Chevalley–Eilenberg equivalence for universal objects

Establish whether the Chevalley–Eilenberg functor from Lie algebras in the chiral monoidal category IndCoh^!_ren(MDisk_Ran) to cocommutative coalgebras restricts to an equivalence between the subcategory of universal chiral algebras Ch^{univ}—those Lie algebras whose underlying sheaf is of the form Δ_{I,!}B_1 for some B_1 ∈ IndCoh^!_ren(MDisk_1)—and the subcategory of universal factorization algebras Fact^{univ}—those cocommutative coalgebras whose comultiplication induces isomorphisms sqcup^!A → A^{⊠ I} for all finite sets I.

Background

The paper constructs a universal chiral monoidal category IndCoh^!_ren(MDisk_Ran) based on the moduli of pointed formal multidisks and defines two notable full subcategories: universal chiral algebras Ch^{univ} and universal factorization algebras Fact^{univ}. In general, Koszul duality yields an equivalence between all Lie algebras and cocommutative coalgebras in the ambient chiral monoidal category.

The authors show the Chevalley–Eilenberg functor gives an equivalence Lie(IndCoh^{!,ch}_ren(MDisk_Ran)) ≅ coComm(IndCoh^{!,ch}_ren(MDisk_Ran)), but it remains unresolved whether this equivalence restricts to the universal subcategories singled out by their geometric conditions (being determined by Δ_{I,!}B_1 and satisfying the universal factorization isomorphisms).