A first step toward higher order chain rules in abelian functor calculus

TL;DR

This paper explores the extension of the chain rule to higher order derivatives within abelian functor calculus, providing a concrete second-order case and a novel proof approach based on cross effects and linearization.

Contribution

It introduces the second higher order chain rule for abelian functors and offers a new proof method distinct from previous work, focusing on properties of cross effects.

Findings

Computed the second higher order directional derivative chain rule for abelian functors

Provided a new proof approach relying on cross effects and linearization

Extended the understanding of higher order chain rules in functor calculus

Abstract

One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when $n=2$. We describe how to obtain the second higher order directional derivative chain rule for abelian functors. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors.