nLab
generalized smooth algebra

Idea

A smooth algebra or $C^{\mathrm{\infty }}$-ring is an algebra $A$ over the reals 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 smooth 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

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 (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 $B$.

Internal smooth algebras

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 forth step is the definition of $mathbL$ 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)

Properties

General properties

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 (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

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

Remark

While a product-preserving co-presheaf $A$ induces the structure of an algebra on each of the sets $A({ℝ}^n)$, with product induced from the componentwise product ${ℝ}^n\times {ℝ}^n\to {ℝ}^n$, it is not a co-presheaf with values in algebras: the co-restriction morphisms assigned to maps ${ℝ}^n\to {ℝ}^m$ which are not just projections or injections will not be algebra homomorphisms.

But conversely this means that restricted to such maps, i.e. restricted along the inclusion $\mathrm{FinSet}\hookrightarrow \mathrm{CartSp}$ of CartSp, $A$ does become a co-presheaf with values in algebras.

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.

Smooth function algebras on 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}) \,.

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).

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)

Synthetic spaces locally isomorphic to smooth loci were discussed in

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

and more recently 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.