Rational Homotopy Calculus of Functors

of “Rational Homotopy Calculus of Functors” by Ben C. Walter, Ph.D., Brown University, 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. Also, we point out that the category DG is equivalent to the stabilizations of the categories DGL and DGC. Derived from our models for homotopy pushouts and pullbacks in DGL and DGC are models for suspensions and loops in these categories. These functors in turn induce natural stabilization and infinite loop functors between the categories DGL (and DGC) and DG. 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.