Model category structures on the category of $L_\infty$-algebras

Question

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?

Answer 1

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. :) ]

Answer 2

I've been discussing this with Jonathan Pridham, who pointed my attention to his Unifying derived deformation theories, where a model category structure is described on a suitable subcategory $DG_{\mathbb{Z}}Sp(k)$ of $dgcu(k)$. (actually, the definition of $DG_{\mathbb{Z}}Sp$ is more general, but on a characteristic zero field $k$ it is naturally a subcategory od $dgcu(k)$).

The category $DG_{\mathbb{Z}}Sp(k)$ has a few remarkable properties: on the one hand it is Quillen equivalent to the larger category $dgcu(k)$ (endowed with the Hinich's model structure); on the other hand $L_\infty$-algebras over $k$ are precisely the fibrant objects in $DG_{\mathbb{Z}}Sp(k)$ and a morphism $\varphi$ between $L_\infty$-algebras is a fibration (resp. a weak equivalence) in $DG_{\mathbb{Z}}Sp(k)$ if its image via the "linearization" functor $L_\infty(k)\to chains(k)$ is a fibration (resp. a weak equivalence) in $chains(k)$, i.e. if the linearization of $\varphi$ is surjective (resp. a quasi-isomorphism).

Answer 3

I think there is no model structure on the category of $L_\infty$-algebras. The category of $L_\infty$-algebra is nevertheless a category of fibrant objects.

Concerning your last question, yes, $L_\infty$-quasi-isomorphisms are weak equivalences between fibrant objects in $dgcu(k)$.