geometry of physics -- coordinate systems

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

> This entry is one chapter of the entry _[[geometry of physics]]_.

> previous chapter: _[[geometry of physics -- categories and toposes|categories and toposes]]_,

> next chapter: _[[geometry of physics -- smooth sets|smooth sets]]_

As discussed in the chapter _[[geometry of physics -- categories and toposes|categories and toposes]]_, every kind of _[[geometry]]_ is modeled on a collection of [[generator|archetypical]] basic [[spaces]] and geometric [[homomorphisms]] between them. In [[differential geometry]] the archetypical spaces are the abstract standard [[Cartesian space|Cartesian coordinate systems]], denoted $\mathbb{R}^n$ (in every [[dimension]] $n \in \mathbb{N}$). The geometric homomorphism between them are [[smooth functions]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$, hence smooth (and possibly degenerate) [[coordinate transformations]].

Here we introduce the basic concept, organizing them in the [[category]] [[CartSp]] of [[Cartesian spaces]] (Prop. \ref{CartSpCategory} below.) We highlight three classical theorems about [[smooth functions]] in Prop. \ref{AlgebraicFactsOfDifferentialGeometry} below, which look innocent but play a decisive role in setting up [[synthetic differential geometry|synthetic differential]] [[supergeometry]] based on the concept of abstract smooth coordinate systems.

At this point these are not yet coordinate systems _on_ some other space. But by applying the general machine of [[geometry of physics -- categories and toposes|categories and toposes]] to these, a concept of [[generalized spaces]] modeled on these abstract coordinate systems is induced. These are the _[[smooth sets]]_ discussed in the next chapter  _[[geometry of physics -- smooth sets]]_.

#Contents#
* table of contents
{:toc}

## Abstract coordinate systems
 {#CoordinateSystemsLayerMod}

In this [Mod Layer ](#LayerMod) we discuss the [[concrete particulars]] of _[[coordinate systems]]_: the [[continuum]] [[real line]] $\mathbb{R}$, the [[Cartesian spaces]] $\mathbb{R}^n$ formed from it and the [[smooth functions]] between these.

### The continuum real line
 {#TheContinuumRealWorldLine}

The fundamental premise of [[differential geometry]] as a model of [[geometry]] in [[physics]] is the following.

**Premise.** _The abstract [[worldline]] of any [[particle]] is modeled by the [[continuum]] [[real line]] $\mathbb{R}$._

This comes down to the following sequence of premises.

1. There is a [[linear ordering]] of the [[points]] on a [[worldline]]: in particular if we pick points at some intervals on the worldline we may label these in an order-preserving way by [[integers]]

   $\mathbb{Z}$.

1. These intervals may each be subdivided into $n$ smaller intervals, for each natural number $n$. Hence we may label points on the [[worldline]] in an order-preserving way by the [[rational numbers]]

   $\mathbb{Q}$.

1. This labeling is dense: every point on the worldline is the [[supremum]] of an [[inhabited set|inhabited]] [[bounded subset]] of such labels. This means that a [[worldline]] [[equivalence|is]] the _[[real line]]_, the [[continuum]] of [[real numbers]]

   $\mathbb{R}$.

The adjective "real" in "[[real number]]" is a historical shadow of the old idea that real numbers are related to _observed reality_, hence to [[physics]] in this way. The experimental success of this assumption shows that it is valid at least to very good approximation.

Speculations are common that in a fully exact theory of [[quantum gravity]], currently unavailable, this assumption needs to be refined.
For instance in _[[p-adic physics]]_ one explores the hypothesis that the relevant [[complete field|completion]] of the rational numbers as above is not the reals, but [[p-adic numbers]] $\mathbb{Q}_p$ for some [[prime number]] $p \in \mathbb{N}$. Or for example in the study of [[QFT on non-commutative spacetime]] one explore the idea that at small scales the smooth [[continuum]] is to be replaced by an object in [[noncommutative geometry]]. Combining these two ideas leads to the notion of [[non-commutative analytic space]] as a potential model for _[[space]]_ in [[physics]]. And so forth.

For the time being all this remains speculation and differential geometry based on the [[continuum]] [[real line]] remains the context of all fundamental [[model (in theoretical physics)|model building]] in physics related to observed [[phenomenology]]. Often it is argued that these speculations are necessitated by the very nature of [[quantum theory]] applied to [[gravity]]. But, at least so far, such statements are not actually supported by the standard theory of [[quantization]]:
we discuss below in _[Geometric quantization](GeometricQuantization)_ how not just [[classical physics]] but also [[quantum theory]], in the best modern version available, is entirely rooted in differential geometry based on the [[continuum]] [[real line]].

This is the motivation for studying models of physics in geometry modeled on the [[continuum]] [[real line]]. On the other hand, in all of what follows our discussion is set up such as to be _maximally independent_ of this specific choice (this is what _[[topos theory]]_ accomplishes for us, discussed below _[Smooth spaces -- Semantic Layer](#SmoothSpacesLayerSem)_). If we do desire to consider another choice of archetypical spaces for the geometry of physics we can simply "change the [[site]]", as discussed [below](#SmoothSpacesLayerSem) and many of the constructions, propositions and theorems in the following will continue to hold. This is notably what we do below in _[Supergeometric coordinate systems](#SupergeometricCoordinateSystems)_ when we generalize the present discussion to a flavor of differential geometry that also formalizes the notion of [[fermion]] [[particles]]: "differential [[supergeometry]]".

### Cartesian spaces and smooth functions
 {#CartesianSpaces}

+-- {: .num_defn #SmoothFunctions}
###### Definition

A [[function]] of [[sets]] $f : \mathbb{R} \to \mathbb{R}$ is called
a **[[smooth function]]** if, [[coinduction|coinductively]]:

1. the [[derivative]] $\frac{d f}{d x} : \mathbb{R} \to \mathbb{R}$ exists;

1. and is itself a smooth function.

We write $C^\infty(\mathbb{R}) \in Set$ for the [[set]] of all smooth functions on $\mathbb{R}$.

=--

+-- {: .num_remark}
###### Remark

The superscript "${}^\infty$" in "$C^\infty(\mathbb{R})$" refers to the order of the [[derivatives]] that exist for smooth functions. More generally for $k \in \mathbb{N}$ one writes $C^k(\mathbb{R})$ for the set of $k$-fold [[differentiable functions]] on $\mathbb{R}$. These will however not play much of a role for our discussion here.

=--

+-- {: .num_defn #CartesianSpaceAndHomomorphism}
###### Definition

For $n \in \mathbb{N}$, the **[[Cartesian space]]** $\mathbb{R}^n$ is the set

$$
  \mathbb{R}^n = \{ (x^1 , \cdots, x^{n}) | x^i \in \mathbb{R} \}
$$

of $n$-[[tuples]] of [[real numbers]]. For $1 \leq k \leq n$ write

$$
  i^k : \mathbb{R} \to \mathbb{R}^n
$$

for the [[function]] such that $i^k(x) = (0, \cdots, 0,x,0,\cdots,0)$ is the [[tuple]] whose $k$th entry is $x$ and all whose other entries are $0 \in \mathbb{R}$; and write

$$
  \mathbb{p}^k : \mathbb{R}^n \to \mathbb{R}
$$

for the function such that $p^k(x^1, \cdots, x^n) = x^k$.

A **[[homomorphism]]** of Cartesian spaces is a [[smooth function]]

$$
  f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}
  \,,
$$

hence a [[function]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ such that all [[partial derivatives]] exist and are [[continuous map|continuous]] (...).

=--

+-- {: .num_example}
###### Example

Regarding $\mathbb{R}^n$ as an $\mathbb{R}$-[[vector space]],
every [[linear function]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$
is in particular a [[smooth function]].

=--

+-- {: .num_remark}
###### Remark

But a homomorphism of Cartesian spaces in def. \ref{CartesianSpaceAndHomomorphism} is *not* required to be a [[linear map]]. We do *not* regard the Cartesian spaces here as [[vector spaces]].

=--

+-- {: .num_defn}
###### Definition

A [[smooth function]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is called a **[[diffeomorphism]]** if there exists another smooth function
$\mathbb{R}^{n_2} \to \mathbb{R}^{n_1}$ such that the underlying functions of sets are [[inverse]] to each other

$$
  f \circ g = id
$$

and

$$
  g \circ f = id
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

There exists a [[diffeomorphism]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ precisely if $n_1 = n_2$.

=--

+-- {: .num_defn #CartesianSpacesAndSmoothFunctions}
###### Definition

We will also say equivalently that

1. a [[Cartesian space]] $\mathbb{R}^n$ is an **[[coordinate system|abstract coordinate system]]**;

1. a [[smooth function]] $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is an **[[coordinate transformation|abstract coordinate transformation]]**;

1. the [[function]] $p^k : \mathbb{R}^{n} \to \mathbb{R}$ is the $k$th **[[coordinate]]** of the coordinate system $\mathbb{R}^n$. We will also write this function as $x^k : \mathbb{R}^{n} \to \mathbb{R}$.

1. for $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[smooth function]], and $1 \leq k \leq n_2$ we write

   1. $f^k \coloneqq p^k\circ f$

   1. $(f^1, \cdots, f^n) \coloneqq f$.

=--

+-- {: .num_remark }
###### Remark

It follows with this notation that

$$
  id_{\mathbb{R}^n} = (x^1, \cdots, x^n) : \mathbb{R}^n \to \mathbb{R}^n
  \,.
$$

Hence an abstract coordinate transformation

$$
  f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}
$$

may equivalently be written as the tuple

$$
  \left(
    f^1 \left( x^1, \cdots, x^{n_1} \right)
    ,
    \cdots,
    f^{n_2}\left( x^1, \cdots, x^{n_1} \right)
  \right)
  \,.
$$

=--

### The magic properties of smooth functions
 {#PropertiesOfSmoothFunctions}

Below we encounter generalizations of ordinary [[differential geometry]] that include explicit "[[infinitesimals]]" in the guise of _[[infinitesimally thickened points]]_, as well as "super-graded infinitesimals", in the guise of _[[superpoints]]_ (necessary for the
description of [[fermion fields]] such as the [[Dirac field]]). As we discuss [below](#FieldBundles), 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:

+-- {: .num_prop #AlgebraicFactsOfDifferentialGeometry}
###### Proposition
**(the three magic algebraic properties of [[differential geometry]])**

1. **[[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 _[[fully faithful functor]]_:

   $$
     C^\infty(-)
     \;\colon\;
     SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op}
     \,.
   $$

   ([Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10](embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93))

1. **[[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]]

   $$
     \Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod
   $$

   ([Nestruev 03, theorem 11.29](smooth+Serre-Swan theorem#Nestruev03))

   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 $C^\infty(X)$.

   ([Nestruev 03, theorem 11.32](smooth+Serre-Swan theorem#Nestruev03))

1. **[[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}

   $$
     \array{
       \Gamma_X(T X)
         &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}&
       Der(C^\infty(X))
       \\
       v &\mapsto& D_v
     }
   $$

   is in fact an [[isomorphism]].

   (This follows directly from the _[[Hadamard lemma]]_.)

=--

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 [below](#FieldBundles).

## The category of abstract coordinate systems
 {#CoordinateSystemsLayerSem}

Here we make explicit the _[[category]]_ formed by abstract coordinate systems (Prop. \ref{CartSpCategory} below) and mention some of its basic properties. This will serve the discussion of [[smooth sets]] as the _[[sheaves]]_ on the category of abstract coordinate systems, in the next chapter _[[geometry of physics -- smooth sets]]_.

$\,$

+-- {: .num_prop #CartSpCategory}
###### Propositions
**(the [[category]] [[CartSp]] of abstract [[coordinate systems]]/[[Cartesian spaces]])**

Abstract coordinate systems according to prop. \ref{CartesianSpacesAndSmoothFunctions} form a _[[category]]_ ([this def.](geometry+of+physics+--+Categories+and+Toposes#Categories))
-- to be denoted _[[CartSp]]_ -- whose

* [[objects]] are the abstract coordinate systems $\mathbb{R}^{n}$ (the [[class]] of objects is the [[set]] $\mathbb{N}$ of [[natural numbers]] $n$);

* [[morphisms]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ are the abstract [[coordinate transformations]] = [[smooth functions]].

Composition of morphisms is given by [[composition]] of [[functions]].

Under this identification

1. The [[identity morphisms]] are precisely the [[identity functions]].

1. The [[isomorphisms]] are precisely the [[diffeomorphisms]].

=--

+-- {: .num_defn #OppositeCategoryOfCartSp}
###### Definition
**([[opposite category]] of [[CartSp]])**

Write [[CartSp]]${}^{op}$ for the [[opposite category]] ([this def.](geometry+of+physics+--+Categories+and+Toposes#OppositeCategory)) of [[CartSp]] (Prop. \ref{CartSpCategory}).

This is the category with the same objects as $CartSp$, but where a morphism $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ in $CartSp^{op}$ is given by a morphism $\mathbb{R}^{n_1} \leftarrow \mathbb{R}^{n_2}$ in $CartSp$.

=--

We will be discussing below the idea of exploring smooth spaces by laying out abstract coordinate systems in them in all possible ways. The reader should begin to think of the sets that appear in the following definition as the _set of ways_ of laying out a given abstract coordinate systems in a given space. This is discussed in more detail below in _[Smooth spaces](#SmoothSpaces)_.

+-- {: .num_defn}
###### Definition

A [[functor]]  $X : CartSp^{op} \to Set$ (a "[[presheaf]]") is

1. for each abstract coordinate system $U$ a [[set]] $X(U)$

1. for each [[coordinate transformation]] $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a [[function]] $X(f) : X(\mathbb{R}^{n_1}) \to X(\mathbb{R}^{n_2})$

such that

1. [[identity]] is respected $X(id_{\mathbb{R}^n}) = id_{X(\mathbb{R}^n)}$;

1. [[composition]] is respected $X(f_2)\circ X(f_1) = X(f_2 \circ f_1)$

=--

+-- {: .num_example #SmoothSetOfDifferentialPForms}
###### Example

Let $\mathcal{C}$ be a [[category]].

1. The following are equivalent:

   1. $\mathcal{C}$ has a [[terminal object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[right adjoint]]

      $$
        \ast
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \mathcal{C}
      $$

    Under this equivalence, the [[terminal object]] is identified with the image under the right adjoint of the unique object of the [[terminal category]].

1. Dually, the following are equivalent:

   1. $\mathcal{C}$ has an [[initial object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[left adjoint]]

      $$
        \mathcal{C}
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \ast
      $$

    Under this equivalence, the [[initial object]] is identified with the image under the left adjoint of the unique object of the [[terminal category]].

=--

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

Since the unique [[hom-set]] in the [[terminal category]] is [[generalized the|the]] [[singleton]], the hom-isomorphism characterizing the [[adjoint functors]] is directly the [[universal property]] of an [[initial object]] in $\mathcal{C}$

$$
  Hom_{\mathcal{C}}( L(\ast) , X )
  \;\simeq\;
  Hom_{\ast}( \ast, R(X) )
  =
  \ast
$$

or of a [[terminal object]]

$$
  Hom_{\mathcal{C}}( X , R(\ast) )
  \;\simeq\;
  Hom_{\ast}( L(X), \ast ) = \ast
  \,,
$$

respectively.

=--

### The algebraic theory of smooth algebras
 {#TheAlgebraicTheoryOfSmoothAlgebras}

+-- {: .num_prop}
###### Propositions

* The [[category]] [[CartSp]] has all [[finite limit|finite]]
[[products]].

* Every object is a finite [[product]] of the object $\mathbb{R}$ (the [[real line]] itself).

* The [[terminal object]] is $\mathbb{R}^0$, the [[point]].

=--

Hence [[CartSp]] is (the [[syntactic category]]) of an [[algebraic theory]] (a [[Lawvere theory]]).

This is called the [[theory]] of _[[smooth algebras]]_.

+-- {: .num_defn}
###### Definition

A [[product]]-preserving [[functor]]

$$
  A : CartSp \to Set
$$

is a _[[smooth algebra]]_. A [[homomorphism]] of smooth algebras is a [[natural transformation]] between the corresponding functors.

=--

The basic example is:

+-- {: .num_example #TheSmoothgAlgebraOfFunctionsOnACartesianSpace}
###### Example

For $n \in \mathbb{N}$, the [[smooth algebra]] $C^\infty(\mathbb{R}^n)$
is the functor $CartSp \to Set$ which is [[representable functor|functor corepresented]] by $\mathbb{R}^n \in $ [[CartSp]]. This means that to $\mathbb{R}^k \in CartSp$ it assigns the set

$$
  Hom_{CartSp}(\mathbb{R}^n , \mathbb{R}^k)
  =
  C^\infty(\mathbb{R}^n, \mathbb{R}^k)
$$

of [[smooth functions]] from $\mathbb{R}^n$ to $\mathbb{R}^k$, and to a [[smooth function]] $f \colon \mathbb{R}^{k_1} \to \mathbb{R}^{k_2}$ it assigns the function

$$
  f\circ (-)
  \colon
  C^\infty(\mathbb{R}^n, \mathbb{R}^{k_1})
  \to
  C^\infty(\mathbb{R}^n, \mathbb{R}^{k_2})
$$

given by postcomposition with $f$.

=--

+-- {: .num_remark}
###### Remark

Example \ref{TheSmoothgAlgebraOfFunctionsOnACartesianSpace} shows how we are to think of a functor $A \colon CartSp \to Set$ as encoding an algebra: such a functor assigns to $\mathbb{R}^n$ a set to be interpreted as a set of "smooth functions on something with values in $\mathbb{R}^n$", only that the "something" here is not pre-defined, but is instead indirectly characterized by this assignment.

Due to this we will often denote smooth algebras as "$C^\infty(X)$", even if "$X$" is not a pre-defined object, and write their value on $\mathbb{R}^n$ as $C^\infty(X,\mathbb{R}^n)$.

=--

This is illustrated by the next example.

+-- {: .num_example #DualNumbers}
###### Example

The **[[ring of dual numbers|smooth algebra of dual numbers]]** $C^\infty(\mathbf{D})$ is the smooth algebra which assigns to $\mathbb{R}^n$ the [[Cartesian product]]

$$
  C^\infty(D,\mathbb{R}^n)

  \coloneqq
  \mathbb{R}^n \times \mathbb{R}^n
$$

of two copies of $\mathbb{R}^n$, which we will write as

$$
  \left\{
    (\epsilon \mapsto (\vec x + \epsilon \vec v))
    |
    \vec x , \vec v \in \mathbb{R}^n
  \right\}
  \,.
$$

Moreover, a [[smooth function]] $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is sent to the function

$$
  C^\infty(D, f)
  \colon
  C^\infty(D, \mathbb{R}^{n_1})
  \to
  C^\infty(D, \mathbb{R}^{n_2})
$$

given by

$$
  \begin{aligned}
  \left(\epsilon \mapsto \left(\vec x + \epsilon \vec v\right)\right)
  \\
  \left(
    \epsilon \mapsto f(\vec x) + (\mathbf{d}f)(\vec v)
  \right)
  &\mapsto
  \left(
    \epsilon
    \mapsto
    \left(
       f\left(\vec x\right)
        +
       \sum_{j = 1}^{n_2}
       \left(\sum_{i = 1}^{n_1}\frac{\partial f^j}{\partial x^i} v^i\right)
       \vec e_j
    \right)
    \right)
  \end{aligned}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

As the notation suggests, we may think of $C^\infty(D)$ as the functions on a first order [[infinitesimal object|infinitesimal neighbourhood]] of the origin in $\mathbb{R}^n$.

=--

### The coverage of differentially good open covers
 {#CoverageOfDifferentiallyGoodOpenCovers}

We discuss a standard structure of a _[[site]]_ on the category [[CartSp]]. Following _[[Sketches of an Elephant|Johnstone -- Sketches of an Elephant]]_, it will be useful and convenient to regard a site as a ([[small site|small]]) category equipped with a _[[coverage]]_. This generates a genuine [[Grothendieck topology]], but need not itself already be one.

+-- {: .num_defn}
###### Definition

For $n \in \mathbb{N}$ the standard [[open ball|open n-ball]] is the subset

$$
  D^n = \{ (x_i)_{i =1}^n \in \mathbb{R}^n | \sum_{i = 1}^n (x_i)^2 \lt 1 \}  \hookrightarrow \mathbb{R}^n
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

There is a [[diffeomorphism]]

$$
  \mathbb{R}^n \stackrel{\simeq}{\to} D^n
  \,.
$$

=--

+-- {: .num_defn #DifferentiallyGoodOpenCover}
###### Definition

A **differentially [[good open cover]]** of a [[Cartesian space]] $\mathbb{R}^n$ is a set $\{U_i \hookrightarrow \mathbb{R}^n\}$ of [[open subset]] inclusions of Cartesian spaces such that these [[open cover|cover]] $\mathbb{R}^n$
and such for each non-empty finite [[intersection]] there exists a [[diffeomorphism]]

$$
  \mathbb{R}^n \stackrel{\simeq}{\to} U_{i_1} \cap \cdots \cap U_{i_k}
$$

that identifies the $k$-fold intersection with a Cartesian space itself.

=--

+-- {: .num_remark}
###### Remark

Differentiably good covers are useful for computations. Their full impact is however on the [[homotopy theory]] of [[simplicial presheaves]] over [[CartSp]]. This we discuss in the chapter on [[geometry of physics -- smooth homotopy types|smooth homotopy types]], around [this prop.](geometry+of+physics+--+smooth homotopy types#DifferentiablyGoodCoverGivesSPlitHyperCoverOverCartSp).

=--

+-- {: .num_prop #DiffGoodOpenCoversRefineOpenCovers}
###### Proposition

Every [[open cover]] refines to a differentially good open cover, def. \ref{DifferentiallyGoodOpenCover}.

=--

A proof is at _[[good open cover]]_.

+-- {: .num_remark}
###### Remark

Despite its appearance, this is not quite a classical statement. The classical statement is only that every open cover is refined by a _topologically_ [[good open cover]]. See the comments _[here in the references-section](http://ncatlab.org/nlab/show/ball#References)_ at _[[open ball]]_ for the situation concerning this statement in the literature.

=--

+-- {: .num_remark}
###### Remark

The _good_ open covers do not yet form a [[Grothendieck topology]] on [[CartSp]]. One of the axioms of a Grothendieck topology is that for every [[covering]] family also its [[pullback]] along any morphism in the category is a covering family. But while the pullback of every [[open cover]] is again an open cover, and hence open covers form a Grothendieck topology on [[CartSp]], not every pullback of a _[[good open cover|good]]_ open cover is again _good_.

=--

+-- {: .num_example #ExampleThatGoodOpenCoversAreNotPullbackStable}
###### Example

Let $\{\mathbb{R}^2\stackrel{\phi_{i}}{\hookrightarrow}\mathbb{R}^2\}_{i \in \{1,2\}}$ be the [[open cover]] of the [[plane]] by an open left half space

$$
   \mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \lt 1 \} \stackrel{\phi_1}{\hookrightarrow} \mathbb{R}^2
$$

and a right open half space

$$
   \mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \gt -1 \} \stackrel{\phi_2}{\hookrightarrow} \mathbb{R}^2
  \,.
$$

The intersection of the two is the open strip

$$
  \mathbb{R}^2 \simeq \{ (x_1, x_2) \in \mathbb{R}^2 | -1 \lt x_1 \lt 1 \}
  \hookrightarrow
  \mathbb{R}^2
  \,.
$$

So this is a [[good open cover]] of $\mathbb{R}^2$.

But consider then the smooth function

$$
  2(\cos(2 \pi (-)), \sin(2 \pi (-))) \colon \mathbb{R}^1 \to \mathbb{R}^2
$$

which sends the line to a curve in the plane that periodically goes around the [[circle]] of radius 2 in the plane.

Then the pullback of the above good open cover on $\mathbb{R}^2$ to $\mathbb{R}^1$ along this function is an [[open cover]] of $\mathbb{R}$ by two open subsets, each being a disjoint union of countably many open [[intervals]] in $\mathbb{R}$.
Each of these open intervals is an open 1-ball hence diffeomorphic to $\mathbb{R}^1$, but their _[[disjoint union]]_ is not [[contractible topological space|contractible]] (it does not contract to the point, but to many points!).

So the pullback of the good open cover that we started with is an open cover which is not _good_ anymore. But it has an evident _refinement_ by a good open cover.

=--

This is a special case of what the following statement says in generality.

+-- {: .num_prop #TheDifferentiallyGoodOpenCoverCoverage}
###### Proposition

The differentially good open covers, def. \ref{DifferentiallyGoodOpenCover}, constitute a [[coverage]] on [[CartSp]].

Hence [[CartSp]] equipped with that coverage is a [[site]].

=--

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

By definition of [[coverage]] we need to check that for $\{U_i \hookrightarrow \mathbb{R}^n\}_{i \in I}$ any [[good open cover]] and $f \colon \mathbb{R}^k \to \mathbb{R}^n$ any smooth function, we can find a good open cover $\{K_j \to \mathbb{R}^k\}_{j \in J}$ and a [[function]] $J \to I$ such that for each $j \in J$ there is a smooth function $\phi \colon K_j \to U_{\rho(j)}$ that makes this [[diagram]]
[[commuting diagram|commute]]:

$$
  \array{

    K_j &\stackrel{\phi}{\to}& U_{i(j)}
    \\
    \downarrow && \downarrow
    \\
    \mathbb{R}^k &\stackrel{f}{\to}& \mathbb{R}^n
  }
  \,.
$$

To obtain this, let $\{f^* U_i \to \mathbb{R}^k\}$ be the pullback of the original covering family, in that

$$
  f^* U_i \coloneqq \{ x \in \mathbb{R}^k | f(x) \in U_i \}
  \hookrightarrow \mathbb{R}^k
  \,.
$$

This is evidently an [[open cover]], albeit not necessarily a _[[good open cover]]_. But by prop. \ref{DiffGoodOpenCoversRefineOpenCovers} there does exist a good open cover $\{\tilde K_{\tilde j} \hookrightarrow \mathbb{R}^k\}_{\tilde j \in \tilde J}$ _refining_ it, which in turn means that for all $\tilde j$ there is

$$
  \array{
    \tilde K_{\tilde j} &\to& K_{j(\tilde j)}
    \\
    \downarrow && \downarrow
    \\
    \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k
  }
  \,.
$$

Therefore then the [[pasting]] composite of these commuting squares

$$
  \array{
    \tilde K_{\tilde j} &\to& K_{j(\tilde j)} &\to& U_{i(j(\tilde j))}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k
   &\stackrel{f}{\to}& \mathbb{R}^n
  }
$$

solves the condition required in the definition of [[coverage]].

=--

By example \ref{ExampleThatGoodOpenCoversAreNotPullbackStable} this [[good open cover]] [[coverage]] is not a [[Grothendieck topology]]. But as any coverage, it uniquely completes to one which has the same [[sheaves]].

+-- {: .num_prop}
###### Proposition

The [[Grothendieck topology]] induced on [[CartSp]] by the differentially good open cover coverage of def. \ref{TheDifferentiallyGoodOpenCoverCoverage} has as covering families the ordinary [[open covers]].

=--

+-- {: .num_remark}
###### Remark

This means that for every sheaf-theoretic construction to follow we can just as well consider the Grothendieck topology of open covers on $CartSp$. The sheaves of the open cover topology are the same as those of the good open cover coverage. But the latter is (more) useful for several computational purposes in the following. It is the _good_ open cover coverage that makes manifest, below, that sheaves on $CartSp$ form a [[locally connected topos]] and in consequence then a [[cohesive topos]]. This kind of argument becomes all the more pronounced as we pass [further below](#SmoothnGroupoids) to [[(∞,1)-sheaves]] on [[CartSp]]. This will be discussed in _[Smooth n-groupoids -- Semantic Layer -- Local Infinity-Connectedness](#InfinityConnectednessOfSmoothInfinityGrpd) below.

=--

### The slice category
 {#SliceCategories}

(...)

* [[slice category]] $CartSp_{\mathbb{R}^n}$

* [[subterminal object]], [[open cover]]

(...)

$\,$

## Syntactic Layer
 {#CoordinateSystemsLayerSyn}

In this [Syn Layer](#LayerSyn) we discuss the [[abstract generals]] of _abstract [[coordinate systems]]_, def. \ref{CartesianSpacesAndSmoothFunctions}: the [[internal language]] of a category with [[products]], which is _[[type theory]]_ with _[[product types]]_.

> This is rough, needs further development.

### Judgments about types and terms
 {#Judgments}

We now introduce a different _notation_ for [[objects]] and [[morphisms]] in a [[category]] (such as the category [[CartSp]] of def. \ref{CartSpCategory}). This notation is designed to, eventually, make more transparent what exactly it is that happens when we [[deductive reasoning|reason deductively]] about [[objects]] and [[morphisms]] of a [[category]].

But before we begin to make any actual [[deductions]] about [[objects]] and [[morphisms]] in a category [below](#NaturalDeductionRulesForProductTypes), in this section here we express the given objects and morphisms at hand in the first place. Such basic statements of the form "There is an object called $A$" are to be called _[[judgments]]_, in order not to confuse these with genuine _[[propositions]]_ that we eventually formalize _within_ this [[metalanguage]].

To express that there is an [[object]]

$$
  X \in \mathcal{C}
$$

in a [[category]] $\mathcal{C}$, we write now equivalently the string of symbols (called a _[[sequent]]_)

$$
  \vdash \; X \colon Type
  \,.
$$

We say that these symbols express the _[[judgment]]_ that $X$ is a _[[type]]_. We also say that $\vdash \; X \colon Type$ is the _[[syntax]]_ of which $X \in \mathcal{C}$ is the _[[categorical semantics]]_.

For instance the [[terminal object]] $* \in \mathcal{C}$ we call the [[categorical semantics]] of the _[[unit type]]_ and write [[syntax|syntactically]] as

$$
  \vdash \; Unit \colon Type
  \,.
$$

If we want to express that we do assume that a terminal object indeed exists, hence that we want to be able to _[[deduction|deduce]]_ the existence of a terminal object from no hypothesis, then we write this [[judgment]] below a horizontal line

$$
  \frac{}{\vdash \; Unit \colon Type}
  \,.
$$

We will see more interesting such horizontal-line statements [below](#NaturalDeductionRulesForProductTypes).

Next, to express an [[element]] of the object $X$ in $\mathcal{C}$, hence a [[morphism]]

$$
  * \stackrel{x}{\to} X
$$

in $\mathcal{C}$ we write equivalently the [[sequent]]

$$
  \vdash \; x \colon X
$$

and call this the [[judgment]] that $x$ is a _[[term]]_ of [[type]] $X$.

Notice that every object $X \in \mathcal{C}$ becomes the [[terminal object]] in the _[[slice category]]_ $\mathcal{C}_{/X}$.
Let $A \to X$ be any morphism in $\mathcal{C}$, regarded as an object in the [[slice category]]

$$
  A \in \mathcal{C}_{/X}
  \,.
$$

We declare that the [[syntax]] of which this is the [[categorical semantics]] is given by the [[sequent]]

$$
  x \colon X \;\vdash \;  A(x) \colon Type
  \,.
$$

We say that this expresses the [[judgement]] that $A$ is an  _$X$-[[dependent type]]_; or a type in the _[[context]] of a [[free variable]]_ $x$ of [[type]] $X$.

Notice that an [[element]] of $A \in \mathcal{C}_{/X}$ is a [[generalized element]] of $A$ in $\mathcal{C}$, namely a morphism $X \to A$ which fits into a [[commuting diagram]]

$$
  \array{
    X &&\stackrel{a}{\to}&& A
    \\
    & {}_{\mathllap{id_X}}\searrow && \swarrow_{}
    \\
    && X
  }
$$

in $\mathcal{C}$.

We declare that the [[syntax]] of which such

$$
  a \in A  \;\;\;\; (in \mathcal{C}_{/X})
$$

is this the [[categorical semantics]] is the [[sequent]]

$$
  x\colon X \;\vdash\; a(x) : A(x)
  \,.
$$

We say that this expresses the [[judgment]] that $a(x)$ is a [[term in context|term depending on]] the [[free variable]] $x$ of [[type]] $X$.

This completes the list of _[[judgment]]_ [[syntax]] to be considered. Notice that if the category $\mathcal{C}$ has [[products]] then, even though it does not explicitly appear above, this is sufficient to express any morphism $X \stackrel{f}{\to} Y$ in $\mathcal{C}$ as the [[semantics]] of a [[term]]: we regard this morphism naturally as being the corresponding morphism in the [[slice category]] $\mathcal{C}_{/X}$ which as a [[commuting diagram]] in $\mathcal{C}$ itself is

$$
  \array{
     X && \stackrel{(f,id_X)}{\to} &&  Y\times X
     \\
     & {}_{\mathllap{id_X}}\searrow && \swarrow_{\mathrlap{p_2}}
     \\
     && X
  }
  \,.
$$

This is the [[categorical semantics]] for which the [[syntax]] is simply

$$
  x \colon X \;\vdash\; y(x) \colon Y
  \,,
$$

being the [[judgment]] which expresses that $y(x)$ is a [[term in context]] of an $X$-[[dependent type]] $Y$ in the special degenerate case that $Y$ does not actually vary with $x \colon X$.

### Natural deduction rules for product types
 {#NaturalDeductionRulesForProductTypes}

With the [above symbolic notation](#Judgments) for making [[judgments]] about the presence of [[objects]] and [[morphisms]] in a [[category]] $\mathcal{C}$, we now consider a system of rule of [[deduction]]  that tells us how we may _process_ these symbols (how to do _[[computations]]_) such that the new symbols we obtain in turn express new objects and new morphisms in $\mathcal{C}$ that we can build out of the given ones by _[[universal constructions]]_ in the sense of [[category theory]].

This way of _deducing_ new expressions from given ones is very basic as well as very natural and hence goes by the technical term _[[natural deduction]]_. For every kind of [[type]] (every [[universal construction]] in [[category theory]]) there is, in [[natural deduction]], one set of rules for how to [[deductive reasoning|deductively reason]] about it. This set of rules, in turn, always consists of four kinds of rules, called the

1. [[type formation rule]]

1. [[term introduction rule]]

1. [[term elimination rule]]

1. [[computation rule]].

These are going to be the [[syntax]] in _[[type theory]]_ of which [[universal constructions]] in [[category theory]] is the [[categorical semantics]].

In our running example where $\mathcal{C} = $ [[CartSp]], the only [[universal construction]] available is that of forming [[products]]. We therefore introduce now the [[natural deduction]] rules by way of example of the special case of [[product types]].

**1. [[type formation rule]]** Let

$$
  A , B \in \mathcal{C}
$$

be two [[objects]] in a [[category]] with [[products]]. Then there exists [[generalized the|the]] [[product]] [[object]]

$$
  A \times B \in \mathcal{C}
  \,.
$$

We now declare that the [[syntax]] of which this state of affairs is the [[categorical semantics]] is the collection of symbols of the form

$$
  \frac{A \colon Type \;\;\;\;\; B \colon Type}{ A \times B \colon Type}
  \,.
$$

Here on top of the horizontal line we have the two [[judgments]] which express that, [[syntax|syntactically]], $A$ is a [[type]] and $B$ is a [[type]], and [[semantics|semantically]]  that $A \in \mathcal{C}$ and $B \in \mathcal{C}$. Below the horizontal line is, in turn, the [[judgment]] which expresses that there is, syntactically,  a [[product type]], which semantically is the [[product]] $A \times B \in \mathcal{C}$. The horizontal line itself is to indicate that if we are given the (symbols of) the collection of judgments on top, then we are entitled to also obtain the judgment on the bottom.

**Remark (Computation)** All this may seem, on first sight, like being a lot of fuss about something of grandiose banality. To see what is gradually being accomplished here despite of this appearance, as we proceed in this discussion, the reader can and should think of this as the first steps in the definition of a [[programming language]]: the notion of judgment is a syntactic rule for strings of symbols that a computer is to read in, and a [[natural deduction]]-step as the [[type formation rule]] above is an operation that this computer validates as being an allowed step of transforming a memory state with a given collection of such strings into a new memory state to which the string below the horizontal line is added. As we add the remaining rules below, what looks like a grandiose banality so far will remain grandiose, but no longer be a banality. The reader feeling in need of more motivational remarks along these lines might want to take a break here and have a look at the entry _[[computational trinitarianism]]_ first, that provides more pointers to the grandiose picture which we are approaching here.

Next, the second [[natural deduction]] rule for [[product types]] is the

**2. [[term elimination rule]]**. The fact that $A \times B \in \mathcal{C}$ is equipped with two [[projection]] [[morphisms]]

$$
  \array{
    A \stackrel{p_1}{\leftarrow} A \times B \stackrel{p_2}{\to} B
  }
$$

means that from every [[element]] $t$ of $A \times B$ we may deduce the existence of [[elements]] $p_1(t)$ and $p_2(t)$ of $A$ and $B$, respectively. We declare now that this is the [[categorical semantics]] of which the [[natural deduction]] [[syntax]] is:

$$
  \frac{\vdash \; t \colon A \times B}{\vdash \; p_1(t) \colon A}
  \;\;\;\;\;\;\;\;\;
  \frac{\vdash \; t \colon A \times B}{\vdash \; p_2(t) \colon B}
  \,.
$$

As before, this is to say that if syntactically we are given strings of symbols expressing [[judgments]] as on the top of these horizontal lines, then we may "[[natural deduction|naturally deduce]]" also the judgment of the string of symbols on the bottom of this line.

**3. [[term introduction rule]]**. The first part of the [[universal property]] of the [[product]] in [[category theory]] is that for  $Q \in \mathcal{C}$ any other [[object]] equipped with morphisms

$$
  \array{
    && Q
    \\
    & {}^{\mathllap{a}}\swarrow && \searrow^{\mathrlap{b}}
    \\
    A  && &&  B
  }
$$

in $\mathcal{C}$, we obtain a canonical morphism

$$
  Q \to A \times B
$$

in $\mathcal{C}$. This is now declared to be the [[categorical semantics]] of which the [[natural deduction]] [[syntax]] is

$$
  \frac{
    \vdash\; a \colon A \;\;\;\;\;\; \vdash\; b \colon B
  }{
   \vdash (a,b) \colon A \times B
  }
  \,.
$$

With the [[elements]] that are the [[semantics]] of the terms appearing here made explicit, this is the syntax for a [[diagram]]

$$
  \array{
     && Q
     \\
     & {}^{\mathllap{a}}\swarrow &\downarrow^{\mathrlap{(a,b)}}& \searrow^{\mathrlap{b}}
     \\
     A && A \times B && B
  }
  \,.
$$

**4. [[computation rule]]**. The next part of the [[universal property]] of the [[product]] in [[category theory]] is that the resulting [[diagram]]

$$
  \array{
    && Q
    \\
    & {}^{\mathllap{a}}\swarrow &\downarrow& \searrow^{\mathrlap{b}}
    \\
    A &\stackrel{p_1}{\leftarrow}& A \times B & \stackrel{p_2}{\to} & B
  }
$$

is in fact a _[[commuting diagram]]_. Syntactically this is, clearly, the rule that the following identifications of strings of symbols are to be enforced

$$
  p_1(a,b) = a \;\;\;\;\;\; p_2(a,b) = b
  \,.
$$

This concluces the description of the [[natural deduction]] about [[objects]], [[morphisms]] and [[products]] in a [[category]] using its [[type theory]] [[syntax]]. We summarize the dictionary between [[category theory]] and [[type theory]] discussed so far [below](#DictionaryTypeTheoryCategoryTheory).

In the [next section](#SmoothSpaces) we promote our running example category $\mathcal{C}$, which admits only very few [[universal constructions]] (just [[products]]), to a richer category, the [[sheaf topos]] over it. That richer category then accordingly comes with a richer [[syntax]] of [[natural deduction]] inside it, namely with full [[dependent type theory]]. This we discuss in the [Syn Layer below](#SmoothSpacesLayerSyn).

### Natural deduction rules for dependent sum types
 {#DependentSumTypes}

(...) [[dependent sum type]] (...)

### Dictionary: type theory / category theory
 {#DictionaryTypeTheoryCategoryTheory}

The dictionary between [[dependent type theory|dependent]] [[type theory]] with [[product types]] and [[category theory]] of categories with [[products]].

[[!include judgements for types and terms - table]]

$\,$

[[!include substitution natural deduction - table]]

$\,$

[[!include product natural deduction - table]]

$\,$

[[!include dependent sum natural deduction - table]]

Below in [Smooth spaces - Syntactic Layer](#SmoothSpacesLayerSyn) we complete this dictionary to one between [[dependent type theory]] with [[dependent products]] and [[toposes]].


geometry of physics -- de Rham coefficients

> This entry is gouing to contain one chapter of _[[geometry of physics]]_.

> previous chapters: _[flat connections](geometry+of+physics#FlatConnections)_

> next chapter: _[principal connections](geometry+of+physics#PrincipalConnections)_

***

## De Rham coefficients

### Model Layer

#### Lie-algebra valued differential 1-forms

+-- {: .num_defn #SheafOfLieAlgebraValuedForms}
###### Definition

Let $G$ be a [[Lie group]], and write $\mathfrak{g}$ for its [[Lie algebra]]. The set of [[Lie algebra valued differential 1-forms]] is the [[tensor product]]

$$
  \Omega^1(U,\mathfrak{g}) = \Omega^1(U) \otimes_{\mathbb{R}} \mathfrak{g}
  \,.
$$

flat forms:

$$
  \Omega^1_{flat}(U, \mathfrak{g}) =
  \left\{
    \omega \in \Omega^1(U,\mathfrak{g}) |
    F_\omega = \mathbf{d} \omega + [\omega, \omega] = 0
  \right\}

  \,.
$$

=--

(...)

This is a [[smooth space]]

$$
  \Omega^1_{flat}(-,\mathfrak{g}) \in Smooth 0 Type
$$

For $\mathfrak{g} = Lie(\mathbb{R})$ we have

$$
  \Omega^1(-,Lie(\mathbb{R})) = \Omega^1
$$

and we write

$$
  \Omega^1_{flat}(-,Lie(\mathbb{R})) = \Omega^1_{cl}
$$

Below we see

$$
  \flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})
$$

#### The de Rham complex

* [[de Rham complex]]

* [[de Rham cohomology]]

* [[de Rham theorem]]

Below we see that

$$
  \flat_{dR}\mathbf{B}^n \mathbb{R}
    \simeq
  \flat_{dR}\mathbf{B}^n U(1)
   \simeq
  DK[ \Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to}\cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)]
 \,.
$$

### Semantic Layer

#### De Rham coefficient objects

+-- {: .num_defn #deRhamCoefficientObject}
###### Definition

For $G \in Gpr(\mathbf{H})$, its **de Rham coefficient object**
is the [[homotopy pullback]]

$$
  \flat_{dR} \mathbf{B}G \coloneqq \flat \mathbf{B}G \times_{\mathbf{B}G} *
$$

in

$$
  \array{
     \flat_{dR} \mathbf{B}G &\stackrel{UnderlyingConnection}{\to}& \flat \mathbf{B}G
     \\
     \downarrow &pb& \downarrow^{\mathrlap{UnderlyingBundle}}
     \\
     * &\to& \mathbf{B}G
  }
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

This pullback diagram expresses that [[generalized element|elements]] of $\flat_{dR}\mathbf{B}G$ are flat $G$-connections $\nabla \colon X \to \flat \mathbf{B}G$, def. \ref{FlatCohesiveConnection} equipped with a trivialization of their underlying $G$-principal bundle, def. \ref{UnderlyingBundleOfFlatConnection}.

=--

#### Recovering smooth differential forms from cohesive de Rham coefficients
 {#OrdinaryDifferentialFormsFromSmoothCohesion}

Let $\mathbf{H} = $ [[Smooth∞Grpd]]. All [[smooth manifolds]] and sheaves on smooth manifolds etc. in the following are canonically regarded as objects in this $\mathbf{H} = Sh_\infty(CartSp)$.

+-- {: .num_prop #DeRhamCoefficientsOfLieGroup}
###### Proposition

For $G$ a [[Lie group]], the de Rham coefficient object $\flat_{dR}\mathbf{B}G$, def. \ref{deRhamCoefficientObject} of its [[delooping]] is given by the [[sheaf]] of flat [[Lie algebra valued differential 1-forms]] $\Omega^1_{flat}(-,\mathfrak{g})$, def. \ref{SheafOfLieAlgebraValuedForms}, for $\mathfrak{g}$ the [[Lie algebra]] of $G$:

$$
  \flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})
  \,.
$$

=--

This is discussed at _[smooth ∞-groupoid - structures - de Rham coefficients for BG with G a Lie group](smooth+infinity-groupoid+--+structures#deRhamWithCoefficientsInBOfLieGroup)_.

Write $U(1)$ for the [[circle group]] regared as a [[Lie group]] in the standard way.

+-- {: .num_prop}
###### Proposition

For $n \in \mathbb{N}$, the de Rham coefficient object $\flat_{dR}\mathbf{B}^n U(1)$, def. \ref{deRhamCoefficientObject}, of the $n$-fold [[delooping]] of $U(1)$ is given by the image under the [[Dold-Kan correspondence]]

$$
  DK \colon : Sh(CartSp, Ch_\bullet) \to Sh(CartSp, sSet)
  \to L_{lwhe} Sh(CartSp, sSet) \simeq \mathbf{H}
$$

of the truncated [[de Rham complex]] of sheaves of differential forms,

$$
  \begin{aligned}
   \flat_{dR}\mathbf{B}^n U(1)
   &\simeq
   \flat_{dR} \mathbf{B}^n \mathbb{R}
   \\
   & \simeq
   DK[\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)]
    \\
   & \simeq DK[\Omega^1_{cl}(-) \to 0 \to  \cdots \to 0]
  \end{aligned}
  \,.
$$

=--

This is discussed at _[smooth ∞-groupoid - structures - de Rham coefficients for the circle n-groups](smooth+infinity-groupoid+--+structures#deRhamCoefficientsInBnU1)_.

### Syntactic Layer

$$
  \begin{aligned}
    \flat_{dR}(\mathbf{B}G \colon Type)\; \colon & Type
    \\
    \coloneqq &
     \;\;
    \sum_{\nabla \colon \flat \mathbf{B}G}
    (  UnderlyingBundle(\nabla) = * )
  \end{aligned}
$$

## **Maurer-Cartan forms**
 {#MaurerCartanForms}

### Model Layer
 {#MaurerCartanLayerMod}

#### Maurer-Cartan form on a Lie group
 {#MaurerCartanFormOnLieGroup}

* [[Maurer-Cartan form]]

$$
  \theta_G \colon G \to \Omega^1_{flat}(-,\mathfrak{g})
$$

Consider

$$
  \flat_{dR} \mathbf{B}\mathbb{R}  = \Omega^1_{cl}
$$

the Maurer-Cartan form on $\mathbb{R}$ is the [[de Rham differential]]

$$
  \theta_{\mathbb{R}} = \mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl} \hookrightarrow \Omega^1
  \,.
$$

### Semantic Layer

#### Maurer-Cartan form on a cohesive $\infty$-group

Let $\mathbf{H}$ be a [[cohesive (infinity,1)-topos]] $(\mathbf{\Pi} \dashv \flat \dashv \sharp)$. We discuss a general formulation of [[Maurer-Cartan forms]] on cohesive [[infinity-groups]]

Let $G \in Grp(\mathbf{H})$ be a [[group object in an (infinity,1)-category|group object]].

Use the [[pasting law]] together with the fact that $\flat$ is
[[right adjoint]] and hence preserves [[limits]] to get $\theta$ in

$$
  \array{
    G &\to& *
    \\
    \downarrow^{\mathrlap{\theta}} & pb & \downarrow
    \\
    \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G
    \\
    \downarrow &pb& \downarrow
    \\
    * &\to& \mathbf{B}G
  }
$$

+-- {: .num_defn #GeneralAbstractMaurerCartanForm}
###### Definition

This is the **[[Maurer-Cartan form]]** on $G$

$$
  \theta \;\colon\; G \to \flat_{dR} \mathbf{B}G
  \,.
$$

=--

+-- {: .num_defn #DifferentiationOfInfinityGroupValuedFunction}
###### Definition

For $S \;\colon\; X \to G$ a morphism, write

$$
  S^{-1} \mathbf{d} S
  \coloneqq S^* \theta_G
  \;\colon\;
  X \stackrel{S}{\to}
  G
  \stackrel{\theta_G}{\to}
  \flat_{dR}\mathbf{B}G
$$

for its composite with the map of def. \ref{GeneralAbstractMaurerCartanForm}, hence the **pullback of the Maurer-Cartan form along $S$**. We also call this the **[[de Rham differential]]** of $S$.

=--

#### Maurer-Cartan forms on smooth $\infty$-groups

+-- {: .num_prop}
###### Proposition

For $G$ a [[Lie group]] canonically regarded  in $\mathbf{H} = $[[Smooth∞Grpd]] the general abstract morphism

$$
  \theta_G \colon G \to \flat_{dR}\mathbf{B}G
$$

is identified, via the identification $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$  of prop. \ref{DeRhamCoefficientsOfLieGroup} and the [[Yoneda lemma]], with the traditional [[Maurer-Cartan form]]

$$
  \theta_G \in \Omega^1_{flat}(G, \mathfrak{g})
  \,.
$$

=--

#### Cohesive differentiation
 {#CohesiveDifferentiation}

The Maurer-Cartan form on the [[line object]]

$$
  \theta_{\mathbb{R}} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})
$$

is the [[de Rham differential]],

$$
  \mathbf{d} = \theta_{\mathbb{R}}
  \,.
$$

#### Universal curvature characteristic forms

For $G = \mathbf{B}^n U(1)$

$$
  curv \colon \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)
$$

sends a circle $n$-bundle to the curvature of a pseudo-connection on it.

### Syntactic Layer

(...)


geometry of physics -- differential forms

> This entry contains one chapter of the material at _[[geometry of physics]]_.

> previous chapter: _[[geometry of physics -- smooth sets|smooth sets]]_

> next chapter: _[[geometry of physics -- integration|integration]]_

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## **Differentiation**
 {#Differentiation}

We introduce the standard concept of _[[nLab:differential forms]]_ in _[Model Layer](#DifferentialFormsLayerMod)_, adding to the traditional discussion a precise version of the statement that differential forms are equivalently "incremental smooth $n$-dimensional measures", which accurately captures the role that they play in [[physics]], notably in [[local action functionals]].

We define differential forms on general smooth spaces seamlessly in terms of the smooth [[moduli space]] $\Omega^\in \in Smooth0Type$ of differential forms. This has the special property that it is, for $n \geq 1$, a _non-[[concrete object|concrete]]_ smooth space. In _[Semantic Layer](#DifferentialFormsLayerSem)_ below we take this as occasion to discuss the notion of [[concrete objects]] in a [[local topos]], such as the topos of smooth spaces. We show how the _concretification_ of the smooth mapping space $[X,\Omega^n]$ for any smooth space $X$ is the _smooth (moduli) space of differential forms on $X$_. Below in _[Action functionals for Chern-Simons type gauge theories](#ActionFunctionalsForChernSimonsTypeGaugeTheories)_ the theory of _concretification_ in a local topos is a central ingredient in the _canonical existence_ of certain [[nLab:action functionals]].

The process of concretification involves the general abstract notion of _[[image|images]]_. The type-theory of this notion we discuss in [Syntactic Layer](#DifferentialFormsLayerSyn) here.

### Model Layer
 {#DifferentialFormsLayerMod}

We have seen above in _[The continuum real (world-)line](TheContinuumRealWorldLine)_ that that [[real line]] $\mathbb{R}$ is the basic [[kinematics|kinematical structure]] in the [[differential geometry]] of [[physics]]. Notably the smooth [[path spaces]] $[\mathbb{R}, X]$ from example \ref{SmoothPathSpace} are to be thought of as the smooth spaces of _trajectories_ (for instance of some [[particle]]) in a [[smooth space]] $X$, hence of smooth maps $\mathbb{R} \to X$.

But moreover, _[[dynamics]]_ in [[physics]]  is encoded by _[[linear functionals|functionals]] on such trajectories_: by "[[action functionals]]". In the simplest case these are for instance homomorphisms of smooth spaces

$$
  S \colon \left[I, X\right] \to \mathbb{R}
  \,,
$$

where $I \hookrightarrow \mathbb{R}$ is the standard unit [[interval]].

Such [[action functionals]] we discuss in their own right in _[Variational calculus](#VariationalCalculus)_ below. Here we first examine in detail a fundamental property they all have: they are supposed to be _[[local action functional|local]]_.

Foremost this means that the value associated to a trajectory is _built up incrementally_ from small contributions associated to small sub-trajectories: if a trajectory $\gamma$ is decomposed as a trajectory $\gamma_1$ followed by a trajectory $\gamma_2$, then the action functional is additive

$$
  S(\gamma) = S(\gamma_1) + S(\gamma_2)
  \,.
$$

As one takes this property to the limit of iterative subdivision, one finds that action functionals are entirely determined by their value on _[[infinitesimal space|infinitesimal]] displacements_ along the worldline. If $\gamma \colon \mathbb{R} \to X$ denotes a path and "$\dot \gamma(x)$" denotes the corresponding "infinitesimal path" at worldline parameter $x$, then the value of the action functional on such an infinitesimal path is traditionally written as

$$
  \mathbf{d}S(\dot \gamma)_x  \in \mathbb{R}
  \,,
$$

to be read as "the small change $\mathbf{d}S$ of $S$ along the infinitesimal path $\dot \gamma_x$".

This function $\mathbf{d}S$ that assigns numbers to infinitesimal paths is called a _[[differential form]]_. Etymologically this originates in the use of "form" as in _[[bilinear form]]_: something that is evaluated. Here it is evaluated on _infinitesimal differences_, referred to as _differentials_.

We define smooth differential forms on [[Cartesian spaces]] in

* _[Differential forms on abstract coordinate sysetms](#DifferentialFormsOnAbstractCoordinateSystem)_

Then we discuss how this induces a notion of smooth differential forms on general [[smooth spaces]] in

* _[Smooth universal moduli space of differential forms](#SmoothUniversalModuliSpaceOfDifferentialForms)_.

Further below we provide a precise version of the statement that "Differential 1-forms are differential measures along paths." in

* _[Differential 1-forms are smooth incremental path measures](#1FormsAsSmoothFunctors)_.

#### Differential forms on abstract coordinate systems
 {#DifferentialFormsOnAbstractCoordinateSystem}

We introduce the basic concept of a smooth [[differential form]] on a [[Cartesian space]] $\mathbb{R}^n$. Below in _[Differential forms on smooth spaces](#SmoothUniversalModuliSpaceOfDifferentialForms)_ we use this to define differential forms on any [[smooth space]].

+-- {: .num_defn #Differential1FormsOnCartesianSpaces}
###### Definition

For $n \in \mathbb{N}$ a **[[smooth differential 1-form]]** $\omega$ on a [[Cartesian space]] $\mathbb{R}^n$ is an $n$-[[tuple]]

$$
  \left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n
$$

of [[smooth functions]], which we think of equivalently as the [[coefficients]] of a [[formal linear combination]]

$$
  \omega = \sum_{i = 1}^n f_i \mathbf{d}x^i
$$

on a [[set]] $\{\mathbf{d}x^1, \mathbf{d}x^2, \cdots, \mathbf{d}x^n\}$ of [[cardinality]] $n$.

Write

$$
  \Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set
$$

for the set of smooth [[differential 1-forms]] on $\mathbb{R}^k$.

=--

+-- {: .num_remark}
###### Remark

We think of $\mathbf{d} x^i$ as a measure for [[infinitesimal space|infinitesimal]] displacements along the $x^i$-[[coordinate]] of a [[Cartesian space]]. This idea is made precise below in _[Differential 1-forms are smooth increnemental path measures](#1FormsAsSmoothFunctors)_.

=--

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.

+-- {: .num_defn #PullbackOfDifferential1FormsOnCartesianSpaces}
###### Definition

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

$$
  \phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})
$$

between sets of differential 1-forms, def. \ref{Differential1FormsOnCartesianSpaces}, which is defined on [[basis]]-elements by

$$
  \phi^* \mathbf{d} x^i \coloneqq \sum_{j = 1}^{\tilde k} \frac{\partial \phi^i}{\partial \tilde x^j} \mathbf{d}\tilde x^j
$$

and then extended linearly by

$$
  \begin{aligned}
    \phi^* \omega & = \phi^* \left( \sum_{i} \omega_i \mathbf{d}x^i \right)
    \\
    & \coloneqq
     \sum_{i = 1}^k \left(\phi^* \omega\right)_i \sum_{j = 1}^{\tilde k} \frac{\partial \phi^i }{\partial \tilde x^j}  \mathbf{d} \tilde x^j
    \\
    & =
     \sum_{i = 1}^k  \sum_{j = 1}^{\tilde k} (\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j}  \mathbf{d} \tilde x^j
  \end{aligned}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

The term "pullback" in _[[pullback of differential forms]]_ is not really related, certainly not historically, to the term _[[pullback]]_ in [[category theory]]. One can relate the pullback of differential forms to categorical pullbacks, but this is not really essential here. The most immediate property that both concepts share is that they take a [[morphism]] going in one direction to a map between structures over domain and codomain of that morphism which goes in the other direction, and in this sense one is "pulling back structure along a morphism" in both cases.

=--

Even if in the above definition we speak only about the [[set]] $\Omega^1(\mathbb{R}^k)$ of differential 1-forms, this set naturally carries further [[structure]].

+-- {: .num_defn #ModuleStructureOn1FormsOnRk}
###### Definition

1. The set $\Omega^1(\mathbb{R}^k)$ is naturally an [[abelian group]] with addition given by componentwise addition

   $$
      \begin{aligned}
         \omega + \lambda & =
          \sum_{i = 1}^k \omega_i \mathbf{d}x^i + \sum_{j = 1}^k \lambda_j \mathbf{d}x^j
         \\
         & = \sum_{i = 1}^k(\omega_i + \lambda_i) \mathbf{d}x^i
      \end{aligned}
      \,,
   $$

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

   $$
     f \cdot \omega = \sum_{i = 1}^k( f \cdot \omega_i) \mathbf{d}x^i
     \,.
   $$

=--

+-- {: .num_defn }
###### Remark

More abstractly, this just says that $\Omega^1(\mathbb{R}^k)$ is the [[free module]] over $C^\infty(\mathbb{R}^k)$ on the set $\{\mathbf{d}x^i\}_{i = 1}^k$.

=--

The following definition captures the idea that if $\mathbf{d} x^i$ is a measure for displacement along the $x^i$-[[coordinate]], and $\mathbf{d}x^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way te get a measure, to be called $\mathbf{d}x^i \wedge \mathbf{d} x^j$, for infinitesimal _surfaces_ (squares) in the $x^i$-$x^j$-plane. And this should keep track of the [[orientation]] of these squares, whith

$$
  \mathbf{d}x^j \wedge \mathbf{d}x^i = - \mathbf{d}x^i \wedge \mathbf{d} x^j
$$

being the same infinitesimal measure with orientation reversed.

+-- {: .num_defn #DifferentialnForms}
###### Definition

For $k,n \in \mathbb{N}$, the **smooth [[differential forms]]** on $\mathbb{R}^k$ is the [[exterior algebra]]

$$
  \Omega^\bullet(\mathbb{R}^k)
   \coloneqq
  \wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k)
$$

over the [[ring]] $C^\infty(\mathbb{R}^k)$ of [[smooth functions]] of the [[module]] $\Omega^1(\mathbb{R}^k)$ of smooth 1-forms, prop. \ref{ModuleStructureOn1FormsOnRk}.

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

=--

+-- {: .num_remark}
###### Remark

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 $C^\infty(\mathbb{R}^k)$ of [[basis]] elements of the form $\mathbf{d} x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n}$ for $i_1 \lt i_2 \lt \cdots \lt i_n$:

$$
  \omega = \sum_{1 \leq i_1 \lt i_2 \lt \cdots \lt i_n \lt k} \omega_{i_1, \cdots, i_n} \mathbf{d}x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n}
  \,.
$$

=--

+-- {: .num_defn #PullbackOfDifferentialForms}
###### Definition

The [[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 $\Omega^n(-)$, given for a smooth function $f \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ on basis elements by

$$
  f^* \left(  \mathbf{d}x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n} \right)
  =
  \left(f^* \mathbf{d}x^{i_1} \wedge \cdots \wedge f^* \mathbf{d}x^{i_n} \right)
  \,.
$$

=--

#### Differential forms on smooth spaces
 {#SmoothUniversalModuliSpaceOfDifferentialForms}

Above we have defined differential $n$-form on abstract coordinate systems. Here we extend this definition to one of differential $n$-forms on arbitrary [[smooth spaces]]. We start by observing that the space of _all_ differential $n$-forms on cordinate systems themselves naturally is a smooth space.

+-- {: .num_prop}
###### Proposition

The assignment of differential $n$-forms

$$
  \Omega^n(-) \colon \mathbb{R}^k \mapsto \Omega^n(\mathbb{R}^k)
$$

of def. \ref{DifferentialnForms} together with the [[pullback of differential forms]]-functions of def. \ref{PullbackOfDifferentialForms}

$$
  \array{
    \mathbb{R}^{k_1} &\mapsto & \Omega^1(\mathbb{R}^{k_1})
    \\
    \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f^*}}
    \\
    \mathbb{R}^{k_2} &\mapsto& \Omega^1(\mathbb{R}^{k_2})
  }
$$

defines a [[smooth space]] in the sense of def. \ref{SmoothSpace}:

$$
  \Omega^n(-) \in Smooth0Type
  \,.
$$

=--

+-- {: .num_defn #SmoothModuliSpaceOfnForms}
###### Definition

We call this

$$
  \Omega^n \colon Smooth0Type
$$

the **universal smooth [[moduli space]]** of differential $n$-forms.

=--

The reason for this terminology is that homomorphisms of smooth spaces into $\Omega^1$ _modulate_ differential $n$-forms on their [[domain]], by prop. \ref{YonedaForSmoothSpaces} (and hence by the [[Yoneda lemma]]):

+-- {: .num_example}
###### Example

For the [[Cartesian space]] $\mathbb{R}^k$ regarded as a smooth space by example \ref{CartesianSpaceAsSmoothSpace}, there is a [[natural bijection]]

$$
  \Omega^n(\mathbb{R}^k) \simeq Hom(\mathbb{R}^k, \Omega^1)
$$

between the set of smooth $n$-forms on $\mathbb{R}^n$ according to def. \ref{Differential1FormsOnCartesianSpaces} and the set of homomorphism of smooth spaces, $\mathbb{R}^k \to \Omega^1$, according to def. \ref{HomomorphismOfSmoothSpaces}.

=--

In view of this we have the following elegant definition of smooth $n$-forms on an arbitrary smooth space.

+-- {: .num_defn #DifferentialnFormOnSmoothSpace}
###### Definition

For $X \in Smooth0Type$ a [[smooth space]], def. \ref{SmoothSpace}, a **[[differential n-form]]** on $X$ is a [[homomorphism]] of [[smooth spaces]] of the form

$$
  \omega \colon X \to \Omega^n(-)
  \,.
$$

Accordingly we write

$$
  \Omega^n(X) \coloneqq Smooth0Type(X,\Omega^n)
$$

for the set of smooth $n$-forms on $X$.

=--

We may unwind this definition to a very explicit description of differential forms on smooth spaces. This we do in a moment in remark \ref{DifferentialFormOnSmoothSpaceAsSystemOfDiffFormsOnCoordinates}.

Notice that differential 0-forms are equivalently smooth $\mathbb{R}$-valued functions.

+-- {: .num_prop #SpaceOf0FormsIsRealLine}
###### Proposition

$\Omega^0 \simeq \mathbb{R}$

=--

+-- {: .num_defn #PullbackOfDifferentialFormsOnSmoothSpaces}
###### Definition

For $f \colon X \to Y$ a [[homomorphism]] of [[smooth spaces]], def. \ref{HomomorphismOfSmoothSpaces}, the **[[pullback of differential forms]]** along $f$ is the [[function]]

$$
  f^* \colon \Omega^n(Y) \to \Omega^n(X)
$$

given by the [[hom-functor]] into the smooth space $\Omega^n$ of def. \ref{SmoothModuliSpaceOfnForms}:

$$
 f^* \coloneqq Hom(-, \Omega^n)
 \,.
$$

This means that it sends an $n$-form $\omega \in \Omega^n(Y)$ which is modulated by a homomorphism $Y \to \Omega^n$ to the $n$-form $f^* \omega \in \Omega^n(X)$ which is modulated by the [[composition|composite]] $X \stackrel{f}{\to} Y \to \Omega^n$.

=--

+-- {: .num_defn}
###### Proposition

For $X = \mathbb{R}^{\tilde k}$ and $Y = \mathbb{R}^{k}$ definition \ref{PullbackOfDifferentialFormsOnSmoothSpaces} reproduces def. \ref{PullbackOfDifferentialForms}.

=--

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

Again by the [[Yoneda lemma]].

=--

+-- {: .num_remark #DifferentialFormOnSmoothSpaceAsSystemOfDiffFormsOnCoordinates}
###### Remark

Using def. \ref{PullbackOfDifferentialFormsOnSmoothSpaces}

Unwinding def. \ref{DifferentialnFormOnSmoothSpace} yields the following explicit description:

a differential $n$-form $\omega \in \Omega^n(X)$ on a smooth space $X$ is

1. for each way $\phi \colon \mathbb{R}^k \to X$ of laying out a coordinate system $\mathbb{R}^k$ in $X$ a differential $n$-form

   $$
     \phi^* \omega \in \Omega^n(\mathbb{R}^k)
   $$

   on the abstract coordinate system, as given by def. \ref{DifferentialnForms};

1. for each abstract [[coordinate transformation]] $f \colon \mathbb{R}^{k_2} \to \mathbb{R}^{k_1}$ a corresponding compatibility condition between local differential forms $\phi_1 \colon \mathbb{R}^{k_1} \to X$ and $\phi_2 \colon \mathbb{R}^{k_2} \to X$ of the form

   $$
     f^* \phi_1^* \omega = \phi_2^* \omega
     \,.
   $$

Hence a differential form on a smooth space is simply a collection of differential forms on all its coordinate systems such that these glue along all possible coordinate transformations.

=--

The following adds further explanation to the role of $\Omega^n \in Smooth0Tye$ as a _[[moduli space]]_. Notice that since $\Omega^n$ is itself a [[smooth space]], we may speak about differential $n$-forms on $\Omega^n$ itsefl.

+-- {: .num_defn #UniversalDifferentialnForm}
###### Definition

The **universal differential $n$-forms** is the differential $n$-form

$$
  \omega^n_{univ} \in \Omega^n(\Omega^n)
$$

which is modulated by the [[identity]] [[homomorphism]] $id \colon \Omega^n \to \Omega^n$.

=--

With this definition we have:

+-- {: .num_prop}
###### Proposition

For $X \in Smooth0Type$ any [[smooth space]], every differential $n$-form on $X$, $\omega \in \Omega^n(X)$ is the pullback of differential forms, def. \ref{PullbackOfDifferentialFormsOnSmoothSpaces}, of the universal differential $n$-form, def. \ref{UniversalDifferentialnForm}, along a homomorphism $f$ from $X$ into the moduli space $\Omega^n$ of differential $n$-forms:

$$
  \omega = f^* \omega^n_{univ}
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

This statement is of course in a way a big tautology. Nevertheless it is a very useful tautology to make explicit. The whole concept of differential forms on smooth spaces here may be thought of as simply a variation of the theme of the [[Yoneda lemma]].

=--

+-- {: bluebox }
######

This ends the Model-layer discussion of differential forms. We now pass to a more advanced discussion of this topic in the [Semantics layer below](#DifferentialFormsLayerSem). The reader wishing to stick to more elementary discussion for the time being should skip ahead to the Model-layer discussion of [differentiation below](#Differentiation).

=--

### Semantic Layer
 {#DifferentialFormsLayerSem}

#### Concrete smooth spaces
 {#ConcreteObjects}

The smooth universal moduli space of differential forms $\Omega^n(-)$
from def. \ref{SmoothModuliSpaceOfnForms}
is noteworthy in that it has a property not shared by
many smooth spaces that one might think of more naively: while evidently being "large" (the space of all differential forms!) it has "very few points" and "very few $k$-dimensional subspaces" for low $k$.  In fact

+-- {: .num_prop}
###### Proposition

For $k \lt n$ the smooth space $\Omega^n$ admits only a unique probe by $\mathbb{R}^k$:

$$
  Hom(\mathbb{R}^k, \Omega^n(-))
  \simeq
  \Omega^n(\mathbb{R}^k)
  =
  \{0\}
  \,.
$$

=--

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

By the [[Yoneda lemma]] a smooth morphism $\mathbb{R}^k \to \Omega^n$ is a [[differential n-form]] $\omega \in \Omega^n(\mathbb{R}^k)$. But for $n \gt k$ there is only the 0 element.

=--

So while $\Omega^n()$ is a large smooth space, it is "not supported on probes" in low dimensions in as much as one might expect, from more naive notions of smooth spaces.

We now formalize this. The formal notion of an smooth space which _is_ supported on its probes is that of a _[[concrete object]]_. There is a univeral map that sends any smooth space to its _concretification_. The universal moduli spaces of differential forms turn out to be _non-concrete_ in that their concetrification is the point.

+-- {: .num_defn #ConcreteObjectsAndConcretification}
###### Definition

Let $\mathbf{H}$ be a [[local topos]]. Write $\sharp \colon \mathbf{H} \to \mathbf{H}$ for the corresponding sharp modality, def. \ref{SharpModalityOfLocalTopos}. Then.

1. An object $X \in \mathbf{H}$ is called a **[[concrete object]]** if

   $$
     DeCoh_X \colon X \to \sharp X
   $$

   is a [[monomorphism]].

1. For $X \in \mathbf{H}$ any object, its  **concretification** $Conc(X) \in \mathbf{H}$ is the [[image]] factorization of $DeCoh_X$, hence the factorization into an [[epimorphism]] followed by a [[monomorphism]]

   $$
     DeCoh_X : X \to Conc(X) \hookrightarrow \sharp X
     \,.
   $$

=--

+-- {: .num_remark}
###### Remark

Hence the concretification $Conc(X)$ of an object $X$ is itself a [[concrete object]] and it is [[universal property|universal]] with this property. (...)

=--

+-- {: .num_prop #DecohesOverASiteWithTerminalObject}
###### Proposition

Let $C$ be a [[site]] of definition for the [[local topos]]
$\mathbf{H}$, with [[terminal object]] $*$. Then for
$X \colon C^{op} \to Set$ a sheaf, $DeCoh_X$
is given over $U \in C$ by

$$
  X(U)
  \underoverset{Yoneda}{\simeq}{\to}
  \mathbf{H}(U, X)
  \stackrel{\Gamma_{U,X}}{\to}
  Set(\Gamma(U),\Gamma(X))
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

For $n \geq 1$ we have

$$
  Conc(\Omega^n) \simeq *
  \,.
$$

=--

In this sense the smooth moduli space of differential $n$-forms is _maximally non-concrete_.

#### Smooth moduli space of differential forms on a smooth space
 {#SmoothModuliSpaceOfDifferentialForms}

We discuss the smooth space of differential forms _on a fixed smooth space_ $X$.

+-- {: .num_remark}
###### Remark

For $X$ a [[smooth space]], the smooth mapping space $[X, \Omega^n] \in Smooth0Type$ is the smooth space whose $\mathbb{R}^k$-plots are differential $n$-forms on the [[product]] $X \times \mathbb{R}^k$

$$
  [X, \Omega^n] \colon \mathbb{R}^k \mapsto \Omega^n(X \times \mathbb{R}^k)
  \,.
$$

This is not _quite_ what one usually wants to regard as an $\mathbb{R}^k$-parameterized of differential forms on $X$. That is instead usually meant to be a differential form $\omega$ on $X \times \mathbb{R}^k$ which has "no leg along $\mathbb{R}^k$". Another way to say this is that the family of forms on $X$ that is represented by some $\omega$ on $X \times \mathbb{R}^k$ is that which over a point $v \colon * \to \mathbb{RR}^k$ has the value $(id_X,v)^* \omega$. Under this [[pullback of differential forms]] any components of $\omega$ with "legs along $\mathbb{R}^k$" are identified with the 0 differential form

=--

This is captured by the following definition.

+-- {: .num_defn #SmoothSpaceOfFormsOnSmoothSpace}
###### Definition

For $X \in Smooth0Type$ and $n \in \mathbb{N}$, the **smooth space of differential $n$-forms** $\mathbf{\Omega}^n(X)$ on $X$ is the concretification, def. \ref{ConcreteObjectsAndConcretification}, of the smooth mapping space $[X, \Omega^n]$, def. \ref{SmoothFunctionSpace}, into the smooth moduli space of differential $n$-forms, def. \ref{SmoothModuliSpaceOfnForms}:

$$
  \mathbf{\Omega}^n(X) \coloneqq Conc([X, \Omega^n])
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

The $\mathbb{R}^k$-plots of $\mathbf{\Omega}^n(\mathbb{R}^k)$
are indeed smooth differential $n$-forms on $X \times \mathbb{R}^k$ which are such that their evaluation on vector fields tangent to $\mathbb{R}^k$ vanish.

=--

+-- {: .proof}
###### Proof (sketch)

By def. \ref{Decohese}, def. \ref{ConcreteObjectsAndConcretification} and
prop. \ref{DecohesOverASiteWithTerminalObject} the set of plots of $\mathbf{\Omega}^n(X)$ over $\mathbb{R}^k$ is the [[image]] of the [[function]]

$$
  \Omega^n(X \times \mathbb{R}^k)
  \simeq
   Hom_{Smooth0Type}(\mathbb{R}^k, [X,\Omega^n])
   \stackrel{\Gamma_{ \mathbb{R}^k, [X,\Omega^n] }}{\to}
   Hom_{Set}(\Gamma(\mathbb{R}^k), \Gamma [X, \Omega^n])
   \simeq
   Hom_{Set}(\mathbb{R}^k_s, \Omega^n(X))
  \,,
$$

where on the right $\mathbb{R}^k_s$ denotes, just for emphasis, the underlying set of $\mathbb{R}^k$. This function manifestly sends a smooth differential form $\omega \in \Omega^n(X \times \mathbb{R}^k)$ to the function from points $v$ of $\mathbb{R}^k$ to differential forms on $X$ given by

$$
  \omega \mapsto \left(v \mapsto  (id_X, v)^* \omega \right)
  \,.
$$

Under this function all components of differential forms with a "leg along" $\mathbb{R}^k$ are sent to the 0-form. Hence the image of this function is the collection of smooth forms on $X \times \mathbb{R}^k$ with "no leg along $\mathbb{R}^k$".

=--

+-- {: .num_remark }
###### Remark

For $n = 0$ we have (for any $X\in Smooth0Type$)

$$
  \begin{aligned}
    \mathbf{\Omega}^0(X)
      & \coloneqq Conc [X, \Omega^0]
    \\
    & \simeq Conc [X, \mathbb{R}]
    \\
    & \simeq [X, \mathbb{R}]
  \end{aligned}
  \,,
$$

by prop. \ref{SpaceOf0FormsIsRealLine}.

=--

### Syntactic Layer
 {#DifferentialFormsLayerSyn}

#### Images

+-- {: .num_prop}
###### Proposition

Let a morphism $f \colon X \to Y$ in $\mathbf{H}$ be the [[categorical semantics]] of the [[syntax]]

$$
  x \colon X \; \vdash \; f(x) \colon Y
  \,.
$$

Then the syntax for the [[infinity-image|image]] $i_f \colon im(f) \hookrightarrow Y$ is

$$
  \vdash\;
   \left(
     \sum_{y \colon Y} \exists_{x \colon X} \left(f\left(x\right) = y\right)
   \right)
   \colon
   Type
  \,.
$$

=--

Here $\exists_{\cdots} \cdots =_{def} \left[ \sum_{\cdots} \cdot\right]$ is the [[bracket type]] of the [[dependent sum type]].

Accordingly the syntax for the smooth moduli space of differential $n$-forms, def. \ref{SmoothSpaceOfFormsOnSmoothSpace},
on a smooth space $X$,

$$
  \sum_{\omega \colon \sharp [X, \Omega^n]}
  \exists_{\lambda \colon [X,\Omega^n]}
  (\omega = DeCoh_X(\lambda))
  \,.
$$

(...)