# Stacks and Maurer-Cartan elements

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 ?

• 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. – Fernando Muro Feb 7 '14 at 8:42
• You're right ! I completely forgot this rectification result, actually available for general homotopy algebras over an operad. – Sinan Yalin Feb 10 '14 at 14:45