A Variation of the Goldman-Millson Theorem for Filtered $L_{\mathrm{\infty }}$ Algebras

In this paper, we extend the Goldman-Millson Theorem for $L_{\mathrm{\infty }}$ algebras. We consider two $L_{\mathrm{\infty }}$ algebras $L$ and $\tilde{L}$ endowed with descending, bounded above and complete filtrations compatible with the $L_{\mathrm{\infty }}$ structures and $U:L\to \tilde{L}$ an $\mathrm{\infty }$-morphism respecting the filtrations. We prove that in the setting of the linear part of $U$, say $\psi$, being a quasi-isomorphism on the r-1st page of the spectral sequences and ${H^1 ((\mathfrak{F}{2^q} L)/(\mathfrak{F}{$\mathrm{m}\mathrm{i}\mathrm{n}$(2^{q+1},r)} L))=0}$forevery$q$with$2^q < r$and${H^i((\mathfrak{F}_1 \tilde{L}) / (\mathfrak{F}_q \tilde{L}))=0}$for$i=0,1$and$q$everypowerof2smallerthan$r$and$q=r$ this induces a weak homotopy equivalence of the simplicial sets ${\mathfrak{M}\mathfrak{C}}_{\bullet }(L)$ and ${\mathfrak{M}\mathfrak{C}}_{\bullet }(\tilde{L})$.