Derivations as Algebras

TL;DR

This paper explores the algebraic structures underlying differentiation in category theory, showing how derivations can be modeled as monads and how the arrow category inherits differential structure, enriching the theoretical framework of differential categories.

Contribution

It demonstrates that the differential modality lifts to a monad on the arrow category and characterizes derivations as its algebras, extending differential structure to arrow categories.

Findings

The differential modality lifts to a monad on the arrow category.

Derivations are precisely the algebras of this monad.

The arrow category of a differential category is itself a differential category.

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.