| Summary: | <p>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.</p>
<p>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 Σn-representations. The derived case requires a further refinement of Koszul duality
to a more genuine setting.</p>
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