The derived Maurer–Cartan locus

Abstract

The derived Maurer-Cartan locus is a functor ${\mathrm{MC}}^{\bullet }$ from differential graded Lie algebras to cosimplicial schemes. If $L$ is a differential graded Lie algebra, let $L_{+}$ be the truncation of $L$ in positive degrees $i>0$. We prove that the differential graded algebra of functions on the cosimplicial scheme ${\mathrm{MC}}^{\bullet }(L)$ is quasi-isomorphic to the Chevalley-Eilenberg complex of $L_{+}$.