Goodwillie Towers of Infinity Categories and Desuspension

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