Derivations as algebras

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 byproducts, 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.