This entry contains one chapter of the material at geometry of physics.
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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
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We introduce the standard concept of differential forms in Model Layer, 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 smooth space. In Semantic Layer 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 the theory of concretification in a local topos is a central ingredient in the canonical existence of certain action functionals.
The process of concretification involves the general abstract notion of images. The type-theory of this notion we discuss in Syntactic Layer here.
We have seen above in The continuum real (world-)line that that real line $\mathbb{R}$ is the basic 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 functionals on such trajectories: by “action functionals”. In the simplest case these are for instance homomorphisms of smooth spaces
where $I \hookrightarrow \mathbb{R}$ is the standard unit interval.
Such action functionals we discuss in their own right in Variational calculus below. Here we first examine in detail a fundamental property they all have: they are supposed to be 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
As one takes this property to the limit of iterative subdivision, one finds that action functionals are entirely determined by their value on 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
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
Then we discuss how this induces a notion of smooth differential forms on general smooth spaces in
Further below we provide a precise version of the statement that “Differential 1-forms are differential measures along paths.” in
We introduce the basic concept of a smooth differential form on a Cartesian space $\mathbb{R}^n$. Below in Differential forms on smooth spaces we use this to define differential forms on any smooth space.
For $n \in \mathbb{N}$ a smooth differential 1-form $\omega$ on a Cartesian space $\mathbb{R}^n$ is an $n$-tuple
of smooth functions, which we think of equivalently as the coefficients of a formal linear combination
on a set $\{\mathbf{d}x^1, \mathbf{d}x^2, \cdots, \mathbf{d}x^n\}$ of cardinality $n$.
Write
for the set of smooth differential 1-forms on $\mathbb{R}^k$.
We think of $\mathbf{d} x^i$ as a measure for 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.
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.
For $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ a smooth function, the pullback of differential 1-forms along $\phi$ is the function
between sets of differential 1-forms, def. 1, which is defined on basis-elements by
and then extended linearly by
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.
The set $\Omega^1(\mathbb{R}^k)$ is naturally an abelian group with addition given by componentwise addition
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
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
being the same infinitesimal measure with orientation reversed.
For $k,n \in \mathbb{N}$, the smooth differential forms on $\mathbb{R}^k$ is the exterior algebra
over the ring $C^\infty(\mathbb{R}^k)$ of smooth functions of the module $\Omega^1(\mathbb{R}^k)$ of smooth 1-forms, prop. 3.
We write $\Omega^n(\mathbb{R}^k)$ for the sub-module of degree $n$ and call its elements the smooth 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 $C^\infty(\mathbb{R}^k)$ of basis elements of the form $\mathbf{d} x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n}$ for $i_1 \lt i_2 \lt \cdots \lt i_n$:
The pullback of differential 1-forms of def. 1 extends as an $C^\infty(\mathbb{R}^k)$-algebra homomorphism to $\Omega^n(-)$, given for a smooth function $f \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ on basis elements by
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.
The assignment of differential $n$-forms
of def. 5 together with the pullback of differential forms-functions of def. 6
defines a smooth space in the sense of def. \ref{SmoothSpace}:
We call this
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):
For the Cartesian space $\mathbb{R}^k$ regarded as a smooth space by example \ref{CartesianSpaceAsSmoothSpace}, there is a natural bijection
between the set of smooth $n$-forms on $\mathbb{R}^n$ according to def. 1 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.
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
Accordingly we write
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 4.
Notice that differential 0-forms are equivalently smooth $\mathbb{R}$-valued functions.
$\Omega^0 \simeq \mathbb{R}$
For $f \colon X \to Y$ a homomorphism of smooth spaces, def. \ref{HomomorphismOfSmoothSpaces}, the pullback of differential forms along $f$ is the function
given by the hom-functor into the smooth space $\Omega^n$ of def. 7:
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 composite $X \stackrel{f}{\to} Y \to \Omega^n$.
For $X = \mathbb{R}^{\tilde k}$ and $Y = \mathbb{R}^{k}$ definition 9 reproduces def. 6.
Again by the Yoneda lemma.
Using def. 9
Unwinding def. 8 yields the following explicit description:
a differential $n$-form $\omega \in \Omega^n(X)$ on a smooth space $X$ is
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
on the abstract coordinate system, as given by def. 5;
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
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.
The universal differential $n$-forms is the differential $n$-form
which is modulated by the identity homomorphism $id \colon \Omega^n \to \Omega^n$.
With this definition we have:
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. 9, of the universal differential $n$-form, def. 11, along a homomorphism $f$ from $X$ into the moduli space $\Omega^n$ of differential $n$-forms:
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.
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. The reader wishing to stick to more elementary discussion for the time being should skip ahead to the Model-layer discussion of differentiation below.
The smooth universal moduli space of differential forms $\Omega^n(-)$ from def. 7 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
For $k \lt n$ the smooth space $\Omega^n$ admits only a unique probe by $\mathbb{R}^k$:
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.
Let $\mathbf{H}$ be a local topos. Write $\sharp \colon \mathbf{H} \to \mathbf{H}$ for the corresponding sharp modality, def. \ref{SharpModalityOfLocalTopos}. Then.
An object $X \in \mathbf{H}$ is called a concrete object if
is a monomorphism.
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
Hence the concretification $Conc(X)$ of an object $X$ is itself a concrete object and it is universal with this property. (…)
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
For $n \geq 1$ we have
In this sense the smooth moduli space of differential $n$-forms is maximally non-concrete.
We discuss the smooth space of differential forms on a fixed smooth space $X$.
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 $X \times \mathbb{R}^k$
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.
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. 12, of the smooth mapping space $[X, \Omega^n]$, def. \ref{SmoothFunctionSpace}, into the smooth moduli space of differential $n$-forms, def. 7:
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.
By def. \ref{Decohese}, def. 12 and prop. 5 the set of plots of $\mathbf{\Omega}^n(X)$ over $\mathbb{R}^k$ is the image of the function
where on the right $\mathbb{R}^k_s$ denotes, just for emphasis, the underlying set of $\mathbb{R}^k_s$. 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
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 $\mathbb{R}^k$”.
For $n = 0$ we have (for any $X\in Smooth0Type$)
by prop. 2.
Let a morphism $f \colon X \to Y$ in $\mathbf{H}$ be the categorical semantics of the syntax
Then the syntax for the image $i_f \colon im(f) \hookrightarrow Y$ is
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. 13, on a smooth space $X$,
(…)