An algebraic model for rational excisive functors
David Barnes, Magdalena Kędziorek, Niall Taggart
TL;DR
The paper develops a new proof of the rational splitting of excisive endofunctors of spectra into a product of homogeneous layers, avoiding reliance on rational Tate vanishing. It builds an algebraic model via the rational Goodwillie-Burnside ring $A_\nQ(d)$, identifies its idempotents, and uses them to split the Goodwillie tower rationally, producing the equivalence $Exc_*^d(Sp^ω,Sp)_\nQ \simeq \prod_{i=1}^d Homog^i(Sp^ω,Sp)_\nQ$. By lifting idempotents to the $\infty$-category and applying Goodwillie’s classification of homogeneous functors, the authors show $Homog^d \simeq \mathrm{Mod}_{\mathrm{Fun}_*(Sp^ω,Sp)_\nQ}(D_d h_S)$ and obtain a concrete algebraic description: $Exc_*^d(Sp^ω,Sp)_\nQ \simeq \prod_{i=1}^d Sp_\nQ^{B\Sigma_i} \simeq \prod_{i=1}^d Ch(\mathbb{Q}[\Sigma_i])$. This provides a formal, algebraic framework for rational excisive functors and links their structure to rational equivariant stable homotopy theory. The results extend the understanding of how rationalization interacts with Goodwillie calculus and offer a potentially broad set of applications in rational homotopy theory and algebraic models of stable phenomena.
Abstract
We provide a new proof of the rational splitting of excisive endofunctors of spectra as a product of their homogeneous layers independent of rational Tate vanishing. We utilise the analogy between endofunctors of spectra and equivariant stable homotopy theory and as a consequence, we obtain an algebraic model for rational excisive functors.