nLab
smooth algebra

Idea

A smooth algebra or $C^{\mathrm{\infty }}$-ring is an algebra $A$ over the reals $ℝ$ for which not only the product operation $\cdot :ℝ\times ℝ\to ℝ$ lifts to the algebra product $A\times A\to A$, but for which every smooth map $f:{ℝ}^n\to {ℝ}^m$ (morphism in Diff) lifts to a map $A(f):A^n\to A^m$ in a compatible way.

In short this means that $A$ is

• a product-preserving co-presheaf on CartSp;

• equivalently: an algebra for the Lawvere theory CartSp;

The smoothness of such $C^{\mathrm{\infty }}$-rings is witnessed by the fact that this Lawvere theory is even a Fermat theory.

The opposite category of the category of $C^{\mathrm{\infty }}$-rings is the category of smooth loci. This and its subcategories play a major role as sites for categories of sheaves that serve as models for synthetic differential geometry.

Motivating example

For $X$ a smooth manifold, the assignment

$${ℝ}^n\mapsto C^{\mathrm{\infty }}(X,{ℝ}^n)={\mathrm{Hom}}_{\mathrm{Diff}}(X,{ℝ}^n)$$\mathbb{R}^n \mapsto C^\infty(X,\mathbb{R}^n) = Hom_{Diff}(X,\mathbb{R}^n)

of the set of smooth ${ℝ}^n$-valued functions on $X$ is clearly covariant and hence yields a co-presheaf on CartSp $\subset$ Diff: a functor

$$C^{\mathrm{\infty }}(X,-):\mathrm{CartSp}\to \mathrm{Set}.$$C^\infty(X,-) : CartSp \to Set \,.

Since the hom-functor sends limits to limits in its second argument this is clearly product preserving.

$$C^{\mathrm{\infty }}(X,{ℝ}^n\times {ℝ}^m)\simeq C^{\mathrm{\infty }}(X,{ℝ}^n)\times C^{\mathrm{\infty }}(X,{ℝ}^m)$$C^\infty(X, \mathbb{R}^n \times \mathbb{R}^m) \simeq C^\infty(X,\mathbb{R}^n) \times C^\infty(X, \mathbb{R}^m)

If as usual we write $C^{\mathrm{\infty }}(X):=C^{\mathrm{\infty }}(X,ℝ)$ for the set of just $ℝ$-valued smooth functions, then the usual pointwise product of functions

$$\cdot :C^{\mathrm{\infty }}(X)\times C^{\mathrm{\infty }}(X)\to C^{\mathrm{\infty }}(X)$$\cdot : C^\infty(X) \times C^\infty(X) \to C^\infty(X)

can be regarded as the image of our co-presheaf under the muliplication map $ℝ\times ℝ\stackrel{-\cdot -}{\to }ℝ$ on the algebra of real numbers:

$$\cdot :C^{\mathrm{\infty }}(X)\times C^{\mathrm{\infty }}(X):=C^{\mathrm{\infty }}(X,ℝ)\times C^{\mathrm{\infty }}(X,ℝ)\simeq C^{\mathrm{\infty }}(X,ℝ\times ℝ)\stackrel{C^{\mathrm{\infty }}(X,-\cdot -)}{\to }{C}^{\mathrm{\infty }}(X,ℝ)=:C^{\mathrm{\infty }}(X).$$\cdot : C^\infty(X) \times C^\infty(X) := C^\infty(X,\mathbb{R}) \times C^\infty(X,\mathbb{R}) \simeq C^\infty(X,\mathbb{R}\times \mathbb{R}) \stackrel{C^\infty(X,-\cdot-)}{\to} C^\infty(X,\mathbb{R}) =: C^\infty(X) \,.

Definitions

Definition

CartSp is the full subcategory of Diff on manifolds of the form ${ℝ}^n$.

Definition

A $C^{\mathrm{\infty }}$-algebra is a finite product-preserving co-presheaf on CartSp, i.e. a finite product preserving functor

$$A:\mathrm{CartSp}\to \mathrm{Set}.$$A : CartSp \to Set \,.

The category of such functors and natural transformations between them we denote by $C^{\mathrm{\infty }}\mathrm{Alg}$.

Remark on terminology

The standard name in the literature for generalized smooth algebras is $C^{\mathrm{\infty }}$-rings. Even though standard, this has the disadvantages for us that it collides badly with the use of $\mathrm{\infty }$- for higher categorical structures.

I don't see why this is a problem; it's not like our ‘$\mathrm{\infty }$’ ever gets into a superscript. I find ‘$C^{\mathrm{\infty }}$-ring’ much more descriptive than ‘generalized smooth algebra’, in fact. —Toby

Urs Schreiber: I’ll see what to do about it here. Over at derived smooth manifolds and related entried before long we’ll have to be talking about ”$\mathrm{\infty }-C^{\mathrm{\infty }}$-rings” or $C_{\mathrm{\infty }}^{\mathrm{\infty }}$-rings or the like, which is not good. Also, the term ”$C^{\mathrm{\infty }}$-ring” hides that it is necessarily an $ℝ$-algebras. Finally, the entire theory is really a special case of algebras over a Fermat theory and hence most other examples of a similar kind will by default be called algebras, not rings. For all these reasons I find ”$C^{\mathrm{\infty }}$-ring” an unfortunate term. But of course I am aware that it is entirely standard.

Tensor product

Definition (smooth tensor product)

The coproduct in $C^{\mathrm{\infty }}\mathrm{Alg}$ we call the smooth tensor product

$${\otimes }_{\mathrm{\infty }}:C^{\mathrm{\infty }}\mathrm{Alg}\times C^{\mathrm{\infty }}\mathrm{Alg}\to C^{\mathrm{\infty }}\mathrm{Alg}.$$\otimes_\infty : C^\infty Alg \times C^\infty Alg \to C^\infty Alg \,.

More generally, for $i:C\to A$ and $j:C\to B$ two morphisms in $C^{\mathrm{\infty }}\mathrm{Alg}$, we call the pushout

$$\begin{array}{ccc}C& \stackrel{i}{\to }& A\\ \downarrow & & \downarrow \\ B& \stackrel{j}{\to }& A{\otimes }_{C}B\end{array}$$\array{ C &\stackrel{i}{\to}& A \\ \downarrow && \downarrow \\ B &\stackrel{j}{\to}& A \otimes_C B }

the smooth tensor product over $C$ of $A$ and $B$.

Finitely generated $C^{\mathrm{\infty }}$-rings

Definition

For $X$ a smooth manifold, the smooth algebra $C^{\mathrm{\infty }}(X)$ is the functor

$$C^{\mathrm{\infty }}(X):={\mathrm{Hom}}_{\mathrm{Diff}}(X,-)$$C^\infty(X) := Hom_{Diff}(X,-)

Definition

(finitely generated and finitely presented $C^{\mathrm{\infty }}$-rings)

For $R$ a $C^{\mathrm{\infty }}$-ring, and $I\in U(R)$ an ideal in the underlying ordinary ring, there is a canonical $C^{\mathrm{\infty }}$-ring structure $R/I$ on the ordinaryy quotient ring $U(R)/I$.

A $C^{\mathrm{\infty }}$-ring $R$ is called finitely generated if it is of the form $C^{\mathrm{\infty }}({ℝ}^n)/I$ for $n\in \mathbb{N}$ and $I$ an ideal in $U(C^{\mathrm{\infty }}({ℝ}^n))$.

It is finitely presented if also $I$ is finitely generated, as an ideal, $I=(i_1,\cdots ,i_k)$ with $i_j\in U(R)$.

This is equivalent to $R$ being a pushout of the form

$$\begin{array}{ccc}{C}^{\mathrm{\infty }}({ℝ}^k)& \to & C^{\mathrm{\infty }}(*)\simeq ℝ\\ \downarrow & & \downarrow \\ C^{\mathrm{\infty }}({ℝ}^n)& \to & R\end{array}.$$\array{ C^\infty(\mathbb{R}^k) &\to& C^{\infty}(*) \simeq \mathbb{R} \\ \downarrow && \downarrow \\ C^\infty(\mathbb{R}^n) &\to& R } \,.

Definition

(germ-determined finitely generated / fair )

For $p\in {ℝ}^n$ let

$${\pi }_p:C^{\mathrm{\infty }}({ℝ}^n)\to C_p^{\mathrm{\infty }}({ℝ}^n)$$\pi_p : C^\infty(\mathbb{R}^n) \to C^\infty_p(\mathbb{R}^n)

be the natural projection onto the smooth algebra of germs of functions at $p$.

A $C^{\mathrm{\infty }}$-ring $C$ is called fair or finitely generated and germ-determined if it is finitely generated $C\simeq C^{\mathrm{\infty }}({ℝ}^n)/I$ and the ideal $I$ has the property that $f\in C^{\mathrm{\infty }}({ℝ}^n)$ is an element of $I$ if (and hence precisely if) for all $p\in {ℝ}^n$ the germ ${\pi }_p(f)\in C_p^{\mathrm{\infty }}(ℝ{)}^n$ is in the germ ${\pi }_p(I)$ of the ideal.

Internal $C^{\mathrm{\infty }}$-rings

For any smooth topos $(𝒯,R)$, there is an internal notion of generalized smooth algebra:

Definition (internal generalized smooth algebra)

For $(𝒯,R)$ a topos equipped with an internal ring object $R$ (possibly but not necessarily a smooth topos), let $\mathrm{CartSp}(𝒯,R)$ be the full subcategory of $𝒯$ on objects of the form $R^n$ for $n\in \mathbb{N}$. Then a $(𝒯,R)$-algebra is a product-preserving functor $A:\mathrm{CartSp}(𝒯,R)\to \mathrm{Set}$.

All constructions on smooth algebras generalize to $(𝒯,R)$-algebras. In particular for $X\in 𝒯$ any object we have the function $(𝒯,R)$-algebra

$$C(X):R^n\mapsto 𝒯(X,R^n).$$C(X) : R^n \mapsto \mathcal{T}(X,R^n) \,.

The following remark asserts that when $𝒯$ is itself a sufficiently nice category of sheaves on formal duals of $(\mathrm{Set},ℝ)$-algebras, then the internal notion of smooth function algebras on formal duals of external smooth algebras reproduces these external smooth algebras.

Proposition

Let $A$ be a finitely generated $C^{\mathrm{\infty }}$-ring, $\ell A$ its incarnation as an object in $𝕃=(C^{\mathrm{\infty }}{\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$ and $Y\ell A$ its incarnation in $\mathrm{Sh}(𝕃)\subset \mathrm{PSh}(𝕃)$, with $Y$ the Yoneda embedding and using the assumption that the Grothendieck topology used to form $\mathrm{Sh}(𝕃)$ is subcanonical.

Also suppose that the line object $R$ is represented by $\ell C^{\mathrm{\infty }}(ℝ)$

Then we have for all $A\in C^{\mathrm{\infty }}{\mathrm{Ring}}^{\mathrm{fin}}$ that

$$C(Y\ell A):R^n\mapsto A(*{)}^n$$C(Y\ell A) : R^n \mapsto A({*})^n

Proof

This is a straightforward manipulation:

$$\begin{array}{rl}{\mathrm{Sh}}_{𝕃}(Y(\ell A),R^n)& ={\mathrm{Sh}}_{𝕃}(Y(\ell A),Y(\ell C^{\mathrm{\infty }}({ℝ}^n)))\\ & ={\mathrm{PSh}}_{𝕃}(Y(\ell A),Y(\ell C^{\mathrm{\infty }}({ℝ}^n)))\\ & \simeq 𝕃(\ell A,\ell C^{\mathrm{\infty }}({ℝ}^n))\\ & \simeq C^{\mathrm{\infty }}{\mathrm{Ring}}^{\mathrm{fin}}(C^{\mathrm{\infty }}({ℝ}^n),A)\\ & \simeq A(*{)}^n\end{array}$$\begin{aligned} Sh_{\mathbb{L}}(Y(\ell A), R^n) & = Sh_{\mathbb{L}}(Y(\ell A), Y(\ell C^\infty(\mathbb{R}^n))) \\ & = PSh_{\mathbb{L}}(Y(\ell A), Y(\ell C^\infty(\mathbb{R}^n))) \\ & \simeq \mathbb{L}(\ell A, \ell C^\infty(\mathbb{R}^n)) \\ & \simeq C^\infty Ring^{fin}(C^\infty(\mathbb{R}^n), A) \\ & \simeq A({*})^n \end{aligned}

Here

1. the first step expresses the nature of the line object in the models under consideration

2. the second step expresses that the embedding $\mathrm{Sh}(𝕃)\to \mathrm{PSh}(𝕃)$ is a full and faithful functor

3. the third step expresses that the Yoneda embedding is a full and faithful functor

4. the fourth step is the definition of $𝕃$ as the opposite category of $C^{\mathrm{\infty }}{\mathrm{Ring}}^{\mathrm{op}}$

5. the fifth step expresses that $C^{\mathrm{\infty }}(R^n)$ is the free generalized smooth algebra on $n$ generators (MSIA, chaper I, prop 1.1)

Local $C^{\mathrm{\infty }}$-algebra

The category CartSp carries a natural Grothendieck topology.

A smooth algebra

$$A:\mathrm{CartSp}\to \mathrm{Set}$$A : CartSp \to Set

is a local algebra if $A$ sends covering families to epimorphism families: for each covering $\{U_i\to U\}$ the morphism

$$\coprod _{i}A(U_i)\to A(U)$$\coprod_i A(U_i) \to A(U)

is an epimorphism.

Proposition

Local smooth algebras are precisely the “local Archimedian” algebras (…).

This is (BungeDubuc, prop. 2.1).

Examples

Functions on smooth manifolds

Definition

For $X$ a smooth manifold, the smooth algebra $C^{\mathrm{\infty }}(X)$ is the functor

$$C^{\mathrm{\infty }}(X):={\mathrm{Hom}}_{\mathrm{Diff}}(X,-)$$C^\infty(X) := Hom_{Diff}(X,-)

Weil algebras: functions on small infintiesimal spaces

A Weil algebra in this context is a finite-dimensional commutative $ℝ$-algebra $W$ with a maximal ideal $I$ such that $W/I\simeq ℝ$ and $I^n=0$ for some $n\in \mathbb{N}$.

Proposition

There is a unique $C^{\mathrm{\infty }}$-ring structure on a Weil alghebra $W$. It makes $W$ a finitely presented $C^{\mathrm{\infty }}$-ring.

Remark

The smooth loci corresponding to Weil algebras are infinitesimal spaces. Weil algebras play a crucial role in the definition of smooth toposes.

Power series

…

Functions on germs of manifolds

For $p\in {ℝ}^n$, the algebra of germs of smooth $ℝ$-valued functions at $p$ carries an evident $C^{\mathrm{\infty }}$-ring structure $C^{\mathrm{\infty }}({ℝ}^n{)}_p$.

With $I_p\subset C^{\mathrm{\infty }}({ℝ}^n)$ the ideal of functions that vanish on a neighbourhood of $p$ we have

$$C_p^{\mathrm{\infty }}({ℝ}^n)\simeq C^{\mathrm{\infty }}({ℝ}^n)/I_p,$$C^\infty_p(\mathbb{R}^n) \simeq C^\infty(\mathbb{R}^n)/I_p \,,

yielding a finitely generated but not (for $n>0$) finitely presented $C^{\mathrm{\infty }}$-ring.

Properties

Limits and colimits

Proposition

All limits and all directed colimits in $C^{\mathrm{\infty }}\mathrm{Ring}$ are computed objectwise in $[\mathrm{CartSp},\mathrm{Set}]$ as limits in Set.

Proof

As discussed at limits and colimits by example, all limits and colimits in $[\mathrm{CartSp},\mathrm{Set}]$ are computed objectwise, so the remaining question is if they preserve the property of functors $\mathrm{CartSp}\to \mathrm{Set}$ to preserved products. The claim follows from the observation that limits and directed colimits do commute with products.

See also MSIA, p. 22.

The underlying ordinary algebra

There is a forgetful functor

$$U:C^{\mathrm{\infty }}\mathrm{Alg}\to \mathrm{Alg}$$U : C^\infty Alg \to Alg

from generalized smooth algebras to ordinary algebras which is given by evaluation on $ℝ$

$$U:A\mapsto A(ℝ)$$U : A \mapsto A(\mathbb{R})

and equipping the set $A(ℝ)$ with the algebra structure induced on it:

the product and sum on $A(ℝ)$ is the image of the corresponding operations on the algebra $ℝ$

$${\cdot }_A:A(ℝ)\times A(ℝ)\stackrel{\simeq }{\to }A(ℝ\times ℝ)\stackrel{A(\cdot )}{\to }A(ℝ).$$\cdot_A : A(\mathbb{R}) \times A(\mathbb{R}) \stackrel{\simeq}{\to} A(\mathbb{R}\times \mathbb{R}) \stackrel{A(\cdot)}{\to} A(\mathbb{R}) \,.

$${+}_A:A(ℝ)\times A(ℝ)\stackrel{\simeq }{\to }A(ℝ+ℝ)\stackrel{A(\cdot )}{\to }A(ℝ).$$+_A : A(\mathbb{R}) \times A(\mathbb{R}) \stackrel{\simeq}{\to} A(\mathbb{R} + \mathbb{R}) \stackrel{A(\cdot)}{\to} A(\mathbb{R}) \,.

Moreover there is canonically a morphism of rings

$$ℝ\to A(ℝ)$$\mathbb{R} \to A(\mathbb{R})

given by

$$(*={ℝ}^0\stackrel{c}{\to }ℝ)\mapsto (*=A({ℝ}^0)\stackrel{A(c)}{\to }A(ℝ)).$$(* = \mathbb{R}^0 \stackrel{c}{\to} \mathbb{R}) \mapsto (* = A(\mathbb{R}^0) \stackrel{A(c)}{\to} A(\mathbb{R})) \,.

This makes $A(ℝ)$ an $ℝ$-algebra.

Proposition

The forgetful functor $U$ fits into an adjunction

$$(F⊣U):C^{\mathrm{\infty }}{\mathrm{Alg}}_{ℝ}\stackrel{\stackrel{F}{\leftarrow }}{\underset{U}{\to }}{\mathrm{Alg}}_{ℝ}.$$(F \dashv U) : C^\infty Alg_{\mathbb{R}} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Alg_{\mathbb{R}} \,.

Proof

This statement may be understood as a special case of the following more general statement:

• If $S$, $T$ are finitary monads and $f:S\to T$ is a monad morphism, then the relative forgetful functor

  $$f^{*}:{\mathrm{Alg}}_T\to {\mathrm{Alg}}_S,$$f^*: Alg_T \to Alg_S,

  which pulls back a $T$-algebra $\xi :TX\to X$ to the $S$-algebra $\xi \circ fX:SX\to X$, admits a left adjoint.

(In the case under discussion, $S$ is the free algebra monad on $\mathrm{Set}$, $T$ is the free smooth algebra monad, and $f:S\to T$ is induced from the obvious inclusion $f(n):S(n)\to T(n)$ which interprets an $n$-ary algebra operation (in the theory ${\mathrm{Th}}_S$) as a smooth operation in the theory ${\mathrm{Th}}_T$. See finitary monad for discussion on the connection between finitary monads $T$ and Lawvere theories ${\mathrm{Th}}_T$.)

The desired left adjoint $f_{!}$ takes an algebra $\theta :SX\to X$ to the reflexive coequalizer exhibited as a diagram

$$TSX\stackrel{\stackrel{({\mu }_T\circ Tf)X}{\to }}{\underset{T\theta }{\to }}TX\stackrel{\pi }{\to }T{\otimes }_{S}X$$T S X \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} T X \stackrel{\pi}{\to} T \otimes_S X

in the category of $T$-algebras, where ${\mu }_T:TT\to T$ is the monad multiplication. The coequalizer is denoted $T{\otimes }_{S}X$ to emphasize the analogy with pushing forward $S$-modules $X$ along a ring homomorphism $f:S\to T$ to get $T$-modules; the proof below is an arrow-theoretic transcription of the usual proof of the adjunction between pushing forward and pulling back in the context of rings and modules.

A finitary monad $T$ preserves reflexive coequalizers, so that there is a canonical isomorphism

$$T(T{\otimes }_{S}X)\cong (TT){\otimes }_{S}X$$T(T \otimes_S X) \cong (T T) \otimes_S X

It follows that $T$-algebra ($T$-module) maps $\overline{g}:T{\otimes }_{S}X\to Y$, i.e., maps that render commutative the diagram

$$\begin{array}{ccc}TT{\otimes }_{S}X& \stackrel{T\overline{g}}{\to }& TY\\ \mu {\otimes }_{S}X\downarrow & & \downarrow \xi \\ T{\otimes }_{S}X& \underset{\overline{g}}{\to }& Y\end{array}$$\array{ T T \otimes_S X & \stackrel{T\bar{g}}{\to} & T Y \\ \mu \otimes_S X \downarrow & & \downarrow \xi \\ T \otimes_S X & \underset{\bar{g}}{\to} & Y }

are in bijection with maps $g:TX\to Y$ that render commutative

$$\begin{array}{ccc}TTX& \stackrel{Tg}{\to }& TY\\ \mu \downarrow & & \downarrow \xi \\ TX& \underset{g}{\to }& Y\end{array}$$\array{ T T X & \stackrel{T g}{\to} & T Y \\ \mu \downarrow & & \downarrow \xi \\ T X & \underset{g}{\to} & Y }

(that is to say, $T$-algebra maps $g:TX\to Y$) which additionally coequalize the parallel pair in the diagram

$$TSX\stackrel{\stackrel{({\mu }_T\circ Tf)X}{\to }}{\underset{T\theta }{\to }}TX\stackrel{g}{\to }Y$$T S X \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} T X \stackrel{g}{\to} Y

Since $f:S\to T$ is a monad morphism, we have commutativity of parallel squares in

$$\begin{array}{ccccc}TSX& \stackrel{\stackrel{({\mu }_T\circ Tf)X}{\to }}{\underset{T\theta }{\to }}& TX& \stackrel{g}{\to }& Y\\ fSX\uparrow & & \uparrow fX& & \\ SSX& \stackrel{\stackrel{{\mu }_{S}X}{\to }}{\underset{S\theta }{\to }}& SX\end{array}$$\array{ T S X & \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} & T X & \stackrel{g}{\to} & Y \\ f S X \uparrow & & \uparrow f X & & \\ S S X & \stackrel{\overset{\mu_S X}{\to}}{\underset{S \theta}{\to}} & S X }

so that $g\circ fX$ coequalizes the bottom pair. However, because $g:TX\to Y$ is a $T$-algebra map, its pullback $g\circ fX:SX\to Y$ defines an $S$-algebra map $SX\to f^{*}Y$. This $S$-algebra map $g\circ fX$ factors through the coequalizer of the bottom pair of maps in ${\mathrm{Alg}}_S$, i.e., factors uniquely through an $S$-algebra map $X\to f^{*}Y$. This establishes the adjunction $f_{!}⊣f^{*}$.

Finitely presented $C^{\mathrm{\infty }}$-rings

Proposition (MSIA, prop. 1.1)

$C^{\mathrm{\infty }}({ℝ}^n)$ is the free smooth algebra on $n$ generators, in that for every $n\in \mathbb{N}$ and every smooth algebra $A$ there is an adjunction isomorphism

$${\mathrm{Hom}}_{C^{\mathrm{\infty }}\mathrm{Alg}}(C^{\mathrm{\infty }}({ℝ}^n),A)\simeq {\mathrm{Hom}}_{\mathrm{Alg}}(ℝ[x_1,...,x_n],A(ℝ)).$$Hom_{C^\infty Alg}(C^\infty(\mathbb{R}^n), A) \simeq Hom_{Alg}(\mathbb{R}[x_1,...,x_n], A(\mathbb{R})) \,.

• Every finitely presented $C^{\mathrm{\infty }}$-ring is fair/germ determined.

We have a chain of inclusions

• finitely presented $C^{\mathrm{\infty }}$-rings

• $\subset$ “good” $C^{\mathrm{\infty }}$-rings

• $\subset$ fair $C^{\mathrm{\infty }}$-rings

• $\subset$ finitely generated $C^{\mathrm{\infty }}$-rings

Points of smooth loci

An $ℝ$-point of a $C^{\mathrm{\infty }}$-ring $C$ is a point $*\to 𝕃(C)$ of the corresponding smooth locus, i.e. a morphism $C\to ℝ\cong C^{\mathrm{\infty }}(*)$.

Proposition

Points of a $C^{\mathrm{\infty }}$-ring are in bijection with points of the underlying $ℝ$-algebra $U(C)$, i.e. with ordinary $ℝ$-algebra morphisms $U(C)\to ℝ$.

In particular every Weil algebra $W$ has a unique point $*\to 𝕃(W)$: every Weil algebra is the algebra of functions on an infinitesimal thickening of an ordinary point.

By the properties of $C^{\mathrm{\infty }}(X)$ for $X$ a smooth manifold discussed below, the $ℝ$-points of $C^{\mathrm{\infty }}(X)$ are precisely the ordinary points of the manifold $X$.

Smooth function algebras on smooth manifolds

Proposition (MSIA, prop. 2.5, 2.6 )

Let $f:X\to Z$ and $g:Y\to Z$ be transversal maps of smooth manifolds. Then the functor $C^{\mathrm{\infty }}(-)$ sends the pullback

$$\begin{array}{ccc}X{\times }_{Z}Y& \to & X\\ \downarrow & & {\downarrow }^f\\ Y& \stackrel{g}{\to }& Z\end{array}$$\array{ X \times_Z Y &\to& X \\ \downarrow && \downarrow^f \\ Y &\stackrel{g}{\to}& Z }

to the pushout

$$\begin{array}{cccc}{C}^{\mathrm{\infty }}(X){\otimes }_{C^{\mathrm{\infty }}(Z)}{C}^{\mathrm{\infty }}(Y)=:& C^{\mathrm{\infty }}(X{\times }_{Z}Y)& \leftarrow & C^{\mathrm{\infty }}(X)\\ & \uparrow & & {\uparrow }^{f^{*}}\\ & C^{\mathrm{\infty }}(Y)& \stackrel{g^{*}}{\leftarrow }& C^{\mathrm{\infty }}(Z)\end{array}$$\array{ C^\infty(X) \otimes_{C^\infty(Z)} C^\infty(Y) =: & C^\infty(X \times_Z Y) &\leftarrow& C^\infty(X) \\ & \uparrow && \uparrow^{f^*} \\ & C^\infty(Y) &\stackrel{g^*}{\leftarrow}& C^\infty(Z) }

In particular this implies (for $Z=*$)that the the smooth tensor product of functions on $X$ and $Y$ is the algebra of functions on the product $X\times Y$:

$$C^{\mathrm{\infty }}(X\times Y)\simeq C^{\mathrm{\infty }}(X){\otimes }_{\mathrm{\infty }}{C}^{\mathrm{\infty }}(Y).$$C^\infty(X \times Y) \simeq C^\infty(X) \otimes_\infty C^\infty(Y) \,.

Remark

The ordinary algebraic tensor product of $C^{\mathrm{\infty }}(X)(ℝ)$ and $C^{\mathrm{\infty }}(Y)(ℝ)$ regarded as ordinary algebras does not in general satisfy this property. Rather one has an inclusion

$$C^{\mathrm{\infty }}(X)(ℝ)\otimes C^{\mathrm{\infty }}(Y)(ℝ)\subset C^{\mathrm{\infty }}(X\times Y)(ℝ).$$C^\infty(X)(\mathbb{R}) \otimes C^\infty(Y)(\mathbb{R}) \subset C^\infty(X \times Y)(\mathbb{R}) \,.

Remark

In the context of geometric function theory the corresponding general statement (without the transversality condition) says that $C^{\mathrm{\infty }}(X)$ is a “good” kind of function. The above equation is one sub-aspect of the one of the fundamental theorems of geometric infinity-function theory.

Turning this inclusion into an equivalence is usually called a completion of the algebraic tensor product. Therefore we see:

The smooth tensor product is automatically the completed tensor product.

In summary this yields the following characterization of smooth function algebras on manifolds.

Theorem (MSIA, theorem 2.8)

The functor $C^{\mathrm{\infty }}(-)={\mathrm{Hom}}_{\mathrm{Diff}}(-,-):\mathrm{Diff}\to C^{\mathrm{\infty }}\mathrm{Alg}$

• is a full and faithful functor

• takes values in finitely presented smooth $C^{\mathrm{\infty }}$-algebras

• sends transversal pullbacks to coproducts (and hence to the smooth tensor product).

Deformation theory of smooth algebras

under construction

For $C$ any category whose objects we think of as “functions algebras on test spaces”, such as $C=C^{\mathrm{\infty }}\mathrm{Ring}$, there is a general intrinsic notion of tangent complex and deformation theory of such objects.

As describe there, the key structure of interest from which all the other structure here is induced is the tangent category

$$TC\to C.$$T C \to C \,.

This is $TC=\mathrm{Ab}(\mathrm{Arr}(C))$ the codomain fibration of $C$ “fiberwise stabilized”, meaning that in each fiber one takes it to consist of $\mathrm{Ab}(C/A)$, the abelian group objects in the overcategory.

We now first recall what this means for ordinary rings and how it induces the ordinary notion of derivations and modules for ordinary rings by setting $C=$ CRing, and then look at what it implies for $C^{\mathrm{\infty }}$-rings by setting $C=C^{\mathrm{\infty }}\mathrm{Rings}$.

By an old argument by Quillen, for $C=$ CRing we have that $TC=\mathrm{Mod}$ is the bifibration of modules over rings, there is a natural equivalence

$${\mathrm{Mod}}_A\stackrel{\simeq }{\to }\mathrm{Ab}(C/A).$$Mod_A \stackrel{\simeq}{\to} Ab(C/A) \,.

This is induced by the functor that sends an $A$-module $N$ to the corresponding object in the square-0-extension $R\oplus N\to R$. (See module).

From this structure alone a lot of further structure follows:

• a derivation $\delta A\to N$ is precisely a section of the corresponding morphism $AolusN\to A$ in $C/A$, in the category $C$ namely a ring homomorphism

  $$\begin{array}{ccccc}A& & \stackrel{\delta }{\to }& & A\oplus N\\ & {}_{=}\searrow & & \swarrow \\ & & A\end{array}.$$\array{ A &&\stackrel{\delta}{\to}&& A \oplus N \\ & {}_ {=}\searrow && \swarrow \\ && A } \,.

• The forgetful functor $T\mathrm{Ring}\simeq \mathrm{Mod}\to \mathrm{Ring}$ has a left adjoint

  $${\Omega }_K^1:\mathrm{Ring}\to \mathrm{Mod}$$\Omega_K^1 : Ring \to Mod

  that sends each ring to its module of Kähler differentials.

  The fact that it is left adjoint is the universal property of the Kähler differentials as te objects co-representing derivations

  ${\mathrm{Hom}}_{\mathrm{Ab}(\mathrm{Ring}/R)}({\Omega }_K^1(A),N)\simeq {\mathrm{Hom}}_{\mathrm{Ring}/A}(A,A\oplus N)$.

  So every derivation $\delta :A\to N$ uniquely corresponds to a module morphism ${\Omega }_K^1(A)\to N$, namely the one that sends $da\mapsto \delta (a)$.

This abstract story remains precisely the same for $C^{\mathrm{\infty }}$-rings (and in fact for everything else!) but what it means concretely changes.

The crucial observation is (as one can show) that an abelian group object in $C^{\mathrm{\infty }}\mathrm{Ring}/A$ is a square-0 extension $(A\oplus N)$ for $N$ an (ordinary) module of the underlying $ℝ$-algebra $A$. This square-0-extension happens to be uniquely equipped with the $C^{\mathrm{\infty }}$-Ring-struncture given by

$$(f\in C^{\mathrm{\infty }}({ℝ}^n)\to ℝ)\mapsto ((a_1,n_1),\cdots ,(a_n,n_n))\mapsto f(a_1,\cdots ,a_n)+\sum _{i=1^n}\frac{\partial f}{\partial x_i}(a_1,\cdots ,a_n)n_i).$$(f \in C^\infty(\mathbb{R}^n) \to \mathbb{R}) \mapsto \left( (a_1, n_1), \cdots, (a_n, n_n)) \mapsto f(a_1, \cdots, a_n) + \sum_{i = 1^n} \frac{\partial f}{\partial x_i}(a_1, \cdots, a_n) n_i \right) \,.

This uniquely induced smooth structure on objects in $\mathrm{Ab}(C^{\mathrm{\infty }}\mathrm{Ring}/A)$ then in turn affects the nature of the notion of derivation and of Kähler differentials, as those are defined by general abstract reasooning from the former.

First of all it follows that a derivation – by general abstract definition a morphism of $C^{\mathrm{\infty }}$-rings $\mathrm{Id}\oplus \delta :A\to A\oplus N$ – is a morphism that satisfies for all $f\in C^{\mathrm{\infty }}({ℝ}^n,ℝ)$ that

$$\delta :f(a_1,\cdots ,a_n)\mapsto \sum _i\frac{\partial f}{\partial x_i}\delta a_i.$$\delta : f(a_1, \cdots, a_n) \mapsto \sum_i \frac{\partial f}{\partial x_i} \delta a_i \,.

For ordinary rings only the compatibility $\delta (a_1\cdot a_2)=\delta (a_1)a_2+a_1\delta (a_2)$ with the single product operation is required. Here, however, compatibility with infinitely more operations $f\in C^{\mathrm{\infty }}({ℝ}^n,ℝ)$ is demanded.

Accordingly, then, the Kähler differentials as defined with respect to such derivations are different from the purely ring-theoretic ones: they produce the right notion of smooth 1-forms here, whereas the ring-theoretic one does not.

Generalizations to higher category theory

The generalization of the notion of smooth algebra fo (∞,1)-category theory is

• smooth (∞,1)-algebra.

References

A standard textbook reference is chapter 1 of

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

The concept of $C^{\mathrm{\infty }}$-rings in particular and that of synthetic differential geometry in general was introduced in

• Bill Lawvere, Categorical dynamics

  in Anders Kock (eds.) Topos theoretic methods in geometry, volume 30 of Various Publ. Ser., pages 1-28, Aarhus Univ. (1997)

but examples of the concept are older. A discussion from the point of view of functional analysis is in

• G. Kainz, A. Kriegl, Peter Michor, $C^{\mathrm{\infty }}$-algebras from the functional analytic view point Journal of pure and applied algebra 46 (1987) (pdf)

A characterization of those $C^{\mathrm{\infty }}$-rings that are algebras of smooth functions on some smooth manifold is given in

• Peter Michor, Jiri Vanzura, Characterizing algebras of $C^{\mathrm{\infty }}$-functions on manifolds (pdf)

Lawvere’s ideas were later developed by Eduardo Dubuc, Anders Kock, Ieke Moerdijk, Gonzalo Reyes, and Gavin Wraith.

Studies of the properties of $C^{\mathrm{\infty }}$-rings include

• Ieke Moerdijk, Gonzalo Reyes, Rings of Smooth Functions and Their Localizations, I , Journal of Algebra 99 (1986)

The notion of the spectrum of a $C^{\mathrm{\infty }}$-ring and that of $C^{\mathrm{\infty }}$-schemes is discussed in

• Eduardo Dubuc, $C^{\mathrm{\infty }}$-schemes Amer. J. Math. 103 (1981) (pdf JSTOR).

and more generally in

• Ieke Moerdijk, Gonzalo Reyes, Rings of smooth functions and their localization II , in Mathematical logic and theoretical computer science page 275 (Google books)

• Marta Bunge, Eduardo Dubuc, Archimedian local $C^{\mathrm{\infty }}$-rings and models of synthetic differential geometry Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 3 (1986), p. 3-22 (numdam).

More recent developments along these lines are in

• Dominic Joyce, Algebraic geometry over $C^{\mathrm{\infty }}$-rings (arXiv:1001.0023)

The higher geometry generalization to a theory of derived smooth manifolds – spaces with structure sheaf taking values in simplicial C∞-rings – was initiated in

• David Spivak, Quasi-Smooth derived manifolds (pdf of original version, arXiv:0810.5174)

based on the general machinery of structured (∞,1)-toposes in

• Jacob Lurie, Structured Spaces

where this is briefly mentioned in the very last paragraph.

See also the references at Fermat theory, of which $C^{\mathrm{\infty }}$-rings are a sepcial case. And the references at smooth locus, the formal dual of a $C^{\mathrm{\infty }}$-ring. And the references at super smooth topos, which involves generalizations of $C^{\mathrm{\infty }}$-rings to supergeometry.