The term Fermat theory seems to have been introduced in (Kock 09) based on (Dubuc-Kock 84). But as the name suggests, it has its roots in an old observation of Fermat. Namely: if is a polynomial function, then
for a unique polynomial function . Clearly
for , but the interesting thing is that
Later Jacques Hadamard generalized this observation from a polynomial function to a continuously differentiable function , where now is unique if required to be continuous. This is the statement of the Hadamard lemma. (For a merely differentiable function , require to be continuous in alone.) The function is thus called a Hadamard quotient. If is to be the same class of function as , then we need smooth functions, and that will be our motivating context from now on.
If we take as a definition of the derivative, we can derive many of the basic rules for derivatives from the formula
using just algebra — no limits! As an exercise, the reader should check these rules:
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 -ary operation there is a unique -ary operation such that
where is a list of variables which act as parameters. (Here we are abusing language by writing the operations and as if they were functions, to avoid unintuitive commutative diagrams.)
with composition of operations defined in the obvious way. An algebra of this Lawvere theory is called a C^∞-ring.
The theory of -rings is a Fermat theory. For any smooth manifold , the algebra of smooth real-valued functions is a ring. More generally, if is any diffeological space, Chen space or Frolicher space, we can define , and this will be a -ring.
that sends each object to the hom-set in the functor category from to the presheaf represented by under the Yoneda embedding. By the canonical right exactness of the hom-functor, this preserves limits and hence in particular products in CartSp.
Let be a Fermat theory and let be an -ary operation, then we may define an operation
This acts like the partial derivative of with respect to its first argument. With a bit of more work we get a list of -ary operations . So, if denotes the set of -ary operations in the algebraic theory , we get maps
Now is automatically an algebra of (this is true for any Lawvere theory: it is the free algebra on generators), whence is a commutative ring. One can check that each map
But the notion of derivation of such modules depends on the -structure:
are equivalent to the condition that for any polynomial and ring elements we have
(It is immediate that the first three axioms imply this one. To see the converse, apply the latter to the polynomials , and .)
The definition of derivations for general -algebras now follows the last expression, using the notion of partial derivatives from above:
For a Fermat theory, a -algebra and an -module, a derivation is a map such that for each and elements we have
Notice that in particular such a derivation of a -algebra is a derivation of the underlying ring. (This follows again by using the above three polynomials and remembering that by definition at least contains all polynomials.)
The sets for canonically have the structures of modules over .
obtained from the partial derivatives is the universal -derivation of .
This means that if is a module of and is a derivation in the above sense, then factors uniquely through the map .
The point of this theorem is that it gives us a version of Kähler differentials for .
We may think of an element as the Kähler differential 1-form and of the derivation as the operation
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.
The original reference is
Parts of the above material are a summary of the following talk:
For more, see:
and the comments on this blog entry.
Refinement to supergeometry and extension to a notion of super Fermat theory is discussed in
Something similar appears in def. 1.1, 1.2 of
For more on this see at synthetic differential supergeometry.