Directional derivatives and higher order chain rules for abelian functor calculus

Abstract

Abstract. In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully construct a category of abelian categories and suitably homotopically defined functors, and show that this category, equipped with the directional derivative, is a cartesian differential category in the sense of Blute, Cockett, and Seely. This provides an abstract framework that makes certain analogies between classical and functor calculus explicit. Inspired by Huang, Marcantognini, and Young's chain rule for higher order directional derivatives of functions, we define a higher order directional derivative for functors of abelian categories. We show that our higher order directional derivative is related to the iterated partial directional derivatives of the second author and McCarthy by a Faà di Bruno style formula. We obtain a higher order chain rule for our directional derivatives using a feature of the cartesian differential category structure, and with this provide a formulation for the nth layers of the Taylor tower of a composition of functors F • G in terms of the derivatives and directional derivatives of F and G, reminiscent of similar formulations for functors of spaces or spectra by Arone and Ching. Throughout, we provide explicit chain homotopy equivalences that tighten previously established quasi-isomorphisms for properties of abelian functor calculus.

A First Step Toward Higher Order Chain Rules in Abelian Functor Calculus

“…, x). Corollary 2.7 of [1] shows that cr n and the diagonal functor are an adjoint pair of functors, so that C n is a comonad. Let ǫ denote the counit of this comonad.…”

“…When we compose functors such as these, we implicitly mean that we use this procedure to compose. This process is made effortless by using the structure of a Kleisli category, which is explained in detail in Section 3 of [1]. For the purposes of this paper, we will suppress the Ch in the codomain of functors, and write D 1 G : A B instead of D 1 G : A → ChB to indicate that we are working in this Kleisli category and to avoid cumbersome Ch's.…”

“…Acknowledgments. The authors would like to thank the Banff International Research Station (host of the second Women in Topology workshop) and the Pacific Institute for Mathematical Sciences for providing us with the opportunity to collaborate along with the other authors of [1]. We would also like to express our gratitude to Kristine Bauer, Brenda Johnson, and Emily Riehl for their guidance and support throughout this project.…”

“…Functor calculus enjoys certain properties that are analogous to results in undergraduate calculus. This paper is a companion to [1], in which many of these analogies are made explicit. In [1], one of the main results [1,Theorem 8.1] provides a chain rule for the nth higher order directional derivative, denoted as ∆ n [1,Definition 7.3], associated to a functor between abelian categories.…”

“…it warranted independent documentation. However, the groundwork -including the definitions and properties of most of the functors we will use -is already documented in [1]. For this reason, we will heavily cite [1] throughout this paper.…”