A smooth algebra or $C^\infty$-ring is an algebra $A$ over the reals $\mathbb{R}$ for which not only the product operation $\cdot : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ lifts to the algebra product $A \times A \to A$, but for which every smooth map $f : \mathbb{R}^n \to \mathbb{R}^m$ (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 $C^\infty$-rings is witnessed by the fact that this Lawvere theory is even a Fermat theory.
The opposite category of the category of $C^\infty$-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.
For $X$ a smooth manifold, the assignment
of the set of smooth $\mathbb{R}^n$-valued functions on $X$ is clearly covariant and hence yields a co-presheaf on CartSp $\subset$ Diff: a functor
Since the hom-functor sends limits to limits in its second argument this is clearly product preserving.
If as usual we write $C^\infty(X) := C^\infty(X,\mathbb{R})$ for the set of just $\mathbb{R}$-valued smooth functions, then the usual pointwise product of functions
can be regarded as the image of our co-presheaf under the muliplication map $\mathbb{R} \times \mathbb{R} \stackrel{-\cdot -}{\to} \mathbb{R}$ on the algebra of real numbers:
A $C^\infty$-algebra is a finite product-preserving co-presheaf on CartSp, i.e. a finite product preserving functor
The category of such functors and natural transformations between them we denote by $C^\infty Alg$.
The standard name in the literature for generalized smooth algebras is $C^\infty$-rings. Even though standard, this has the disadvantages for us that it collides badly with the use of $\infty$- for higher categorical structures.
The coproduct in $C^\infty Alg$ we call the smooth tensor product
More generally, for $i : C \to A$ and $j : C \to B$ two morphisms in $C^\infty Alg$, we call the pushout
the smooth tensor product over $C$ of $A$ and $B$.
For $X$ a smooth manifold, the smooth algebra $C^\infty(X)$ is the functor
(finitely generated and finitely presented $C^\infty$-rings)
For $R$ a $C^\infty$-ring, and $I \in U(R)$ an ideal in the underlying ordinary ring, there is a canonical $C^\infty$-ring structure $R/I$ on the ordinaryy quotient ring $U(R)/I$.
A $C^\infty$-ring $R$ is called finitely generated if it is of the form $C^\infty(\mathbb{R}^n)/I$ for $n \in \mathbb{N}$ and $I$ an ideal in $U(C^\infty(\mathbb{R}^n))$.
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
(germ-determined finitely generated / fair )
For $p \in \mathbb{R}^n$ let
be the natural projection onto the smooth algebra of germs of functions at $p$.
A $C^\infty$-ring $C$ is called fair or finitely generated and germ-determined if it is finitely generated $C \simeq C^\infty(\mathbb{R}^n)/I$ and the ideal $I$ has the property that $f \in C^\infty(\mathbb{R}^n)$ is an element of $I$ if (and hence precisely if) for all $p \in \mathbb{R}^n$ the germ $\pi_p(f) \in C^\infty_p(\mathbb{R})^n$ is in the germ $\pi_p(I)$ of the ideal.
For any smooth topos $(\mathcal{T}, R)$, there is an internal notion of generalized smooth algebra:
For $(\mathcal{T}, R)$ a topos equipped with an internal ring object $R$ (possibly but not necessarily a smooth topos), let $CartSp(\mathcal{T},R)$ be the full subcategory of $\mathcal{T}$ on objects of the form $R^n$ for $n \in \mathbb{N}$. Then a $(\mathcal{T},R)$-algebra is a product-preserving functor $A : CartSp(\mathcal{T}, R) \to Set$.
All constructions on smooth algebras generalize to $(\mathcal{T},R)$-algebras. In particular for $X \in \mathcal{T}$ any object we have the function $(\mathcal{T},R)$-algebra
The following remark asserts that when $\mathcal{T}$ is itself a sufficiently nice category of sheaves on formal duals of $(Set,\mathbb{R})$-algebras, then the internal notion of smooth function algebras on formal duals of external smooth algebras reproduces these external smooth algebras.
Let $A$ be a finitely generated $C^\infty$-ring, $\ell A$ its incarnation as an object in $\mathbb{L} = (C^\infty Ring^{fin})^{op}$ and $Y\ell A$ its incarnation in $Sh(\mathbb{L}) \subset PSh(\mathbb{L})$, with $Y$ the Yoneda embedding and using the assumption that the Grothendieck topology used to form $Sh(\mathbb{L})$ is subcanonical.
Also suppose that the line object $R$ is represented by $\ell C^\infty(\mathbb{R})$
Then we have for all $A \in C^\infty Ring^{fin}$ that
This is a straightforward manipulation:
Here
the first step expresses the nature of the line object in the models under consideration
the second step expresses that the embedding $Sh(\mathbb{L}) \to PSh(\mathbb{L})$ is a full and faithful functor
the third step expresses that the Yoneda embedding is a full and faithful functor
the fourth step is the definition of $\mathbb{L}$ as the opposite category of $C^\infty Ring^{fin}$
the fifth step expresses that $C^\infty(R^n)$ is the free generalized smooth algebra on $n$ generators (MSIA, chaper I, prop 1.1)
The category CartSp carries a natural Grothendieck topology.
A smooth algebra
is a local algebra if $A$ sends covering families to epimorphism families: for each covering $\{U_i \to U\}$ the morphism
is an epimorphism.
Local smooth algebras are precisely the “local Archimedian” algebras (…).
This is (BungeDubuc, prop. 2.1).
For $X$ a smooth manifold, the smooth algebra $C^\infty(X)$ is the functor
A Weil algebra in this context is a finite-dimensional commutative $\mathbb{R}$-algebra $W$ with a maximal ideal $I$ such that $W/I \simeq \mathbb{R}$ and $I^n = 0$ for some $n \in \mathbb{N}$.
There is a unique $C^\infty$-ring structure on a Weil algebra $W$. It makes $W$ a finitely presented $C^\infty$-ring.
The smooth loci corresponding to Weil algebras are infinitesimal spaces. Weil algebras play a crucial role in the definition of smooth toposes.
…
For $p \in \mathbb{R}^n$, the algebra of germs of smooth $\mathbb{R}$-valued functions at $p$ carries an evident $C^\infty$-ring structure $C^\infty(\mathbb{R}^n)_p$.
With $I_p \subset C^\infty(\mathbb{R}^n)$ the ideal of functions that vanish on a neighbourhood of $p$ we have
yielding a finitely generated but not (for $n \gt 0$) finitely presented $C^\infty$-ring.
All limits and all directed colimits in $C^\infty Ring$ are computed objectwise in $[CartSp,Set]$ as limits in Set.
As discussed at limits and colimits by example, all limits and colimits in $[CartSp,Set]$ are computed objectwise, so the remaining question is if they preserve the property of functors $CartSp \to Set$ to preserved products. The claim follows from the observation that limits and directed colimits do commute with products.
See also MSIA, p. 22.
There is a forgetful functor
from generalized smooth algebras to ordinary algebras which is given by evaluation on $\mathbb{R}$
and equipping the set $A(\mathbb{R})$ with the algebra structure induced on it:
the product and sum on $A(\mathbb{R})$ is the image of the corresponding operations on the algebra $\mathbb{R}$
Moreover there is canonically a morphism of rings
given by
This makes $A(\mathbb{R})$ an $\mathbb{R}$-algebra.
The forgetful functor $U$ fits into an adjunction
This statement may be understood as a special case of the following more general statement:
which pulls back a $T$-algebra $\xi: T X \to X$ to the $S$-algebra $\xi \circ f X: S X \to X$, admits a left adjoint.
(In the case under discussion, $S$ is the free algebra monad on $Set$, $T$ 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 $Th_S$) as a smooth operation in the theory $Th_T$. See finitary monad for discussion on the connection between finitary monads $T$ and Lawvere theories $Th_T$.)
The desired left adjoint $f_!$ takes an algebra $\theta: S X \to X$ to the reflexive coequalizer exhibited as a diagram
in the category of $T$-algebras, where $\mu_T: T T \to T$ 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
It follows that $T$-algebra ($T$-module) maps $\bar{g}: T \otimes_S X \to Y$, i.e., maps that render commutative the diagram
are in bijection with maps $g: T X \to Y$ that render commutative
(that is to say, $T$-algebra maps $g: T X \to Y$) which additionally coequalize the parallel pair in the diagram
Since $f: S \to T$ is a monad morphism, we have commutativity of parallel squares in
so that $g \circ f X$ coequalizes the bottom pair. However, because $g: T X \to Y$ is a $T$-algebra map, its pullback $g \circ f X: S X \to Y$ defines an $S$-algebra map $S X \to f^* Y$. This $S$-algebra map $g \circ f X$ factors through the coequalizer of the bottom pair of maps in $Alg_S$, i.e., factors uniquely through an $S$-algebra map $X \to f^*Y$. This establishes the adjunction $f_! \dashv f^*$.
$C^\infty(\mathbb{R}^n)$ 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
We have a chain of inclusions
finitely presented $C^\infty$-rings
$\subset$ “good” $C^\infty$-rings
$\subset$ fair $C^\infty$-rings
$\subset$ finitely generated $C^\infty$-rings
An $\mathbb{R}$-point of a $C^\infty$-ring $C$ is a point $* \to \mathbb{L}(C)$ of the corresponding smooth locus, i.e. a morphism $C \to \mathbb{R} \cong C^\infty(*)$.
Points of a $C^\infty$-ring are in bijection with points of the underlying $\mathbb{R}$-algebra $U(C)$, i.e. with ordinary $\mathbb{R}$-algebra morphisms $U(C) \to \mathbb{R}$.
In particular every Weil algebra $W$ has a unique point $* \to \mathbb{L}(W)$: every Weil algebra is the algebra of functions on an infinitesimal thickening of an ordinary point.
By the properties of $C^\infty(X)$ for $X$ a smooth manifold discussed below, the $\mathbb{R}$-points of $C^\infty(X)$ are precisely the ordinary points of the manifold $X$.
Let $f : X \to Z$ and $g : Y \to Z$ be transversal maps of smooth manifolds. Then the functor $C^\infty(-)$ sends the pullback
to the pushout
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 $X \times Y$:
The ordinary algebraic tensor product of $C^\infty(X)(\mathbb{R})$ and $C^\infty(Y)(\mathbb{R})$ regarded as ordinary algebras does not in general satisfy this property. Rather one has an inclusion
In the context of geometric function theory the corresponding general statement (without the transversality condition) says that $C^\infty(X)$ 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.
The functor $C^\infty(-) = Hom_{Diff}(-,-) : Diff \to C^\infty Alg$
under construction
For $C$ any category whose objects we think of as “functions algebras on test spaces”, such as $C = C^\infty Ring$, 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
This is $T C = Ab(Arr(C))$ the codomain fibration of $C$ “fiberwise stabilized”, meaning that in each fiber one takes it to consist of $Ab(C/A)$, 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 $C^\infty$-rings by setting $C = C^\infty Rings$.
By an old argument by Quillen, for $C =$ CRing we have that $T C = Mod$ is the bifibration of modules over rings, there is a natural equivalence
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 $A \oplus N \to A$ in $C/A$, in the category $C$ namely a ring homomorphism
The forgetful functor $T Ring \simeq Mod \to Ring$ has a left adjoint
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 the objects co-representing derivations
$Hom_{Ab(Ring/R)}(\Omega_K^1(A),N) \simeq Hom_{Ring/A}(A,A \oplus N)$.
So every derivation $\delta : A \to N$ uniquely corresponds to a module morphism $\Omega^1_K(A) \to N$, namely the one that sends $d a \mapsto \delta(a)$.
This abstract story remains precisely the same for $C^\infty$-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 $C^\infty Ring/A$ is a square-0 extension $(A \oplus N)$ for $N$ an (ordinary) module of the underlying $\mathbb{R}$-algebra $A$. This square-0-extension happens to be uniquely equipped with the $C^\infty$-Ring-structure given by
This uniquely induced smooth structure on objects in $Ab(C^\infty Ring/A)$ then in turn affects the nature of the notion of derivation and of Kähler differentials, as those are defined by general abstract reasoning from the former.
First of all it follows that a derivation – by general abstract definition a morphism of $C^\infty$-rings $Id \oplus \delta : A \to A \oplus N$ – is a morphism that satisfies for all $f \in C^\infty(\mathbb{R}^n, \mathbb{R})$ that
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 $f \in C^\infty(\mathbb{R}^n, \mathbb{R})$ 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.
The generalization of the notion of smooth algebra for (∞,1)-category theory is
The generalization to supergeometry is
A standard textbook reference is chapter 1 of
The concept of $C^\infty$-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
A characterization of those $C^\infty$-rings that are algebras of smooth functions on some smooth manifold is given in
Lawvere’s ideas were later developed by Eduardo Dubuc, Anders Kock, Ieke Moerdijk, Gonzalo Reyes, and Gavin Wraith.
Studies of the properties of $C^\infty$-rings include
The notion of the spectrum of a $C^\infty$-ring and that of $C^\infty$-schemes is discussed in
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 $C^\infty$-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
The higher geometry generalization to a theory of derived smooth manifolds – spaces with structure sheaf taking values in simplicial C∞-rings – was initiated in
based on the general machinery of structured (∞,1)-toposes in
where this is briefly mentioned in the very last paragraph.
See also the references at Fermat theory, of which $C^\infty$-rings are a special case. And the references at smooth locus, the formal dual of a $C^\infty$-ring. And the references at super smooth topos, which involves generalizations of $C^\infty$-rings to supergeometry.