Unstable $1$-semiadditivity as classifying Goodwillie towers
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911412584972288 |
|---|---|
| author | Malin, Connor |
| author_facet | Malin, Connor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unstable $1$-semiadditivity as classifying Goodwillie towers Malin, Connor Algebraic Topology Category Theory 18F50 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. |
| title | Unstable $1$-semiadditivity as classifying Goodwillie towers |
| topic | Algebraic Topology Category Theory 18F50 |
| url | https://arxiv.org/abs/2506.11245 |