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Compositional Taylor expansion in cartesian differential categories

Aymeric Walch

TL;DR

This work develops a compositional theory of Taylor expansion inside cartesian differential categories. It defines a hierarchy of order-$n$ Taylor-expansion functors $\mathsf{T}_{n}$ built from the tangent bundle, and shows they are functors and monads via distributive laws, thereby modeling higher-order dual numbers and jet-bundle structures categorically. The framework extends to an infinitary, coherent expansion $\mathsf{T}_{\omega}$ and establishes an equivalence between Taylor axioms and analytic behavior in categories with countable sums, connecting to Ehrhard–Walch and Manzonetto’s differential lambda-calculus. Together these results unify derivative calculus with category-theoretic constructs, offering a semantic foundation for automated differentiation and higher-order differential programming, and suggest directions toward reverse derivatives and broader tangent-category generalizations.

Abstract

This paper provides a compositional approach to Taylor expansion, in the setting of cartesian differential categories. Taylor expansion is captured here by a functor that generalizes the tangent bundle functor to higher order derivatives. The fundamental properties of Taylor expansion then boils down to naturality equations that turns this functor into a monad. This monad provides a categorical approach to higher order dual numbers and the jet bundle construction used in automated differentiation.

Compositional Taylor expansion in cartesian differential categories

TL;DR

This work develops a compositional theory of Taylor expansion inside cartesian differential categories. It defines a hierarchy of order- Taylor-expansion functors built from the tangent bundle, and shows they are functors and monads via distributive laws, thereby modeling higher-order dual numbers and jet-bundle structures categorically. The framework extends to an infinitary, coherent expansion and establishes an equivalence between Taylor axioms and analytic behavior in categories with countable sums, connecting to Ehrhard–Walch and Manzonetto’s differential lambda-calculus. Together these results unify derivative calculus with category-theoretic constructs, offering a semantic foundation for automated differentiation and higher-order differential programming, and suggest directions toward reverse derivatives and broader tangent-category generalizations.

Abstract

This paper provides a compositional approach to Taylor expansion, in the setting of cartesian differential categories. Taylor expansion is captured here by a functor that generalizes the tangent bundle functor to higher order derivatives. The fundamental properties of Taylor expansion then boils down to naturality equations that turns this functor into a monad. This monad provides a categorical approach to higher order dual numbers and the jet bundle construction used in automated differentiation.

Paper Structure

This paper contains 16 sections, 46 theorems, 103 equations, 2 figures.

Table of Contents

  1. Cartesian differential categories
  2. The tangent bundle construction
  3. The iterated tangent bundle monad
  4. Taylor expansion as a functor
  5. Alternative definition of Taylor expansion
  6. Functoriality of the Taylor expansion
  7. The monadic structure of Taylor expansion
  8. The natural transformations of Taylor expansion
  9. An infinitary Taylor expansion
  10. The monad structure of Taylor expansion
  11. A direct axiomatization of Taylor expansion
  12. Taylor expansion and countable sums
  13. Conclusion
  14. Proofs of \ref{['sec:tangent-bundle']}
  15. Proof of \ref{['prop:Slift-natural']}
  16. ...and 1 more sections

Key Result

Proposition 1

For any operator $\mathsf{D}$ that satisfies def:cdc-projections and def:cdc-chain, $\mathsf{D} \left\langle f, g \right\rangle = \left\langle \mathsf{D} f, \mathsf{D} g \right\rangle$.

Figures (2)

  • Figure 1: Graphical representation of $\Stree[3]$
  • Figure 2: The cone of Taylor expansions

Theorems & Definitions (63)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1: Lemma 2.6 of Lemay18
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 2: Lemma 3.2 of Garner21
  • Example 1
  • ...and 53 more