  We construct a derived generalization of the pure spinor superfield formalism
and prove that it exhibits an equivalence of dg-categories between multiplets
for a supertranslation algebra and equivariant modules over its
Chevalley-Eilenberg cochains. This equivalence is closely linked to Koszul
duality for the supertranslation algebra. After introducing and describing the
category of supermultiplets, we define the derived pure spinor construction
explicitly as a dg-functor. We then show that the functor that takes the
derived supertranslation invariants of any supermultiplet is a quasi-inverse to
the pure spinor construction, using an explicit calculation. Finally, we
illustrate our findings with examples and use insights from the derived
formalism to answer some questions regarding the ordinary (underived) pure
spinor superfield formalism.
