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

Statement

Theorem

(“Milnor‘s exercise”)

The functor

$$C^\infty(-) \colon SmoothMfd \longrightarrow Alg_{\mathbb{R}}^{op}$$

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

In other words, for two smooth manifolds $X,Y$ there is a natural bijection between the smooth functions $X \to Y$ and the $\mathbb{R}$-algebra homomorphisms $C^\infty(X)\leftarrow C^\infty(Y)$.

The statement originates with Milnor-Stasheff 74, Problem 1-C (p. 11). Proof is in (Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$	$\phantom{A}$fin. gen.$\phantom{A}$ $\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$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 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)

Statement and proof for isomorphisms is 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 Kolar, Jan Slovák, Peter Michor, Natural operations in differential geometry, Springer (1993) (book webpage, doi:10.1007/978-3-662-02950-3)

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)