Goodwillie towers of ∞-categories and desuspension

Abstract: We reconceptualize the process of forming $n$-excisive approximations to $\infty$-categories, in the sense of Heuts [1], 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–Schwänzl–Vogt [2]: every 2-connected cogroup-like $A_\infty$-space admits a desuspension.

This is joint work with Klaus O. Johnson and Michael River.

References:

[1] G. Heuts, Goodwillie approximations to higher categories, arXiv:1510.03304, 2018.

[2] J. Klein, R. Schwänzl, and R. Vogt, Comultiplication and suspension, Topology and its Applications, 1997.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)