Goodwillie Towers of $\infty$-Categories and Desuspension

TL;DR

This paper reinterprets the construction of n-excisive approximations of ∞-categories as inverting a lifted suspension functor, characterizing n-excisive categories by desuspension properties of A_n-cogroup objects.

Contribution

It provides a new conceptual framework for understanding n-excisive approximations via suspension inversion and desuspension of A_n-cogroup objects.

Findings

Characterizes n-excisive ∞-categories through desuspension of A_n-cogroup objects.

Reproves a theorem that 2-connected cogroup-like A_∞-spaces admit desuspension.

Offers a new perspective on Goodwillie calculus in the context of ∞-categories.

Abstract

We reconceptualize the process of forming $n$-excisive approximations to $\infty$-categories, in the sense of Heuts, as inverting the suspension functor lifted to $A_n$-cogroup objects. We characterize $n$-excisive $\infty$-categories as those $\infty$-categories in which $A_n$-cogroup objects admit desuspensions. Applying this result to pointed spaces we reprove a theorem of Klein-Schw\"anzl-Vogt: every 2-connected cogroup-like $A_\infty$-space admits a desuspension.