Algebraic Goodwillie Spectral Sequence

Let s L s\mathcal {L} be the ∞ \infty -category of simplicial restricted Lie algebras over F = F ¯ p \mathbf {F} = \overline {\mathbf {F}}_p , the algebraic closure of a finite field F p \mathbf {F}_p . By the work of A. K. Bousfield et al. on the unstable Adams spectral sequence, the category s L s\mathcal {L} can be viewed as an algebraic approximation of the ∞ \infty -category of pointed p p -complete spaces. We study the functor calculus in the category s L s\mathcal {L} . More specifically, we consider the Taylor tower for the functor L r : M o d F ≥ 0 → s L L^r\colon \mathcal {M}\mathrm {od}^{\geq 0}_{\mathbf {F}} \to s\mathcal {L} of a free simplicial restricted Lie algebra together with the associated Goodwillie spectral sequence. We show that this spectral sequence evaluated at Σ l F