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The spectrum of excisive functors

Gregory Arone, Tobias Barthel, Drew Heard, Beren Sanders

TL;DR

Arone, Barthel, Heard, and Sanders develop a tensor triangular-geometry framework for Goodwillie calculus by studying the category of $d$-excisive functors Exc$_d({ m Sp}^c,{ m Sp})$. They construct a smashing Day-convolution monoidal structure, prove compact generation by $P_d h_{ m S}(i)$, and connect derivatives with a new Goodwillie–Burnside ring $A(d)$ to organize the Balmer spectrum. The core result is a complete tt-classification of compact $d$-excisive functors in terms of a prime spectrum described by a non-abelian blueshift function, yielding a disjoint union structure of primes analogous to equivariant spectra and generalizing classical thick-subcategory theorems. The work also develops functor-calculus analogues of transchromatic Smith–Floyd-type phenomena and opens a path to coefficient changes and broader categorical contexts via the Lurie tensor product. Together, these results offer a cohesive geometry-driven understanding of excisive functors and establish a robust bridge to equivariant and chromatic homotopy theory.

Abstract

We prove a thick subcategory theorem for the category of $d$-excisive functors from finite spectra to spectra. This generalizes the Hopkins-Smith thick subcategory theorem (the $d=1$ case) and the $C_2$-equivariant thick subcategory theorem (the $d=2$ case). We obtain our classification theorem by completely computing the Balmer spectrum of compact $d$-excisive functors. A key ingredient is a non-abelian blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. Also important are the techniques of tensor triangular geometry and striking analogies between functor calculus and equivariant homotopy theory. In particular, we introduce a functor calculus analogue of the Burnside ring and describe its Zariski spectrum à la Dress. The analogy with equivariant homotopy theory is strengthened further through two applications: We explain the effect of changing coefficients from spectra to ${\mathrm{H}\mathbb{Z}}$-modules and we establish a functor calculus analogue of transchromatic Smith-Floyd theory as developed by Kuhn-Lloyd. Our work offers a new perspective on functor calculus which builds upon the previous approaches of Arone-Ching and Glasman.

The spectrum of excisive functors

TL;DR

Arone, Barthel, Heard, and Sanders develop a tensor triangular-geometry framework for Goodwillie calculus by studying the category of -excisive functors Exc. They construct a smashing Day-convolution monoidal structure, prove compact generation by , and connect derivatives with a new Goodwillie–Burnside ring to organize the Balmer spectrum. The core result is a complete tt-classification of compact -excisive functors in terms of a prime spectrum described by a non-abelian blueshift function, yielding a disjoint union structure of primes analogous to equivariant spectra and generalizing classical thick-subcategory theorems. The work also develops functor-calculus analogues of transchromatic Smith–Floyd-type phenomena and opens a path to coefficient changes and broader categorical contexts via the Lurie tensor product. Together, these results offer a cohesive geometry-driven understanding of excisive functors and establish a robust bridge to equivariant and chromatic homotopy theory.

Abstract

We prove a thick subcategory theorem for the category of -excisive functors from finite spectra to spectra. This generalizes the Hopkins-Smith thick subcategory theorem (the case) and the -equivariant thick subcategory theorem (the case). We obtain our classification theorem by completely computing the Balmer spectrum of compact -excisive functors. A key ingredient is a non-abelian blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. Also important are the techniques of tensor triangular geometry and striking analogies between functor calculus and equivariant homotopy theory. In particular, we introduce a functor calculus analogue of the Burnside ring and describe its Zariski spectrum à la Dress. The analogy with equivariant homotopy theory is strengthened further through two applications: We explain the effect of changing coefficients from spectra to -modules and we establish a functor calculus analogue of transchromatic Smith-Floyd theory as developed by Kuhn-Lloyd. Our work offers a new perspective on functor calculus which builds upon the previous approaches of Arone-Ching and Glasman.
Paper Structure (16 sections, 113 theorems, 336 equations, 5 figures)

This paper contains 16 sections, 113 theorems, 336 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Categorical foundations
  3. The tensor triangulated category of d-excisive functors
  4. Cross-effects and idempotents
  5. The cross-effects of representable functors
  6. Compact generation of the category of d-excisive functors
  7. Tensor idempotents and Goodwillie derivatives
  8. The primes of d-excisive functors
  9. The Balmer spectrum of n-excisive functors as a set
  10. The Goodwillie--Burnside ring
  11. The comparison map to the Goodwillie--Burnside ring
  12. The spectrum of d-excisive functors
  13. Blueshift of Tate-derivatives
  14. The topology of the Balmer spectrum
  15. Applications
  16. ...and 1 more sections

Key Result

Theorem A

The category of $d$-excisive functors $\mathop{\mathrm{Exc}}\nolimits_{d}({\mathscr{S}p}^c,{\mathscr{S}p})$ is a compactly generated presentably symmetric monoidal stable $\infty$-category whose compact and dualizable objects coincide. A set of compact generators is given by the functors $P_dh_{\mat

Figures (5)

  • Figure 1: The Balmer spectrum of 3-excisive functors, along with the comparison map to the Zariski spectrum of $A(3)$.
  • Figure 2: The Balmer spectrum $\mathop{\mathrm{Spc}}\nolimits({\mathscr{S}p}^c)$ of the category of finite spectra
  • Figure 3: The Zariski spectrum of the ring $A(3)$.
  • Figure 4: The $p$-local part of the Balmer spectrum of 4-excisive functors from finite spectra to spectra.
  • Figure 5: The Balmer spectrum $\mathop{\mathrm{Spc}}\nolimits(\mathop{\mathrm{Exc}}\nolimits_{3}({\mathscr{S}p}^c,\mathop{\mathrm{Mod}}\nolimits_{{\mathop{\mathrm{H}}\nolimits \space\mathbb{Z}}})^c)$.

Theorems & Definitions (408)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Theorem H
  • Definition 2.1
  • Example 2.2
  • ...and 398 more