On the universality of multiexcisive functors

Abstract: We provide a multiplicative classification of polynomial endofunctors on spectra in terms of their Mackey functors of cross--effects. More precisely, we prove that various categories of multivariable excisive functors from spectra to spectra are symmetric monoidally equivalent to the corresponding variants of spectral Mackey functors. The symmetric monoidal structures appearing here are the Day convolutions on both sides, and the Mackey functors we consider involve variations on the category of finite sets and surjections. The method is first to introduce certain multivariable functors we call subdiagonal functors. By considering them all at once using parametrised category theory, we prove inductively that they all admit Mackey functor descriptions as symmetric monoidal categories, endowing them with a universal property along the way. In particular, specialising this to univariate functors gives a new proof and strengthening of Glasman's result about d-excisive endofunctors on spectra. As application of our perspective, we prove a ``Segal conjecture'' in the context of Goodwillie calculus when d is a prime number.

• The paper establishes a canonical symmetric monoidal equivalence between excisive functors and spectral Mackey functors over Epi_d using Day convolution.
• It introduces subdiagonal multivariable functors defined via Goodwillie towers to manage cross-effects and enable recursive handling of functor stratifications.
• Key implications include a parametrized universal property and a version of the Segal conjecture for Goodwillie calculus, advancing links to equivariant stable homotopy theory.

Universality of Multiexcisive Functors: A Technical Analysis

Introduction and Context

The paper "On the universality of multiexcisive functors" (2604.02232) establishes a symmetric monoidal, universal classification of polynomial (specifically, multiexcisive) endofunctors on spectra. It does so by relating these, via cross-effects, to spectral Mackey functors indexed by categories $Epi_d$ of finite sets up to size $d$ and surjections. This framework merges multiplicative structure (via Day convolution) with deep connections to Goodwillie calculus, category theory, and spectral Mackey functors. The core results provide parametrized (i.e., indexed) symmetric monoidal equivalences and a unifying stratification structure for excisive functor categories, extending classical parallels with equivariant homotopy theory.

Main Contributions

Multiplicative Classification of Excisive Functors

At the heart of the paper is the following symmetric monoidal equivalence: $Exc_d \simeq \mackey(Epi_d; Sp)$ where $Exc_d$ is the category of reduced $d$-excisive endofunctors on $Sp$, and $\mackey(Epi_d; Sp)$ denotes spectral Mackey functors over $Epi_d$ with Day convolution.

The authors extend Glasman’s equivalence from a mere symmetric monoidal equivalence to an equivalence with a universal property in the context of parametrised category theory—a significant strengthening, as previous approaches did not respect multiplicative structures canonically. The multiplicative structure is essential for applications to stratifications, localizations, and comparisons with equivariant stable homotopy theory.

Subdiagonal Functors and Parametrised Categories

A principal technical innovation is the introduction of subdiagonal multivariable functors. These are defined via the structure of their Goodwillie towers: a functor is subdiagonal if its cube of multivariable excisive approximations is determined (via right Kan extension) by "subdiagonal terms", indexed by tuples $(k_1, ..., k_r)$ with $\sum k_i \leq d$. This descent condition carves out a subcategory $d$0 amenable to Mackey description, resolving obstacles in handling cross-effects for general multivariable polynomial functors.

Cross-Effects, Day Convolution, and Universal Properties

The parametrised category framework is exploited to inductively handle all cross-effects and restriction/extension functors compatible with the full range of surjections in $d$1 categories. The paper proves that:

• All relevant restriction and extension functors between subdiagonal functor categories are symmetric monoidal and admit right and left adjoints that preserve the required exactness.
• Cross-effects and (left/right) Kan extensions interact compatibly with excisive approximations and reduction functors under Day convolution.

A key technical advance is the universal property: $d$2 is initial among $d$3-presentably symmetric monoidal stable categories with appropriately compatible restriction/extension operations.

Theoretical Implications

Strengthened Mackey Description

The equivalence is not only symmetric monoidal but parametrized: it organizes $d$4 as an $d$5-indexed family compatible with cross-effects and Kan extensions indexed by surjections and finite sets. This deeper structure:

• Enables an explicit "stable recollement" and stratification of functor categories akin to the isotropy separation in equivariant topology.
• Provides a dictionary between Goodwillie calculus and genuine equivariant stable homotopy theory, allowing the transport of stratification and localization phenomena.

Universality and Stratifications

By viewing the categories $d$6 as assembling into a parametrized symmetric monoidal category over $d$7, the paper realizes multiexcisive functors as initial (or universal) for canonical functors out of the category of spectra, up to Day convolution. This equips $d$8 with a universal property analogous to that of $d$9-spectra in equivariant settings, delineating its structural role within homotopical algebra.

Application to the Calculus Version of the Segal Conjecture

The stratified, Mackey-centric perspective yields new structural results, including a prime degree ($Exc_d \simeq \mackey(Epi_d; Sp)$0) version of the Segal conjecture in Goodwillie calculus: the completion map from $Exc_d \simeq \mackey(Epi_d; Sp)$1-adic completion of $Exc_d \simeq \mackey(Epi_d; Sp)$2 to $Exc_d \simeq \mackey(Epi_d; Sp)$3 is a $Exc_d \simeq \mackey(Epi_d; Sp)$4-adic equivalence. The Mackey formalism and imported stratification methods from equivariant theory are essential in reducing the new Segal conjecture in calculus to its classical counterpart by tracking the action of the Goodwillie–Burnside ideal.

Strong Results and Claims

• Canonical symmetric monoidal equivalence between $Exc_d \simeq \mackey(Epi_d; Sp)$5 and $Exc_d \simeq \mackey(Epi_d; Sp)$6, upgrading and streamlining all known equivalences by internalizing compatibility with Day convolution and all cross-effect manipulations.
• Inductive parametrized structure for all subcategories $Exc_d \simeq \mackey(Epi_d; Sp)$7, yielding a recursive Mackey description for all multivariable excisive functor categories.
• Numerical Goldilocks zone: subdiagonality is shown, via pigeonhole principle arguments, to capture the precise descent condition enabling unification of excisiveness data across variables and targets.
• A Segal conjecture for Goodwillie calculus: for $Exc_d \simeq \mackey(Epi_d; Sp)$8 prime, the canonical map between $Exc_d \simeq \mackey(Epi_d; Sp)$9-completed and homotopy fixed point functors is a $Exc_d$0-adic equivalence, a result unattainable for composite $Exc_d$1 or without $Exc_d$2-completion.

Potential Future Developments

This framework opens several avenues:

• Extension to generalized parametrized settings: similar techniques may classify other highly structured functor categories, such as analytic or piecewise polynomial functors, and their multiplicative or equivariant enhancements.
• Explicit models for cross-effects in unstable or spectral settings, refined by this parametrized universal viewpoint.
• Interactions with chromatic homotopy theory: the stratified setup is ideal for tracking Picard spectra, localizations, and Balmer spectra (as in [ABHS24]), possibly leading to new invariants and computational tools.
• Potential for a broader Segal-type theory, exploring the relationship between Mackey stratification and completion techniques in polynomial and functor calculus outside the classical stable field.

Conclusion

The paper provides a robust, categorical foundation for the study of multiexcisive functors, demonstrating a canonical symmetric monoidal equivalence with spectral Mackey functors, organizing all cross-effects and excisive approximations via parametrized category theory. Its structural elucidation of stable recollements, universal properties, and application to the Segal conjecture establishes a technical standard and template for future research in higher-categorical and parametrized functor calculus.