Goodwillie Towers of $\infty$-Categories and Desuspension

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.