Compositional Taylor expansion in cartesian differential categories

TL;DR

This paper introduces a categorical framework for Taylor expansion within cartesian differential categories, generalizing tangent bundles and providing a monadic structure for higher order derivatives and jet bundles.

Contribution

It presents a novel functor-based approach to Taylor expansion that generalizes tangent bundles and formalizes higher order derivatives categorically.

Findings

Defines a functor capturing Taylor expansion in cartesian differential categories

Shows this functor forms a monad, enabling compositional differentiation

Connects the framework to higher order dual numbers and jet bundle constructions

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.