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

Abstract

In this paper, we extend the Goldman-Millson Theorem for $L_{\infty }$ algebras. We consider two $L_{\infty }$ algebras $L$ and $\tilde{L}$ endowed with descending, bounded above and complete filtrations compatible with the $L_{\infty }$ structures and $U:L\to \tilde{L}$ an $\infty$-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 $H^1(({\mathfrak{F}}_{2^q}L)/({\mathfrak{F}}_{\min (2^{q+1},r)}L))=0$ for every $q$ with $2^q<r$ and $H^i(({\mathfrak{F}}_1\tilde{L})/({\mathfrak{F}}_q\tilde{L}))=0$ for $i=0,1$ and $q$ every power of 2 smaller than $r$ and $q=r$ this induces a weak homotopy equivalence of the simplicial sets ${\mathfrak{MC}}_{\bullet }(L)$ and ${\mathfrak{MC}}_{\bullet }(\tilde{L})$.

Extended Theorem 1.4, Clarified Proof of Injectivity, Removed Typos

References in corpus (4)

• A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras
• On an enhancement of the category of shifted L-infinity algebras
• Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of $E_n$-operads
• Operads and Maurer-Cartan spaces

Cited by in corpus (1)

• Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of $E_n$-operads