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One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$. For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, there is a dg Lie algebra whose Maurer-Cartan elements are the $P$-algebras structures on a given vector space (or the $P_{\infty}$-structures on a chain complex). The set of Maurer-Cartan elements $MC(g)$ forms an algebraic variety.

Now, considering the gauge group $G$ associated to the degree zero part of $g$, which acts on $MC(g)$, one can define the moduli set of Maurer-Cartan elements $MC(g)/G$. This moduli set forms an algebraic stack (more precisely one can make it into a functor which is the functor of points of an algebraic stack).

My question is concerned with the $L_{\infty}$ generalization of these facts. Indeed, a lot of deformation theoretic problems are actually controled by nilpotent, at least complete, dg $L_{\infty}$-algebras. One can still define Maurer-Cartan elements, and also a simplicial set of Maurer-Cartan elements via the following way: for every commutative differential graded algebra $A$, one forms the extension $g\otimes A$ and endow it with a nilpotent/complete $L_{\infty}$-algebra structure. Then, one defines $MC_{\bullet}(g)=MC(g\otimes\Omega_{\bullet})$, where $\Omega_n$ stands for Sullivan's construction of de Rham polynomial forms on the standard simplex $\Delta^n$. The set of connected components $\pi_0MC_{\bullet}(g)$ defines the moduli set of Maurer-Cartan elements in the $L_{\infty}$ case, since there is no gauge group anymore. When $g$ is a dg Lie algebra we have $\pi_0MC_{\bullet}(g)\cong MC(g)/G$.

Does the Maurer-Cartan moduli set of a dg $L_{\infty}$-algebra form, under suitable assumptions, an algebraic stack ? If this property holds, are there any references about it ?

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    $\begingroup$ These stacks etc. are supposed to be invariant under quasi-isomorphisms, right? And any L-infinity algebras is quasi-isomorphic to an honest dg Lie algebra. $\endgroup$ Feb 7, 2014 at 8:42
  • $\begingroup$ You're right ! I completely forgot this rectification result, actually available for general homotopy algebras over an operad. $\endgroup$ Feb 10, 2014 at 14:45

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It is not completely true that the set of gauge equivalence classes of Maurer-Cartan elements of a dg Lie algebra is invariant under quasi-isomorphisms. Here is an example (which I learned from Professor Andrey Lazarev): Let $\mathbb L(x)$ be the graded free Lie algebra generated by an element $x$ of degree $-1$. Then $\mathbb L(x)$ is 2-dimensional generated by $x$ and $[x,x]$ (by the graded Jacobi identity it follows that $[x,[x,x]]=0$). Consider now the dg Lie algebra $L=(\mathbb L(x),d(x)=-1/2[x,x])$, which has trivial homology.

We have that $x$ is a Maurer-Cartan element of $L$, but since $L$ is trivial in degree zero, it has a trivial gauge group and consequently $x$ and $0$ are not gauge-equivalent. However, $L$ is quasi-isomorphic to the trivial dg Lie algebra, which has a single Maurer-Cartan element.

Regarding the question of the thread: I have written a note regarding a possible candidate for gauge groups for nilpotent $L_\infty$-algebras. If there are no mistakes there, it is given by the degree 0 part of the $L_\infty$-algebra with the underlying additive structure. In particular, it is not a generalization of the gauge group of a dg Lie algebra, but have the same underlying set.

People I've talked to about this are pretty skeptical, so read at your own risk :).

https://drive.google.com/file/d/1M8WmINGHXIRAktr6yNy5a_qCdrlsyiwB/view?usp=sharing

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