Cartesian Differential Kleisli Categories

TL;DR

This paper investigates when the Kleisli category of a monad forms a Cartesian differential category, introducing Cartesian differential monads and characterizing their properties with examples like tangent bundle and reader monads.

Contribution

It introduces Cartesian differential monads and characterizes when their Kleisli categories are Cartesian differential categories, expanding the understanding of differential structures in category theory.

Findings

Kleisli categories of Cartesian differential monads are Cartesian differential categories

Examples include tangent bundle monads and reader monads

Eilenberg-Moore categories of these monads are tangent categories

Abstract

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.