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\begin{document}

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\section*{smooth algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{synthetic_differential_geometry}{}\paragraph*{{Synthetic differential geometry}}\label{synthetic_differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivating_example}{Motivating example}\dotfill \pageref*{motivating_example} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{tensor_product}{Tensor product}\dotfill \pageref*{tensor_product} \linebreak
\noindent\hyperlink{finitely_generated_rings}{Finitely generated $C^\infty$-rings}\dotfill \pageref*{finitely_generated_rings} \linebreak
\noindent\hyperlink{internal_rings}{Internal $C^\infty$-rings}\dotfill \pageref*{internal_rings} \linebreak
\noindent\hyperlink{local_algebra}{Local $C^\infty$-algebra}\dotfill \pageref*{local_algebra} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{functions_on_smooth_manifolds}{Functions on smooth manifolds}\dotfill \pageref*{functions_on_smooth_manifolds} \linebreak
\noindent\hyperlink{weil_algebras_functions_on_small_infinitesimal_spaces}{Weil algebras: functions on small infinitesimal spaces}\dotfill \pageref*{weil_algebras_functions_on_small_infinitesimal_spaces} \linebreak
\noindent\hyperlink{power_series}{Power series}\dotfill \pageref*{power_series} \linebreak
\noindent\hyperlink{Germs}{Functions on germs of manifolds}\dotfill \pageref*{Germs} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{limits_and_colimits}{Limits and colimits}\dotfill \pageref*{limits_and_colimits} \linebreak
\noindent\hyperlink{UnderlyingAlgebra}{The underlying ordinary algebra}\dotfill \pageref*{UnderlyingAlgebra} \linebreak
\noindent\hyperlink{finitely_presented_rings}{Finitely presented $C^\infty$-rings}\dotfill \pageref*{finitely_presented_rings} \linebreak
\noindent\hyperlink{points_of_smooth_loci}{Points of smooth loci}\dotfill \pageref*{points_of_smooth_loci} \linebreak
\noindent\hyperlink{SmoothFunctionAlgebrasOnSmoothManifolds}{Smooth function algebras on smooth manifolds}\dotfill \pageref*{SmoothFunctionAlgebrasOnSmoothManifolds} \linebreak
\noindent\hyperlink{deformation_theory_of_smooth_algebras}{Deformation theory of smooth algebras}\dotfill \pageref*{deformation_theory_of_smooth_algebras} \linebreak
\noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{smooth algebra} or \textbf{$C^\infty$-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 \emph{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

\begin{itemize}%
\item a [[product]]-preserving [[presheaf|co-presheaf]] on [[CartSp]];


\item equivalently: an algebra for the [[Lawvere theory]] [[CartSp]];



\end{itemize}
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 [[site]]s for [[category of sheaves|categories of sheaves]] that serve as [[Models for Smooth Infinitesimal Analysis|models]] for [[synthetic differential geometry]].

\hypertarget{motivating_example}{}\subsection*{{Motivating example}}\label{motivating_example}

For $X$ a [[smooth manifold]], the assignment

\begin{displaymath}
\mathbb{R}^n \mapsto C^\infty(X,\mathbb{R}^n)
   = Hom_{Diff}(X,\mathbb{R}^n)
\end{displaymath}
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]]

\begin{displaymath}
C^\infty(X,-) : CartSp \to Set
  \,.
\end{displaymath}
Since the [[hom-functor]] sends [[limit]]s to [[limit]]s in its second argument this is clearly [[product]] preserving.

\begin{displaymath}
C^\infty(X, \mathbb{R}^n \times \mathbb{R}^m)
  \simeq
  C^\infty(X,\mathbb{R}^n)
  \times
  C^\infty(X, \mathbb{R}^m)
\end{displaymath}
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

\begin{displaymath}
\cdot :  C^\infty(X) \times C^\infty(X) \to C^\infty(X)
\end{displaymath}
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:

\begin{displaymath}
\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)
  \,.
\end{displaymath}
\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
[[CartSp]] is the full subcategory of [[Diff]] on manifolds of the form $\mathbb{R}^n$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A $C^\infty$-algebra is a finite [[product]]-preserving [[presheaf|co-presheaf]] on [[CartSp]], i.e. a finite [[product]] preserving [[functor]]

\begin{displaymath}
A : CartSp \to Set
  \,.
\end{displaymath}
The category of such functors and [[natural transformation]]s between them we denote by $C^\infty Alg$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The standard name in the literature for generalized smooth algebras is \textbf{$C^\infty$-rings}. Even though standard, this has the disadvantages for us that it collides badly with the use of $\infty$- for [[higher category theory|higher categorical]] structures.

\end{remark}
\hypertarget{tensor_product}{}\subsubsection*{{Tensor product}}\label{tensor_product}

\begin{defn}
\label{}\hypertarget{}{}
The [[coproduct]] in $C^\infty Alg$ we call the \textbf{smooth tensor product}

\begin{displaymath}
\otimes_\infty : C^\infty Alg \times C^\infty Alg
   \to C^\infty Alg
  \,.
\end{displaymath}
More generally, for $i : C \to A$ and $j : C \to B$ two morphisms in $C^\infty Alg$, we call the [[pushout]]

\begin{displaymath}
\itexarray{
    C &\stackrel{i}{\to}& A
    \\
    \downarrow && \downarrow
    \\
    B &\stackrel{j}{\to}& A \otimes_C B
  }
\end{displaymath}
the \textbf{smooth tensor product over $C$} of $A$ and $B$.

\end{defn}
\hypertarget{finitely_generated_rings}{}\subsubsection*{{Finitely generated $C^\infty$-rings}}\label{finitely_generated_rings}

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[smooth manifold]], the smooth algebra $C^\infty(X)$ is the functor

\begin{displaymath}
C^\infty(X) := Hom_{Diff}(X,-)
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(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 ordinary quotient ring $U(R)/I$.

A $C^\infty$-ring $R$ is called \textbf{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 \textbf{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{displaymath}
\itexarray{
    C^\infty(\mathbb{R}^k) &\to& C^{\infty}(*) \simeq \mathbb{R}
    \\
    \downarrow && \downarrow
    \\
    C^\infty(\mathbb{R}^n) &\to& R
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(germ-determined finitely generated / fair )}

For $p \in \mathbb{R}^n$ let

\begin{displaymath}
\pi_p : C^\infty(\mathbb{R}^n) \to C^\infty_p(\mathbb{R}^n)
\end{displaymath}
be the natural projection onto the \hyperlink{Germs}{smooth algebra of germs} of functions at $p$.

A $C^\infty$-ring $C$ is called \textbf{fair} or \textbf{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.

\end{defn}
\hypertarget{internal_rings}{}\subsubsection*{{Internal $C^\infty$-rings}}\label{internal_rings}

For any [[smooth topos]] $(\mathcal{T}, R)$, there is an [[internalization|internal]] notion of [[generalized smooth algebra]]:

\begin{defn}
\label{}\hypertarget{}{}
For $(\mathcal{T}, R)$ a [[topos]] equipped with an [[internalization|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 \textbf{$(\mathcal{T},R)$-algebra} is a product-preserving functor $A : CartSp(\mathcal{T}, R) \to Set$.

\end{defn}
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

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

\begin{prop}
\label{}\hypertarget{}{}
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 coverage|subcanonical]].

Also suppose that the line object $R$ is [[representable functor|represented]] by $\ell C^\infty(\mathbb{R})$

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

\begin{displaymath}
C(Y\ell A) : R^n \mapsto A({*})^n
\end{displaymath}
\end{prop}
\begin{proof}
This is a straightforward manipulation:

\begin{displaymath}
\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}
\end{displaymath}
Here

\begin{enumerate}%
\item the first step expresses the nature of the line object in the models under consideration


\item the second step expresses that the embedding $Sh(\mathbb{L}) \to PSh(\mathbb{L})$ is a [[full and faithful functor]]


\item the third step expresses that the [[Yoneda embedding]] is a [[full and faithful functor]]


\item the fourth step is the definition of $\mathbb{L}$ as the [[opposite category]] of $C^\infty Ring^{fin}$


\item the fifth step expresses that $C^\infty(R^n)$ is the free [[generalized smooth algebra]] on $n$ generators ([[Models for Smooth Infinitesimal Analysis|MSIA, chaper I, prop 1.1]])



\end{enumerate}
\end{proof}
\hypertarget{local_algebra}{}\subsubsection*{{Local $C^\infty$-algebra}}\label{local_algebra}

The category [[CartSp]] carries a natural [[Grothendieck topology]].

A smooth algebra

\begin{displaymath}
A : CartSp \to Set
\end{displaymath}
is a [[locally algebra-ed topos|local algebra]] if $A$ sends [[covering]] families to epimorphism families: for each covering $\{U_i \to U\}$ the morphism

\begin{displaymath}
\coprod_i A(U_i) \to A(U)
\end{displaymath}
is an [[epimorphism]].

\begin{prop}
\label{}\hypertarget{}{}
Local smooth algebras are precisely the ``local Archimedian'' algebras (\ldots{}).

\end{prop}
This is (\hyperlink{BungeDubuc}{Bunge-Dubuc, prop. 2.1}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{functions_on_smooth_manifolds}{}\subsubsection*{{Functions on smooth manifolds}}\label{functions_on_smooth_manifolds}

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[smooth manifold]], the smooth algebra $C^\infty(X)$ is the functor

\begin{displaymath}
C^\infty(X) := Hom_{Diff}(X,-)
\end{displaymath}
\end{defn}
\hypertarget{weil_algebras_functions_on_small_infinitesimal_spaces}{}\subsubsection*{{Weil algebras: functions on small infinitesimal spaces}}\label{weil_algebras_functions_on_small_infinitesimal_spaces}

A \emph{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 $n \in \mathbb{N}$.

\begin{prop}
\label{}\hypertarget{}{}
There is a unique $C^\infty$-ring structure on a Weil algebra $W$. It makes $W$ a finitely presented $C^\infty$-ring.

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
The [[smooth loci]] corresponding to Weil algebras are [[infinitesimal space]]s. Weil algebras play a crucial role in the definition of [[smooth topos]]es.

\end{remark}
\hypertarget{power_series}{}\subsubsection*{{Power series}}\label{power_series}

\ldots{}

\hypertarget{Germs}{}\subsubsection*{{Functions on germs of manifolds}}\label{Germs}

For $p \in \mathbb{R}^n$, the algebra of [[germ]]s 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

\begin{displaymath}
C^\infty_p(\mathbb{R}^n) \simeq C^\infty(\mathbb{R}^n)/I_p
  \,,
\end{displaymath}
yielding a finitely generated but not (for $n \gt 0$) finitely presented $C^\infty$-ring.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{limits_and_colimits}{}\subsubsection*{{Limits and colimits}}\label{limits_and_colimits}

\begin{prop}
\label{}\hypertarget{}{}
All [[limit]]s and all [[directed colimit]]s in $C^\infty Ring$ are computed objectwise in $[CartSp,Set]$ as limits in [[Set]].

\end{prop}
\begin{proof}
As discussed at [[limits and colimits by example]], all [[limit]]s and [[colimit]]s 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 [[limit]]s and [[directed colimit]]s do commute with products.

\end{proof}
See also [[Models for Smooth Infinitesimal Analysis|MSIA, p. 22]].

\hypertarget{UnderlyingAlgebra}{}\subsubsection*{{The underlying ordinary algebra}}\label{UnderlyingAlgebra}

There is a [[stuff, structure, property|forgetful functor]]

\begin{displaymath}
U : C^\infty Alg \to Alg
\end{displaymath}
from generalized smooth algebras to ordinary algebras which is given by evaluation on $\mathbb{R}$

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

\begin{displaymath}
\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})
  \,.
\end{displaymath}
\begin{displaymath}
+_A :
  A(\mathbb{R}) \times A(\mathbb{R})
  \stackrel{\simeq}{\to}
  A(\mathbb{R} + \mathbb{R})
  \stackrel{A(\cdot)}{\to}
  A(\mathbb{R})
  \,.
\end{displaymath}
Moreover there is canonically a morphism of rings

\begin{displaymath}
\mathbb{R} \to A(\mathbb{R})
\end{displaymath}
given by

\begin{displaymath}
(* = \mathbb{R}^0 \stackrel{c}{\to} \mathbb{R})
  \mapsto
  (* = A(\mathbb{R}^0) \stackrel{A(c)}{\to} A(\mathbb{R}))
  \,.
\end{displaymath}
This makes $A(\mathbb{R})$ an $\mathbb{R}$-algebra.

\begin{prop}
\label{}\hypertarget{}{}
The [[forgetful functor]] $U$ fits into an [[adjunction]]

\begin{displaymath}
(F \dashv U)
  :
  C^\infty Alg_{\mathbb{R}}
  \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  Alg_{\mathbb{R}}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This statement may be understood as a special case of the following more general statement:

\begin{itemize}%
\item If $S$, $T$ are [[finitary monad]]s and $f: S \to T$ is a monad morphism, then the relative forgetful functor\begin{displaymath}
f^*: Alg_T \to Alg_S,
\end{displaymath}
which [[pullback|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]].



\end{itemize}
(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

\begin{displaymath}
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
\end{displaymath}
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 $T$ \href{http://ncatlab.org/nlab/show/reflexive+coequalizer#applications_4}{preserves reflexive coequalizers}, so that there is a canonical isomorphism

\begin{displaymath}
T(T \otimes_S X) \cong (T T) \otimes_S X
\end{displaymath}
It follows that $T$-algebra ($T$-module) maps $\bar{g}: T \otimes_S X \to Y$, i.e., maps that render commutative the diagram

\begin{displaymath}
\itexarray{
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
}
\end{displaymath}
are in bijection with maps $g: T X \to Y$ that render commutative

\begin{displaymath}
\itexarray{
T T X & \stackrel{T g}{\to} & T Y \\
\mu \downarrow & & \downarrow \xi \\
T X & \underset{g}{\to} & Y
}
\end{displaymath}
(that is to say, $T$-algebra maps $g: T X \to Y$) which additionally coequalize the parallel pair in the diagram

\begin{displaymath}
T S X \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} T X \stackrel{g}{\to} Y
\end{displaymath}
Since $f: S \to T$ is a monad morphism, we have commutativity of parallel squares in

\begin{displaymath}
\itexarray{
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
}
\end{displaymath}
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 $f_! \dashv f^*$.

\end{proof}
\hypertarget{finitely_presented_rings}{}\subsubsection*{{Finitely presented $C^\infty$-rings}}\label{finitely_presented_rings}

\begin{prop}
\label{}\hypertarget{}{}
$C^\infty(\mathbb{R}^n)$ 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

\begin{displaymath}
Hom_{C^\infty Alg}(C^\infty(\mathbb{R}^n), A)
  \simeq
  Hom_{Alg}(\mathbb{R}[x_1,...,x_n], A(\mathbb{R}))
  \,.
\end{displaymath}
\end{prop}
\begin{itemize}%
\item Every finitely presented $C^\infty$-ring is fair/germ determined.

\end{itemize}
We have a chain of inclusions

\begin{itemize}%
\item finitely presented $C^\infty$-rings


\item $\subset$ ``good'' $C^\infty$-rings


\item $\subset$ fair $C^\infty$-rings


\item $\subset$ finitely generated $C^\infty$-rings



\end{itemize}
\hypertarget{points_of_smooth_loci}{}\subsubsection*{{Points of smooth loci}}\label{points_of_smooth_loci}

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

\begin{prop}
\label{}\hypertarget{}{}
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 $U(C) \to \mathbb{R}$.

\end{prop}
In particular every Weil algebra $W$ has a unique point $* \to \mathbb{L}(W)$: every Weil algebra is the algebra of functions on an \emph{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 $X$.

\hypertarget{SmoothFunctionAlgebrasOnSmoothManifolds}{}\subsubsection*{{Smooth function algebras on smooth manifolds}}\label{SmoothFunctionAlgebrasOnSmoothManifolds}

\begin{prop}
\label{}\hypertarget{}{}
Let $f : X \to Z$ and $g : Y \to Z$ be [[transversal maps]] of [[smooth manifold]]s. Then the functor $C^\infty(-)$ sends the [[pullback]]

\begin{displaymath}
\itexarray{
    X \times_Z Y &\to& X
    \\
    \downarrow && \downarrow^f
    \\
    Y &\stackrel{g}{\to}& Z
  }
\end{displaymath}
to the [[pushout]]

\begin{displaymath}
\itexarray{
    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)
  }
\end{displaymath}
\end{prop}
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]] $X \times Y$:

\begin{displaymath}
C^\infty(X \times Y) \simeq C^\infty(X) \otimes_\infty C^\infty(Y)
  \,.
\end{displaymath}
\begin{remark}
\label{}\hypertarget{}{}
The ordinary algebraic tensor product of $C^\infty(X)(\mathbb{R})$ and $C^\infty(Y)(\mathbb{R})$ regarded as ordinary algebras does \emph{not} in general satisfy this property. Rather one has an inclusion

\begin{displaymath}
C^\infty(X)(\mathbb{R}) \otimes C^\infty(Y)(\mathbb{R})
  \subset
  C^\infty(X \times Y)(\mathbb{R})
  \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
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]].

\end{remark}
Turning this inclusion into an equivalence is usually called a \emph{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.

\begin{theorem}
\label{}\hypertarget{}{}
The functor $C^\infty(-) = Hom_{Diff}(-,-) : Diff \to C^\infty Alg$

\begin{itemize}%
\item is a [[full and faithful functor]]


\item takes values in finitely presented smooth $C^\infty$-algebras


\item sends transversal [[pullback]]s to [[coproduct]]s (and hence to the smooth tensor product).



\end{itemize}
\end{theorem}
\hypertarget{deformation_theory_of_smooth_algebras}{}\subsection*{{Deformation theory of smooth algebras}}\label{deformation_theory_of_smooth_algebras}

\begin{quote}%
\textbf{under construction}


\end{quote}
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 \textbf{tangent category}

\begin{displaymath}
T C \to C
 \,.
\end{displaymath}
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 object]]s 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 [[module]]s over rings, there is a natural equivalence

\begin{displaymath}
Mod_A \stackrel{\simeq}{\to} Ab(C/A)
  \,.
\end{displaymath}
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:

\begin{itemize}%
\item 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

\begin{displaymath}
\itexarray{
    A &&\stackrel{\delta}{\to}&& A \oplus N
    \\
    & {}_ {=}\searrow && \swarrow
    \\
    && A
  }
  \,.
\end{displaymath}

\item The forgetful functor $T Ring \simeq Mod \to Ring$ has a [[left adjoint]]

\begin{displaymath}
\Omega_K^1 : Ring \to Mod
\end{displaymath}
that sends each ring to its module of [[Kähler differential]]s.

The fact that it is left adjoint is the universal property of the K\"a{}hler 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 $d a \mapsto \delta(a)$.



\end{itemize}
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 \emph{underlying} $\mathbb{R}$-algebra $A$. This square-0-extension happens to be \emph{uniquely} equipped with the $C^\infty$-Ring-structure given by

\begin{displaymath}
(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)
  \,.
\end{displaymath}
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 K\"a{}hler 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 \emph{for all} $f \in C^\infty(\mathbb{R}^n, \mathbb{R})$ that

\begin{displaymath}
\delta : f(a_1, \cdots, a_n) \mapsto 
  \sum_i \frac{\partial f}{\partial x_i}
  \delta a_i
  \,.
\end{displaymath}
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 K\"a{}hler differentials as defined with respect to such derivations are different from the purely ring-theoretic ones: they produce the \emph{right} notion of smooth 1-forms here, whereas the ring-theoretic one does not.

\hypertarget{variants}{}\subsection*{{Variants}}\label{variants}

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

\begin{itemize}%
\item [[smooth (∞,1)-algebra]].

\end{itemize}
The generalization to [[supergeometry]] is

\begin{itemize}%
\item [[smooth superalgebra]].

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[embedding of smooth manifolds into formal duals of R-algebras]]


\item [[derivations of smooth functions are vector fields]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

A standard textbook reference is chapter 1 of

\begin{itemize}%
\item [[Ieke Moerdijk]] and [[Gonzalo Reyes]], \emph{[[Models for Smooth Infinitesimal Analysis]]} Springer (1991)

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

\begin{itemize}%
\item [[Bill Lawvere]], \emph{Categorical dynamics}

in [[Anders Kock]] (eds.) \emph{Topos theoretic methods in geometry}, volume 30 of \emph{Various Publ. Ser.}, pages 1-28, Aarhus Univ. (1979)



\end{itemize}
but examples of the concept are older. A discussion from the point of view of [[functional analysis]] is in

\begin{itemize}%
\item G. Kainz, A. Kriegl, [[Peter Michor]], \emph{$C^\infty$-algebras from the functional analytic view point} Journal of pure and applied algebra 46 (1987) (\href{http://www.mat.univie.ac.at/~michor/c-oo-alg.pdf}{pdf})

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

\begin{itemize}%
\item [[Peter Michor]], [[Jiri Vanzura]], \emph{Characterizing algebras of $C^\infty$-functions on manifolds} (\href{http://www.emis.de/journals/CMUC/pdf/cmuc9603/michor.pdf}{pdf})

\end{itemize}
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

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Gonzalo Reyes]], \emph{[[RingsOfSmoothFunctionsI.pdf:file]]} , Journal of Algebra 99 (1986)

\end{itemize}
The notion of the spectrum of a $C^\infty$-ring and that of $C^\infty$-[[scheme]]s is discussed in

\begin{itemize}%
\item [[Eduardo Dubuc]], \emph{$C^\infty$-schemes} Amer. J. Math. 103 (1981) (\href{http://www.math.ist.utl.pt/~jroquet/Dubuc.pdf}{pdf} \href{http://www.jstor.org/stable/select/2374046}{JSTOR}).

\end{itemize}
and more generally in

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Gonzalo Reyes]], \emph{Rings of smooth functions and their localization II} , in \emph{Mathematical logic and theoretical computer science} page 275 (\href{http://books.google.nl/books?id=w02XaOa_0HAC&pg=PA262&lpg=PA262&dq=smooth+zariski+topos&source=bl&ots=Hj0AN3Lz3y&sig=TqRwTXz_u5R30AN2udLNOMQ2Ly0&hl=nl&ei=mCWZS-eLJcaD-QalyZlS&sa=X&oi=book_result&ct=result&resnum=2&ved=0CA4Q6AEwAQ#v=onepage&q=smooth%20zariski%20topos&f=false}{Google books})


\item [[Marta Bunge]], [[Eduardo Dubuc]], \emph{Archimedian local $C^\infty$-rings and models of synthetic differential geometry} Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 27 no. 3 (1986), p. 3-22 (\href{http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0}{numdam}).



\end{itemize}
More recent developments along these lines are in

\begin{itemize}%
\item [[Dominic Joyce]], \emph{Algebraic geometry over $C^\infty$-rings} (\href{http://arxiv.org/abs/1001.0023}{arXiv:1001.0023})

\end{itemize}
The [[higher geometry]] generalization to a theory of [[derived smooth manifold]]s -- [[space]]s with [[structure sheaf]] taking values in [[simplicial C∞-ring]]s -- was initiated in

\begin{itemize}%
\item [[David Spivak]], \emph{Quasi-Smooth derived manifolds} (\href{http://www.uoregon.edu/~dspivak/thesis2.pdf}{pdf of original version}, \href{http://arxiv.org/abs/0810.5174}{arXiv:0810.5174})

\end{itemize}
based on the general machinery of [[structured (∞,1)-topos]]es in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Structured Spaces]]}

\end{itemize}
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]].

[[!redirects smooth algebras]] [[!redirects generalized smooth algebra]] [[!redirects generalized smooth algebras]] [[!redirects C-infinity-ring]] [[!redirects C-infinity ring]] [[!redirects C-infinity-rings]] [[!redirects C-infinity rings]] [[!redirects C-∞-ring]] [[!redirects C-∞ ring]] [[!redirects C-∞-rings]] [[!redirects C-∞ rings]]



\end{document}