Homology, Homotopy and Applications

Volume 18 (2016)

Number 2

Goodwillie calculus via adjunction and LS cocategory

Pages: 31 – 58

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n2.a2


Rosona Eldred (Fachbereich Mathematik, University of Münster, Germany)


In this paper, we establish a new monadic structure on the intermediate constructions, $\mathrm{T}_n F$, of Goodwillie’s calculus of functors. We show that as a result these functors take values in spaces of Hopkins’ symmetric Lusternik–Schnirelmann (LS) cocategory $\leqslant n$, which is an upper bound on the homotopy nilpotence class of the space. This property allows us to extend results of Biedermann–Dwyer linking Goodwillie calculus to homotopy nilpotence and of Chorny–Scherer on the vanishing of Whitehead products for spaces which are values of $n$-excisive functors. We also use a dual form of our adjunction to give a rigorous formulation of homotopy functor analog of McCarthy’s Dual Calculus, where $n$-co-excisive functors take certain pullback cubes to pushout cubes, and dualize our results of calculus and LS cocategory to dual calculus and LS category.


LS category, LS cocategory, Goodwillie calculus, homotopy limit, nilpotence

2010 Mathematics Subject Classification

Primary 55P99. Secondary 55P45, 55Q15, 55U30.

Published 29 November 2016