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geometry of physics -- smooth sets

Context

Physics

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theory (physics), model (physics)

experiment, measurement, computable physics

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

This entry one chapter of geometry of physics.

previous chapters: categories and toposes

next chapters: supergeometry, smooth homotopy types

\,

\,

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. 10 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. 3 below) and as such form a nice category – a topos – and this contains as full subcategories the more “tame” objects such as smooth manifolds and diffeological spaces.

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

\,
\;\;\;\; {\{coordinate systems}\}
\hookrightarrow {\{smooth manifolds}\}
\hookrightarrow {\{Hilbert manifolds}\}
\hookrightarrow {\{Banach manifolds}\}
\hookrightarrow {\{Fréchet manifolds}\}
\hookrightarrow {\{diffeological spaces}\}
\hookrightarrow {\{smooth sets}\}
\hookrightarrow {\{smooth orbifolds}\}
\hookrightarrow {\{smooth groupoids}\}
\hookrightarrow {\{smooth 2-groupoids}\}
\hookrightarrow \cdots
\hookrightarrow {\{smooth ∞-groupoids}\}
\hookrightarrow {\{formal smooth ∞-groupoids}\}
\hookrightarrow {\{super formal smooth ∞-groupoids}\}

\,

Contents

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 n\mathbb{R}^n (in every dimension nn \in \mathbb{N}). The geometric homomorphism between them are smooth functions n 1 n 2\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. 3 below.) We highlight three classical theorems about smooth functions in Prop. 2 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 nn smaller intervals, for each natural number nn. 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:f : \mathbb{R} \to \mathbb{R} is called a smooth function if, coinductively:

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

  2. and is itself a smooth function.

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

Remark

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

Definition

For nn \in \mathbb{N}, the Cartesian space n\mathbb{R}^n is the set

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

of nn-tuples of real numbers. For 1kn1 \leq k \leq n write

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

for the function such that i k(x)=(0,,0,x,0,,0)i^k(x) = (0, \cdots, 0,x,0,\cdots,0) is the tuple whose kkth entry is xx and all whose other entries are 00 \in \mathbb{R}; and write

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

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

A homomorphism of Cartesian spaces is a smooth function

f: n 1 n 2, f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} \,,

hence a function f: n 1 n 2f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} such that all partial derivatives exist and are continuous.

Example

Regarding n\mathbb{R}^n as an \mathbb{R}-vector space, every linear function n 1 n 2\mathbb{R}^{n_1} \to \mathbb{R}^{n_2} is in particular a smooth function.

Remark

But a homomorphism of Cartesian spaces in def. 2 is not required to be a linear map. We do not regard the Cartesian spaces here as vector spaces.

Definition

A smooth function f: n 1 n 2f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} is called a diffeomorphism if there exists another smooth function n 2 n 1\mathbb{R}^{n_2} \to \mathbb{R}^{n_1} such that the underlying functions of sets are inverse to each other

fg=id f \circ g = id

and

gf=id. g \circ f = id \,.
Proposition

There exists a diffeomorphism n 1 n 2\mathbb{R}^{n_1} \to \mathbb{R}^{n_2} precisely if n 1=n 2n_1 = n_2.

Definition

We will also say equivalently that

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

  2. a smooth function n 1 n 2\mathbb{R}^{n_1} \to \mathbb{R}^{n_2} is an abstract coordinate transformation;

  3. the function p k: np^k : \mathbb{R}^{n} \to \mathbb{R} is the kkth coordinate of the coordinate system n\mathbb{R}^n. We will also write this function as x k: nx^k : \mathbb{R}^{n} \to \mathbb{R}.

  4. for f: n 1 n 2f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} a smooth function, and 1kn 21 \leq k \leq n_2 we write

    1. f kp kff^k \coloneqq p^k\circ f

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

Remark

It follows with this notation that

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

Hence an abstract coordinate transformation

f: n 1 n 2 f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}

may equivalently be written as the tuple

(f 1(x 1,,x n 1),,f n 2(x 1,,x n 1)). \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 XX and YY two Cartesian spaces, the smooth functions f:XYf \colon X \longrightarrow Y between them (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}) are in natural bijection with their induced algebra homomorphisms C (X)f *C (Y)C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y) (example \ref{AlgebraOfSmoothFunctionsOnCartesianSpaces}), so that one may equivalently handle Cartesian spaces entirely via their \mathbb{R}-algebras of smooth functions.

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

    C ():SmthMfdAAAAAlg op. 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 1vb 1XE_1 \overset{vb_1}{\to} X and E 2vb 2XE_2 \overset{vb_2}{\to} X two vector bundle (def. \ref{VectorBundle}) there is then a natural bijection between vector bundle homomorphisms f:E 1E 2f \colon E_1 \to E_2 and the homomorphisms of modules f *:Γ X(E 1)Γ X(E 2)f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2) that these induces between the spaces of sections (example \ref{ModuleOfSectionsOfAVectorBundle}).

    More abstractly this means that the functor Γ X()\Gamma_X(-) is a fully faithful functor

    Γ X():VectBund XAAAAC (X)Mod \Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod

    (Nestruev 03, theorem 11.29heorem#Nestruev03))

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

    (Nestruev 03, theorem 11.32heorem#Nestruev03))

  3. vector fields are derivations of smooth functions.

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

    Γ X(TX) AA Der(C (X)) v D v \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. 2 hold not just for Cartesian spaces, but generally for smooth manifolds (def./prop. \ref{SmoothManifoldInsideDiffeologicalSpaces} below; if only we generalize in the second statement from free modules to projective modules. However for our development here it is useful to first focus on just Cartesian spaces and then bootstrap the theory of smooth manifolds and much more from that, which we do below.

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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. 3 below). Equipped with the information of how one abstract coordinate system may be covered by other coordinate systems (Def. 6 below), this becomes a site (Prop. 5) below.

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Propositions

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

Abstract coordinate systems according to prop. 4 form a category (this def.) – to be denoted CartSp – whose

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

D n={(x i) i=1 n n| i=1 n(x i) 2<1} n. 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

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

(differentially good open covers)

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

nU i 1U i k \mathbb{R}^n \stackrel{\simeq}{\to} U_{i_1} \cap \cdots \cap U_{i_k}

that identifies the kk-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. 6.

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

Remark

Lemma 1 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 { 2ϕ i 2} i{1,2}\{\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

2{(x 1,x 2) 2|x 1<1}ϕ 1 2 \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

2{(x 1,x 2) 2|x 1>1}ϕ 2 2. \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

2{(x 1,x 2) 2|1<x 1<1} 2. \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 2\mathbb{R}^2.

But consider then the smooth function

2(cos(2π()),sin(2π())): 1 2 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 2\mathbb{R}^2 to 1\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 1\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. 6, constitute a coverage (this Def.) on the category CartSp (from Prop. 3).

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

Proof

By definition of coverage we need to check that for {U i n} iI\{U_i \hookrightarrow \mathbb{R}^n\}_{i \in I} any good open cover and f: k nf \colon \mathbb{R}^k \to \mathbb{R}^n any smooth function, we can find a good open cover {K j k} jJ\{K_j \to \mathbb{R}^k\}_{j \in J} and a function JIJ \to I such that for each jJj \in J there is a smooth function ϕ:K jU ρ(j)\phi \colon K_j \to U_{\rho(j)} that makes this diagram commute:

K j ϕ U i(j) k f n. \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 k}\{f^* U_i \to \mathbb{R}^k\} be the pullback of the original covering family, in that

f *U i{x k|f(x)U i} k. 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 1 there does exist a good open cover {K˜ j˜ k} j˜J˜\{\tilde K_{\tilde j} \hookrightarrow \mathbb{R}^k\}_{\tilde j \in \tilde J} refining it, which in turn means that for all j˜\tilde j there is

K˜ j˜ K j(j˜) k = k. \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

K˜ j˜ K j(j˜) U i(j(j˜)) k = k f n \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 2 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. 5 has as covering families the ordinary open covers.

Hence if we explicitly write CartSp goodCartSp_{good} and CartSp GrothCartSp_{Groth} for CartSpCartSp 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 good open cov)Sh(CartSp all open cov)\. Sh(CartSp_{\text{good open cov}}) \;\simeq\; Sh(CartSp_{\text{all open cov}}) \.
Remark

Prop. 6 means that for every sheaf-theoretic construction to follow we may just as well consider the Grothendieck topology of open covers on CartSpCartSp, 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 CartSpCartSp 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

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

Proposition

(equivalent sites for CartSp)

Consider the canonical full subcategory-inclusion functors

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

which regard, in turn, a Cartesian space (Def. 3) as an open subset of itself, and regard every open subset of Euclidean space (Def. 7) 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:

Sh(CartSp)Sh(EuclOp)Sh(SmthMfd). Sh(CartSp) \;\simeq\; Sh(EuclOp) \;\simeq\; Sh(SmthMfd) \,.
Proof

By Prop. 6 we may identify Sh(CartSp)=Sh(CartSp good open cov)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.).

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 XX. Instead, we define smooth sets XX (Def. 10, below) entirely operationally as something about which we may ask “Which ways are there to lay out n\mathbb{R}^n inside XX?” 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 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 XX is

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

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

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

  2. for each smooth function f: n 1 n 2f : \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):X( n 2)X( n 1) 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 XX by n 2\mathbb{R}^{n_2} to the correspondingly transformed plot by n 1\mathbb{R}^{n_1} induced by laying out n 1\mathbb{R}^{n_1} inside n 2\mathbb{R}^{n_2}.

such that this is compatible with coordinate transformations:

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

    X(id n)=id X( n), X(id_{\mathbb{R}^n}) = id_{X(\mathbb{R}^n)} \,,
  2. changing plots along two consecutive coordinate transformations f 1: n 1 n 2f_1 \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} and f 2: n 2 n 3f_2 \colon \mathbb{R}^{n_2} \to \mathbb{R}^{n_3} is the same as changing them along the composite coordinate transformation f 2f 1f_2 \circ f_1:

    X(f 1)X(f 2)=X(f 2f 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 n\mathbb{R}^n a Cartesian space, then

  1. an open cover

    {U iX} iI \{U_i \hookrightarrow X\}_{i \in I}

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

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

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

Definition

(glued plots)

Let XX be a pre-smooth set, def. 8. For {U i n} iI\{U_i \to \mathbb{R}^n\}_{i \in I} a differentially good open cover (def. 6) let

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

be the set of II-tuples of U iU_i-plots of XX which coincide on all double intersections

U iU j ι i ι j U i U j n \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 XX over the given cover):

GluedPlots({U i n},X){(p iX(U i)) iI| i,jI:X(ι i)(p i)=X(ι j)(p j)}. 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. 9 the equation

X(ι i)(p i)=X(ι j)(p j) X(\iota_i)(p_i) = X(\iota_j)(p_j)

says in words:

The plot p ip_i of XX by the coordinate system U iU_i inside the bigger coordinate system n\mathbb{R}^n coincides with the plot p jp_j of XX by the other coordinate system U jU_j inside XX when both are restricted to the intersection U iU jU_i \cap U_j of U iU_i with U jU_j inside n\mathbb{R}^n.

Remark

(comparing global plots to glued plots)

For each differentially good open cover {U i n} iI\{U_i \to \mathbb{R}^n\}_{i \in I} (def. 6) and each pre-smooth set XX, def. 8, there is a canonical function

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

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

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

If XX 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 n\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 n}\{U_i \to \mathbb{R}^n\} as above. Therefore:

Definition

(smooth set)

A pre-smooth set XX, def. 8 is a smooth set if for all differentially good open covers {U i n}\{U_i \to \mathbb{R}^n\} (def. 6) the canonical comparison function of remark 9 from plots to glued plots is a bijection

(2)X( n)GluedPlots({U i n},X). 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 10 explicitly says that a smooth set is something that is consistently probeable by such local models; by a general abstract fact – which we discuss in more detail below in smooth sets - Semantic Layer – that is sometimes called the co-Yoneda lemma it follows in fact that smooth sets are precisely the objects that are obtained by gluing coordinate systems together.

For instance we will see that two open 2-balls 2D 2\mathbb{R}^2 \simeq D^2 along a common rim yields the smooth set version of the sphere S 2S^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 nn \in \mathbb{R}^n, there is a smooth set, def. 10, whose set of plots over the abstract coordinate systems k\mathbb{R}^k is the set

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

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

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

n: kCartSp( k, 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 XX is; these become smooth sets by taking their plots to be the ordinary smooth functions between smooth manifolds, from Cartesian spaces:

X( n)Hom SmthMfd( n,X). 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 kk \in \mathbb{N}. Then there is a smooth set (def. 3) to be denoted Ω k\mathbf{\Omega}^k given as follows:

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

    Ω k( n)Ω k( n) \mathbf{\Omega}^k(\mathbb{R}^n) \coloneqq \Omega^k(\mathbb{R}^n)
  2. for f: n 1 n 2f \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 ff:

    Ω k( n 2)f *Ω k( n 1). \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 the chapter geometry of physics -- differential forms.

A smooth set (def. 10) need not have an underlying set, for instance the smooth set Ω k\mathbf{\Omega}^k from example 4 for k1k \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:

Example

(diffeological spaces)

A smooth set XX (def. 10) is called a concrete smooth set or a diffeological space if there exists

  1. a set X sSetX_s \in Set;

  2. for each nn \in \mathbb{N} a natural identification of the set of plots X( n)X(\mathbb{R}^n) with a subset of the set of functions s nX s\mathbb{R}^n_s \to X_s from the underlying set s n\mathbb{R}^n_s of n\mathbb{R}^n (forgetting all smooth structure) to that set X sX_s:

    X( n)Hom Set( s n,X s). X(\mathbb{R}^n) \hookrightarrow Hom_{Set}(\mathbb{R}^n_s, X_s) \,.

Key examples of diffeological spaces are mapping spaces between smooth manifolds, which we turn to below (def. 16).

Example

(discrete smooth set)

For SS \in Set a set, write

DiscSSmoothSet Disc S \in SmoothSet

for the smooth set whose set of UU-plots for every UCartSpU \in CartSp is always SS.

DiscS:US Disc S \colon U \mapsto S

and which sends every coordinate transformation f: n 1 n 2f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} to the identity function on SS.

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. 10, 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:XYf : X \to Y between two smooth sets is, it has to take the plots of XX by n\mathbb{R}^n to a corresponding plot of YY, such that this respects coordinate transformations.

Definition

Let XX and YY be two smooth sets, def. 10. Then a homomorphism f:XYf \colon X \to Y is

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

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

    that sends n\mathbb{R}^n-plots of XX to n\mathbb{R}^n-plots of YY

such that

  • for each smooth function ϕ: n 1 n 2\phi : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} we have

    Y(ϕ)f n 1=f n 2X(ϕ) Y(\phi) \circ f_{\mathbb{R}^{n_1}} = f_{\mathbb{R}^{n_2}} \circ X(\phi)

    hence a commuting diagram

    X( n 1) f n 1 Y( n 1) X(ϕ) Y(ϕ) X( n 2) f n 2 Y( n 1). \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:XYf_1 : X \to Y and f 2:XYf_2 : X \to Y two homomorphisms of smooth sets, their composition f 2f 1:XYf_2 \circ f_1 \colon X \to Y is defined to be the homomorphism whose component over n\mathbb{R}^n is the composite of functions of the components of f 1f_1 and f 2f_2:

(f 2f 1) nf 2 nf 1 n. (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 SmoothSetSmoothSet for the category (this def.) whose objects are smooth sets, def. 10, and whose morphisms are homomorphisms of smooth sets according to def. 11.

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 XX: on the hand, XX comes by definition with a rule for what the set X( n)X(\mathbb{R}^n) of its n\mathbb{R}^n-plots is. On the other hand, we can now regard the abstract coordinate system n\mathbb{R}^n itself as a smooth set, by example 3, and then say that an n\mathbb{R}^n-plot of XX should be a homomorphism of smooth sets of the form nX\mathbb{R}^n \to X.

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

Proposition

Let XX be any smooth set, def. 10, and regard the abstract coordinate system n\mathbb{R}^n as a smooth set, by example 3. There is a natural bijection

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

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

Proof

This is a special case of the Yoneda lemma, 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. 11 to deduce that a homomorphism f: nXf : \mathbb{R}^n \to X is uniquely fixed by the image of the identity element in n( n)CartSp( n, n)\mathbb{R}^n(\mathbb{R}^n) \coloneqq CartSp(\mathbb{R}^n, \mathbb{R}^n) under the component function f n: n( n)X( n)f_{\mathbb{R}^n} : \mathbb{R}^n(\mathbb{R}^n) \to X(\mathbb{R}^n).

Example

Let SmoothSet\mathbb{R} \in SmoothSet denote the real line, regarded as a smooth set by example 3. Then for XSmoothSetX \in SmoothSet any smooth set, a homomorphism of smooth sets

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

is a smooth function on XX. Prop. 9 says here that when XX happens to be an abstract coordinate system regarded as a smooth set by def. 3, 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 0\mathbb{R}^0 we also call the point and regarded as a smooth set we will often write it as

*SmoothSet. * \in SmoothSet \,.

For any XSmoothSetX \in SmoothSet, we say that a homomorphism

x:*X x \colon * \to X

is a point of XX.

Remark

By prop. 9 the points of a smooth set XX are naturally identified with its 0-dimensional plots, hence with the “ways of laying out a 0-dimensional coordinate system” in XX:

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

Products and fiber products of smooth sets

Definition

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

X×Y( n)X( n)×Y( n)Set. 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:X×YX p_1 \colon X \times Y \to X

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

p 1 n:X( n)×Y( n)p 1X( n). {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:X×YY. p_2 \colon X \times Y \to Y \,.
Proposition

Let *= 0* = \mathbb{R}^0 be the point, regarded as a smooth set, def. 13. Then for XSmoothSetX \in SmoothSet any smooth set the canonical projection homomorphism

X×*X X \times * \to X

is an isomorphism.

Definition

Let f:XZf \colon X \to Z and g:YZg \colon Y \to Z be two homomorphisms of smooth sets, def. 11. There is then a new smooth set to be denoted

X× ZYSmoothSet X \times_Z Y \in SmoothSet

(with ff and gg understood), called the fiber product of XX and YY along ff and gg, and defined as follows:

the set of n\mathbb{R}^n-plots of X× ZYX \times_Z Y is the set of pairs of plots of XX and YY which become the same plot of ZZ under ff and gg, respectively:

(X× ZY)( n)={(p XX( n),p YY( n))|f n(p X)=g n(p Y)}. (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 Σ,XSmoothSet\Sigma, X \in SmoothSet be two smooth sets, def. 10. Then the smooth mapping space

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

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

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

Here in Σ× n\Sigma \times \mathbb{R}^n we first regard the abstract coordinate system n\mathbb{R}^n as a smooth set by example 3 and then we form the product smooth set by def. 14.

Remark

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

Proposition

There is a natural bijection

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

for every smooth set KK.

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 KK into the mapping space [Σ,X][\Sigma,X] is equivalently a smooth function from K×ΣK \times \Sigma to XX. The latter we may regard as a KK-parameterized smooth family of smooth functions ΣX\Sigma \to X. Therefore in view of the previous remark 12 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 [Σ,X][\Sigma,X] is the smooth moduli space of smooth functions from ΣX\Sigma \to X, because it is such that smooth maps K[Σ,X]K \to [\Sigma,X] into it modulate, as we move around on KK, a family of smooth functions ΣX\Sigma\to X, depending on KK.

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 nn-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. 13, of a smooth mapping space [Σ,X][\Sigma,X] is the bare set of homomorphism ΣX\Sigma \to X: there is a natural isomorphism

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

Combine prop. 11 with prop. 10.

Example

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

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

By prop. 12 the points of PXP X are indeed precisely the smooth trajectories 1X\mathbb{R}^1 \to X. But PXP 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 XX is a model for spacetime, then PXP X may notably be interpreted as the smooth set of worldlines in XX, hence the smooth set of paths or trajectories of a particle in XX.

Example

If in the above example 8 the path is constraind to be a loop in XX, one obtains the smooth loop space

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

In example 7 we saw that a smooth function on a general smooth set XX is a homomorphism of smooth sets, def. 11

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

The collection of these forms the hom-set Hom SmoothSet(X,)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 XSmoothSetX \in SmoothSet a smooth set, we say that the moduli space of smooth functions on XX is the smooth mapping space (def. 16), from XX into the standard real line \mathbb{R}

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

We will also denote this by

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

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

Remark

We call this a moduli space because by prop. 11 above and in the sense of remark 13 it is such that smooth functions into it modulate smooth functions XX \to \mathbb{R}.

By prop. 12 a point *[X, 1]* \to [X,\mathbb{R}^1] of the moduli space is equivalently a smooth function X 1X \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. 13 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. 14 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. 12, 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. 6)

SmoothSetSh(CartSp). SmoothSet \;\simeq\; Sh(CartSp) \,.
Proof

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

\,

Proposition

(smooth sets form a cohesive topos)

The site CartSp (Prop. 5) is a cohesive site this Def., hence its sheaf topos is a cohesive topos (by this Prop.). Under the identification of Prop. 13, this means that:

The category SmoothSetSmoothSet of smooth sets (Def. 12) is a cohesive topos (this Def.)

(3)SmoothSetAAAΠ 0AAA AADiscAA AAAΓAAA AAcoDiscAASet 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
Proof

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

*= 0 \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:

n 1× n 2 n 1+n 2. \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 n}\{U_i \overset{}{\to} \mathbb{R}^n\} (Def. 6). 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)(i,jy(U i nU j) iy(U i)). 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)× Xy(U j)y(U i XU j)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)limCartSp opC({U i} i) (limCartSp opi,jy(U i nU j) limCartSp opiy(U i)) (i,jlimCartSp opy(U i nU j) ilimCartSp opy(U i)) (i,j* i*). \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 n\mathbb{R}^n, regarded as a topological space. But this is evidently a connected topological space, which finally shows that, indeed

π 0limCartSp opC({U i} i)*. \pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,.

The second item of the second clause in Def. \ref{OneCohesiveSite} 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 CartSpCartSp has a terminal object *= 0\ast = \mathbb{R}^0, which is hence an initial object in the opposite category CartSp opCartSp^{op}, limits over CartSp opCartSp^{op} yield simply the evaluation on that object:

(6)limCartSp opC({U i} i) (limCartSp opi,jy(U i nU j) limCartSp opiy(U i)A) (i,jHom CartSp(*,U i nU j) iHom CartSp(*,U i)). \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 yy.

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

π 0limCartSp opC({U i} i)Hom CartSp(*, n). \pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,.

\,

Concrete smooth sets: Diffeological spaces

The cohesiveness of smooth sets (Prop. 14) implies in particular that there is a concept of concrete objects among the smooth sets (this Def.). We show here (Prop. 15 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. 18 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 (…), 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. 30 below.

\,

Definition

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

A diffeological space X\mathbf{X} is

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

  2. for each open subset UE nU \subset E^n of some Euclidean space E nE^n, for any nn \in \mathbb{N}, a subset

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

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

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

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

    Xconst UX(U)Hom Set(U,X), X \overset{const_U}{\hookrightarrow} \mathbf{X}(U) \hookrightarrow Hom_{Set}(U,X) \,,
  2. for every function ϕ:UX\phi \;\colon\; U \to X and for every open cover {U iι iU}\{U_i \overset{\iota_i}{\to} U\}, if each restriction is a plot, ϕ| U iX(U i)\phi\vert_{U_i} \in \mathbf{X}(U_i), then ϕ\phi itself is a plot: ϕX(U)\phi \in \mathbf{X}(U);

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

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

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

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

is a function of underlying sets

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

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

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

This defines a category DiffeologicalSpaceDiffeologicalSpace (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. 18) is a full subcategory of the category of smooth sets (Def. 12).

Moreover, in terms of the cohesive structure on the category of smooth sets from Prop. 14, the diffeological spaces are precisely the concrete objects (this def.) among the smooth sets:

DiffeologicalSpaceSmoothSet concAAAASmoothSet. DiffeologicalSpace \;\simeq\; SmoothSet_{conc} \overset{\phantom{AAAA}}{\hookrightarrow} SmoothSet \,.
Proof

First observe that the assignment of sets of plots

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

of a diffeological space X\mathbf{X}, according to Def. 18 constitutes a sheaf (this Def.) on the site EuclOpEuclOp of open subsets of Euclidean spaces, by the same unwinding of Definitions as in Prop. 13:

  • the third clause in the list of properties in Def. 18 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. 18 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

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

But by Prop. 7 the restriction from the site EuclOpEuclOp 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

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

where the last equivalence is Prop. 13.

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 XSmoothSet\mathbf{X} \in SmoothSet is concrete, precisely if its sharp-unit is a monomorphism

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

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

XAη XAcoDiscΓX. \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 UCartSpU \in CartSp in the site, its component function

(6)X(U)Aη X (U)A(coDiscΓX)(U) \mathbf{X}(U) \overset{\phantom{A} \eta_X^\sharp(U) \phantom{A}}{\hookrightarrow} (coDisc \Gamma \mathbf{X})(U)

is an injective function (this Prop.). The codomain of this function may be re-identified as follows:

(coDiscΓX(U)) Hom SmoothSet(y(U),coDiscΓX) Hom Set(Γy(U),ΓX) Hom Set(U,X), \begin{aligned} (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 ) \,, \end{aligned}

where we first used the Yoneda lemma (this Prop.), then the adjunction isomorphism (here) of (ΓcoDisc)(\Gamma \dashv coDisc). In the final step we used that the cohesive structure on SmoothSetSmoothSet comes from CartSpCartSp being a cohesive site (Prop. 14) and that in this case Γ\Gamma is given by evaluation on the point (here), and we wrote

(7)XΓX=X(*) X \;\coloneqq\; \Gamma \mathbf{X} = \mathbf{X}(\ast)

for the set of points in X\mathbf{X}. Notice that if X\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. 18.

This shows that (6) being an injection means equivalently that we have an injection of the form

(8)X(U)Aη X (U)AHom Set(U,X). \mathbf{X}(U) \overset{\phantom{A} \eta_X^\sharp(U) \phantom{A}}{\hookrightarrow} Hom_{Set}(U, X) \,.

Hence that X(U)\mathbf{X}(U) is always a subset of the function set from UU to the set XX, as in the second clause in Def. 18.

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 X\mathbf{X} with prescribed inclusion function

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

then (8) indeed reproduces this inclusion.

To see this, first notice that, by the Yoneda lemma and the definition of smooth maps between diffeological spaces, this inclusion function equals the function

Γ y(U),X:Hom SmoothSet(y(U),X) Hom Set(Γy(U),ΓX) f Γ(f) \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) }

which is the component function of the functor Γ:SmoothSetSet\Gamma \;\colon\; SmoothSet \to Set, that acts by point evaluation.

Hence we need to show that this function is equal to that given by postcomposition with the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction unit, followed by forming the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunct:

(9)Γ y(U),X=η X()˜. \Gamma_{y(U), \mathbf{X}} \;=\; \widetilde{\eta_{\mathbf{X}} \circ (-)} \,.

For this, let fHom SmoothSet(y(U),X)f \in Hom_{SmoothSet}( y(U), \mathbf{X} ) be any element, and consider the following naturality square for the adjunction hom-isomorphism (this Def.):

Hom Set(ΓX,ΓX) ()˜ Hom SmoothSet(X,coDiscΓX) Hom SmoothSet(Γ(f),X) Hom SmoothSet(f,coDiscΓX) Hom Set(Γy(U),ΓX) ()˜ Hom SmoothSet(y(U),coDiscΓX)AAAAA{ΓXidΓX} {Xη XcoDiscΓX} {Γy(U)Γ(f)ΓX} {y(U)η Xf} \array{ Hom_{Set}( \Gamma \mathbf{X} , \Gamma \mathbf{X} ) &\overset{\widetilde {(-)}}{\longrightarrow}& Hom_{SmoothSet}( \mathbf{X}, coDisc \Gamma \mathbf{X} ) \\ {}^{\mathllap{ Hom_{SmoothSet}(\Gamma(f), \mathbf{X}) }} \big\downarrow && \!\!\!\!\! \big\downarrow^{\mathrlap{ Hom_{SmoothSet}( f, coDisc \Gamma \mathbf{X} ) }} \\ Hom_{Set}( \Gamma y(U), \Gamma \mathbf{X} ) &\overset{\widetilde{ (-) }}{\longleftarrow}& Hom_{SmoothSet}( y(U), coDisc \Gamma \mathbf{X} ) } \phantom{AAAAA} \array{ \{ \Gamma \mathbf{X} \overset{id}{\to} \Gamma \mathbf{X}\} &\longrightarrow& \{ \mathbf{X} \overset{\eta_{\mathbf{X}}}{\to} coDisc \Gamma \mathbf{X} \} \\ \big\downarrow && \big\downarrow \\ \{ \Gamma y(U) \overset{\Gamma(f)}{\to} \Gamma \mathbf{X} \} &\longleftarrow& \{ y(U) \overset{\eta_{\mathbf{X}} \circ f}{\longrightarrow} \} }

and chase the identity morphism in the top left set through this diagram, as shown on the right. This gives rise to the adjunction unit as shown in the top right (by this def.), and hence to the composite η Xf\eta_{\mathbf{X}}\circ f in the bottom right. The image of that in the bottom left is, by definition, the adjunct, and hence the commutativity of this square yields (9).

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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 [,X][\mathbb{R}, X] from example 8 are to be thought of as the smooth spaces of trajectories (for instance of some particle) in a smooth space XX, hence of smooth maps X\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:[I,X], S \colon \left[I, X\right] \to \mathbb{R} \,,

where II \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 γ\gamma is decomposed as a trajectory γ 1\gamma_1 followed by a trajectory γ 2\gamma_2, then the action functional is additive

S(γ)=S(γ 1)+S(γ 2). 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 γ:X\gamma \colon \mathbb{R} \to X denotes a path and “γ˙(x)\dot \gamma(x)” denotes the corresponding “infinitesimal path” at worldline parameter xx, then the value of the action functional on such an infinitesimal path is traditionally written as

dS(γ˙) x, \mathbf{d}S(\dot \gamma)_x \in \mathbb{R} \,,

to be read as “the small change dS\mathbf{d}S of SS along the infinitesimal path γ˙ x\dot \gamma_x”.

This function dS\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.

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Definition

For nn \in \mathbb{N} a smooth differential 1-form ω\omega on a Cartesian space n\mathbb{R}^n is an nn-tuple

(ω iCartSp( n,)) i=1 n \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

ω= i=1 nf idx i \omega = \sum_{i = 1}^n f_i \mathbf{d}x^i

on a set {dx 1,dx 2,,dx n}\{\mathbf{d}x^1, \mathbf{d}x^2, \cdots, \mathbf{d}x^n\} of cardinality nn.

Write

Ω 1( k)CartSp( k,) ×kSet \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 k\mathbb{R}^k.

Remark

We think of dx i\mathbf{d} x^i as a measure for infinitesimal displacements along the x ix^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 n\mathbb{R}^n and a smooth function f: n˜ nf \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n, then this induces a measure for infinitesimal displacement on n˜\mathbb{R}^{\tilde n} by sending whatever happens there first with ff to n\mathbb{R}^n and then applying the given measure there. This is captured by the following definition.

Definition

For ϕ: k˜ k\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k a smooth function, the pullback of differential 1-forms along ϕ\phi is the function

ϕ *:Ω 1( k)Ω 1( k˜) \phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})

between sets of differential 1-forms, def. 19, which is defined on basis-elements by

ϕ *dx i j=1 k˜ϕ ix˜ jdx˜ j \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

ϕ *ω =ϕ *( iω idx i) i=1 k(ϕ *ω) i j=1 k˜ϕ ix˜ jdx˜ j = i=1 k j=1 k˜(ω iϕ)ϕ ix˜ jdx˜ j. \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 Ω 1( k)\Omega^1(\mathbb{R}^k) of differential 1-forms, this set naturally carries further structure.

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

    ω+λ = i=1 kω idx i+ j=1 kλ jdx j = i=1 k(ω i+λ i)dx i, \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 Ω 1( k)\Omega^1(\mathbb{R}^k) is naturally equipped with the structure of a module over the ring C ( k,)=CartSp( k,)C^\infty(\mathbb{R}^k,\mathbb{R}) = CartSp(\mathbb{R}^k, \mathbb{R}) of smooth functions, where the action C ( k,)×Ω 1( k)Ω 1( k)C^\infty(\mathbb{R}^k,\mathbb{R}) \times\Omega^1(\mathbb{R}^k) \to \Omega^1(\mathbb{R}^k) is given by componentwise multiplication

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

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

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

dx jdx i=dx idx j \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,nk,n \in \mathbb{N}, the smooth differential forms on k\mathbb{R}^k is the exterior algebra

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

over the ring C ( k)C^\infty(\mathbb{R}^k) of smooth functions of the module Ω 1( k)\Omega^1(\mathbb{R}^k) of smooth 1-forms, prop. 21.

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

Remark

Explicitly this means that a differential n-form ωΩ n( k)\omega \in \Omega^n(\mathbb{R}^k) on k\mathbb{R}^k is a formal linear combination over C ( k)C^\infty(\mathbb{R}^k) of basis elements of the form dx i 1dx i n\mathbf{d} x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n} for i 1<i 2<<i ni_1 \lt i_2 \lt \cdots \lt i_n:

ω= 1i 1<i 2<<i n<kω i 1,,i ndx i 1dx 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. 19 extends as an C ( k)C^\infty(\mathbb{R}^k)-algebra homomorphism to Ω n()\Omega^n(-), given for a smooth function f: k˜ kf \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k on basis elements by

f *(dx i 1dx i n)=(f *dx i 1f *dx i n). 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) \,.

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

Proposition

The assignment of differential nn-forms

Ω n(): kΩ n( k) \Omega^n(-) \colon \mathbb{R}^k \mapsto \Omega^n(\mathbb{R}^k)

of def. 23 together with the pullback of differential forms-functions of def. 24

k 1 Ω 1( k 1) f f * k 2 Ω 1( k 2) \array{ \mathbb{R}^{k_1} &\mapsto & \Omega^1(\mathbb{R}^{k_1}) \\ \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f^*}} \\ \mathbb{R}^{k_2} &\mapsto& \Omega^1(\mathbb{R}^{k_2}) }

defines a smooth set in the sense of def. 10:

Ω n()Smooth0Type. \Omega^n(-) \in Smooth0Type \,.
Definition

We call this

Ω n:Smooth0Type \Omega^n \colon Smooth0Type

the universal smooth moduli space of differential nn-forms.

The reason for this terminology is that homomorphisms of smooth sets into Ω 1\Omega^1 modulate differential nn-forms on their domain, by prop. 9 (and hence by the Yoneda lemma):

Example

For the Cartesian space k\mathbb{R}^k regarded as a smooth set by example 3, there is a natural bijection

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

between the set of smooth nn-forms on n\mathbb{R}^n according to def. 19 and the set of homomorphism of smooth set, kΩ 1\mathbb{R}^k \to \Omega^1, according to def. 11.

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

Definition

For XSmoothSetX \in SmoothSet a smooth set, def. 10, a differential n-form on XX is a homomorphism of smooth sets of the form

ω:XΩ n(). \omega \colon X \to \Omega^n(-) \,.

Accordingly we write

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

for the set of smooth nn-forms on XX.

We may unwind this definition to a very explicit description of differential forms on smooth sets. This we do in a moment in remark 19.

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

Proposition

Ω 0\Omega^0 \simeq \mathbb{R}

Definition

For f:XYf \colon X \to Y a homomorphism of smooth sets, def. 11, the pullback of differential forms along ff is the function

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

given by the hom-functor into the smooth set Ω n\Omega^n of def. 25:

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

This means that it sends an nn-form ωΩ n(Y)\omega \in \Omega^n(Y) which is modulated by a homomorphism YΩ nY \to \Omega^n to the nn-form f *ωΩ n(X)f^* \omega \in \Omega^n(X) which is modulated by the composite XfYΩ nX \stackrel{f}{\to} Y \to \Omega^n.

Proposition

For X= k˜X = \mathbb{R}^{\tilde k} and Y= kY = \mathbb{R}^{k} definition 27 reproduces def. 24.

Proof

Again by the Yoneda lemma.

Remark

Using def. 27

Unwinding def. 26 yields the following explicit description:

a differential nn-form ωΩ n(X)\omega \in \Omega^n(X) on a smooth set XX is

  1. for each way ϕ: kX\phi \colon \mathbb{R}^k \to X of laying out a coordinate system k\mathbb{R}^k in XX a differential nn-form

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

    on the abstract coordinate system, as given by def. 23;

  2. for each abstract coordinate transformation f: k 2 k 1f \colon \mathbb{R}^{k_2} \to \mathbb{R}^{k_1} a corresponding compatibility condition between local differential forms ϕ 1: k 1X\phi_1 \colon \mathbb{R}^{k_1} \to X and ϕ 2: k 2X\phi_2 \colon \mathbb{R}^{k_2} \to X of the form

    f *ϕ 1 *ω=ϕ 2 *ω. 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 Ω nSmooth0Tye\Omega^n \in Smooth0Tye as a moduli space. Notice that since Ω n\Omega^n is itself a smooth set, we may speak about differential nn-forms on Ω n\Omega^n itsefl.

Definition

The universal differential nn-forms is the differential nn-form

ω univ nΩ n(Ω n) \omega^n_{univ} \in \Omega^n(\Omega^n)

which is modulated by the identity homomorphism id:Ω nΩ nid \colon \Omega^n \to \Omega^n.

With this definition we have:

Proposition

For XSmooth0TypeX \in Smooth0Type any smooth set, every differential nn-form on XX, ωΩ n(X)\omega \in \Omega^n(X) is the pullback of differential forms, def. 27, of the universal differential nn-form, def. 29, along a homomorphism ff from XX into the moduli space Ω n\Omega^n of differential nn-forms:

ω=f *ω univ n. \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 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.

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

Remark

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

[X,Ω n]: kΩ n(X× 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 k\mathbb{R}^k-parameterized of differential forms on XX. That is instead usually meant to be a differential form ω\omega on X× kX \times \mathbb{R}^k which has “no leg along k\mathbb{R}^k”. Another way to say this is that the family of forms on XX that is represented by some ω\omega on X× kX \times \mathbb{R}^k is that which over a point v:*ℝℝ kv \colon * \to \mathbb{RR}^k has the value (id X,v) *ω(id_X,v)^* \omega. Under this pullback of differential forms any components of ω\omega with “legs along k\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 XSmoothSetX \in SmoothSet any smooth set and nn \in \mathbb{N}, the smooth space of differential nn-forms Ω n(X)\mathbf{\Omega}^n(X) on XX is the concretification, def. \ref{ConcreteObjectsAndConcretification}, of the smooth mapping space [X,Ω n][X, \Omega^n], def. 16, into the smooth moduli space of differential nn-forms, def. 25:

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

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

Proof (sketch)

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

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

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

ω(v(id X,v) *ω). \omega \mapsto \left(v \mapsto (id_X, v)^* \omega \right) \,.

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

Remark

For n=0n = 0 we have (for any XSmooth0TypeX\in Smooth0Type)

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

by prop. 17.

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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.

Last revised on June 16, 2018 at 11:04:54. See the history of this page for a list of all contributions to it.