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

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?

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How do you define your model structure on dgcu(k)? – Kevin Lin May 18 2011 at 18:06
Hi Kevin, the model structure on $dgcu(k)$ is a bit less explicit than the one on $dgca(k)$. It is described, for instance, in section 3 of Vladimir Hinich's "DG coalgebras as formal stacks", but I guess it actually dates back to Dan Quillen's "Rational homotopy theory" paper. – domenico fiorenza May 18 2011 at 18:37
mathoverflow.net/questions/40772/… maybe useful to understand weak equivalences for coalgebras. – Daniel Pomerleano May 19 2011 at 7:08
To make sure that weak equivalences agree is exactly the reason the notion of weak equivalences for coalgebras is stronger than quasi-isomorphism. In the simply-connected, finite type case that Quillen was considering, rational homotopy theory tells you that weak equivalence is enough... – Daniel Pomerleano May 19 2011 at 7:21

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).

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Dear Domenico, you are answering your own question ? I am right ? ;) – Bruno V. May 26 2011 at 21:53
Eager to get the self-learner badge :) (no, really: that was to report here Jon's suggestion, I thought that was best suited as an answer rather than as a comment to my question) – domenico fiorenza Jun 7 2011 at 12:11
I am collecting some of that material from Jonathan Pridham's arcticle here: ncatlab.org/nlab/show/… – Urs Schreiber Feb 6 at 18:52

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. :) ]

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 Dear Bruno, thanks a lot for this answer and for the pointer to your book! – domenico fiorenza Jun 7 2011 at 12:13 Wow, the first two paragraphs just cleared something up for me which I'd been quite confused by over on this thread: mathoverflow.net/questions/62386/…. Thanks for writing such a clear answer two years ago! – David White Feb 26 at 21:40

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 $dgcu(k)$.

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yes, it is the category of fibrant objects in $DG_\mathbf{Z}Sp(k)$, as I'm saying in my self-answer above. Actually in that answer I'm leaving quite implicit the fact that "category of fibrant objects" rather than "model category" is the right answer to my original question: thanks for having stressed this. – domenico fiorenza May 20 2011 at 14:01