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Cartesian Differential Kleisli Categories

Jean-Simon Pacaud Lemay

TL;DR

This work identifies Cartesian differential monads as the right framework for lifting differential structure to Kleisli categories, proving that the Kleisli category KL(S) of a Cartesian differential monad S on a Cartesian k differential category X is itself a Cartesian differential category with the same differential operator as the base. It provides a precise characterization of when a Cartesian differential category arises as a Kleisli category of a Cartesian differential monad via Cartesian differential abstract Kleisli categories, and it shows that the Eilenberg–Moore category EM(S) carries a tangent category structure, with differential objects corresponding to D linear algebras. The results unify several standard constructions such as tangent bundle monads and reader monads, and they illuminate how effects interact with differential structure in a principled categorical way. The work also clarifies the role of Kleisli differential combinators and abstract Kleisli formalisms, offering a robust foundation for differential calculus in effectful settings with potential applications to tangent programming language design.

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.

Cartesian Differential Kleisli Categories

TL;DR

This work identifies Cartesian differential monads as the right framework for lifting differential structure to Kleisli categories, proving that the Kleisli category KL(S) of a Cartesian differential monad S on a Cartesian k differential category X is itself a Cartesian differential category with the same differential operator as the base. It provides a precise characterization of when a Cartesian differential category arises as a Kleisli category of a Cartesian differential monad via Cartesian differential abstract Kleisli categories, and it shows that the Eilenberg–Moore category EM(S) carries a tangent category structure, with differential objects corresponding to D linear algebras. The results unify several standard constructions such as tangent bundle monads and reader monads, and they illuminate how effects interact with differential structure in a principled categorical way. The work also clarifies the role of Kleisli differential combinators and abstract Kleisli formalisms, offering a robust foundation for differential calculus in effectful settings with potential applications to tangent programming language design.

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.
Paper Structure (8 sections, 18 theorems, 59 equations)

This paper contains 8 sections, 18 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Cartesian Differential Categories
  3. Cartesian Differential Monads
  4. Kleisli Categories of Cartesian Differential Monads
  5. Cartesian Differential Abstract Kleisli Categories
  6. Kleisli Differential Combinators
  7. Eilenberg-Moore Categories of Cartesian Differential Monads
  8. Conclusion

Key Result

lemma 1

blute2009cartesian Let $\mathsf{F}: \mathbb{X} \to \mathbb{Y}$ be a strong Cartesian $k$-linear functor. Then $\mathsf{F}$ preserves $k$-linear maps, that is, if $f$ is a $k$-linear map in $\mathbb{X}$, then $\mathsf{F}(f)$ is a $k$-linear map in $\mathbb{Y}$.

Theorems & Definitions (40)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1
  • definition 5
  • lemma 2
  • proof
  • definition 6
  • lemma 3
  • ...and 30 more