Identify dg Lie∞ Chevalley–Eilenberg cohomology with double-complex homology

Establish that the Chevalley–Eilenberg cohomology groups defined for a differential graded Lie∞ algebra L with coefficients in an L-module V by homotopy classes of ∞-morphisms are naturally isomorphic to the homology groups of the Chevalley–Eilenberg double complex, namely H^n_{CE}(L,V) ≃ H^n(Tot(C^{*,*}_{CE}(L,V))).

Background

The paper defines Chevalley–Eilenberg cohomology for a non-negatively graded Lie∞ algebra L and an L-module V using homotopy classes of ∞-morphisms in a slice category. In the differential graded case, a cocycle consists not of a single multilinear map but of a collection of maps satisfying coherence conditions.

The conjecture proposes that this homotopy-theoretic definition agrees with the more classical cohomology obtained from the totalization of the Chevalley–Eilenberg double complex described by Manetti. Establishing this comparison would connect the paper’s representability-based construction with the standard double-complex formalism.