Relative differential categories, differential clones and Fermat theories

A differential category [2] is a symmetric monoidal category $(𝒞,\otimes ,I)$ enriched over commutative monoids, together with a monad $S$ on $𝒞$ and a differentiation operator $\mathrm{d}:SA\to SA\otimes A$ satisfying standard axioms. Relative monads are a generalization of monads where the underlying functor is not necessarily an endofunctor [1]. Our main contribution in this talk is the introduction of the notion of a relative differential category where the monad is replaced by a relative monad. Every differential category is a relative differential category but the converse is false. An example is ${\mathrm{𝖵𝖾𝖼}}_k$ with its usual tensor product and with $S(n)=k[x_1,\mathrm{\dots },x_n]$. Another one is ${\mathrm{𝖵𝖾𝖼}}_{ℝ}$ with its usual tensor product and with $S(n)=C^{\mathrm{\infty }}({ℝ}^n,ℝ)$. This framework allows one to work directly with polynomials and smooth functions, instead of using coordinate-free constructions such as the symmetric algebra.

These two examples can be generalized using clones [3]. A rig clone $𝒪$ is given by a commutative rig $𝒪(n)$ for every $n\ge 0$ together with projections and composition operations satisfying familiar identities. We introduce the notion of a differential clone as a rig clone together with partial derivative operations ${\partial }_i:𝒪(n)\to 𝒪(n)$. Differential clones are equivalent to the differential theories of [2]. For every differential clone, the symmetric monoidal category $({\mathrm{Mod}}_{𝒪(0)},\otimes ,𝒪(0))$ is a relative differential category. In particular, we recover our two previous examples of relative differential categories by choosing $𝒪(n)=k[x_1,\mathrm{\dots },x_n]$ or $𝒪(n)={𝒞}^{\mathrm{\infty }}({ℝ}^n,ℝ)$.

A Fermat theory [4] is a ring clone where differentiation is axiomatized algebraically through difference quotients. We show that every Fermat theory is a differential clone. The differential clones $𝒪(n)=k[x_1,\mathrm{\dots },x_n]$ and $𝒪(n)=C^{\mathrm{\infty }}({ℝ}^n,ℝ)$ are Fermat theories. However, not every differential clone is a Fermat theory and we provide a counterexample inspired from differential Galois theory [5]. Finally, given any topological field $k$ whose topology is nontrivial and nondiscrete, we define smooth functions from $k^n$ to $k$ following [6]. We show that $𝒪(n)=C^{\mathrm{\infty }}(k^n,k)$ gives a Fermat theory. This provides the first connection between smooth functions on topological fields and differential categories.