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Unstable $1$-semiadditivity as classifying Goodwillie towers

Connor Malin

TL;DR

This work develops a framework connecting unstable $1$-semiadditivity to the classification of Goodwillie towers via operadic (co)modules. It introduces infinitesimal and differential variants of $1$-semiadditivity, showing they provide complete obstructions to equipping Goodwillie derivatives with right or divided-power right module structures and thus to classifying the towers; it then develops a Koszul-duality-based machinery that ties derivative data to coend- and endomorphism- operads. The results yield structural insights across several domains, including Morita theory for operads, Poincaré/Koszul duality for $E_d$-algebras, and Lie-algebra models of $v_h$-periodic spaces, and they formalize when polynomial/Goodwillie data can be recovered from operadic right-module information. The framework also introduces notions of differential algebraicity and shows these properties behave well under localizations, with concrete examples such as rational spaces illustrating the nuanced landscape where divided powers either classify or fail to classify Goodwillie towers. Overall, the paper provides a unifying lens to view Goodwillie calculus through operadic (co)modules, Tate vanishing, and Koszul duality, linking unstable homotopy theory to algebraic models and local-to-global classification phenomena.

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 Poincaré/Koszul duality of $E_d$-algebras.

Unstable $1$-semiadditivity as classifying Goodwillie towers

TL;DR

This work develops a framework connecting unstable -semiadditivity to the classification of Goodwillie towers via operadic (co)modules. It introduces infinitesimal and differential variants of -semiadditivity, showing they provide complete obstructions to equipping Goodwillie derivatives with right or divided-power right module structures and thus to classifying the towers; it then develops a Koszul-duality-based machinery that ties derivative data to coend- and endomorphism- operads. The results yield structural insights across several domains, including Morita theory for operads, Poincaré/Koszul duality for -algebras, and Lie-algebra models of -periodic spaces, and they formalize when polynomial/Goodwillie data can be recovered from operadic right-module information. The framework also introduces notions of differential algebraicity and shows these properties behave well under localizations, with concrete examples such as rational spaces illustrating the nuanced landscape where divided powers either classify or fail to classify Goodwillie towers. Overall, the paper provides a unifying lens to view Goodwillie calculus through operadic (co)modules, Tate vanishing, and Koszul duality, linking unstable homotopy theory to algebraic models and local-to-global classification phenomena.

Abstract

A stable -category is -semiadditive if the norms for all finite group actions are equivalences. In the presence of -semiadditivity, Goodwillie calculus simplifies drastically. We introduce two variants of -semiadditivity for an -category and study their relation to the Goodwillie calculus of functors . We demonstrate that these variations of -semiadditivity are complete obstructions to the problem of endowing with either a right module or a divided power right module structure which completely classifies the Goodwillie tower of . 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 -periodic homotopy theory, and the Poincaré/Koszul duality of -algebras.

Paper Structure

This paper contains 19 sections, 63 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Acknowledgments
  4. Conventions
  5. Right (co)modules over an operad
  6. Right modules over an operad
  7. Divided power right modules over an operad
  8. Right modules in Goodwillie calculus
  9. Right module structures on Goodwillie derivatives
  10. Comparison with the operad $\partial_\ast{\mathsf{Id}}_C$
  11. Goodwillie calculus of spaces
  12. Goodwillie calculus of algebras over an operad
  13. Poincaré/Koszul duality of $E_d$-algebras
  14. Lie algebra models of $v_h$-periodic spaces
  15. Classification of Goodwillie towers
  16. ...and 4 more sections

Key Result

Theorem 1.1.1

If $C$ is differentially dualizable, the Goodwillie derivatives lift to a symmetric monoidal functor with respect to the pointwise smash product of functors and Day convolution of right modules Given $F\in {\mathsf{Fun}}^\omega(C,L{\mathsf{Sp}})$ such that $\partial_\ast F$ is levelwise dualizable, there is an equivalence of right modules where $\mathop{{\mathsf{\mathsf{nat}}}}\nolimits(F,(\Sigm

Theorems & Definitions (154)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Proposition 1.1.4: cf. ACClassification, crosseffectsclassification
  • Theorem 1.1.5
  • Theorem 1.1.6
  • Theorem 1.1.7
  • Corollary 1.1.8
  • Definition 2.1.1
  • Proposition 2.1.2
  • ...and 144 more