The Kriz theorem provides an algebraic model for $p$-adic homotopy
theory in terms of cosimplicial $p$-Boolean algebras, i.e.
$\mathbb{F}_p$-algebras such that $x^p=x$ for any element $x$.
Unfortunately, the Koszul duality for $p$-Boolean algebras is unknown,
so a Lie model for $p$-homotopy theory remains unknown as well. However,
Itamar Mor recently constructed the category $Syn$, which degenerates
unstable homotopy theory to simplicial restricted Lie algebras. The
category $Syn$ categorifies the lower central series spectral sequence.
In my talk, I will present an algebraic model for the category $Syn$
which degenerates the Kriz theorem into the Koszul duality between
restricted Lie algebras and $\mathbb{F}_p$-algebras with trivial
Frobenius. If time permits, I will also explain how the synthetic Kriz
theorem reveals some interactions between different versions of the
unstable Adams spectral sequence.