TITLE: Model category structures on the category of $L_\infty$-algebras
QUESTION [7 upvotes]: Let $k$ be a characteristic zero field. Then it is known that the forgetful functor $dgla(k)\to chain(k)$ from differential graded Lie algebras (over $k$) to cochain complexes induces a model category structure on $dgla(k)$ with "the same" fibrations and weak equivalences as on $chain(k)$, i.e., fibrations are surjective dgla morphisms and weak-equivalences are quasi-isomorphisms. 
Also on the category of $L_\infty$-algebras over $k$ there is a forgetful functor $L_\infty(k)\to chain(k)$, which picks the linear part of an $L_\infty$-morphism. 
 Then, on $L_\infty(k)$ we have two natural functors: the forgetful functor $L_\infty(k)\to chain(k)$ and the embedding $L_\infty(k)\hookrightarrow dgcu(k)$, where $dgcu(k)$ is the category of differential graded counitary cocommutative coalgebras over $k$. 
This suggests we could have two natural model category structures on $L_\infty(k)$, and my question is: how are they related? do they coincide? in particular, is a morphism of $L_\infty$-algebras whose linear part is surjective a fibration in the $dgcu(k)$ model structure? is a quasi-isomorphism of $L_\infty$-algebras (i.e., a morphism of $L_\infty$-algebras whose linear part is a quasi-isomorphism) a weak-equivalence in the the $dgcu(k)$ model structure?

REPLY [7 votes]: If you start with the category of $L_\infty$-algebras with "strict morphisms" ($L_\infty$-morphisms such that the higher components vanish), then you can put a model category structure by the classical operadic means: this category is the category of algebras over the operad $L_\infty:=\Omega( \text{Koszul dual of}\ Lie)$. 
Now if you consider the category of $L_\infty$-algebras with $L_\infty$-morphisms, this is not encoded by an operad, but rather by the Koszul dual cooperad of Lie, which is equal to the linear dual of $Com$ up to suspension. One can prove that it is the category of fibrant-cofibrant objects of a certain model category on dg cocommutative coalgebras. 
In this case, an $L_\infty$-morphism is an $L_\infty$-quasi-isomorphism if and only if its image under the "bar construction" between $L_\infty$-algebras and dg cocommutative coalgebras is a weak equivalence.  
[You can find all the constructions and functors in Chapter 11 of http://math.unice.fr/~brunov/Operads.html. For the model category structure on dg coalgebras over the Koszul dual cooperad of an operad, please wait a little bit; I am typing this these days. :) ]