Derivations as Algebras
Jean-Simon Pacaud Lemay, Chiara Sava
TL;DR
This work characterizes $\mathsf{S}$-derivations as algebras of a lifted monad on the arrow category of a differential category, giving a unifying categorical perspective on differentiation. It shows that the differential modality $\mathsf{S}$ lifts to a monad $\overline{\mathsf{S}}$ with algebras precisely the $\mathsf{S}$-derivations, and, when biproducts exist, that the arrow category inherits a differential modality, making it a differential category. Consequently, $\mathsf{S}$-derivations form a tangent category, and the derivations on free $\mathsf{S}$-algebras form a cartesian differential category, linking tangent/cotangent structures to classical notions of derivations. The construction yields a systematic source of differential categories and unifies ordinary derivations (via $\mathsf{Sym}$) and $\mathcal{C}^{\infty}$-derivations (via $\mathsf{S}^{\infty}$) within a single framework.
Abstract
Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular, derivations. In this paper, we show that the differential modality of a differential category lifts to a monad on the arrow category and, moreover, that the algebras of this monad are precisely derivations. Furthermore, in the presence of finite biproducts, the differential modality in fact lifts to a differential modality on the arrow category. In other words, the arrow category of a differential category is again a differential category. As a consequence, derivations also form a tangent category, and derivations on free algebras form a cartesian differential category.