Deriving calculus with cotriples

Abstract

We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie’s Taylor tower for functors of spaces, and is related to Dold and Puppe’s stable derived functors and Mac Lane’s Q Q -construction. We study the layers, D n F = fiber ⁡ ( P n F → P n − 1 F ) D_{n}F = \operatorname {fiber}(P_{n}F\rightarrow P_{n-1}F) , and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.