Unstable $1$-semiadditivity as classifying Goodwillie towers

TL;DR

This paper explores how variations of 1-semiadditivity in stable $mbda$-categories serve as obstructions to classifying Goodwillie towers, with applications across algebraic topology and operad theory.

Contribution

It introduces two variants of 1-semiadditivity and shows they fully classify Goodwillie towers via module structures, connecting to multiple advanced topics in homotopy theory.

Findings

Variations of 1-semiadditivity act as obstructions to classifying Goodwillie towers.

These obstructions relate to module and divided power structures on derivatives of functors.

Applications include algebraic localizations, operad Morita theory, and duality in $E_d$-algebras.

Abstract

A stable $\infty$-category is $1$-semiadditive if the norms for all finite group actions are equivalences. In the presence of $1$-semiadditivity, Goodwillie calculus simplifies drastically. We introduce two variants of $1$-semiadditivity for an $\infty$-category $C$ and study their relation to the Goodwillie calculus of functors $C \rightarrow \s(C)$. We demonstrate that these variations of $1$-semiadditivity are complete obstructions to the problem of endowing $\partial_\ast F$ with either a right module or a divided power right module structure which completely classifies the Goodwillie tower of $F$. We find applications to algebraic localizations of spaces, the Morita theory of operads, and bar-cobar duality of algebras. Along the way, we address several milestones in these areas including: Lie structures in the Goodwillie calculus of spaces, spectral Lie algebra models of $v_h$-periodic homotopy theory, and the Poincar\'e/Koszul duality of $E_d$-algebras.