Rational Homotopy Calculus of Functors

TL;DR

This paper develops a homotopy calculus of functors specifically for rational homotopy theory categories, introducing a rational Taylor tower with simple models and demonstrating their computational usefulness.

Contribution

It constructs a rational homotopy calculus of functors with explicit models for the Taylor tower in categories like DG, DGL, and DGC, extending Goodwillie's framework.

Findings

Existence of a rational Taylor tower for homotopy functors

Development of simple models for objects in the tower

Application of models to compute rational homotopy invariants

Abstract

This is a (slightly edited) version of the PhD dissertation of the author, submitted to Brown University in July 2005. We construct a homotopy calculus of functors in the sense of Goodwillie for the categories of rational homotopy theory. More precisely, given a homotopy functor between any of the categories of differential graded vector spaces (DG), reduced differential graded vector spaces, differential graded Lie algebras (DGL), and differential graded coalgebras (DGC), we show that there is an associated approximating "rational Taylor tower" of excisive functors. The fibers in this tower are homogeneous functors which factor as homogeneous endomorphisms of the category of differential graded vector spaces. Furthermore, we develop very straightforward and simple models for all of the objects in this tower. Constructing these models entails first building very simple models for homotopy pushouts and pullbacks in the categories DG, DGL, and DGC. We end with a short example of the usefulness of our computationally simple models for rational Taylor towers, as well as a preview of some further results dealing with the structure of rational (and non-rational) Taylor towers.