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.

Comment: For the proceedings of MFPS2023

• 18C20 - Eilenberg-Moore and Kleisli constructions for monads
• 18F40 - Synthetic differential geometry, tangent categories, differential categories