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Cartesian differential categories come equipped with a differentialcombinator which axiomatizes the fundamental properties of the total derivativefrom differential calculus. The objective of this paper is to understand whenthe Kleisli category of a monad is a Cartesian differential category. Weintroduce Cartesian differential monads, which are monads whose Kleislicategory is a Cartesian differential category by way of lifting thedifferential combinator from the base category. Examples of Cartesiandifferential monads include tangent bundle monads and reader monads. We give aprecise characterization of Cartesian differential categories which are Kleislicategories of Cartesian differential monads using abstract Kleisli categories.We also show that the Eilenberg-Moore category of a Cartesian differentialmonad is a tangent category.

Cartesian Differential Kleisli Categories