# nLab Fermat theory

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

A ‘Fermat theory’ is a Lawvere theory that extends the usual theory of commutative rings by permitting differentiation.

The idea of a Fermat theory seems to have been invented by Anders Kock. But as the name suggests, it has its roots in an old observation of Fermat. Namely: if $f:ℝ\to ℝ$ is a polynomial function, then

$f\left(x+y\right)=f\left(x\right)+y\stackrel{˜}{f}\left(x,y\right)$f (x+y) = f(x) + y \tilde{f}(x,y)

for a unique polynomial function $\stackrel{˜}{f}:{ℝ}^{2}\to ℝ$. Clearly

$\stackrel{˜}{f}\left(x,y\right)=\frac{f\left(x+y\right)-f\left(x\right)}{y}$\tilde{f}(x,y) = \frac{f(x+y) - f(x)}{y}

for $y\ne 0$, but the interesting thing is that

$\stackrel{˜}{f}\left(x,0\right)=f\prime \left(x\right)$\tilde{f}(x,0) = f'(x)

So, the function $\stackrel{˜}{f}$ knows about the derivative of $f$! (This can be done for polynomials over any commutative ring, although Fermat wasn't working in that generality.)

Jacques Hadamard? generalized this observation from a polynomial function $f$ to a continuously differentiable function $f$, where now $\stackrel{˜}{f}$ is unique if required to be continuous. (For a merely differentiable function $f$, require $\stackrel{˜}{f}$ to be continuous in $y$ alone.) The function $\stackrel{˜}{f}$ is thus called a Hadamard quotient. If $\stackrel{˜}{f}$ 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 $\stackrel{˜}{f}\left(x,0\right)=f\prime \left(x\right)$ as a definition of the derivative, we can derive many of the basic rules for derivatives from the formula

$f\left(x+y\right)=f\left(x\right)+y\stackrel{˜}{f}\left(x,y\right)$f(x+y) = f(x) + y \tilde{f}(x,y)

using just algebra — no limits! As an exercise, the reader should check these rules:

$\left(f+g\right)\prime =f\prime +g\prime$(f + g)' = f' + g'
$\left(fg\right)\prime =f\prime g+fg\prime$(f g)' = f' g + f g'
$\left(f\circ g\right)\prime =\left(f\prime \circ g\right)g\prime$(f \circ g)' = (f' \circ g) g'

All these ideas also work if $f$ is parametrized, that is, a smooth function from $ℝ×{ℝ}^{n}$ to $ℝ$, where the $n$ extra variables serve as parameter?s.

## Definition

The above observations suggest defining the following kind of Lawvere theory. A Fermat theory is an extension of the algebraic theory of commutative rings, such that for any $\left(n+1\right)$-ary operation $f$ there is a unique $\left(n+2\right)$-ary operation $\stackrel{˜}{f}$ such that

$f\left(x+y,\stackrel{⇀}{z}\right)=f\left(x,\stackrel{⇀}{z}\right)+y\stackrel{˜}{f}\left(x,y,\stackrel{⇀}{z}\right)$f(x + y, \vec{z}) = f(x, \vec{z}) + y \tilde{f}(x,y,\vec{z})

where $\stackrel{⇀}{z}$ is a list of $n$ variables which act as parameters. (Here we are abusing language by writing the operations $f$ and $\stackrel{˜}{f}$ as if they were functions, to avoid unintuitive commutative diagrams.)

## Examples

### ${C}^{\infty }$-rings

There is a Lawvere theory called the theory of ${C}^{\infty }$-rings, whose $n$-ary operations are the smooth maps $f:{ℝ}^{n}\to ℝ$,

$T\left(n\right)≔{C}^{\infty }\left({ℝ}^{n},ℝ\right)\phantom{\rule{thinmathspace}{0ex}},$T(n) \coloneqq C^\infty(\mathbb{R}^n, \mathbb{R}) \,,

with composition of operations defined in the obvious way. An algebra of this Lawvere theory is called a C^∞-ring.

The theory of ${C}^{\infty }$-rings is a Fermat theory. For any smooth manifold $M$, the algebra of smooth real-valued functions ${C}^{\infty }\left(M\right)$ is a ${C}^{\infty }$ ring. More generally, if $M$ is any diffeological space, Chen space or Frolicher space, we can define ${C}^{\infty }\left(M\right)$, 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

${C}^{\infty }\left(X\right):{ℝ}^{n}↦\left[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Set}\right]\left(X,Y\left({ℝ}^{n}\right)\right)$C^\infty(X) : \mathbb{R}^n \mapsto [CartSp^{op},Set](X,Y(\mathbb{R}^n))

that sends each object ${ℝ}^{n}\in \mathrm{CartSp}$ to the hom-set in the functor category $\left[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Set}\right]$ from $X$ to the presheaf represented by ${ℝ}^{n}$ under the Yoneda embedding. By the canonical right exactness of the hom-functor, this preserves limits and hence in particular products in CartSp.

## Partial derivatives

Let $T$ be a Fermat theory and let $f$ be an $\left(n+1\right)$-ary operation, then we may define an operation ${\partial }_{1}f$

by

${\partial }_{1}\left(x,\stackrel{⇀}{z}\right)=\stackrel{˜}{f}\left(x,0,\stackrel{⇀}{z}\right)$\partial_1(x, \vec{z}) = \tilde{f}(x,0,\vec{z})

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 $T\left(n\right)$ denotes the set of $n$-ary operations in the algebraic theory $T$, we get maps

${\partial }_{i}:T\left(n\right)\to T\left(n\right)$\partial_i : T(n) \to T(n)

for $1\le i\le n$.

Now $T\left(n\right)$ is automatically an algebra of $T$ (this is true for any Lawvere theory: it is the free algebra on $n$ generators), whence $T\left(n\right)$ is a commutative ring. One can check that each map

${\partial }_{i}:T\left(n\right)\to T\left(n\right)$\partial_i : T(n) \to T(n)

is a derivation of this ring — this is really just the chain rule.

## Modules and derivations

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 $\delta :A\to N$

1. $\delta \left(a+b\right)=\delta \left(a\right)+\delta \left(b\right)$

2. $\delta \left(\lambda a\right)=\lambda \delta \left(a\right)$

3. $\delta \left(a\cdot b\right)=a\delta \left(a\right)+b\delta \left(b\right)$

are equivalent to the condition that for any polynomial $p\in ℝ\left[{x}_{1},\cdots ,{x}_{n}\right]$ and ring elements ${a}_{i}$ we have

$\delta \left(p\left({a}_{1},\cdots ,{a}_{n}\right)\right)=\sum _{i=1}^{n}\frac{\partial p}{\partial {x}_{i}}\left({a}_{1},\cdots ,{a}_{n}\right)\delta \left({a}_{i}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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) \,.

(It is immediate that the first three axioms imply this one. To see the converse, apply the latter to the polynomials ${p}_{1}\left(x,y\right)=x+y$, ${p}_{2}\left(x\right)=\lambda a$ and ${p}_{3}\left(x,y\right)=xy$.)

The definition of derivations for general $T$-algebras now follows the last expression, using the notion of partial derivatives from above:

###### Definition

For $T$ a Fermat theory, $A$ a $T$-algebra and $N$ an $A$-module, a derivation $\delta :A\to N$ is a map such that for each $f\in T\left(n\right)$ and elements $\left({a}_{i}\in A\right)$ we have

$\delta \left(f\left({a}_{1},\cdots ,{a}_{n}\right)\right)=\sum _{i=1}^{n}\frac{\partial f}{\partial {x}_{i}}\left({a}_{1},\cdots ,{a}_{n}\right)\delta \left({a}_{i}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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) \,.

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 $T\left(n\right)$ at least contains all polynomials.)

### Kähler differentials

The sets $T\left(n\right)$ for $n\in ℕ$ canonically have the structures of modules over $T\left(n\right)$.

###### Theorem

The map

$d:=⟨{\partial }_{1},\dots ,{\partial }_{n}⟩:T\left(n\right)\to \prod _{i=1}^{n}T\left(n\right)$d := \langle \partial_1, \dots, \partial_n \rangle : T(n) \to \prod_{i = 1}^{n} T(n)

obtained from the partial derivatives is the universal $T$-derivation of $T\left(n,1\right)$.

This means that if $N$ is a module of $T\left(n\right)$ and $\delta :T\left(n\right)\to N$ is a derivation in the above sense, then $\delta$ factors uniquely through the map $⟨{\partial }_{1},\dots ,{\partial }_{n}⟩$.

The point of this theorem is that it gives us a version of Kähler differentials for $T\left(n\right)$.

We may think of an element $\left({f}_{i}\right)\in {\prod }_{i=1}^{n}T\left(n\right)$ 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 $d:=⟨{\partial }_{1},\dots ,{\partial }_{n}⟩$ as the operation

$d:f↦\sum _{i}\frac{\partial {f}_{i}}{\partial {x}^{i}}d{x}^{i}\phantom{\rule{thinmathspace}{0ex}}.$d : f \mapsto \sum_i \frac{\partial f_i}{ \partial x^i} d x^i \,.

Indeed, when the Fermat theory is that of C-infinity rings, then this notion of Kähler differentials does coincide with the ordinary notion of smooth 1-form. The same is not true, in general, if one instead forms ring-theoretic Kähler differentials.

## References

The original reference is

Parts of the above material are a summary of the following talk:

• Anders Kock, Kähler differentials for Fermat theories, talk at Fields Workshop on Smooth Structures in Logic, Category Theory and Physics, May 1, 2009, University of Ottawa. (abstract)

For more, see:

and the comments on this blog entry.

refinement to supergeometry and extension to a notion of super Fermat theory is disucssed in

Something similar appears in def. 1.1, 1.2 of

For more on this see at synthetic differential supergeometry.

Revised on December 4, 2012 12:18:20 by Toby Bartels (64.134.165.128)