Cartesian differential categories come equipped with a differential combinator which axiomatizes the fundamental properties of the total derivative from differential calculus. The objective of this paper is to understand when the Kleisli category of a monad is a Cartesian differential category. We introduce Cartesian differential monads, which are monads whose Kleisli category is a Cartesian differential category by way of lifting the differential combinator from the base category. Examples of Cartesian differential monads include tangent bundle monads and reader monads. We give a precise characterization of Cartesian differential categories which are Kleisli categories of Cartesian differential monads using abstract Kleisli categories. We also show that the Eilenberg-Moore category of a Cartesian differential monad is a tangent category.

翻译：笛卡尔微分范畴配备了一个微分组合子，该组合子公理化了微分学中全微分的基本性质。本文旨在探究何时一个单子的Kleisli范畴能成为笛卡尔微分范畴。我们引入了笛卡尔微分单子，这类单子通过从基范畴提升微分组合子，使其Kleisli范畴成为笛卡尔微分范畴。笛卡尔微分单子的实例包括切丛单子和阅读器单子。我们利用抽象Kleisli范畴，精确刻画了那些作为笛卡尔微分单子的Kleisli范畴的笛卡尔微分范畴。此外，我们还证明了笛卡尔微分单子的Eilenberg-Moore范畴是一个切触范畴。