the ``[[Jacobian]]'' of $f$) is everywhere non-vanishing

\begin{displaymath}
det
  \left(
   \itexarray{
     \frac{\partial f^1}{\partial x^1}(x)
     &\cdots&
     \frac{\partial f^n}{\partial x^1}(x)
     \\
     \vdots && \vdots
     \\
     \frac{\partial f^1}{\partial x^n}(x)
     &\cdots&
     \frac{\partial f^n}{\partial x^n}(x)
   }
  \right)
  \;\neq\;
  0
  \phantom{AAAA}
  \text{for all} \, x \in \mathbb{R}^n
  \,.
\end{displaymath}
If the function $f$ is both a [[local diffeomorphism]], as above, as well as an [[injective function]] then we call it an \emph{[[open embedding]]}, denoted

\begin{displaymath}
f \;\colon\;
  \mathbb{R}^n \overset{\phantom{A}et\phantom{A}}{\hookrightarrow} \mathbb{R}^n
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{DifferentiablyGoodOpenCover}\hypertarget{DifferentiablyGoodOpenCover}{}
\textbf{([[good open cover]] of [[Cartesian spaces]])}

For $\mathbb{R}^n$ a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), a \emph{[[differentiably good open cover]]} is

\begin{itemize}%
\item an [[indexed set]]

\begin{displaymath}
\left\{
     \mathbb{R}^n
    \underoverset{et}{\phantom{AA}f_i\phantom{AA}}{\hookrightarrow}
     \mathbb{R}^n
   \right\}_{i \in I}
\end{displaymath}
of [[open embeddings]] (def. \ref{LocalDiffeomorphismBetweenCartesianSpaces})



\end{itemize}
such that the [[images]]

\begin{displaymath}
U_i \coloneqq im(f_i) \subset \mathbb{R}^n
\end{displaymath}
satisfy:

\begin{enumerate}%
\item ([[open cover]]) every point of $\mathbb{R}^n$ is contained in at least one of the $U_i$;


\item ([[good open cover|good]]) all [[finite set|finite]] [[intersections]] $U_{i_1} \cap \cdots \cap U_{i_k} \subset \mathbb{R}^n$ are either [[empty set]] or themselves images of [[open embeddings]] according to def. \ref{LocalDiffeomorphismBetweenCartesianSpaces}.



\end{enumerate}
The inclined reader may notice that the concept of [[differentiably good open covers]] from def. \ref{DifferentiablyGoodOpenCover} is a \emph{[[coverage]]} on the [[category]] \emph{[[CartSp]]} of [[Cartesian spaces]] with [[smooth functions]] between them, making it a \emph{[[site]]}, but the reader not so inclined may ignore this.

\end{defn}
([[schreiber:Cech Cocycles for Differential characteristic Classes|Fiorenza-Schreiber-Stasheff 12, def. 6.3.9]])

$\,$

$\,$

\textbf{[[fiber bundles]]}

Given any context of [[objects]] and [[morphisms]] between them, such as the [[Cartesian spaces]] and [[smooth functions]] from def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem} it is of interest to fix one [[object]] $X$ and consider other objects \emph{[[dependent type|parameterized over]]} it. These are called \emph{[[bundles]]} (def. \ref{BundlesAndFibers}) below. For reference, we briefly discuss here the basic concepts related to [[bundles]] in the context of [[Cartesian spaces]].

Of course the theory of bundles is mostly trivial over Cartesian spaces; it gains its main interest from its generalization to more general [[smooth manifolds]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below). It is still worthwhile for our development to first consider the relevant concepts in this simple case first.

For more exposition see at \emph{[[fiber bundles in physics]]}.

$\,$

\begin{defn}
\label{BundlesAndFibers}\hypertarget{BundlesAndFibers}{}
\textbf{([[bundles]])}

We say that a [[smooth function]] $E \overset{fb}{\to} X$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is a \emph{[[bundle]]} just to amplify that we think of it as exhibiting $E$ as being a ``space over $X$'':

\begin{displaymath}
\itexarray{
    E
    \\
    \downarrow\mathrlap{fb}
    \\
    X
  }
  \,.
\end{displaymath}
For $x \in X$ a point, we say that the \emph{[[fiber]]} of this [[bundle]] over $x$ is the [[pre-image]]

\begin{equation}
E_x \coloneqq fb^{-1}(\{x\}) \subset E
\label{FiberOfAFiberBundle}\end{equation}
of the point $x$ under the smooth function. We think of $fb$ as exhibiting a ``smoothly varying'' set of [[fiber]] spaces over $X$.

Given two [[bundles]] $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ over $X$, a \emph{[[homomorphism]] of bundles} between them is a [[smooth function]] $f \colon E_1 \to E_2$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) between their total spaces which respects the bundle projections, in that

\begin{displaymath}
fb_2  \circ f = fb_1
  \phantom{AAAA}
  \text{i.e.}
  \phantom{AAA}
  \itexarray{
    E_1 && \overset{f}{\longrightarrow} && E_2
    \\
    & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}}
    \\
    && X
  }
  \,.
\end{displaymath}
Hence a bundle homomorphism is a smooth function that sends [[fibers]] to [[fibers]] over the same point:

\begin{displaymath}
f\left(
    (E_1)_x
  \right)
    \;\subset\;
  (E_2)_x
  \,.
\end{displaymath}
The inclined reader may notice that this defines a [[category]] of [[bundles]] over $X$, which is in fact just the \emph{[[slice category]]} $CartSp_{/X}$; the reader not so inclined may ignore this.

\end{defn}
\begin{defn}
\label{Sections}\hypertarget{Sections}{}
\textbf{([[sections]])}

Given a [[bundle]] $E \overset{fb}{\to} X$ (def. \ref{BundlesAndFibers}) a \emph{[[section]]} is a [[smooth function]] $s \colon X \to E$ such that

\begin{displaymath}
fb \circ s = id_X
  \phantom{AAAAA}
  \itexarray{
    && E
    \\
    &
    {}^{\mathllap{s}}\nearrow
    & \downarrow\mathrlap{fb}
    \\
    X &=& X
  }
  \,.
\end{displaymath}
This means that $s$ sends every point $x \in X$ to an element in the [[fiber]] over that point

\begin{displaymath}
s(x) \in E_x
  \,.
\end{displaymath}
We write

\begin{displaymath}
\Gamma_X(E) \coloneqq
  \left\{
     \itexarray{
        && E
        \\
        & {}^{\mathllap{s}}\nearrow & \downarrow^\mathrlap{fb}
        \\
        X &=& X
     }
     \phantom{fb}
  \right\}
\end{displaymath}
for the [[space of sections|set of sections]] of a bundle.

For $E_1 \overset{f_1}{\to} X$ and $E_2 \overset{f_2}{\to} X$ two [[bundles]] and for

\begin{displaymath}
\itexarray{
    E_1
      && \overset{f}{\longrightarrow} &&
    E_2
    \\
    & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}}
    \\
    && X
  }
\end{displaymath}
a bundle [[homomorphism]] between them (def. \ref{BundlesAndFibers}), then [[composition]] with $f$ sends [[sections]] to [[sections]] and hence yields a [[function]] denoted

\begin{displaymath}
\itexarray{
    \Gamma_X(E_1)
      &\overset{f_\ast}{\longrightarrow}&
    \Gamma_X(E_2)
    \\
    s &\mapsto& f \circ s
  }
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{TrivialBundleOnCartesianSpace}\hypertarget{TrivialBundleOnCartesianSpace}{}
\textbf{([[trivial bundle]])}

For $X$ and $F$ [[Cartesian spaces]], then the [[Cartesian product]] $X \times F$ equipped with the [[projection]]

\begin{displaymath}
\itexarray{
    X \times F
    \\
    \downarrow^\mathrlap{pr_1}
    \\
    X
  }
\end{displaymath}
to $X$ is a [[bundle]] (def. \ref{BundlesAndFibers}), called the \emph{[[trivial bundle]]} with [[fiber]] $F$. This represents the \emph{constant} smoothly varying set of [[fibers]], constant on $F$

If $F = \ast$ is the point, then this is the identity bundle

\begin{displaymath}
\itexarray{
    X
    \\
    \downarrow\mathrlap{id}
    \\
    X
  }
  \,.
\end{displaymath}
Given any [[bundle]] $E \overset{fb}{\to} X$, then a bundle homomorphism (def. \ref{BundlesAndFibers}) from the identity bundle to $E \overset{fb}{\to} X$ is equivalently a [[section]] of $E \overset{fb}{\to} X$ (def. \ref{Sections})

\begin{displaymath}
\itexarray{
     X && \overset{s}{\longrightarrow} && E
     \\
     & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{fb}}
     \\
     && X
  }
\end{displaymath}
\end{example}
\begin{defn}
\label{FiberBundle}\hypertarget{FiberBundle}{}
\textbf{([[fiber bundle]])}

A [[bundle]] $E \overset{fb}{\to} X$ (def. \ref{BundlesAndFibers}) is called a \emph{[[fiber bundle]]} with \emph{typical fiber} $F$ if there exists a [[differentiably good open cover]] $\{U_i \hookrightarrow X\}_{i \in I}$ (def. \ref{DifferentiablyGoodOpenCover}) such that the restriction of $fb$ to each $U_i$ is [[isomorphism|isomorphic]] to the [[trivial fiber bundle]] with fiber $F$ over $U_i$. Such [[diffeomorphisms]] $f_i \colon U_i \times F \overset{\simeq}{\to} E\vert_{U_i}$ are called \emph{[[local trivializations]]} of the fiber bundle:

\begin{displaymath}
\itexarray{
    U_i \times F &\underoverset{\simeq}{f_i}{\longrightarrow}& E\vert_{U_i}
    \\
    & {}_{\mathllap{pr_1}}\searrow & \downarrow\mathrlap{fb\vert_{U_i}}
    \\
    && U_i
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{VectorBundle}\hypertarget{VectorBundle}{}
\textbf{([[vector bundle]])}

A \emph{[[vector bundle]]} is a [[fiber bundle]] $E \overset{vb}{\to} X$ (def. \ref{FiberBundle}) with typical fiber a [[vector space]] $V$ such that there exists a [[local trivialization]] $\{U_i \times V \underoverset{\simeq}{f_i}{\to} E\vert_{U_i}\}_{i \in I}$ whose \emph{gluing functions}

\begin{displaymath}
U_i \cap U_j \times V
    \overset{f_i\vert_{U_i \cap U_j}}{\longrightarrow}
  E\vert_{U_i \cap U_j}
    \overset{f_j^{-1}\vert_{U_i \cap U_j}}{\longrightarrow}
  U_i \cap U_j \times V
\end{displaymath}
for all $i,j \in I$ are [[linear functions]] over each point $x \in U_i \cap U_j$.

A [[homomorphism]] of [[vector bundle]] is a bundle morphism $f$ (def. \ref{BundlesAndFibers}) such that there exist [[local trivializations]] on both sides with respect to which $g$ is [[fiber]]-wise a [[linear map]].

The inclined reader may notice that this makes vector bundles over $X$ a [[category]] (denoted $Vect_{/X}$); the reader not so inclined may ignore this.

\end{defn}
\begin{example}
\label{ModuleOfSectionsOfAVectorBundle}\hypertarget{ModuleOfSectionsOfAVectorBundle}{}
\textbf{([[module]] of [[sections]] of a [[vector bundle]])}

Given a [[vector bundle]] $E \overset{vb}{\to} X$ (def. \ref{VectorBundle}), then its [[space of sections|set of sections]] $\Gamma_X(E)$ (def. \ref{BundlesAndFibers}) becomes a [[real vector space]] by [[fiber]]-wise multiplication with [[real numbers]]. Moreover, it becomes a [[module]] over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(X)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) by the same [[fiber]]-wise multiplication:

\begin{displaymath}
\itexarray{
     C^\infty(X) \otimes_{\mathbb{R}} \Gamma_X(E) &\longrightarrow& \Gamma_X(E)
     \\
     (f,s) &\mapsto& (x \mapsto f(x) \cdot s(x))
  }
  \,.
\end{displaymath}
For $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ two [[vector bundles]] and

\begin{displaymath}
\itexarray{
    E_1 && \overset{f}{\longrightarrow} && E_2
    \\
    & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}}
    \\
    && X
  }
\end{displaymath}
a vector bundle homomorphism (def. \ref{VectorBundle}) then the induced function on sections (def. \ref{Sections})

\begin{displaymath}
f_\ast \;\colon\; \Gamma_X(E_1) \longrightarrow \Gamma_X(E_2)
\end{displaymath}
is compatible with this [[action]] by smooth functions and hence constitutes a [[homomorphism]] of $C^\infty(X)$-[[modules]].

The inclined reader may notice that this means that taking [[spaces of sections]] yields a [[functor]]

\begin{displaymath}
\Gamma_X(-)
  \;\colon\;
  Vect_{/X}
    \longrightarrow
  C^\infty(X) Mod
\end{displaymath}
from the [[category of vector bundles]] over $X$ to that over [[modules]] over $C^\infty(X)$.

\end{example}
\begin{example}
\label{TangentVectorFields}\hypertarget{TangentVectorFields}{}
\textbf{([[tangent vector fields]] and [[tangent bundle]])}

For $\mathbb{R}^n$ a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) the [[trivial vector bundle]] (example \ref{TrivialBundleOnCartesianSpace}, def. \ref{VectorBundle})

\begin{displaymath}
\itexarray{
    T \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times \mathbb{R}^n
    \\
    \mathllap{tb}\downarrow && \downarrow\mathrlap{pr_1}
    \\
    \mathbb{R}^n &=& \mathbb{R}^n
  }
\end{displaymath}
is called the \emph{[[tangent bundle]]} of $\mathbb{R}^n$. With $(x^a)_{a = 1}^n$ the [[coordinate functions]] on $\mathbb{R}^n$ (def. \ref{CoordinateFunctionsAreSmoothFunctions}) we write $(\partial_a)_{a = 1}^n$ for the corresponding [[linear basis]] of $\mathbb{R}^n$ regarded as a [[vector space]]. Then a general [[section]] (def. \ref{Sections})

\begin{displaymath}
\itexarray{
    && T \mathbb{R}^n
    \\
    & {}^{\mathllap{v}}\nearrow& \downarrow\mathrlap{tb}
    \\
    \mathbb{R}^n &=& \mathbb{R}^n
  }
\end{displaymath}
of the [[tangent bundle]] has a unique expansion of the form

\begin{displaymath}
v = v^a \partial_a
\end{displaymath}
where a sum over indices is understood ([[Einstein summation convention]]) and where the components $(v^a \in C^\infty(\mathbb{R}^n))_{a = 1}^n$ are [[smooth functions]] on $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}).

Such a $v$ is also called a smooth \emph{[[tangent vector field]]} on $\mathbb{R}^n$.

Each tangent vector field $v$ on $\mathbb{R}^n$ determines a [[partial derivative]] on [[smooth functions]]

\begin{displaymath}
\itexarray{
    C^\infty(\mathbb{R}^n) &\overset{D_v}{\longrightarrow}& C^\infty(\mathbb{R}^n)
    \\
    f &\mapsto& \mathrlap{ D_v f \coloneqq v^a \partial_a (f) \coloneqq  \sum_a v^a \frac{\partial f}{\partial x^a} }
  }
  \,.
\end{displaymath}
By the [[product law]] of [[differentiation]], this is a [[derivation]] on the [[algebra of functions|algebra of]] [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) in that

\begin{enumerate}%
\item it is an $\mathbb{R}$-[[linear map]] in that

\begin{displaymath}
D_v( c_1 f_1 + c_2 f_2 ) = c_1 D_v f_1 + c_2 D_v f_2
\end{displaymath}

\item it satisfies the [[Leibniz rule]]

\begin{displaymath}
D_v(f_1 \cdot f_2) = (D_v f_1) \cdot f_2 + f_1 \cdot (D_v f_2)
\end{displaymath}


\end{enumerate}
for all $c_1, c_2 \in \mathbb{R}$ and all $f_1, f_2 \in C^\infty(\mathbb{R}^n)$.

Hence regarding [[tangent vector fields]] as [[partial derivatives]] constitutes a [[linear function]]

\begin{displaymath}
D \;\colon\; \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \longrightarrow Der(C^\infty(\mathbb{R}^n))
\end{displaymath}
from the [[space of sections]] of the [[tangent bundle]]. In fact this is a [[homomorphism]] of $C^\infty(\mathbb{R}^n)$-[[modules]] (example \ref{ModuleOfSectionsOfAVectorBundle}), in that for $f \in C^\infty(\mathbb{R}^n)$ and $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ we have

\begin{displaymath}
D_{f v}(-) = f \cdot D_v(-)
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{VerticalTangentBundle}\hypertarget{VerticalTangentBundle}{}
\textbf{([[vertical tangent bundle]])}

Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. Then its \emph{[[vertical tangent bundle]]} $T_\Sigma E \overset{T fb}{\to} \Sigma$ is the [[fiber bundle]] (def. \ref{FiberBundle}) over $\Sigma$ whose [[fiber]] over a point is the [[tangent bundle]] (def. \ref{TangentVectorFields}) of the fiber of $E \overset{fb}{\to}\Sigma$ over that point:

\begin{displaymath}
(T_\Sigma E)_x \coloneqq T(E_x)
  \,.
\end{displaymath}
If $E \simeq \Sigma \times F$ is a [[trivial fiber bundle]] with [[fiber]] $F$, then its vertical vector bundle is the trivial fiber bundle with fiber $T F$.

\end{example}
\begin{defn}
\label{DualVectorBundle}\hypertarget{DualVectorBundle}{}
\textbf{([[dual vector bundle]])}

For $E \overset{vb}{\to} \Sigma$ a [[vector bundle]] (def. \ref{VectorBundle}), its \emph{[[dual vector bundle]]} is the vector bundle whose [[fiber]] \eqref{FiberOfAFiberBundle} over $x \in \Sigma$ is the [[dual vector space]] of the corresponding fiber of $E \to \Sigma$:

\begin{displaymath}
(E^\ast)_x \;\coloneqq\; (E_x)^\ast
  \,.
\end{displaymath}
The defining pairing of [[dual vector spaces]] $(E_x)^\ast \otimes E_x \to \mathbb{R}$ applied pointwise induces a pairing on the [[modules]] of [[sections]] (def. \ref{ModuleOfSectionsOfAVectorBundle}) of the original vector bundle and its dual with values in the [[smooth functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}):

\begin{equation}
\itexarray{
    \Gamma_\Sigma(E) \otimes_{C^\infty(X)} \Gamma_\Sigma(E^\ast)
      &\longrightarrow&
    C^\infty(\Sigma)
    \\
    (v,\alpha) &\mapsto& (v \cdot \alpha \colon x \mapsto \alpha_x(v_x) )
  }
\label{PairingOfDualSections}\end{equation}
\end{defn}
$\,$

\textbf{[[synthetic differential geometry]]}

Below we encounter generalizations of ordinary [[differential geometry]] that include explicit ``[[infinitesimals]]'' in the guise of \emph{[[infinitesimally thickened points]]}, as well as ``super-graded infinitesimals'', in the guise of \emph{[[superpoints]]} (necessary for the description of [[fermion fields]] such as the [[Dirac field]]). As we discuss \hyperlink{FieldBundles}{below}, these structures are naturally incorporated into [[differential geometry]] in just the same way as [[Grothendieck]] introduced them into [[algebraic geometry]] (in the guise of ``[[formal schemes]]''), namely in terms of [[formal dual|formally dual]] [[rings of functions]] with [[nilpotent ideals]]. That this also works well for [[differential geometry]] rests on the following three basic but important properties, which say that [[smooth functions]] behave ``more algebraically'' than their definition might superficially suggest:

\begin{prop}
\label{AlgebraicFactsOfDifferentialGeometry}\hypertarget{AlgebraicFactsOfDifferentialGeometry}{}
\textbf{(the three magic algebraic properties of [[differential geometry]])}

\begin{enumerate}%
\item \textbf{[[embedding of smooth manifolds into formal duals of R-algebras|embedding of Cartesian spaces into formal duals of R-algebras]]}

For $X$ and $Y$ two [[Cartesian spaces]], the [[smooth functions]] $f \colon X \longrightarrow Y$ between them (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) are in [[natural bijection]] with their induced algebra [[homomorphisms]] $C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), so that one may equivalently handle [[Cartesian spaces]] entirely via their $\mathbb{R}$-algebras of smooth functions.

Stated more [[category theory|abstractly]], this means equivalently that the [[functor]] $C^\infty(-)$ that sends a [[smooth manifold]] $X$ to its $\mathbb{R}$-[[associative algebra|algebra]] $C^\infty(X)$ of [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) is a \emph{[[fully faithful functor]]}:

\begin{displaymath}
C^\infty(-)
  \;\colon\;
  SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op}
  \,.
\end{displaymath}
(\href{embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93}{Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10})


\item \textbf{[[smooth Serre-Swan theorem|embedding of smooth vector bundles into formal duals of R-algebra modules]]}

For $E_1 \overset{vb_1}{\to} X$ and $E_2 \overset{vb_2}{\to} X$ two [[vector bundle]] (def. \ref{VectorBundle}) there is then a [[natural bijection]] between vector bundle [[homomorphisms]] $f \colon E_1 \to E_2$ and the [[homomorphisms]] of [[modules]] $f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2)$ that these induces between the [[spaces of sections]] (example \ref{ModuleOfSectionsOfAVectorBundle}).

More [[category theory|abstractly]] this means that the [[functor]] $\Gamma_X(-)$ is a [[fully faithful functor]]

\begin{displaymath}
\Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod
\end{displaymath}
(\href{smooth+Serre-Swan+theorem#Nestruev03}{Nestruev 03, theorem 11.29})

Moreover, the [[modules]] over the $\mathbb{R}$-algebra $C^\infty(X)$ of [[smooth functions]] on $X$ which arise this way as [[sections]] of [[smooth vector bundles]] over a [[Cartesian space]] $X$ are precisely the [[finitely generated module|finitely generated]] [[free modules]] over $C^\infty(X)$.

(\href{smooth+Serre-Swan+theorem#Nestruev03}{Nestruev 03, theorem 11.32})


\item \textbf{[[derivations of smooth functions are vector fields|vector fields are derivations of smooth functions]]}.

For $X$ a [[Cartesian space]] (example \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), then any [[derivation]] $D \colon  C^\infty(X) \to C^\infty(X)$ on the $\mathbb{R}$-[[associative algebra|algebra]] $C^\infty(X)$ of [[smooth functions]] (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) is given by [[differentiation]] with respect to a uniquely defined smooth [[tangent vector field]]: The function that regards [[tangent vector fields]] with [[derivations]] from example \ref{TangentVectorFields}

\begin{displaymath}
\itexarray{
    \Gamma_X(T X)
      &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}&
    Der(C^\infty(X))
    \\
    v &\mapsto& D_v
  }
\end{displaymath}
is in fact an [[isomorphism]].

(This follows directly from the \emph{[[Hadamard lemma]]}.)



\end{enumerate}
\end{prop}
Actually all three statements in prop. \ref{AlgebraicFactsOfDifferentialGeometry} hold not just for [[Cartesian spaces]], but generally for [[smooth manifolds]] (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below; if only we generalize in the second statement from [[free modules]] to [[projective modules]]. However for our development here it is useful to first focus on just [[Cartesian spaces]] and then bootstrap the theory of [[smooth manifolds]] and much more from that, which we do \hyperlink{FieldBundles}{below}.

$\,$

$\,$

\textbf{[[differential forms]]}

We introduce and discuss [[differential forms]] on [[Cartesian spaces]].

\begin{defn}
\label{Differential1FormsOnCartesianSpaces}\hypertarget{Differential1FormsOnCartesianSpaces}{}
\textbf{([[differential 1-forms]] on [[Cartesian spaces]] and the [[cotangent bundle]])}

For $n \in \mathbb{N}$ a \emph{[[smooth differential 1-form]]} $\omega$ on a [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is an [[n-tuple]]

\begin{displaymath}
\left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n
\end{displaymath}
of [[smooth functions]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), which we think of equivalently as the [[coefficients]] of a [[formal linear combination]]

\begin{displaymath}
\omega = \omega_i d x^i
\end{displaymath}
on a [[set]] $\{d x^1, d x^2, \cdots, d x^n\}$ of [[cardinality]] $n$.

Here a sum over repeated indices is tacitly understood ([[Einstein summation convention]]).

Write

\begin{displaymath}
\Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set
\end{displaymath}
for the set of smooth [[differential 1-forms]] on $\mathbb{R}^k$.

We may think of the expressions $(d x^a)_{a = 1}^n$ as a [[linear basis]] for the [[dual vector space]] $\mathbb{R}^n$. With this the [[differential 1-forms]] are equivalently the [[sections]] (def. \ref{Sections}) of the [[trivial vector bundle]] (example \ref{TrivialBundleOnCartesianSpace}, def. \ref{VectorBundle})

\begin{displaymath}
\itexarray{
    T^\ast \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times (\mathbb{R}^n)^\ast
    \\
    \mathllap{cb}\downarrow && \downarrow\mathrlap{pr_1}
    \\
    \mathbb{R}^n &=& \mathbb{R}^n
  }
\end{displaymath}
called the \emph{[[cotangent bundle]]} of $\mathbb{R}^n$ (def. \ref{Differential1FormsOnCartesianSpaces}):

\begin{displaymath}
\Omega^1(\mathbb{R}^n) = \Gamma_{\mathbb{R}^n}(T^\ast \mathbb{R}^n)
  \,.
\end{displaymath}
This amplifies via example \ref{ModuleOfSectionsOfAVectorBundle} that $\Omega^1(\mathbb{R}^n)$ has the [[structure]] of a [[module]] over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(\mathbb{R}^n)$, by the evident multiplication of [[differential 1-forms]] with [[smooth functions]]:

\begin{enumerate}%
\item The set $\Omega^1(\mathbb{R}^k)$ of [[differential 1-forms]] in a [[Cartesian space]] (def. \ref{Differential1FormsOnCartesianSpaces}) is naturally an [[abelian group]] with addition given by componentwise addition

\begin{displaymath}
\begin{aligned}
      \omega + \lambda & =
    \omega_i d x^i + \lambda_i d x^i
      \\
      & = (\omega_i + \lambda_i) d x^i
   \end{aligned}
   \,,
\end{displaymath}

\item The abelian group $\Omega^1(\mathbb{R}^k)$ is naturally equipped with the structure of a [[module]] over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(\mathbb{R}^k)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), where the [[action]] $C^\infty(\mathbb{R}^k) \times\Omega^1(\mathbb{R}^k) \to \Omega^1(\mathbb{R}^k)$ is given by componentwise multiplication

\begin{displaymath}
f \cdot \omega = ( f \cdot \omega_i) d x^i
  \,.
\end{displaymath}


\end{enumerate}
Accordingly there is a canonical pairing between [[differential 1-forms]] and [[tangent vector fields]] (example \ref{TangentVectorFields})

\begin{equation}
\itexarray{
    \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)
    \otimes_{\mathbb{R}}
    \Gamma_{\mathbb{R}^n}(T \ast \mathbb{R}^n)
      &\overset{\iota_{(-)}(-) }{\longrightarrow}&
    C^\infty(\mathbb{R}^n)
    \\
    (v,\omega)
    &\mapsto&
    \mathrlap{
      \iota_v \omega \coloneqq v^a \omega_a
    }
  }
\label{PairingVectorFieldsWithDifferential1Forms}\end{equation}
With [[differential 1-forms]] in hand, we may collect all the first-order [[partial derivatives]] of a [[smooth function]] into a single object: the \emph{[[exterior derivative]]} or \emph{[[de Rham differential]]} is the $\mathbb{R}$-[[linear function]]

\begin{equation}
\itexarray{
    C^\infty(\mathbb{R}^n) &\overset{d}{\longrightarrow}& \Omega^1(\mathbb{R}^n)
    \\
    f &\mapsto& \mathrlap{ d f \coloneqq \frac{\partial f}{ \partial x^a} d x^a }
  }
  \,.
\label{deRhamDifferentialOnFunctionsOnCartesianSpace}\end{equation}
Under the above pairing with [[tangent vector fields]] $v$ this yields the particular [[partial derivative]] along $v$:

\begin{displaymath}
\iota_v d f = D_v f = v^a \frac{\partial f}{\partial x^a}
  \,.
\end{displaymath}
\end{defn}
We think of $d  x^i$ as a measure for [[infinitesimal space|infinitesimal]] displacements along the $x^i$-[[coordinate]] of a [[Cartesian space]]. If we have a measure of infintesimal displacement on some $\mathbb{R}^n$ and a smooth function $f \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n$, then this induces a measure for infinitesimal displacement on $\mathbb{R}^{\tilde n}$ by sending whatever happens there first with $f$ to $\mathbb{R}^n$ and then applying the given measure there. This is captured by the following definition:

\begin{defn}
\label{PullbackOfDifferential1FormsOnCartesianSpaces}\hypertarget{PullbackOfDifferential1FormsOnCartesianSpaces}{}
\textbf{([[pullback of differential forms|pullback of differential 1-forms]])}

For $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ a [[smooth function]], the \textbf{[[pullback of differential forms|pullback of differential 1-forms]]} along $\phi$ is the [[function]]

\begin{displaymath}
\phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})
\end{displaymath}
between sets of differential 1-forms, def. \ref{Differential1FormsOnCartesianSpaces}, which is defined on [[basis]]-elements by

\begin{displaymath}
\phi^* d  x^i
    \;\coloneqq\;
  \frac{\partial \phi^i}{\partial \tilde x^j} d \tilde x^j
\end{displaymath}
and then extended linearly by

\begin{displaymath}
\begin{aligned}
    \phi^* \omega & = \phi^* \left(  \omega_i d x^i \right)
    \\
    & \coloneqq
      \left(\phi^* \omega\right)_i  \frac{\partial \phi^i }{\partial \tilde x^j}  d  \tilde x^j
    \\
    & =
     (\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j}  d  \tilde x^j
  \end{aligned}
  \,.
\end{displaymath}
This is compatible with [[identity morphisms]] and [[composition]] in that

\begin{equation}
(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)}
  \phantom{AAAA}
  (g \circ f)^\ast = f^\ast \circ g^\ast
  \,.
\label{PullbackOfDifferentialFormsCompatibleWithComposition}\end{equation}
Stated more [[category theory|abstractly]], this just means that [[pullback of differential forms|pullback of differential 1-forms]] makes the assignment of sets of differential 1-forms to [[Cartesian spaces]] a [[contravariant functor|contravariant]] [[functor]]

\begin{displaymath}
\Omega^1(-) \;\colon\; CartSp^{op} \longrightarrow Set
  \,.
\end{displaymath}
\end{defn}
The following definition captures the idea that if $d  x^i$ is a measure for displacement along the $x^i$-[[coordinate]], and $d x^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way to get a measure, to be called $d x^i \wedge d  x^j$, for [[infinitesimal]] \emph{[[surfaces]]} (squares) in the $x^i$-$x^j$-plane. And this should keep track of the [[orientation]] of these squares, with

\begin{displaymath}
d x^j \wedge d x^i = - d x^i \wedge d  x^j
\end{displaymath}
being the same infinitesimal measure with orientation reversed.

\begin{defn}
\label{DifferentialnForms}\hypertarget{DifferentialnForms}{}
\textbf{([[exterior algebra]] of [[differential n-forms]])}

For $k,n \in \mathbb{N}$, the \textbf{smooth [[differential forms]]} on a [[Cartesian space]] $\mathbb{R}^k$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is the [[exterior algebra]]

\begin{displaymath}
\Omega^\bullet(\mathbb{R}^k)
   \coloneqq
  \wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k)
\end{displaymath}
over the [[algebra of functions|algebra of]] [[smooth functions]] $C^\infty(\mathbb{R}^k)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) of the [[module]] $\Omega^1(\mathbb{R}^k)$ of smooth 1-forms.

We write $\Omega^n(\mathbb{R}^k)$ for the sub-module of degree $n$ and call its elements the \emph{[[differential n-forms]]}.

Explicitly this means that a [[differential n-form]] $\omega \in \Omega^n(\mathbb{R}^k)$ on $\mathbb{R}^k$ is a [[formal linear combination]] over $C^\infty(\mathbb{R}^k)$ (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}) of [[basis]] elements of the form $d  x^{i_1} \wedge \cdots \wedge d x^{i_n}$ for $i_1 \lt i_2 \lt \cdots \lt i_n$:

\begin{displaymath}
\omega = \omega_{i_1, \cdots, i_n} d x^{i_1} \wedge \cdots \wedge d x^{i_n}
  \,.
\end{displaymath}
\end{defn}
Now all the constructions for [[differential 1-forms]] above extent naturally to [[differential n-forms]]:

\begin{defn}
\label{deRhamDifferential}\hypertarget{deRhamDifferential}{}
\textbf{([[exterior derivative]] or [[de Rham differential]])}

For $\mathbb{R}^n$ a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) the [[de Rham differential]] $d \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$ \eqref{deRhamDifferentialOnFunctionsOnCartesianSpace} uniquely extended as a [[derivation]] of degree +1 to the [[exterior algebra]] of [[differential forms]] (def. \ref{DifferentialnForms})

\begin{displaymath}
d
    \;\colon\;
  \Omega^\bullet(\mathbb{R}^n)
    \longrightarrow
  \Omega^\bullet(\mathbb{R}^n)
\end{displaymath}
meaning that for $\omega_i \in \Omega^{k_i}(\mathbb{R})$ then

\begin{displaymath}
d(\omega_1 \wedge \omega_2)
  \;=\;
  (d \omega_1) \wedge \omega_2
  +
  \omega_1 \wedge d \omega_2
  \,.
\end{displaymath}
In components this simply means that

\begin{displaymath}
\begin{aligned}
    d \omega
    & =
    d \left(\omega_{i_1 \cdots i_k} d x^{i_1} \wedge \cdots \wedge d x^{i_k}\right)
    \\
    & =
    \frac{\partial \omega_{i_1 \cdots i_k}}{\partial x^{a}}
    d x^a \wedge d x^{i_1} \wedge \cdots \wedge d x^{i_k}
  \end{aligned}
  \,.
\end{displaymath}
Since [[partial derivatives]] commute with each other, while differential 1-form anti-commute, this implies that $d$ is nilpotent

\begin{displaymath}
d^2 = d \circ d = 0
  \,.
\end{displaymath}
We say hence that [[differential forms]] form a \emph{[[cochain complex]]}, the \emph{[[de Rham complex]]} $(\Omega^\bullet(\mathbb{R}^n), d)$.

\end{defn}
\begin{defn}
\label{ContractionOfFormsWithVectorFields}\hypertarget{ContractionOfFormsWithVectorFields}{}
\textbf{(contraction of [[differential n-forms]] with [[tangent vector fields]])}

The pairing $\iota_v \omega = \omega(v)$ of [[tangent vector fields]] $v$ with [[differential 1-forms]] $\omega$ \eqref{PairingVectorFieldsWithDifferential1Forms} uniquely [[extension|extends]] to the [[exterior algebra]] $\Omega^\bullet(\mathbb{R}^n)$ of [[differential forms]] (def. \ref{DifferentialnForms}) as a [[derivation]] of degree -1

\begin{displaymath}
\iota_v \;\colon\; \Omega^{\bullet+1}(\mathbb{R}^n) \longrightarrow \Omega^\bullet(\mathbb{R}^n)
  \,.
\end{displaymath}
In particular for $\omega_1, \omega_2 \in \Omega^1(\mathbb{R}^n)$ two [[differential 1-forms]], then

\begin{displaymath}
\iota_{v} (\omega_1 \wedge \omega_2)
   \;=\;
  \omega_1(v) \omega_2 - \omega_2(v) \omega_1
  \;\in\;
  \Omega^1(\mathbb{R}^n)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{PullbackOfDifferentialForms}\hypertarget{PullbackOfDifferentialForms}{}
\textbf{([[pullback of differential forms|pullback of differential n-forms]])}

For $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[smooth function]] between [[Cartesian spaces]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) the operationf of [[pullback of differential forms|pullback of differential 1-forms]] of def. \ref{Differential1FormsOnCartesianSpaces} extends as an $C^\infty(\mathbb{R}^k)$-[[associative algebra|algebra]] [[homomorphism]] to the [[exterior algebra]] of [[differential forms]] (def. \ref{DifferentialnForms}),

\begin{displaymath}
f^\ast
  \;\colon\;
  \Omega^\bullet(\mathbb{R}^{n_2})
    \longrightarrow
  \Omega^\bullet(\mathbb{R}^{n_1})
\end{displaymath}
given on basis elements by

\begin{displaymath}
f^* \left(  dx^{i_1} \wedge \cdots \wedge dx^{i_n} \right)
  =
  \left(f^* dx^{i_1} \wedge \cdots \wedge f^* dx^{i_n} \right)
  \,.
\end{displaymath}
This commutes with the [[de Rham differential]] $d$ on both sides (def. \ref{deRhamDifferential}) in that

\begin{displaymath}
d \circ f^\ast
  =
  f^\ast \circ d
  \phantom{AAAAA}
  \itexarray{
    \Omega^\bullet(X) &\overset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y)
    \\
    \mathllap{d}\downarrow && \downarrow\mathrlap{d}
    \\
    \Omega^\bullet(X) &\underset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y)
  }
\end{displaymath}
hence that [[pullback of differential forms]] is a \emph{[[chain map]]} of [[de Rham complexes]].

This is still compatible with [[identity morphisms]] and [[composition]] in that

\begin{equation}
(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)}
  \phantom{AAAA}
  (g \circ f)^\ast = f^\ast \circ g^\ast
  \,.
\label{PullbackOfDiffereentialFormsCompatibleWithComposition}\end{equation}
Stated more [[category theory|abstractly]], this just means that [[pullback of differential forms|pullback of differential n-forms]] makes the assignment of sets of [[differential n-forms]] to [[Cartesian spaces]] a [[contravariant functor|contravariant]] [[functor]]

\begin{displaymath}
\Omega^n(-) \;\colon\; CartSp^{op} \longrightarrow Set
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{CartanHomotopyFormula}\hypertarget{CartanHomotopyFormula}{}
\textbf{([[Cartan's homotopy formula]])}

Let $X$ be a [[Cartesian space]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}), and let $v \in \Gamma(T X)$ be a smooth [[tangent vector field]] (example \ref{TangentVectorFields}).

For $t \in \mathbb{R}$ write $\exp(t v) \colon X \overset{\simeq}{\to} X$ for the [[flow]] by [[diffeomorphisms]] along $v$ of parameter length $t$.

Then the [[derivative]] with respect to $t$ of the [[pullback of differential forms]] along $\exp(t v)$, hence the [[Lie derivative]] $\mathcal{L}_v \colon \Omega^\bullet(X) \to \Omega^\bullet(X)$, is given by the [[anticommutator]] of the contraction derivation $\iota_v$ (def. \ref{ContractionOfFormsWithVectorFields}) with the [[de Rham differential]] $d$ (def. \ref{deRhamDifferential}):

\begin{displaymath}
\begin{aligned}
    \mathcal{L}_v
      &\coloneqq
    \frac{d}{d t } \exp(t v)^\ast \omega \vert_{t = 0}
    \\
    & =
    \iota_v d \omega + d \iota_v \omega
    \,.
  \end{aligned}
\end{displaymath}
\end{prop}
Finally we turn to the concept of [[integration of differential forms]] (def. \ref{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces} below). First we need to say what it is that differential forms may be integrated over:

\begin{defn}
\label{SingularSimplicesInCartesianSpaces}\hypertarget{SingularSimplicesInCartesianSpaces}{}
\textbf{(smooth [[singular simplicial chains]] in [[Cartesian spaces]])}

For $n \in \mathbb{N}$, the \emph{standard [[n-simplex]]} in the [[Cartesian space]] $\mathbb{R}^n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) is the [[subset]]

\begin{displaymath}
\Delta^n
  \;\coloneqq\;
  \left\{
    (x^i)_{i = 1}^n
    \;\vert\;
    0 \leq x^1 \leq \cdots \leq x^n
  \right\}
  \;\subset\;
  \mathbb{R}^n
  \,.
\end{displaymath}
More generally, a \emph{smooth [[singular simplicial complex|singular n-simplex]]} in a [[Cartesian space]] $\mathbb{R}^k$ is a [[smooth function]] (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem})

\begin{displaymath}
\sigma
    \;\colon\;
  \mathbb{R}^n
    \longrightarrow
  \mathbb{R}^k
  \,,
\end{displaymath}
to be thought of as a smooth extension of its restriction

\begin{displaymath}
\sigma\vert_{\Delta^n}
    \;\colon\;
  \Delta^n
    \longrightarrow
  \mathbb{R}^k
  \,.
\end{displaymath}
(This is called a \emph{[[singular simplicial complex|singular]]} simplex because there is no condition that $\Sigma$ be an [[embedding]] in any way, in particular $\sigma$ may be a [[constant function]].)

A [[singular chain]] in $\mathbb{R}^k$ of [[dimension]] $n$ is a [[formal linear combination]] of singular $n$-simplices in $\mathbb{R}^k$.

In particular, given a singular $n+1$-simplex $\sigma$, then its \emph{[[boundary of a simplex|boundary]]} is a [[singular chain]] of singular $n$-simplices $\partial \sigma$.

\end{defn}
\begin{defn}
\label{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces}\hypertarget{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces}{}
\textbf{([[fiber integration|fiber]]-[[integration of differential forms]]) over smooth [[singular chains]] in [[Cartesian spaces]])}

For $n \in \mathbb{N}$ and $\omega \in \Omega^n(\mathbb{R}^n)$ a [[differential n-form]] (def. \ref{DifferentialnForms}), which may be written as

\begin{displaymath}
\omega = f d x^1 \wedge \cdots d x^n
  \,,
\end{displaymath}
then its [[integration of differential forms|integration]] over the standard [[n-simplex]] $\Delta^n \subset \mathbb{R}^n$ (def. \ref{SingularSimplicesInCartesianSpaces}) is the ordinary [[integral]] (e.g. [[Riemann integral]])

\begin{displaymath}
\int_{\Delta^n} \omega
  \;\coloneqq\;
  \underset{0 \leq x^1 \leq \cdots \leq x^n \leq 1}{\int} f(x^1, \cdots, x^n) \, d x^1 \cdots d x^n
  \,.
\end{displaymath}
More generally, for

\begin{enumerate}%
\item $\omega \in \Omega^n(\mathbb{R}^k)$ a [[differential n-forms]];


\item $C = \underset{i}{\sum} c_i \sigma_i$ a singular $n$-chain (def. \ref{SingularSimplicesInCartesianSpaces})



\end{enumerate}
in any [[Cartesian space]] $\mathbb{R}^k$. Then the \emph{[[integration of differential forms|integration]]} of $\omega$ over $x$ is the [[sum]] of the integrations, as above, of the [[pullback of differential forms]] (def. \ref{PullbackOfDifferentialForms}) along all the singular [[n-simplices]] in the chain:

\begin{displaymath}
\int_C \omega
  \;\coloneqq\;
  \underset{i}{\sum}
  c_i
  \int_{\Delta^n} (\sigma_i)^\ast \omega
  \,.
\end{displaymath}
Finally, for $U$ another Cartesian space, then \emph{[[fiber integration]] of differential forms along $U \times C \to U$} is the linear map

\begin{displaymath}
\int_C \;\colon\; \Omega^{\bullet + dim(C)}(U \times C) \longrightarrow \Omega^\bullet(U)
\end{displaymath}
which on differential forms of the form $\omega_U \wedge \omega$ is given by

\begin{displaymath}
\int_C \omega_U \wedge \omega
   \;\coloneqq\;
  (-1)^{\vert \omega_U\vert} \int_C \omega
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{StokesTheorem}\hypertarget{StokesTheorem}{}
\textbf{([[Stokes theorem]] for [[fiber integration|fiber]]-[[integration of differential forms]])}

For $\Sigma$ a smooth [[singular simplicial chain]] (def. \ref{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces}) the operation of [[fiber integration|fiber]]-[[integration of differential forms]] along $U \times \Sigma \overset{pr_1}{\longrightarrow} U$ (def. \ref{IntegrationOfDifferentialFormsOverSmoothSingularChainsInCartesianSpaces}) is compatible with the [[exterior derivative]] $d_U$ on $U$ (def. \ref{deRhamDifferential}) in that

\begin{displaymath}
\begin{aligned}
    d \int_\Sigma \omega
     & =
    (-1)^{dim(\Sigma)} \int_\Sigma d_U \omega
    \\
    & =
    (-1)^{dim(\Sigma)}
    \left(
      \int_\Sigma d \omega
      -
      \int_{\partial \Sigma} \omega
    \right)
  \end{aligned}
  \,,
\end{displaymath}
where $d = d_U + d_\Sigma$ is the [[de Rham differential]] on $U \times \Sigma$ (def. \ref{deRhamDifferential}) and where the second equality is the \emph{[[Stokes theorem]]} along $\Sigma$:

\begin{displaymath}
\int_\Sigma d_\Sigma \omega = \int_{\partial \Sigma} \omega
  \,.
\end{displaymath}
\end{prop}
$\,$

This concludes our review of the basics of ([[synthetic differential geometry|synthetic]]) [[differential geometry]] on which the following development of quantum field theory is based. In the \hyperlink{Spacetime}{next chapter} we consider \emph{[[spacetime]]} and \emph{[[spin]]}.



\end{document}