  Infinitesimal deformations are governed by partition Lie algebras. In
characteristic $0$, these higher categorical structures are modelled by
differential graded Lie algebras, but in characteristic $p$, they are more
subtle. We give explicit models for partition Lie algebras over general
coherent rings, both in the setting of spectral and derived algebraic geometry.
For the spectral case, we refine operadic Koszul duality to a functor from
operads to divided power operads, by taking refined linear duals of
$\Sigma_n$-representations. The derived case requires a further refinement of
Koszul duality to a more genuine setting.
