A First Step Toward Higher Order Chain Rules in Abelian 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.

Formal differential variables and an abstract chain rule

“…Faà di Bruno's formula, named after the 19th century Italian priest Francesco Faà di Bruno, is a formula for the higher derivatives of f (g(x)). See [8] and [3] for the history of Faà di Bruno's formula (see also [10] for related work in category theory by another ACMS presenter). The formula can be stated combinatorially:…”