Cartesian Differential Kleisli 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.

For the proceedings of MFPS2023

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• A Simple Differentiable Programming Language
• Cartesian Differential Storage Categories
• Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits
• Tangent bundles in differential lambda-categories
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Cited by in corpus (1)

• Normed modules, integral sequences, and integrals with variable upper limits