Goodwillie Calculus via Adjunction and LS Cocategory

In this paper, we show that for reduced homotopy endofunctors of spaces, F, and for all \(n \geq 1\) there are adjoint functors \(R_n, L_n\) with \(T_n F \simeq R_n F L_n\), where \(P_n F\) is the \(n\)-excisive approximation to \(F\), constructed by taking the homotopy colimit over iterations of \(T_n F\). This then endows \(T_n\) of the identity with the structure of a monad and the \(T_n F\)'s are the functor version of bimodules over that monad. It follows that each \(T_n F\) (and \(P_nF\)) takes values in spaces of symmetric Lusternik-Schnirelman cocategory \(n\), as defined by Hopkins. This also recovers recent results of Chorny-Scherer. The spaces \(T_n F(X)\) are in fact classically nilpotent (in the sense of Berstein-Ganea) but not nilpotent in the sense of Biedermann and Dwyer. We extend the original constructions of dual calculus to our setting, establishing the \(n\)-co-excisive approximation for a functor, and dualize our constructions to obtain analogous results concerning constructions \(T^n\), \(P^n\),and LS category.