Koszul duality for simplicial restricted Lie algebras

Let ${\mathsf{s}}_0{\mathsf{L}\mathsf{i}\mathsf{e}}^r$ be the category of $0$-reduced simplicial restricted Lie algebras over a fixed perfect field of positive characteristic $p$. We prove that there is a full subcategory $\mathrm{H}\mathrm{o}({\mathsf{s}}_0{\mathsf{L}\mathsf{i}\mathsf{e}}_{\xi }^r)$ of the homotopy category $\mathrm{H}\mathrm{o}({\mathsf{s}}_0{\mathsf{L}\mathsf{i}\mathsf{e}}^r)$ and an equivalence $\mathrm{H}\mathrm{o}({\mathsf{s}}_0{\mathsf{L}\mathsf{i}\mathsf{e}}_{\xi }^r)\simeq \mathrm{H}\mathrm{o}({\mathsf{s}}_1{\mathsf{C}\mathsf{o}\mathsf{A}\mathsf{l}\mathsf{g}}^{tr})$. Here ${\mathsf{s}}_1{\mathsf{C}\mathsf{o}\mathsf{A}\mathsf{l}\mathsf{g}}^{tr}$ is the category of $1$-reduced simplicial truncated coalgebras; informally, a coaugmented cocommutative coalgebra $C$ is truncated if $x^p=0$ for any $x$ from the augmentation ideal of the dual algebra $C^{\ast }$. Moreover, we provide a sufficient and necessary condition in terms of the homotopy groups ${\pi }_{\ast }(L_{\bullet })$ for $L_{\bullet }\in \mathrm{H}\mathrm{o}({\mathsf{s}}_0{\mathsf{L}\mathsf{i}\mathsf{e}}^r)$ to lie in the full subcategory $\mathrm{H}\mathrm{o}({\mathsf{s}}_0{\mathsf{L}\mathsf{i}\mathsf{e}}_{\xi }^r)$. As an application of the equivalence above, we construct and examine an analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in the category $\mathsf{s}{\mathsf{L}\mathsf{i}\mathsf{e}}^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra.