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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Fermat theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{rings}{$C^\infty$-rings}\dotfill \pageref*{rings} \linebreak \noindent\hyperlink{PartialDerivative}{Partial derivatives}\dotfill \pageref*{PartialDerivative} \linebreak \noindent\hyperlink{ModulesDerivations}{Modules and derivations}\dotfill \pageref*{ModulesDerivations} \linebreak \noindent\hyperlink{KählerDiffs}{K\"a{}hler differentials}\dotfill \pageref*{KählerDiffs} \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 `Fermat theory' is a [[Lawvere theory]] that extends the usual theory of [[commutative rings]] by permitting [[differentiation]]. The term \emph{Fermat theory} seems to have been introduced in (\hyperlink{Kock09}{Kock 09}) based on (\hyperlink{DubucKock84}{Dubuc-Kock 84}). But as the name suggests, it has its roots in an old observation of [[Fermat]]. Namely: if $f \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}$ is a [[polynomial]] function, then \begin{displaymath} f (x+y) = f(x) + y \tilde{f}(x,y) \end{displaymath} for a unique polynomial function $\tilde{f} \colon \mathbb{R}^2 \to \mathbb{R}$. Clearly \begin{displaymath} \tilde{f}(x,y) = \frac{f(x+y) - f(x)}{y} \end{displaymath} for $y \ne 0$, but the interesting thing is that \begin{displaymath} \tilde{f}(x,0) = f'(x) \end{displaymath} So, the function $\tilde{f}$ knows about the [[derivative]] of $f$! (This can be done for polynomials over any [[commutative ring]], although Fermat wasn't working in that generality.) Later [[Jacques Hadamard]] generalized this observation from a polynomial function $f$ to a [[continuously differentiable function]] $f$, where now $\tilde{f}$ is unique if required to be [[continuous map|continuous]]. This is the statement of the \emph{[[Hadamard lemma]]}. (For a merely [[differentiable function]] $f$, require $\tilde{f}$ to be continuous in $y$ alone.) The function $\tilde{f}$ is thus called a [[Hadamard quotient]]. If $\tilde{f}$ is to be the same class of function as $f$, then we need [[smooth functions]], and that will be our motivating context from now on. If we take $\tilde{f}(x,0) = f'(x)$ as a \emph{definition} of the derivative, we can derive many of the basic rules for derivatives from the formula \begin{displaymath} f(x+y) = f(x) + y \tilde{f}(x,y) \end{displaymath} using just algebra --- no [[limit of a sequence|limits]]! As an exercise, the reader should check these rules: \begin{displaymath} (f + g)' = f' + g' \end{displaymath} \begin{displaymath} (f g)' = f' g + f g' \end{displaymath} \begin{displaymath} (f \circ g)' = (f' \circ g) g' \end{displaymath} These ideas continue to work if $f$ is a smooth function from $\mathbb{R}^n$ to $\mathbb{R}$; focussing on one variable and treating the others as [[parameter]]s, we have [[partial derivative|partial differentiation]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} The above observations suggest defining the following kind of [[Lawvere theory]]. A \textbf{Fermat theory} is an extension of the [[algebraic theory]] of [[commutative rings]], such that for any $(n+1)$-ary operation $f$ there is a unique $(n+2)$-ary operation $\tilde{f}$ such that \begin{displaymath} f(x + y, \vec{z}) = f(x, \vec{z}) + y \tilde{f}(x,y,\vec{z}) \end{displaymath} where $\vec{z}$ is a list of $n$ variables which act as parameters. (Here we are abusing language by writing the operations $f$ and $\tilde{f}$ as if they were functions, to avoid unintuitive commutative diagrams.) \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{rings}{}\subsubsection*{{$C^\infty$-rings}}\label{rings} There is a [[Lawvere theory]] called \textbf{the theory of $C^\infty$-rings}, whose $n$-ary operations are the [[smooth maps]] $f: \mathbb{R}^n \to \mathbb{R}$, \begin{displaymath} T(n) \coloneqq C^\infty(\mathbb{R}^n, \mathbb{R}) \,, \end{displaymath} with composition of operations defined in the obvious way. An algebra of this Lawvere theory is called a \textbf{[[generalized smooth algebra|C{\tt \symbol{94}}∞-ring]]}. The theory of $C^\infty$-rings is a Fermat theory. For any [[smooth manifold]] $M$, the algebra of smooth real-valued functions $C^\infty(M)$ is a $C^\infty$ ring. More generally, if $M$ is any [[diffeological space]], [[Chen space]] or [[Frolicher space]], we can define $C^\infty(M)$, and this will be a $C^\infty$-ring. In formulas, and even more generally: for any generalized [[space]] given by a [[presheaf]] $X$ on [[CartSp]], the corresponding $C^\infty$-ring is the copresheaf \begin{displaymath} C^\infty(X) : \mathbb{R}^n \mapsto [CartSp^{op},Set](X,Y(\mathbb{R}^n)) \end{displaymath} that sends each object $\mathbb{R}^n \in CartSp$ to the [[hom-set]] in the [[functor category]] $[CartSp^{op},Set]$ from $X$ to the presheaf [[representable functor|represented]] by $\mathbb{R}^n$ under the [[Yoneda embedding]]. By the canonical right [[exact functor|exactness]] of the [[hom-functor]], this preserves [[limit]]s and hence in particular [[product]]s in [[CartSp]]. \hypertarget{PartialDerivative}{}\subsection*{{Partial derivatives}}\label{PartialDerivative} Let $T$ be a Fermat theory and let $f$ be an $(n+1)$-ary operation, then we may define an operation $\partial_1 f$ by \begin{displaymath} \partial_1(x, \vec{z}) = \tilde{f}(x,0,\vec{z}) \end{displaymath} This acts like the [[partial derivative]] of $f$ with respect to its first argument. With a bit of more work we get a list of $n$-ary operations $\partial_i f$. So, if $T(n)$ denotes the set of $n$-ary operations in the algebraic theory $T$, we get maps \begin{displaymath} \partial_i : T(n) \to T(n) \end{displaymath} for $1 \le i \le n$. Now $T(n)$ is automatically an algebra of $T$ (this is true for any Lawvere theory: it is the free algebra on $n$ generators), whence $T(n)$ is a commutative ring. One can check that each map \begin{displaymath} \partial_i : T(n) \to T(n) \end{displaymath} is a [[derivation]] of this ring --- this is really just the [[chain rule]]. \hypertarget{ModulesDerivations}{}\subsection*{{Modules and derivations}}\label{ModulesDerivations} Let $T$ be a Fermat theory, and let $A$ be a $T$-algebra. A [[module]] $N$ over $A$ is simply a module for the underlying [[ring]] of $A$. But the notion of [[derivation]] $\delta : A \to N$ of such modules depends on the $T$-structure: To motivate the concept, let first $A$ be an ordinary [[ring]] and $N$ an ordinary [[module]]. Then the three axioms of an ordinary derivation $\delta : A \to N$ \begin{enumerate}% \item $\delta(a + b ) = \delta(a) + \delta(b)$ \item $\delta(\lambda a) = \lambda \delta(a)$ \item $\delta(a \cdot b) = a \delta(a) + b \delta(b)$ \end{enumerate} are equivalent to the condition that for any polynomial $p \in \mathbb{R}[x_1, \cdots, x_n]$ and ring elements $a_i$ we have \begin{displaymath} \delta\left( p(a_1, \cdots, a_n) \right) = \sum_{i= 1 }^{n} \frac{\partial p}{\partial x_i} \left( a_1, \cdots, a_n \right) \delta(a_i) \,. \end{displaymath} (It is immediate that the first three axioms imply this one. To see the converse, apply the latter to the polynomials $p_1(x,y) = x + y$, $p_2(x) = \lambda a$ and $p_3(x,y) = x y$.) The definition of derivations for general $T$-algebras now follows the last expression, using the notion of \hyperlink{PartialDerivative}{partial derivatives} from above: \begin{defn} \label{}\hypertarget{}{} For $T$ a Fermat theory, $A$ a $T$-algebra and $N$ an $A$-module, a \textbf{derivation} $\delta : A \to N$ is a map such that for each $f \in T(n)$ and elements $(a_i \in A)$ we have \begin{displaymath} \delta\left( f(a_1, \cdots, a_n) \right) = \sum_{i= 1 }^{n} \frac{\partial f}{\partial x_i} \left( a_1, \cdots, a_n \right) \delta(a_i) \,. \end{displaymath} \end{defn} Notice that in particular such a derivation of a $T$-algebra $A$ is a derivation of the underlying ring. (This follows again by using the above three polynomials and remembering that by definition $T(n)$ at least contains all polynomials.) \hypertarget{KählerDiffs}{}\subsubsection*{{K\"a{}hler differentials}}\label{KählerDiffs} The sets $T(n)$ for $n \in \mathbb{N}$ canonically have the structures of [[module]]s over $T(n)$. \begin{theorem} \label{}\hypertarget{}{} The map \begin{displaymath} d := \langle \partial_1, \dots, \partial_n \rangle : T(n) \to \prod_{i = 1}^{n} T(n) \end{displaymath} obtained from the partial derivatives is the universal $T$-derivation of $T(n,1)$. \end{theorem} This means that if $N$ is a module of $T(n)$ and $\delta : T(n) \to N$ is a derivation in the above sense, then $\delta$ factors uniquely through the map $\langle \partial_1, \dots, \partial_n \rangle$. The point of this theorem is that it gives us a version of [[Kähler differentials]] for $T(n)$. We may think of an element $(f_i) \in \prod_{i = 1}^{n} T(n)$ as the [[Kähler differential]] 1-form $f_1 d x^1 + f_2 d x^2 + \cdots + f_n d x^n$ and of the derivation $d := \langle \partial_1, \dots, \partial_n \rangle$ as the operation \begin{displaymath} d : f \mapsto \sum_i \frac{\partial f_i}{ \partial x^i} d x^i \,. \end{displaymath} Indeed, when the Fermat theory is that of [[generalized smooth algebra|C-infinity rings]], then this notion of K\"a{}hler differentials does coincide with the ordinary notion of smooth [[differential form|1-form]]. The same is not true, in general, if one instead forms ring-theoretic K\"a{}hler differentials. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[model structure on differential graded-commutative superalgebras]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The original reference is \begin{itemize}% \item [[Eduardo Dubuc]], [[Anders Kock]], \emph{On 1-form classifiers}, Comm. Alg. \textbf{12} (1984), no. 11-12, 1471--1531 (\href{http://home.imf.au.dk/kock/DK84.pdf}{pdf}) \end{itemize} Parts of the above material are a summary of the following talk: \begin{itemize}% \item [[Anders Kock]], \emph{K\"a{}hler differentials for Fermat theories}, talk at \emph{Fields Workshop on Smooth Structures in Logic, Category Theory and Physics}, May 1, 2009, University of Ottawa. (\href{http://aix1.uottawa.ca/~scpsg/Fields09/smooth.abstracts.html}{abstract}) \end{itemize} For more, see: \begin{itemize}% \item [[Alex Hoffnung]], \emph{Smooth Structure in Ottawa II} (\href{http://golem.ph.utexas.edu/category/2009/05/smooth_structures_in_ottawa_ii.html#c030745}{blog entry}) \end{itemize} and the comments on this blog entry. Refinement to [[supergeometry]] and extension to a notion of \emph{super Fermat theory} is discussed in \begin{itemize}% \item [[David Carchedi]], [[Dmitry Roytenberg]], \emph{On theories of superalgebras of differentiable functions}, (\href{http://arxiv.org/abs/1211.6134}{arxiv:1211.6134}) \end{itemize} Something similar appears in def. 1.1, 1.2 of \begin{itemize}% \item [[David Yetter]], \emph{Models for synthetic supergeometry}, Cahiers, 29, 2 (1988) (\href{http://www.numdam.org/item?id=CTGDC_1988__29_2_87_0}{NUMDAM}) \end{itemize} For more on this see at \emph{[[synthetic differential supergeometry]]}. [[!redirects Fermat theory]] [[!redirects Fermat theories]] [[!redirects fermat theory]] \end{document}