nLab geometry of physics -- smooth sets

This entry one chapter of geometry of physics.

previous chapters: categories and toposes

next chapters: supergeometry, smooth homotopy types

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This chapter introduces a generalized kind of sets equipped with smooth structure, to be called smooth sets or, with an eye towards their generalization to smooth homotopy types, smooth h-sets or smooth 0-types¹.

The definition (Def. below) subsumes that of smooth manifolds, Fréchet manifolds and diffeological spaces but is both simpler and more powerful: smooth sets are simply sheaves on the gros site of Cartesian Spaces (Prop. below) and as such form a nice category – a topos – and this contains as full subcategories the more “tame” objects such as smooth manifolds (Prop. below) and diffeological spaces (Prop. below).

In fact smooth sets are an early stage in a long sequence of generalized smooth spaces appearing in higher differential geometry:

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$\;\;\;\;$ $\{$coordinate systems$\}$
$\hookrightarrow$ $\{$smooth manifolds$\}$ $\phantom{AA}$ (Prop. below)
$\hookrightarrow$ $\{$Hilbert manifolds$\}$
$\hookrightarrow$ $\{$Banach manifolds$\}$
$\hookrightarrow$ $\{$Fréchet manifolds$\}$ $\phantom{AA}$ (Prop. below)
$\hookrightarrow$ $\{$diffeological spaces$\}$ $\phantom{AA}$ (Prop. below)
$\hookrightarrow$ $\{$smooth sets$\}$ $\hookrightarrow$ $\{$formal smooth sets$\}$ $\hookrightarrow$ $\{$super formal smooth sets$\}$ $\phantom{AA}$ (chapter on supergeometry)

$\hookrightarrow$ $\{$smooth orbifolds$\}$
$\hookrightarrow$ $\{$smooth groupoids$\}$
$\hookrightarrow$ $\{$smooth 2-groupoids$\}$
$\hookrightarrow$ $\cdots$
$\hookrightarrow$ $\{$smooth ∞-groupoids$\}$ $\phantom{AA}$ (chapter on smooth ∞-groupoids)
$\hookrightarrow$ $\{$formal smooth ∞-groupoids$\}$
$\hookrightarrow$ $\{$super formal smooth ∞-groupoids$\}$

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Abstract coordinate systems

As discussed in the chapter categories and toposes, every kind of geometry is modeled on a collection of archetypical basic spaces and geometric homomorphisms between them. In differential geometry the archetypical spaces are the abstract standard 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. below.) We highlight three classical theorems about smooth functions in Prop. below, which look innocent but play a decisive role in setting up 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 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.

The continuum real line

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

   $\mathbb{Z}$.

2. 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

   $\mathbb{Q}$.

3. This labeling is dense: every point on the worldline is the supremum of an inhabited bounded subset of such labels. This means that a worldline 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.

Cartesian spaces and smooth functions

Definition

(smooth function)

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

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

2. and is itself a smooth function.

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

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.

Definition

(Cartesian spaces and smooth functions)

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.

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.

Remark

But a homomorphism of Cartesian spaces in def. is not required to be a linear map. We do not regard the Cartesian spaces here as vector spaces.

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

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

Proposition

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

Definition

We will also say equivalently that

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

2. a smooth function $\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ is an abstract coordinate transformation;

3. 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 $x^k : \mathbb{R}^{n} \to \mathbb{R}$.

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

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

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

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

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, 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 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:

Proposition

(the three magic algebraic properties of differential geometry)

1. 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. ) are in natural bijection with their induced algebra homomorphisms $C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)$ (example ), so that one may equivalently handle Cartesian spaces entirely via their $\mathbb{R}$-algebras of smooth functions.

   Stated more abstractly, this means equivalently that the functor $C^\infty(-)$ that sends a smooth manifold $X$ to its $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions (example ) 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)

2. 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 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.

   More 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)

   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 free modules over $C^\infty(X)$.

   (Nestruev 03, theorem 11.32)

3. vector fields are derivations of smooth functions.

   For $X$ a Cartesian space (Def. ), then any derivation $D \colon C^\infty(X) \to C^\infty(X)$ on the $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions is given by differentiation with respect to a uniquely defined smooth tangent vector field: The function that regards tangent vector fields as derivations

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

   is in fact an isomorphism.

   (This follows directly from the Hadamard lemma.)

Actually all three statements in prop. hold not just for Cartesian spaces, but generally for smooth manifolds (def./prop. 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.

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The site of abstract coordinate systems

Much of the above disucssion is usefully summarized by saying that abstract coordinate systems with smooth functions between them form a category (Prop. below). Equipped with the information of how one abstract coordinate system may be covered by other coordinate systems (Def. below), this becomes a site (Prop. ) below.

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Propositions

(the category CartSp of abstract coordinate systems/Cartesian spaces)

Abstract coordinate systems according to prop. form a category (this def.) – 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.

2. The isomorphisms are precisely the diffeomorphisms.

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

Definition

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

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

Proposition

There is a diffeomorphism

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

Definition

(differentially good open covers)

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 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.

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 smooth homotopy types, around this prop.omotopy types#DifferentiablyGoodCoverGivesSPlitHyperCoverOverCartSp).

Lemma

(every open cover has refinement by a differentially good open cover)

Every open cover of a smooth manifold has a refinement by a differentially good open cover, according to Def. .

For proof see FSS10, Prop. A1, or see at good open cover.

Remark

Lemma 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 at open ball for the situation concerning this statement in the literature.

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 is again good.

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 (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.

Proposition

(the site of Cartesian spaces with differentially good open covers)

The differentially good open covers, Def. , constitute a coverage (this Def.) on the category CartSp (from Prop. ).

Hence CartSp equipped with this coverage is a site (this def.).

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 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 Lemma 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 this good open cover coverage is not a Grothendieck topology. But as any coverage, it uniquely completes to one which has the same sheaves:

Proposition

(completing good open covers to all open covers)

The Grothendieck topology induced on CartSp by the differentially good open cover coverage of def. has as covering families the ordinary open covers.

Hence if we explicitly write $CartSp_{good}$ and $CartSp_{Groth}$ for $CartSp$ equipped with the coverage of differentially good open covers as and that of all open covers, respectively, then there is an equivalence of categories (this Def.) between their categories of sheaves (this Def.)

(1)$$Sh(CartSp_{\text{good open cov}}) \;\simeq\; Sh(CartSp_{\text{all open cov}}) \.$$

Remark

Prop. means that for every sheaf-theoretic construction to follow we may just as well consider the Grothendieck topology of open covers on $CartSp$, and hence we may and will suppress the subscripts in (1).

While 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 to (∞,1)-sheaves on CartSp. This will be discussed in Smooth n-groupoids – Semantic Layer – Local Infinity-Connectedness below.

There are further sites in use, which induce the same categories of sheaves:

Definition

(sites of smooth manifolds and open subsets of Euclidean spaces)

We write

• SmthMfd for the category of smooth manifolds of any finite dimension, with smooth functions between them;

• $EuclOp$ for the category of open subsets of Euclidean spaces of any finite dimension, with smooth functions between them.

Both of these carry the respective coverage of good open covers and as such become sites (this Def.)

Smooth sets

In the section Coordinate systems we have set up the archetypical spaces of differential geometry. Here we now define in terms of these the most general smooth sets that differential geometry can deal with.

Plots of smooth sets and their gluing

The general kind of “smooth space” that we want to consider is a something that can be probed by laying out coordinate systems inside it, as in this definition, and which may be reconstructed by gluing all the possible coordinate systems in it together.

At this point we want to impose no further conditions on a “space” than this. In particular we do not assume that we know beforehand a set of points underlying $X$. Instead, we define smooth sets $X$ (Def. , below) entirely operationally as something about which we may ask “Which ways are there to lay out $\mathbb{R}^n$ inside $X$?” and such that there is a self-consistent answer to this question.

By the discussion in the chapter categories and toposes, this means that we should define a smooth set to be a sheaf on a site of Cartesian spaces. The following definitions spell this out.

The idea of the following definitions may be summarized like this:

1. a generalized smooth set is something that may be probed by laying out coordinate systems into it, in a way that respects transformation of coordinate patches and gluing of coordinate patches;

2. the Yoneda lemma (this prop) says that this is consistent in that coordinate systems themselves as well as smooth manifolds may naturally be regarded as generalized smooth sets themselves and that under this identification “laying out a coordinate system” in a smooth set means having a map of smooth sets from the coordinate system to the smooth set.

The first set of consistency conditions on plots of a space is that they respect coordinate transformations. This is what the following definition formalises.

Definition

(pre-smooth set)

A pre-smooth set $X$ is

1. a collection of sets: for each Cartesian space $\mathbb{R}^n$ (hence for each natural number $n$) a set

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

   – to be thought of as the set of ways of laying out $\mathbb{R}^n$ inside $X$, also called the set of plots of $X$, for short;

2. for each smooth function $f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ (to be thought of as an abstract coordinate transformation) a function between the corresponding sets of plots

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

   – to be thought of as the function that sends a plot of $X$ by $\mathbb{R}^{n_2}$ to the correspondingly transformed plot by $\mathbb{R}^{n_1}$ induced by laying out $\mathbb{R}^{n_1}$ inside $\mathbb{R}^{n_2}$.

such that this is compatible with coordinate transformations:

1. the identity coordinate transformation does not change the plots:

   $$X(id_{\mathbb{R}^n}) = id_{X(\mathbb{R}^n)} \,,$$

2. changing plots along two consecutive coordinate transformations $f_1 \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ and $f_2 \colon \mathbb{R}^{n_2} \to \mathbb{R}^{n_3}$ is the same as changing them along the composite coordinate transformation $f_2 \circ f_1$:

   $$X(f_1) \circ X(f_2) = X(f_2 \circ f_1) \,.$$

But there is one more consistency condition for a collection of plots to really be probes of some space: it must be true that if we glue small coordinate systems to larger ones, then the plots by the larger ones are the same as the plots by the collection of smaller ones that agree where they overlap. We first formalize this idea of “plots that agree where their coordinate systems overlap.”

Definition

(differentially good open cover)

For $\mathbb{R}^n$ a Cartesian space, then

1. an open cover

   $$\{U_i \hookrightarrow X\}_{i \in I}$$

   is a set of open subsets of $X$, such that their union is $X$; hence $\underset{i \in I}{\cup} U_i = X$;

2. this is a good open cover if all the $U_i$ as well as all their non-empty finite intersections are homeomorphic to open balls, hence to $\mathbb{R}^n$ itself;

3. this is a differentially good open cover if all the $U_i$ as well as all their non-empty finite intersections are diffeomorphic to open balls, hence to $\mathbb{R}^n$ itself.

Definition

(glued plots)

Let $X$ be a pre-smooth set, def. . For $\{U_i \to \mathbb{R}^n\}_{i \in I}$ a differentially good open cover (def. ) let

$$GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \in Set$$

be the set of $I$-tuples of $U_i$-plots of $X$ which coincide on all double intersections

$$\array{ && U_i \cap U_j \\ & {}^{\mathllap{\iota_i}}\swarrow && \searrow^{\mathrlap{\iota_j}} \\ U_i &&&& U_j \\ & \searrow && \swarrow \\ && \mathbb{R}^n }$$

(also called the matching families of $X$ over the given cover):

$$GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \;\;\coloneqq\;\; \left\{ \; \left(p_i \in X(U_i)\right)_{i \in I} \;|\;\; \forall_{i,j \in I} \;:\; X(\iota_i)(p_i) = X(\iota_j)(p_j) \; \right\} \,.$$

Remark

(interpretation of the gluing condition)

In def. the equation

$$X(\iota_i)(p_i) = X(\iota_j)(p_j)$$

says in words:

The plot $p_i$ of $X$ by the coordinate system $U_i$ inside the bigger coordinate system $\mathbb{R}^n$ coincides with the plot $p_j$ of $X$ by the other coordinate system $U_j$ inside $X$ when both are restricted to the intersection $U_i \cap U_j$ of $U_i$ with $U_j$ inside $\mathbb{R}^n$.

Remark

(comparing global plots to glued plots)

For each differentially good open cover $\{U_i \to \mathbb{R}^n\}_{i \in I}$ (def. ) and each pre-smooth set $X$, def. , there is a canonical function

$$X(\mathbb{R}^n) \longrightarrow GluedPlots(\{U_i \to \mathbb{R}^n\}, X)$$

from the set of $\mathbb{R}^n$-plots of $X$ to the set of tuples of glued plots, which sends a plot $p \in X(\mathbb{R}^n)$ to its restriction to all the $\phi_i \colon U_i \hookrightarrow \mathbb{R}^n$:

$$p \mapsto (X(\phi_i)(p))_{i \in I} \,.$$

If $X$ is supposed to be consistently probable by coordinate systems, then it must be true that the set of ways of laying out a coordinate system $\mathbb{R}^n$ inside it coincides with the set of ways of laying out tuples of glued coordinate systems inside it, for each good cover $\{U_i \to \mathbb{R}^n\}$ as above. Therefore:

Definition

(smooth set)

A pre-smooth set $X$, def. is a smooth set if for all differentially good open covers $\{U_i \to \mathbb{R}^n\}$ (def. ) the canonical comparison function of remark from plots to glued plots is a bijection

(2)$$X(\mathbb{R}^n) \stackrel{\simeq}{\longrightarrow} GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \,.$$

Remark

We may think of a smooth set as being a kind of space whose local models (in the general sense discussed at geometry) are Cartesian spaces:

While definition explicitly says that a smooth set is something that is consistently probeable by such local models, a general abstract fact sometimes called the co-Yoneda lemma – which we discuss in more detail below in smooth sets - Semantic Layer – implies that smooth sets are also precisely the objects that are obtained by amalgamating coordinate systems, in some sense.

For instance, we will see that two open 2-balls $\mathbb{R}^2 \simeq D^2$ along a common rim yields the smooth set version of the sphere $S^2$, a basic example of a smooth manifold. But before we examine such explicit constructions, we discuss here for the moment more general properties of smooth sets. The reader instead wishing to see more of these concrete examples at this point should jump ahead to smooth sets - Outlook.

But the following most basic example we consider right now:

Example

(Cartesian spaces and smooth manifold as smooth sets)

For $n \in \mathbb{R}^n$, there is a smooth set, def. , whose set of plots over the abstract coordinate systems $\mathbb{R}^k$ is the set

$$CartSp(\mathbb{R}^k, \mathbb{R}^n) \in Set$$

of smooth functions from $\mathbb{R}^k$ to $\mathbb{R}^n$.

Clearly this is the rule for plots that characterize $\mathbb{R}^n$ itself as a smooth set, and so we will just denote this smooth set by the same symbols “$\mathbb{R}^n$”:

$$\mathbb{R}^n \colon \mathbb{R}^k \mapsto CartSp(\mathbb{R}^k, \mathbb{R}^n) \,.$$

In particular the real line $\mathbb{R}$ is this way itself a smooth set.

More generally, if the reader already knows what a smooth manifold $X$ is; these become smooth sets by taking their plots to be the ordinary smooth functions between smooth manifolds, from Cartesian spaces:

$$X(\mathbb{R}^n) \coloneqq Hom_{SmthMfd}(\mathbb{R}^n, X) \,.$$

Some smooth sets are far from being like smooth manifolds:

Example

(smooth moduli space of differential forms)

Let $k \in \mathbb{N}$. Then there is a smooth set (def. ) to be denoted $\mathbf{\Omega}^k$ given as follows:

1. The set $\mathbf{\Omega}^k(\mathbb{R}^n)$ of plots from $\mathbb{R}^n$ is the set of smooth differential k-forms on $\mathbb{R}^n$

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

2. for $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a smooth function, then the corresponding change-of-plots function is the operation of pullback of differential forms along $f$:

   $$\Omega^{k}(\mathbb{R}^{n_2}) \overset{f^\ast}{\longrightarrow} \Omega^k(\mathbb{R}^{n_1}) \,.$$

We introduce and discuss this example in detail in more detail below

A smooth set (def. ) need not have an underlying set, for instance the smooth set $\mathbf{\Omega}^k$ from example for $k \geq 1$ has only a single plot from the point (corresponding to the zero differential form on the point), and yet it is far from being the point. If a smooth set does have an underlying set, then it is called a diffeological space, see around Prop. below.

Example

(discrete smooth set)

For $S \in$ Set a set, write

$$Disc S \in SmoothSet$$

for the smooth set whose set of $U$-plots for every $U \in CartSp$ is always $S$.

$$Disc S \colon U \mapsto S$$

and which sends every coordinate transformation $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ to the identity function on $S$.

A smooth set of this form we call a discrete smooth set.

More examples of smooth sets can be built notably by intersecting images of two smooth sets inside a bigger one. In order to say this we first need a formalization of homomorphism of smooth sets. This we turn to now.

Homomorphisms of smooth sets

We discuss “functions” or “maps” between smooth sets, def. , which preserve the smooth set structure in a suitable sense. As with any notion of function that preserves structure, we refer to them as homomorphisms.

The idea of the following definition is to say that whatever a homomorphism $f : X \to Y$ between two smooth sets is, it has to take the plots of $X$ by $\mathbb{R}^n$ to a corresponding plot of $Y$, such that this respects coordinate transformations.

Definition

(homomorphisms of smooth sets – smooth functions)

Let $X$ and $Y$ be two smooth sets, def. . Then a homomorphism of smooth function $f \colon X \to Y$ between them is

• for each abstract coordinate system $\mathbb{R}^n$ (hence for each $n \in \mathbb{N}$) a function

  $f_{\mathbb{R}^n} : X(\mathbb{R}^n) \to Y(\mathbb{R}^n)$

  that sends $\mathbb{R}^n$-plots of $X$ to $\mathbb{R}^n$-plots of $Y$

such that

• for each smooth function $\phi : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ we have

  $$Y(\phi) \circ f_{\mathbb{R}^{n_1}} = f_{\mathbb{R}^{n_2}} \circ X(\phi)$$

  hence a commuting diagram

  $$\array{ X(\mathbb{R}^{n_1}) &\stackrel{f_{\mathbb{R}^{n_1}}}{\to}& Y(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{X(\phi)}} && \downarrow^{\mathrlap{Y(\phi)}} \\ X(\mathbb{R}^{n_2}) &\stackrel{f_{\mathbb{R}^{n_2}}}{\to}& Y(\mathbb{R}^{n_1}) } \,.$$

For $f_1 : X \to Y$ and $f_2 : X \to Y$ two homomorphisms of smooth sets, their composition $f_2 \circ f_1 \colon X \to Y$ is defined to be the homomorphism whose component over $\mathbb{R}^n$ is the composite of functions of the components of $f_1$ and $f_2$:

$$(f_2\circ f_1)_{\mathbb{R}^n} \coloneqq {f_2}_{\mathbb{R}^n} \circ {f_1}_{\mathbb{R}^n} \,.$$

Definition

(the category SmoothSet of smooth sets with smooth functions)

Write SmoothSet or $SmoothSet$ for the category (this def.) whose objects are smooth sets, def. , and whose morphisms are homomorphisms of smooth sets according to def. .

At this point it may seem that we have now two different notions for how to lay out a coordinate system in a smooth set $X$: on the hand, $X$ comes by definition with a rule for what the set $X(\mathbb{R}^n)$ of its $\mathbb{R}^n$-plots is. On the other hand, we can now regard the abstract coordinate system $\mathbb{R}^n$ itself as a smooth set, by example , and then say that an $\mathbb{R}^n$-plot of $X$ should be a homomorphism of smooth sets of the form $\mathbb{R}^n \to X$.

The following proposition says that these two superficially different notions actually naturally coincide.

Proposition

Let $X$ be any smooth set, def. , and regard the abstract coordinate system $\mathbb{R}^n$ as a smooth set, by example . There is a natural bijection

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

between the postulated $\mathbb{R}^n$-plots of $X$ and the actual $\mathbb{R}^n$-plots given by homomorphism of smooth sets $\mathbb{R}^n \to X$.

Proof

This is a special case of the Yoneda lemma (this prop.), as will be made more explicit below in The topos of smooth sets. The reader unfamiliar with this should write out the simple proof explicitly: use the defining commuting diagrams in def. to deduce that a homomorphism $f : \mathbb{R}^n \to X$ is uniquely fixed by the image of the identity element in $\mathbb{R}^n(\mathbb{R}^n) \coloneqq CartSp(\mathbb{R}^n, \mathbb{R}^n)$ under the component function $f_{\mathbb{R}^n} : \mathbb{R}^n(\mathbb{R}^n) \to X(\mathbb{R}^n)$.

Example

Let $\mathbb{R} \in SmoothSet$ denote the real line, regarded as a smooth set by example . Then for $X \in SmoothSet$ any smooth set, a homomorphism of smooth sets

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

is a smooth function on $X$. Prop. says here that when $X$ happens to be an abstract coordinate system regarded as a smooth set by def. , then this general notion of smooth functions between smooth sets reproduces the basic notion of this def.ystems#CartesianSpaceAndHomomorphism)

Definition

The 0-dimensional abstract coordinate system $\mathbb{R}^0$ we also call the point and regarded as a smooth set we will often write it as

$$* \in SmoothSet \,.$$

For any $X \in SmoothSet$, we say that a homomorphism

$$x \colon * \to X$$

is a point of $X$.

Remark

By prop. the points of a smooth set $X$ are naturally identified with its 0-dimensional plots, hence with the “ways of laying out a 0-dimensional coordinate system” in $X$:

$$Hom(*, X) \simeq X(\mathbb{R}^0) \,.$$

Products and fiber products of smooth sets

Definition

Let $X, Y \in SmoothSet$ by two smooth sets. Their product is the smooth set $X \times Y \in SmoothSet$ whose plots are pairs of plots of $X$ and $Y$:

$$X\times Y (\mathbb{R}^n) \coloneqq X(\mathbb{R}^n) \times Y(\mathbb{R}^n) \;\; \in Set \,.$$

The projection on the first factor is the homomorphism

$$p_1 \colon X \times Y \to X$$

which sends $\mathbb{R}^n$-plots of $X \times Y$ to those of $X$ by forming the projection of the cartesian product of sets:

$${p_1}_{\mathbb{R}^n} \colon X(\mathbb{R}^n) \times Y(\mathbb{R}^n) \stackrel{p_1}{\to} X(\mathbb{R}^n) \,.$$

Analogously for the projection to the second factor

$$p_2 \colon X \times Y \to Y \,.$$

Proposition

Let $* = \mathbb{R}^0$ be the point, regarded as a smooth set, def. . Then for $X \in SmoothSet$ any smooth set the canonical projection homomorphism

$$X \times * \to X$$

is an isomorphism.

Definition

Let $f \colon X \to Z$ and $g \colon Y \to Z$ be two homomorphisms of smooth sets, def. . There is then a new smooth set to be denoted

$$X \times_Z Y \in SmoothSet$$

(with $f$ and $g$ understood), called the fiber product of $X$ and $Y$ along $f$ and $g$, and defined as follows:

the set of $\mathbb{R}^n$-plots of $X \times_Z Y$ is the set of pairs of plots of $X$ and $Y$ which become the same plot of $Z$ under $f$ and $g$, respectively:

$$(X \times_Z Y)(\mathbb{R}^n) = \left\{ (p_X \in X(\mathbb{R}^n), p_Y \in Y(\mathbb{R}^n)) \; |\; f_{\mathbb{R}^n}(p_X) = g_{\mathbb{R}^n}(p_Y) \right\} \,.$$

Smooth mapping spaces and smooth moduli spaces

Definition

(smooth mapping space)

Let $\Sigma, X \in SmoothSet$ be two smooth sets, def. . Then the smooth mapping space

$$[\Sigma,X] \in SmoothSet$$

is the smooth set defined by saying that its set of $\mathbb{R}^n$-plots is

$$[\Sigma, X](\mathbb{R}^n) \coloneqq Hom(\Sigma \times \mathbb{R}^n, X) \,.$$

Here in $\Sigma \times \mathbb{R}^n$ we first regard the abstract coordinate system $\mathbb{R}^n$ as a smooth set by example and then we form the product smooth set by def. .

Remark

This means in words that a $\mathbb{R}^n$-plot of the mapping space $[\Sigma,X]$ is a smooth $\mathbb{R}^n$-parameterized family of homomorphisms $\Sigma \to X$.

Proposition

There is a natural bijection

$$Hom(K, [\Sigma, X]) \simeq Hom(K \times \Sigma, X)$$

for every smooth set $K$.

Proof

With a bit of work this is straightforward to check explicitly by unwinding the definitions. It follows however from general abstract results once we realize that $[-,-]$ is of course the internal hom of smooth sets. This we come to below in Smooth sets - Semantic Layer.

Remark

This says in words that a smooth function from any $K$ into the mapping space $[\Sigma,X]$ is equivalently a smooth function from $K \times \Sigma$ to $X$. The latter we may regard as a $K$-parameterized smooth family of smooth functions $\Sigma \to X$. Therefore in view of the previous remark this says that smooth mapping spaces have a universal property not just over abstract coordinate systems, but over all smooth sets.

We will therefore also say that $[\Sigma,X]$ is the smooth moduli space of smooth functions from $\Sigma \to X$, because it is such that smooth maps $K \to [\Sigma,X]$ into it modulate, as we move around on $K$, a family of smooth functions $\Sigma\to X$, depending on $K$.

First interesting examples of such smooth moduli spaces are discussed in Differential forms – Model Layer below. Many more interesting examples follow once we pass from smooth 0-types to smooth $n$-types below in Smooth n-groupoids.

We will see many more examples of smooth moduli spaces, starting below in Differential forms - Model Layer.

Proposition

The set of points, def. , of a smooth mapping space $[\Sigma,X]$ is the bare set of homomorphism $\Sigma \to X$: there is a natural isomorphism

$$Hom(*, [\Sigma, X]) \simeq Hom(\Sigma, X) \,.$$

Proof

Combine prop. with prop. .

Example

Given a smooth set $X \in SmoothSet$, its smooth path space is the smooth mapping space

$$\mathbf{P}X \coloneqq [\mathbb{R}^1, X] \,.$$

By prop. the points of $P X$ are indeed precisely the smooth trajectories $\mathbb{R}^1 \to X$. But $P X$ also knows how to smoothly vary such smooth trajectories.

This is central for variational calculus which determines equations of motion in physics. This we turn to below in Variational calculus.

Remark

In physics, if $X$ is a model for spacetime, then $P X$ may notably be interpreted as the smooth set of worldlines in $X$, hence the smooth set of paths or trajectories of a particle in $X$.

Example

If in the above example the path is constraind to be a loop in $X$, one obtains the smooth loop space

$$\mathbf{L}X \coloneqq [S^1, X] \,.$$

In example we saw that a smooth function on a general smooth set $X$ is a homomorphism of smooth sets, def.

$$f \colon X \to \mathbb{R} \,.$$

The collection of these forms the hom-set $Hom_{SmoothSet}(X, \mathbb{R})$. But by the discussion in Smooth mapping spaces such hom-sets are naturally refined to smooth sets themselves.

Definition

For $X \in SmoothSet$ a smooth set, we say that the moduli space of smooth functions on $X$ is the smooth mapping space (def. ), from $X$ into the standard real line $\mathbb{R}$

$$[X, \mathbb{R}] \in SmoothSet \,.$$

We will also denote this by

$$\mathbf{C}^\infty(X) \coloneqq [X, \mathbb{R}] \,,$$

since in the special case that $X$ is a Cartesian space this is the smooth refinement of the set $C^\infty(X)$ of smooth functions on $X$.

Remark

We call this a moduli space because by prop. above and in the sense of remark it is such that smooth functions into it modulate smooth functions $X \to \mathbb{R}$.

By prop. a point $* \to [X,\mathbb{R}^1]$ of the moduli space is equivalently a smooth function $X \to \mathbb{R}^1$.

The cohesive topos of smooth sets

In the language of categories and toposes, we may summarize the concept of smooth sets by saying that they form the sheaf topos over the site of Cartesian spaces (Prop. below).

This perspective allows to see good abstract properties enjoyed by the smooth sets. The key such property is that the topos which they form is a cohesive topos (Prop. below).

$\,$

Proposition

(equivalence of categories between smooth sets and sheaves on CartSp)

There is a canonical equivalence of categories (this def.) between the category SmoothSet of smooth sets from def. , and the category of sheaves (this def.) on the category CartSp (this def.) equipped with the coverage (this def.) of differentiably good open covers (def. )

$$SmoothSet \;\simeq\; Sh(CartSp) \,.$$

Proof

This is a straightforward matter of matching definitions. We spell it out:

• A pre-smooth set, def. is equivalently a presheaf (this Example) on CartSp (this prop.), hence a functor (this def.) $X : CartSp^{op} \to Set$ from the category CartSp to the category of sets (this Example);

• a smooth set, def. , is equivalently a presheaf on CartSp (this prop.) which is a sheaf (this def.) with respect to the coverage (this def.) of differentially good open cover (def. ):

  • the set of “glued plots” (def. ) is the set of matching families (this def.)

  • the comparison morphism from global plots to glued plots of remark is the comparison map from to matching families (here);

  • the condition (2) that this be a bijection is the sheaf condition (here).

Proposition

(equivalent sites for CartSp)

Consider the canonical full subcategory-inclusion functors

$$CartSp \overset{\phantom{AAA}}{\hookrightarrow} EuclOp \overset{\phantom{AAA}}{\hookrightarrow} SmthMfd$$

which regard, in turn, a Cartesian space (Def. ) as an open subset of itself, and regard every open subset of Euclidean space (Def. ) as a smooth manifold (this Example[this Example] and (differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds)).

Then the induced pre-composition functors induce equivalences of categories (this Def.) between the corresponding categories of sheaves:

$$SmoothSet \;\simeq\; Sh(CartSp) \;\simeq\; Sh(EuclOp) \;\simeq\; Sh(SmthMfd) \,.$$

Proof

By Prop. we may identify $Sh(CartSp) = Sh(CartSp_{\text{good open cov}})$. With that, both inclusions are evidently dense subsite-inclusions (this Def.). Therefore the statement follows by the comparison lemma (this prop.).

As a corollary we obtain:

Proposition

(smooth manifolds fully faithful inside smooth sets)

Regarding smooth manifolds as smooth sets via Example yields a full subcategory-inclusion

$$SmoothManifold \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow} SmoothSet \,,$$

meaning that for $X, Y \in SmoothManifold$ any two smooth manifolds, this functor induces a bijection between the smooth functions $X \to Y$ regarded in the sense of smooth manifolds, and regarded in the sense of smooth sets (Def. ):

$$Hom_{SmoothManifold}(X,Y) \;\simeq\; Hom_{SmoothSet}(\iota(X), \iota(Y)) \,.$$

Proof

By Prop. we have an equivalence of categories

$$SmoothSet \;\simeq\; Sh(SmoothManifold) \,.$$

With this the statement is given by the Yoneda lemma (this prop.).

$\,$

Proposition

(smooth sets form a cohesive topos)

The site CartSp (Prop. ) is a cohesive site (this Def.), hence its sheaf topos is a cohesive topos (by this Prop.). Under the identification of Prop. , this means that:

The category SmoothSet of smooth sets (Def. ) is a cohesive topos (this Def.):

(3)$$SmoothSet \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } Set$$

Moreover, this cohesive topos satisfies the following equivalent conditions (from this Prop.):

1. pieces have points,

2. discrete objects are concrete.

Proof

The category $CartSp$ clearly has finite products: The terminal object is the point, given by the 0-dimensional Cartesian space

$$\ast = \mathbb{R}^0$$

and the Cartesian product of two Cartesian spaces is the Cartesian space whose dimension is the sum of the two separate dimensions:

$$\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,.$$

This establishes the first clause in the definition of cohesive site (this def.)

For the second clause, consider a differentiably-good open cover $\{U_i \overset{}{\to} \mathbb{R}^n\}$ (Def. ). This being a good cover implies that its Cech groupoid is, as an internal groupoid (via this remark), of the form

(4)$$C(\{U_i\}_i) \;\simeq\; \left( \array{ \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} y(U_i) } \right) \,.$$

where we used the defining property of good open covers to identify $y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )$.

The colimit of (4), regarded just as a presheaf of reflexive directed graphs (hence ignoring composition for the moment), is readily seen to be the graph of the colimit of the components (the universal property follows immediately from that of the component colimits):

(5)$$\begin{aligned} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \ast \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \ast } \right) \end{aligned} \,.$$

Here we first used that colimits commute with colimits, hence in particular with coproducts (this prop.) and then that the colimit of a representable presheaf is the singleton set (this Lemma).

This colimiting graph carries a unique composition structure making it a groupoid, since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point.

Precisely this operation on Cech groupoids of good open covers of topological spaces is what Borsuk's nerve theorem is about, a classical result in topology/homotopy theory. This theorem implies directly that the set of connected components of the groupoid (6) is in bijection with the set of connected components of the Cartesian space $\mathbb{R}^n$, regarded as a topological space. But this is evidently a connected topological space, which finally shows that, indeed

$$\pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,.$$

The second item of the second clause in Def. follows similarly, but more easily: The limit of the Cech groupoid is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since $CartSp$ has a terminal object $\ast = \mathbb{R}^0$, which is hence an initial object in the opposite category $CartSp^{op}$, limits over $CartSp^{op}$ yield simply the evaluation on that object:

(6)$$\begin{aligned} \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \phantom{A} \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} Hom_{CartSp}\left( \ast, U_i \underset{\mathbb{R}^n}{\cap} U_j \right) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} Hom_{CartSp}( \ast, U_i ) } \right) \end{aligned} \,.$$

Here we used that colimits (here coproducts) of presheaves are computed objectwise, and then the definition of the Yoneda embedding $y$.

But the equivalence relation induced by this graph on its set of objects $\underset{i}{\coprod} Hom_{CartSp}( \ast, U_i )$ precisely identifies pairs of points, one in $U_i$ the other in $U_j$, that are actually the same point of the $\mathbb{R}^n$ being covered. Hence the set of equivalence classes is the set of points of $\mathbb{R}^n$, which is just what remained to be shown:

$$\pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,.$$

Finally to see that pieces have points and discrete objects are concrete is satisfied, it is sufficient to observe, by this prop. that for each $n \in \mathbb{N}$, the hom set $Hom_{CartSp}(\ast, \mathbb{R}^n)$ is non-empty.

The following statement is a 0-truncated shadow of the differential geometric analog of the concept of A1-localization:

Proposition

(shape modality on smooth sets is $\mathbb{R}^1$-localization)

The reflective subcategory-inclusion from Prop.

$$Set \underoverset {\underset{ \phantom{A}Disc\phantom{A} }{\hookrightarrow}} {\overset{\Pi}{\longleftarrow}} {\bot} SmoothSet$$

of sets as the discrete smooth sets, exhibits the reflection on $\mathbb{R}^1$-local objects (this Def.) for $\mathbb{R}^1$ the real line, in the sense of localization at an object:

$$\Pi \;\simeq\; L_{\mathbb{R}^1} \,.$$

Proof

Since we already know that the left adjoint $\Pi$ exists, we need to show that the full subcategory of discrete smooth sets $Set \overset{Disc}{\hookrightarrow} SmoothSet$ is equivalently that of local objects (this Def.) with respect to the class of morphisms of the form

$$X \times \mathbb{R}^1 \overset{ p_1 }{\longrightarrow} X$$

for $X$ any object of $SmoothSet$.

We claim that a smooth set $A$ is a local object with respect to this class, already if it is a local object with respect to just the small set

(7)$$\left\{ \mathbb{R}^n \times \mathbb{R}^1 \overset{p_1}{\longrightarrow} \mathbb{R}^n \right\}_{n \in \mathbb{N}}$$

for $X = \mathbb{R}^n$ a Cartesian space.

To see this, we identify $SmoothSet \simeq Sh(CartSp)$ via Prop. , and then use the co-Yoneda lemma (this Prop.) to express any $X \in SmoothSet$ as a colimit of Cartesian spaces: $X \simeq \int^{\mathbb{R}^n \in Cartsp} \mathbb{R}^n \cdot X(\mathbb{R}^n)$.

Assuming then that $A$ is local with respect to the small set (7) we obtain

$$\begin{aligned} Hom_{SmoothSet}\big( X \times \mathbb{R}^1 \overset{p_1}{\to} X \;\,,\; A \big) & \simeq \int_{\mathbb{R}^n \in CartSp} Hom_{Set}\left( X(\mathbb{R}^n), \, \underset{\text{bijection}}{ \underbrace{ Hom_{SmoothSet}\big( \mathbb{R}^n \times \mathbb{R}^1 \overset{p_1}{\to} \mathbb{R}^n \;\,,\; A \big) }} \right) \end{aligned}$$

where we used that

1. the Cartesian product $(-) \times \mathbb{R}^1$ preserves colimits, since it is a left adjoint (this Prop.) and since left adjoints preserve colimits (this Prop.),

2. the hom-functor preserves limits (this Prop.), hence sends colimits in the first argument to limits.

But on the right this is now a morphisms of limits induced by morphism of diagrams which is objectwise an bijection, as shown, and hence is a bijection) itself.

Hence we are reduced to showing that the local objects with respect to (7) are the discrete smooth sets. But by induction over $n \in \mathbb{N}$, locality with respect to (7) means equivalently locality with respect to

$$\left\{ \mathbb{R}^n \overset{\exists !}{\to} \mathbb{R}^0 = \ast \right\}$$

The local objects with respect to this set are manifestly exactly the constant presheaves. But by the proof of Prop. (proceeding via the proof of this Prop.) these are indeed exactly the objects in the image of $Set \overset{Disc}{\hookrightarrow} SmoothSet$.

$\,$

Concrete smooth sets: Diffeological spaces

The cohesiveness of smooth sets (Prop. ) implies in particular that there is a concept of concrete objects among the smooth sets (this Def.). We show here (Prop. below) that these concrete smooth sets are equivalent to a kind of generalized smooth spaces that are known as Chen smooth spaces (Chen 77) or diffeological spaces (Souriau 79, Iglesias-Zemmour 85), which we recall as Def. below.

A comprehensive development of differential geometry in terms of diffeological spaces is spelled out in (Iglesias-Zemmour 13).

Since the concrete objects in any cohesive topos, and hence the diffeological spaces among all smooth sets, form a reflective subcategory (This prop.), every smooth set has a concretification to a concrete smooth set, hence to a diffeological space. An important example of this construction are moduli spaces of differential forms, this we turn to in Def. below.

$\,$

Definition

(diffeological space (e.g. Iglesias Zemmour 18, def. 2 and 5))

A diffeological space $\mathbf{X}$ is

1. a set $X \in Set$, called the underlying set of the diffeological space;

2. for each open subset $U \subset E^n$ of some Euclidean space $E^n$, for any $n \in \mathbb{N}$, a subset

   $$\mathbf{X}(U) \subset Hom_{Set}(U, X)$$

   of the function set of functions of sets from $U$ to $X$, called the set of plots of $\mathbf{X}$ by $U$;

such that for every open subset $U$ as above the following conditions hold:

1. the set of plots $\mathbf{X}(U)$ contains all the constant functions

   $$X \overset{const_U}{\hookrightarrow} \mathbf{X}(U) \hookrightarrow Hom_{Set}(U,X) \,,$$

2. for every function $\phi \;\colon\; U \to X$ and for every open cover $\{U_i \overset{\iota_i}{\to} U\}$, if each restriction is a plot, $\phi\vert_{U_i} \in \mathbf{X}(U_i)$, then $\phi$ itself is a plot: $\phi \in \mathbf{X}(U)$;

3. for all plots $\phi \in \mathbf{X}(U)$, all open subsets $V$ of any Euclidean space, and all smooth functions $V \overset{f}{\to} U$, we have, the composition is again a plot

   $$\phi \circ f \;\in\; \mathbf{X}(V) \,.$$

Moreover, for $\mathbf{X}$ and $\mathbf{Y}$ two diffeological spaces, as above, then a smooth map between them

$$\mathbf{X} \overset{f}{\longrightarrow} \mathbf{Y}$$

is a function of underlying sets

$$X \overset{\phantom{A} f \phantom{A}}{\longrightarrow} Y \;\;\in\;\; Set$$

such that for each plot $\phi \in \mathbf{X}(U)$ of $\mathbf{X}$ the composition with that function is a plot of $\mathbf{Y}$:

$$f \circ \phi \;\in\; \mathbf{X}(U) \,.$$

This defines a category $DiffeologicalSpace$ (this def.) whose objects are the diffeological spaces, whose morphisms are the smooth maps between them, with composition of morphisms the ordinary composition of functions of underlying sets.

Proposition

(diffeological spaces are the concret smooth sets)

The category of diffeological spaces (Def. ) is a full subcategory of the category of smooth sets (Def. ).

Moreover, in terms of the cohesive structure on the category of smooth sets from Prop. , the diffeological spaces are precisely the concrete objects (this def.) among the smooth sets:

$$DiffeologicalSpace \;\simeq\; SmoothSet_{conc} \overset{\phantom{AAAA}}{\hookrightarrow} SmoothSet \,.$$

Proof

First observe that the assignment of sets of plots

$$U \mapsto \mathbf{X}(U)$$

of a diffeological space $\mathbf{X}$, according to Def. constitutes a sheaf (this Def.) on the site $EuclOp$ of open subsets of Euclidean spaces, by the same unwinding of Definitions as in Prop. :

• the third clause in the list of properties in Def. says that the assignment of sets of plots is a presheaf,

• the second clause in the list of properties says that this presheaf satisfies the sheaf condition,

• while the first clause in the list of properties is an extra condition, singling out diffeological spaces among all sheaves.

Under this identification, the definition of a smooth map of diffeological spaces in Def. says that it is equivalently a morphism of presheaves of sets of plots between sheaves of sets of plots, and hence a morphism of sheaves. This establishe a full subcategory-inclusion

$$DiffeologicalSpace \hookrightarrow Sh(EuclOp) \,.$$

But by Prop. the restriction from the site $EuclOp$ of all open subsets of Euclidean spaces to that of just the site CartSp of Cartesian spaces is an equivalence of categories between the corresponding sheaf toposes. This yields the full subcategory inclusion

$$DiffeologicalSpace \hookrightarrow Sh(EuclOp) \simeq Sh(CartSp) \simeq SmoothSet \,,$$

where the last equivalence is Prop. .

It remains to see that under this inclusion, the diffeological spaces are identified with the concrete objects among the smooth set.

By definition (this Def.), a smooth set $\mathbf{X} \in SmoothSet$ is concrete, precisely if its sharp-unit is a monomorphism

$$\mathbf{X} \overset{\phantom{A} \eta_X^\sharp \phantom{A}}{\hookrightarrow} \sharp \mathbf{X} \,,$$

which is the adjunction unit (this Def.) of the $(\Gamma \dashv coDisc)$-adjunction

$$\mathbf{X} \overset{\phantom{A} \eta_X \phantom{A}}{\hookrightarrow} coDisc \Gamma \mathbf{X} \,.$$

Now a morphism of sheaves is a monomorphism, precisely if for each object $U \in CartSp$ in the site, its component function

(8)$$\mathbf{X}(U) \overset{\phantom{A} \eta_X^\sharp(U) \phantom{A}}{\hookrightarrow} (coDisc \Gamma \mathbf{X})(U)$$

is an injective function (this Prop.). Unde the Yoneda lemma, this function may be re-identified as follows:

(9)$$\array{ \mathbf{X}(U) &\simeq& Hom_{SmoothSet}(y(U), \mathbf{X}) \\ {}^{\mathllap{ \eta_X^\sharp(U) }}\big\downarrow && {}^{\mathllap{ Hom_{SmoothSet}(y(U), \eta_X^\sharp ) }}\big\downarrow &\searrow^{\mathrlap{ \widetilde{\eta^\sharp_X \circ (-)} }}& \\ (coDisc \Gamma \mathbf{X}(U)) &\simeq& Hom_{SmoothSet}( y(U), coDisc \Gamma \mathbf{X} ) &\simeq& Hom_{Set}( \Gamma y(U), \Gamma \mathbf{X} ) &\simeq& Hom_{Set}( U, X ) \\ && \phi &\mapsto& \widetilde \phi \,, }$$

where we first used the Yoneda lemma (this Prop.), then the adjunction isomorphism (here) of $(\Gamma \dashv coDisc)$. In the final step we used that the cohesive structure on $SmoothSet$ comes from $CartSp$ being a cohesive site (Prop. ) and that in this case $\Gamma$ is given by evaluation on the point (here), and we wrote

(10)$$X \;\coloneqq\; \Gamma \mathbf{X} = \mathbf{X}(\ast)$$

for the set of points in $\mathbf{X}$. Notice that if $\mathbf{X}$ is indeed a diffeological space, then this set is indeed its underlying set, by the first clause in the list of conditions on a diffeological space in Def. .

This shows that (8) being an injection means equivalently that we have an injection of the form

(11)$$\mathbf{X}(U) \overset{\phantom{A} \eta_X^\sharp(U) \phantom{A}}{\hookrightarrow} Hom_{Set}(U, X) \,.$$

Hence that $\mathbf{X}(U)$ is always a subset of the function set from $U$ to the set $X$, as in the second clause in Def. .

This shows that every concrete smooth set is a diffeological space. For the converse, it remains to check that if we start with a diffeological space $\mathbf{X}$ with prescribed inclusion function

(12)$$\mathbf{X}(U) \hookrightarrow Hom_{Set}(U,X)$$

then (11) indeed reproduces this inclusion.

To see this, first notice that, by the Yoneda lemma (this prop.) and the definition of smooth maps between diffeological spaces, the inclusion function (12) equals the component function of the functor $\Gamma \;\colon\; SmoothSet \to Set$, that acts by point evaluation:

$$\Gamma_{y(U),\mathbf{X}} \;\colon\; \array{ Hom_{SmoothSet}( y(U), \mathbf{X} ) &\hookrightarrow& Hom_{Set}( \Gamma y(U), \Gamma \mathbf{X} ) \\ f &\mapsto& \Gamma(f) }$$

Hence, by (8), we need to show that

(13)$$\Gamma_{y(U), \mathbf{X}} \;=\; \widetilde{\eta_{\mathbf{X}} \circ (-)} \,.$$

But this holds as a general fact about adjunctions (a special case of this Example).

By this prop. it follows that

Proposition

(reflection of diffeological spaces in cohesive topos of smooth sets)

The category of diffeological spaces (Def. ) is “in between” the category of sets (this Example) and the category of smooth sets (Def. ) as exhibited by the following system of adjoint functors:

$$\array{ \\ \phantom{a} \\ \phantom{A} \\ \Gamma \;\dashv\; coDisc } \;\;\colon\;\; SmoothSet \array{ \phantom{\overset{ \phantom{AA} \Pi_0 \phantom{AA} }{\longrightarrow}} \\ \phantom{\overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow}} \\ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } DiffeologicalSpace \array{ \overset{ \phantom{AA} \Pi_0 \phantom{AA} }{\longrightarrow} \\ \overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set$$

where on the left we have a reflective subcategory with reflector being concretification (this prop.), and on the right we have the corestriction of the adjoint quadruple of cohesion from (3).

Proposition

(Fréchet manifolds fully faithful among smooth sets)

Write $FrechetMfd$ for the category of Fréchet manifolds and smooth functions between these, which generalizes smooth manifolds to possibly infinite-dimensional smooth manifolds

$$CartSp \overset{\phantom{AAAA}}{\hookrightarrow} SmoothMfd \overset{\phantom{AAAA}}{\hookrightarrow} FrechetMfd \,.$$

For $X,Y$ two Fréchet manifolds, write again $C^\infty(X,Y)$ for the set of smooth functions between them. Then the same kind of construction as for smooth manifolds, sending a Fréchet manifold to the presheaf

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

defines a fully faithful functor (this Example)

$$FrechetMfd \overset{\phantom{AAAA}}{\hookrightarrow} SmoothSet \,,$$

hence a full subcategory inclusion.

Proof

The construction clearly factors through diffeological spaces (Def. ), identified as a full subcategory of smooth sets via Prop. .

With this it is now sufficient to see that Fréchet manifolds are fully faithful among diffeological spaces. This is due to (Losik 94, theorem 3.1.1),

$\,$

Differential forms

We have seen above in The continuum real 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 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

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

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

Such action functionals we discuss in their own right in Variational calculus 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

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

As one takes this property to the limit of iterative subdivision, one finds that action functionals are entirely determined by their value on infinitesimal 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.

$\,$

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 $\mathbb{R}^k$.

Remark

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.

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

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

between sets of differential 1-forms, def. , 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

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

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.

Definition

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

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

2. 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

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

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, with

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

being the same infinitesimal measure with orientation reversed.

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. .

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

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 $i_1 \lt i_2 \lt \cdots \lt i_n$:

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

Definition

The pullback of differential 1-forms of def. 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

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

$\,$

So far we have defined differential $n$-form on abstract coordinate systems. Here we extend this definition to one of differential $n$-forms on arbitrary smooth sets. We start by observing that the space of all differential $n$-forms on cordinate systems themselves naturally is a smooth set.

Proposition

(smooth moduli space of differential forms)

The assignment of differential $n$-forms

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

of def. together with the pullback of differential forms-functions of def.

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

constitutes a smooth set in the sense of def. , which we denote by

$$\mathbf{\Omega}^n(-) \;\in\; SmoothSet \,.$$

We call this the universal smooth moduli space of differential $n$-forms.

The reason for this terminology is that homomorphisms of smooth sets into $\Omega^1$ modulate differential $n$-forms on their domain, by prop. (and hence by the Yoneda lemma, this prop):

Example

For the Cartesian space $\mathbb{R}^k$ regarded as a smooth set by example , 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. and the set of homomorphism of smooth set, $\mathbb{R}^k \to \Omega^1$, according to def. .

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

Definition

For $X \in SmoothSet$ a smooth set, def. , a differential n-form on $X$ is a homomorphism of smooth sets of the form

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

Accordingly we write

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

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

We may unwind this definition to a very explicit description of differential forms on smooth sets. This we do in a moment in remark .

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

Examples

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

Definition

For $f \colon X \to Y$ a homomorphism of smooth sets, def. , 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 set $\Omega^n$ of def. :

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

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 composite $X \stackrel{f}{\to} Y \to \Omega^n$.

Proposition

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

Proof

Again by the Yoneda lemma (this prop.)

Remark

Using def.

Unwinding def. yields the following explicit description:

a differential $n$-form $\omega \in \Omega^n(X)$ on a smooth set $X$ 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. ;

2. 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 set 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 set, we may speak about differential $n$-forms on $\Omega^n$ itsefl.

Definition

The universal differential $n$-form 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:

Proposition

For $X \in SmoothSet$ any smooth set, every differential $n$-form on $X$, $\omega \in \Omega^n(X)$ is the pullback of differential forms, def. , of the universal differential $n$-form, def. , along a homomorphism $f$ from $X$ into the moduli space $\Omega^n$ of differential $n$-forms:

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

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 sets here may be thought of as simply a variation of the theme of the Yoneda lemma (this prop.).

$\,$

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

Remark

For $X$ a smooth space, the smooth mapping space $[X, \Omega^n] \in SmoothSet$ 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.

Definition

(concrete moduli space of differential forms on a smooth set)

For $X \in SmoothSet$ any smooth set and $n \in \mathbb{N}$, the smooth space of differential $n$-forms $\mathbf{\Omega}^n(X)$ on $X$ is the concretification (Prop. ) of the smooth mapping space $[X, \Omega^n]$, def. , into the smooth moduli space of differential $n$-forms, def. :

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

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 (sketch)

By the proof of this Prop. spring the set of plots of $\mathbf{\Omega}^n(X)$ over $\mathbb{R}^k$ is the image of the function

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

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 $\mathbb{R}^k$”.

Remark

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

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

by prop. .

$\,$

Integration and transgression

The traditional concept of integration of differential forms over a compact smooth manifold $\Sigma$ applies in smooth families of differential forms and hence constitutes in fact a smooth function from the smooth moduli space of differential forms on the given manifold, this is Def. below.

Using this, transgression of differential forms may be defined as the application of the mapping space-functor out of $\Sigma$ to the modulating morphisms of differential forms and applying integration of differential forms to the result (Def. below). This simple construction turns out to be equivalent to the traditional definition (Prop. below).

Definition

(parameterized integration of differential forms)

Let

1. $X$ be a smooth set;

2. $\Sigma_k$ be a compact smooth manifold of dimension $k$, regarded as a smooth set via Example .

3. $n \geq k \in \mathbb{N}$ a natural number;

Consider the smooth mapping space $[\Sigma_k, \mathbf{\Omega}^n]$ (Def. ) out of $\Sigma_k$ into the universal smooth moduli space $\mathbf{\Omega}^n$ of differential n-forms (Prop. ).

Then we write

$$\int_{\Sigma} \;\colon\; [\Sigma_k, \mathbf{\Omega}^n] \longrightarrow \mathbf{\Omega}^{n-k}$$

for the smooth function (Def. ) which takes a plot $\omega_{(-)} \colon U \to [\Sigma, \mathbf{\Omega}^k]$, hence equivalently a differential $n$-form $\omega_{(-)}(-)$ on $U \times \Sigma$, to the result of integration of differential forms over $\Sigma$:

$$\int_{\Sigma} \omega_{(-)}(-) \coloneqq \int_\Sigma \omega_{(-)} \,.$$

Definition

(transgression of differential forms to mapping spaces)

Let

1. $X$ be a smooth set;

2. $\Sigma_k$ be a compact smooth manifold of dimension $k$, regarded as a smooth set via Example .

3. $n \geq k \in \mathbb{N}$ a natural number;

Consider the smooth mapping space $[\Sigma_k, \mathbf{\Omega}^n]$ (Def. ) out of $\Sigma_k$ into the universal smooth moduli space $\mathbf{\Omega}^n$ of differential n-forms (Prop. ).

Then the operation of transgression of differential n-forms on $X$ with respect to $\Sigma_k$ is the function

$$\tau_\Sigma \;\coloneqq\; \int_\Sigma [\Sigma,-] \;\colon\; \Omega^n(X) \to \Omega^{n-k}([\Sigma,X])$$

from differential n-forms on $X$ to differential $n-k$-forms on the smooth mapping space $[\Sigma_k,X]$ spring which takes the differential form corresponding to the smooth function

$$(X \stackrel{\omega}{\to} \Omega^n) \in \Omega^n(X)$$

to the differential form corresponding to the following composite smooth function:

$$\tau_\Sigma \omega \coloneqq \int_{\Sigma} [\Sigma,\omega] \;\colon\; [\Sigma, X] \stackrel{[\Sigma, \omega]}{\longrightarrow} [\Sigma, \Omega^n] \stackrel{\int_{\Sigma}}{\longrightarrow} \Omega^{n-k} \,,$$

where $[\Sigma,\omega]$ is the mapping space functor on morphisms and $\int_{\Sigma}$ is the parameterized integration of differential forms from def. .

More explicitly in terms of plots this means equivalently the following

A plot of the mapping space

$$\phi_{(-)} \;\colon\; U \to [\Sigma, X]$$

is equivalently a smooth function of the form

$$\phi_{(-)}(-) \;\colon\; U \times \Sigma \to X \,.$$

The smooth function $[\Sigma,\omega]$ takes this smooth function to the plot

$$U \times \Sigma \to X \overset{\phi_{(-)}(-)}{\longrightarrow} X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^{n}$$

which is equivalently a differential form

$$(\phi_{(-)}(-))^\ast \omega \in \Omega^n(U \times \Sigma) \,.$$

Finally the smooth function $\int_\Sigma$ takes this to the result of integration of differential forms over $\Sigma$:

$$\tau_{\Sigma}\omega\vert_{\phi_{(-)}} \;=\; \int_\Sigma (\phi_{(-)}(-))^\ast \omega \;\in\; \Omega^{n-k}(U) \,.$$

Definition

(transgression of differential forms to mapping space via evaluation map)

Let

1. $X$ be a smooth set;

2. $n \geq k \in \mathbb{N}$;

3. $\Sigma_k$ be a compact smooth manifold of dimension $k$.

Then the operation of transgression of differential $n$-forms on $X$ with respect to $\Sigma$ is the function

$$\tau_\Sigma \coloneqq \int_\Sigma ev^\ast \;\colon\; \Omega^n(X) \overset{ev^\ast}{\longrightarrow} \Omega^n(\Sigma \times [\Sigma, X]) \overset{\int_\Sigma}{\longrightarrow} \Omega^{n-k}([\Sigma,X])$$

from differential $n$-forms on $X$ to differential $n-k$-forms on the mapping space $[\Sigma,X]$ (Def. ) which is the composite of forming the pullback of differential forms along the evaluation map $ev \colon [\Sigma, X] \times \Sigma \to X$ with integration of differential forms over $\Sigma$.

This construction manifestly extends to the smooth set of differential forms

Proposition

The two definitions of transgression of differential forms to mapping spaces from def. and def. are equivalent.

Proof

We need to check that for all plots $\gamma \colon U \to [\Sigma, X]$ the pullbacks of the two forms to $U$ coincide.

For def. we get

$$\gamma^\ast \int_\Sigma \mathrm{ev}^\ast A = \int_\Sigma (\gamma,\mathrm{id}_\Sigma)^\ast \mathrm{ev}^\ast A \; \in \Omega^n(U)$$

Here we recognize in the integrand the pullback along the $( (-)\times \Sigma \dashv [\Sigma,-])$-adjunct $\tilde \gamma : U \times \Sigma \to \Sigma$ of $\gamma$, which is given by applying the left adjoint $(-)\times \Sigma$ and then postcomposing with the adjunction counit $\mathrm{ev}$:

$$\array{ U \times \Sigma & \overset{(\gamma, \mathrm{id}_\Sigma)}{\longrightarrow} & [\Sigma,X] \times \Sigma & \overset{\mathrm{ev}}{\longrightarrow} & X } \,.$$

Hence the integral is now

$$\cdots = \int_{\Sigma} \tilde \gamma^\ast A \,.$$

This is the operation of the top horizontal composite in the following naturality square for adjuncts, and so the claim follows by its commutativity:

$$\array{ \tilde \gamma \in & Hom(U \times\Sigma, X) & \overset{Hom(U \times \Sigma,A)}{\longrightarrow} & Hom(U \times \Sigma, \mathbf{\Omega}^{n+k}) & \overset{\int_\Sigma(U)}{\longrightarrow} & \Omega^n(U) \\ & {}^{\mathllap{\simeq}}\big\downarrow && {}^{\mathllap{\simeq}}\big\downarrow && {}^{\mathllap{\simeq}}\big\downarrow \\ \gamma \in & Hom(U,[\Sigma,X]) & \overset{Hom(U,[\Sigma,A])}{\longrightarrow} & Hom(U,[\Sigma,\mathbf{\Omega}^{n+k}]) & \overset{Hom(U,\int_\Sigma)}{\longrightarrow} & Hom(U,\mathbf{\Omega}^n) }$$

(here we write $Hom(-,-) \coloneqq Hom_{SmoothSet}(-,-)$ for the hom functor of smooth sets).

Proposition

(relative transgression over manifolds with boundary)

1. $X$ be a smooth set;

2. $\Sigma_k$ be a compact smooth manifold of dimension $k$ with boundary $\partial \Sigma$

3. $n \geq k \in \mathbb{N}$;

4. $\omega \in \Omega^n_{X}$ a closed differential form.

Write

$$(-)\vert_{\partial \Sigma} \;\coloneqq\; [\partial \Sigma \hookrightarrow \Sigma, X] \;\colon\; [\Sigma, X] \longrightarrow [\partial \Sigma, X]$$

for the smooth function that restricts smooth functions on $\Sigma$ to smooth functions on the boundary $\partial \Sigma$.

Then the operations of transgression of differential forms (def. ) to $\Sigma$ and to $\partial \Sigma$, respectively, are related by

$$d \left( \tau_{\Sigma}(\omega) \right) = (-1)^{k+1} ((-)\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma}(\omega) \phantom{AAAAAAAA} \array{ [\Sigma, X] &\overset{ \tau_{\Sigma}(\omega) }{\longrightarrow}& \mathbf{\Omega}^{n-k} \\ {}^{\mathllap{(-)\vert_{\partial \Sigma} }}\downarrow && \downarrow^{\mathrlap{ (-1)^{k+1} d}} \\ [\partial \Sigma, X] &\underset{ \tau_{\partial\Sigma}(\omega) }{\longrightarrow}& \mathbf{\Omega}^{n-k+1} } \,.$$

In particular this means that if the compact manifold $\Sigma$ happens to have no boundary (is a closed manifold) then transgression over $\Sigma$ takes closed differential forms to closed differential forms.

Proof

Let $\phi_{(-)}(-) \colon U \times \Sigma \to X$ be a plot of the mapping space $[\Sigma, X]$. Notice that the de Rham differential on the Cartesian product $U \times \Sigma$ decomposes as

$$d = d_U + d_\Sigma \,.$$

Now we compute as follows:

$$\begin{aligned} d \tau_{\Sigma}\omega\vert_{\phi_(-)} & = d_U \int_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma d_U (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma (d - d \Sigma) (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma d (\phi_{(-)}(-))^\ast \omega - (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma (\phi_{(-)}(-))^\ast \underset{= 0}{\underbrace{d \omega}} - (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = - (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = -(-1)^k \int_{\partial \Sigma} (\phi_{(-)}(-))^\ast \omega \\ & = -(-1)^k \tau_{\partial \Sigma} \omega \vert_{\phi_{(-)}} \end{aligned}$$

where in the second but last step we used Stokes' theorem.

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1. In view of the smooth homotopy types to be discussed in geometry of physics – smooth homotopy types, the structures discussed now are properly called smooth 0-types or maybe smooth h-sets or just smooth sets. While this subsumes smooth manifolds which are indeed sets equipped with (particularly nice) smooth structure, it is common in practice to speak of manifolds as “spaces” (indeed as topological spaces equipped with smooth structure). Historically the Cartesian space and Euclidean space of Newtonian physics are the archetypical examples of smooth manifolds and modern differential geometry developed very much via motivation by the study of the spaces in general relativity, namely spacetimes. Unfortunately, in a parallel development the word “space” has evolved in homotopy theory to mean (just) the homotopy types represented by an actual topological space (their fundamental infinity-groupoids). Ironically, with this meaning of the word “space” the original Euclidean spaces become equivalent to the point, signifying that the modern meaning of “space” in homotopy theory is quite orthogonal to the original meaning, and that in homotopy theory therefore one should better stick to “homotopy types”. Since historically grown terminology will never be fully logically consistent, and since often the less well motivated terminology is more widely understood, we will follow tradition here and take the liberty to use “smooth sets” and “smooth spaces” synonymously, the former when we feel more formalistic, the latter when we feel more relaxed. ↩