nLab embedding of smooth manifolds into formal duals of R-algebras

Idea

The functor which sends

1. smooth manifolds to their algebras of smooth functions with values in the real numbers, regarded just as $\mathbb{R}$-algebras (instead of, say smooth algebras aka “$C^\infty$-rings”),

2. smooth functions between these manifolds to the corresponding pullback/precomposition algebra homomorphisms between these function algebras

turns out to be fully faithful, hence a full subcategory embedding of SmthMfd into the opposite of $\mathbb{R}Alg$.

This is remarkable, because such a relation between spaces and their plain algebras of functions is (more) manifest only for affine varieties in algebraic geometry, where however it holds essentially by construction. In contrast, nothing in the usual definition of smooth manifolds manifestly suggests that they behave to some extent similarly to affine varieties with respect to $\mathbb{R}$-algebras of smooth functions. (See also at duality between algebra and geometry.)

Related “miracles” in differential geometry, revealing a maybe surprising algebro-geometric-nature, are the facts that:

• derivations of smooth functions are vector fields,

• Kähler C^∞-differentials of smooth functions are differential 1-forms,

• smooth differential forms form the free C^∞-DGA on smooth functions

• finitely generated projective modules over smooth functions are smooth vector bundles.

Accordingly, the embedding of smooth manifolds into formal duals of $\mathbb{R}$-algebras allows to import some constructions and tools from algebraic geometry into differential geometry without strengthening the notion of “algebra” to something like smooth algebras.

This is useful and is used particularly for discussions in synthetic differential geometry — cf. e.g. the emphasis on “Weil algebras”, hence of would-be function algebras on infinitesimally thickened points, in Kolář, Michor & Slovák 1993 §35.

Statement

Lemma

(Milnor’s exercise)
For a smooth manifold $X \in SmthMfd$, the evaluation map (from its underlying set to the hom-set)

$$\begin{array}{l} X &\overset{ev}{\longrightarrow}& Hom_{Alg_{\mathbb{R}}} \big( C^\infty(X) ,\, \mathbb{R} \big) \\ x &\mapsto& \big( \phi \,\mapsto\, \phi(x) \big) \end{array}$$

is a bijection.

(due to Milnor & Stasheff (1974) Prob. 1-C (p. 11), detailed proof in Kolář, Michor & Slovák 1993 Lem. 35.8 & Cor. 35.9)

Proof

We provide a simplified proof using Hadamard's lemma. Suppose $M$ is a smooth manifold and $\phi\colon C^\infty(M)\to\mathbf{R}$ is a homomorphism of real algebras.

If $M=\mathbf{R}^n$ for some $n\ge0$, then set $y_i=\phi(x_i)$, where $x_i\colon\mathbf{R}^n\to\mathbf{R}$ is the $i$th coordinate function. We have

$$\phi(f)=\phi(f(y)+\sum_i (x_i-y_i)\cdot g_i)=f(y)+\sum_i (\phi(x_i)-y_i)\cdot \phi(g_i)=f(y),$$

where the functions $g_i$ are provided by Hadamard's lemma.

For a general $M$, use Whitney's embedding theorem to embed $M$ into some $\mathbf{R}^n$. Without loss of generality we can assume the embedding to be proper so that the subset $M\subset\mathbf{R}^n$ is closed.

Consider the composition

$$C^\infty(\mathbf{R}^n)\to C^\infty(M)\to \mathbf{R},$$

where the first homomorphism $\rho$ is given by restricting along the embedding. The composition is given by evaluating at some point $p\in\mathbf{R}^n$.

If $p\notin M$, we can construct a smooth function $b$ that vanishes on $M$ and does not vanish at $p$. For example, we can take the smooth function with zero locus $M$ constructed by the smooth Tietze extension theorem, or simply use smooth bump functions. The function $b$ vanishes on $M$ and therefore belongs to the kernel of the map $\rho$. The evaluation homomorphism at $p\notin M$ does not vanish on the element $b$, hence cannot factor through the map $\rho$, a contradiction. Therefore, we must have $p\in M$.

Milnor’s exercise (Lem. ) implies — and also is the special case for $X = \ast$ (the point) of:

Theorem

The functor

$$\begin{array}{ccc} SmthMfd &\longrightarrow& Alg_{\mathbb{R}}^{op} \\ X &\mapsto& C^\infty(X) \\ \mathllap{{}^f}\Big\downarrow && \Big\uparrow\mathrlap{{}^{{f^\ast}}} \\ Y &\mapsto& C^\infty(Y) \end{array}$$

which sends a smooth manifold (finite dimensional, paracompact, second countable) to (the formal dual of) its $\mathbb{R}$-algebra of smooth functions, $C^\infty(X) \coloneqq C^\infty(X, \mathbb{R})$, is a full and faithful functor.

In other words, given a pair of smooth manifolds $X,Y$ then the operation of precomposition (pullback of functions) $f^\ast(g) \coloneqq g \circ f$ constitutes a natural bijection

$$\begin{array}{l} C^\infty(X,Y) &\overset{\sim}{\longrightarrow}& Hom_{Alg_{\mathbb{R}}}\big( C^\infty(Y) ,\, C^\infty(X) \big) \\ f &\mapsto& f^\ast \end{array}$$

between

1. the smooth functions $X \to Y$,

2. the $\mathbb{R}$-algebra homomorphisms $C^\infty(X)\leftarrow C^\infty(Y)$.

In particular, the diffeomorphisms between smooth manifolds are in natural bijection to the isomorphisms between their algebras of functions.

(Kolář, Slovák & Michor 1993 Cor. 35.10)

Proof

It is clear that the functor is faithful; we need to show that it is full, hence that for any $\mathbb{R}$-algebra homomorphism

$$\phi \;\colon\; C^\infty(Y, \mathbb{R}) \longrightarrow C^\infty(X, \mathbb{R})$$

there exists a smooth function $f \colon X \to Y$ such that $\phi = f^\ast$.

To that end, observe that given a point $x \in X$, the postcomposition of $\phi$ with the evaluation map at $x$

$$C^\infty(Y, \mathbb{R}) \overset{ \phi }{\longrightarrow} C^\infty(X, \mathbb{R}) \overset{ ev_x }{\longrightarrow} \mathbb{R}$$

is an algebra homomorphism of the form assumed in Milnor’s exercise (Lem. ) and thus given uniquely by evaluation at some point $f(x) \in Y$

$$ev_x \circ \phi \;=\; ev_{f(x)} \,.$$

This implies that $\phi$ acts on any $g \in C^\infty(Y)$ by

$$\begin{array}{rcl} g &\overset{\phi}{\mapsto}& \big( x \mapsto \phi(g)(x) \big) \\ &=& \Big( x \mapsto ev_x(\phi(g)) \Big) \\ &=& \Big( x \mapsto ev_{f(x)}(g) \Big) \\ &=& \Big( x \mapsto g\big(f(x)\big) \Big) \\ &=& \Big( x \mapsto (g \circ f)(x) \Big) \mathrlap{\,,} \end{array}$$

hence that it acts by precomposition with the assignment $f$:

$$\phi(g) \;=\; g \circ f \,.$$

It just remains to observe that this $f$ is necessarily smooth. But since $\phi$ is given on all $g \in C^\infty(Y)$ this way, to see that $f$ is smooth at some $x \in X$, choose any coordinate chart $U \subset Y$ around $f(x) \in Y$ and consider a $g$ which restricts to one of the coordinate functions $x^i$ on $U$. Then $(g \circ f)_{\vert f^{-1}(U)} = f^i_{\vert f^{-1}(U)}$ is the $i$th coordinate component of $f$ restricted to a neighbourhood of $x$, and this being smooth for all $i$ means that $f$ is smooth around $x$.

Remark

(attribution)
For the case of diffeomorphisms, Thm. was proven by Pursell (1952), following an announcement by Shanks (1951). This is the case that most reviews focus on, e.g. Grabowski (1978), Marsden, Ratiu & Abraham (2002), Grabowski (2005).

For the case that the domain is a point the statement is left as an exercise (without reference to Pursell) in Milnor & Stasheff (1974) Problem 1-C (p. 11), sometimes now referred to as “Milnor’s exercise” (Lem. ), with a detailed proof given in Kolář, Slovák & Michor (1993) 35.8-9

The general statement of Thm. appears as Kolář, Slovák & Michor (1993) 35.10.

Remark

The statement of theorem serves as the stepping-stone for generalizations of differential geometry such as to supergeometry. On the other hand, for transporting various applications familiar from algebraic geometry to differential geometry (such as Kähler differentials, see there) the above embedding is insufficient, and instead of just remembering the associative algebra structure, one needs to remember the smooth algebra-structure on algebras of smooth functions. See also at synthetic differential geometry.

Remark

If one drops standard regularity assumptions on manifolds then theorem may break. For instance allowing uncountably many connected components, then there are counterexamples (MO discussion).

Related theorems

• Borel's theorem

• Hadamard's lemma

• the Tietze extension theorem

• the Whitney extension theorem

• the Steenrod-Wockel approximation theorem

• smooth Serre-Swan theorem

• derivations of smooth functions are vector fields

The analogous statement in topology is:

• Gelfand duality

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

The case of the category of smooth manifolds with (just) diffeomorphisms between them is proved in

• Lyle Eugene Pursell: Algebraic structures associated with smooth manifolds, PhD dissertation, Purdue University (1952) 93 pp. [ISBN:978-1392-88143-9, proquest:2327629257, pdf]

following an announcement in

• M. E. Shanks, Rings of functions on locally compact spaces, 469th meeting of the AMS (1951) [pdf]

The statement for domain a point is due to:

• John Milnor, James Stasheff, Problem 1-C (p. 11) in: Characteristic Classes, Annals of Mathematics Studies, Princeton University Press (1974) [ISBN:9780691081229, pdf]

with a proof spelled out in:

• Eduardo Dubuc, Prop. 0.7 in: Sur les modèles de la géométrie différentielle synthétique, Cahiers de Topologie et Géométrie Différentielle Catégoriques 20 3 (1979) 231-279 [numdam:CTGDC_1979__20_3_231_0]

  (in defining the Cahiers topos)

Expository accounts for the case of isomorphisms are in

• Janusz Grabowski, Isomorphisms and ideals of the Lie algebras of vector fields, Inventiones mathematicae volume 50, pages 13–33 (1978) (doi:10.1007/BF01406466)

• Jerrold Marsden, J. Ratiu, R. Abraham, Theorem 4.2.36 in: Manifolds, tensor analysis, and applications, Springer 2003 (ISBN:978-1-4612-1029-0)

• Janusz Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. 85 (2005), 190-196 (arXiv:math/0310295)

The general statement and its proof is discussed in:

• Ivan Kolář, Peter Michor, Jan Slovák, chapter 35 of: Natural operations in differential geometry, Springer (1993) [book webpage, doi:10.1007/978-3-662-02950-3, pdf]

Discussion that takes the dual algebraic formulation as the very definition of smooth functions is in

• Jet Nestruev, section 6 of: Smooth manifolds and Observables, Graduate Texts in Mathematics 218, Springer 2003 (doi:10.1007/b98871)

The analog of the statement for real algebras refined to smooth algebras is theorem 2.8 in

• Ieke Moerdijk, Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis Springer (1991)