The derived Maurer-Cartan locus

TL;DR

This paper investigates the derived Maurer-Cartan locus associated with differential graded Lie algebras, establishing a quasi-isomorphism between its function algebra and the Chevalley-Eilenberg complex of a truncated algebra.

Contribution

It proves a quasi-isomorphism linking the functions on the derived Maurer-Cartan locus to the Chevalley-Eilenberg complex of the positive truncation of a dg Lie algebra.

Findings

The derived Maurer-Cartan locus is a functor to cosimplicial schemes.

The algebra of functions on this locus is quasi-isomorphic to the Chevalley-Eilenberg complex.

This links geometric and algebraic structures in dg Lie theory.

Abstract

The derived Maurer-Cartan locus $\text{MC}^\bullet(L)$ is a functor from differential graded Lie algebras to cosimplicial schemes. If L is 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 $\text{MC}^\bullet(L)$ is quasi-isomorphic to the Chevalley-Eilenberg complex of $L_+$.