If $S_0, S_1$ are two solutions of MaurerCartan equation $dS+\frac{1}{2}{S,S}=0$ for a dgLie algebra $g$, do we have a suitable concept of homotopy between $S_0$ and $S_1$?

To continue what Kevin Lin mentioned, a discussion on the relation between homotopy and gauge equivalence for solutions of the MaurerCartan equation is in the last section of Manetti's "Deformation theory via differential graded Lie algebras". To sum up, you can define two solutions $S_0,S_1\in MC_{\mathfrak{g}}(A)$ to be homotopic iff there exists an element $S\in MC_{\mathfrak{g}[t,dt]}(A)$ with $S(0)=S_0$ and $S(1)=S_1$. Now it turns out that in a certain sense this homotopy equivalence and the gauge equivalence are the same thing, however the homotopy definition is preferable since it extends to $L_\infty$ algebras. 


$S_0$ and $S_1$ are said to be homotopy equivalent if there is a MaurerCartan element $S(t,dt)$ in the dgla $g[t,dt]$ such that $S(0)=S_0$ and $S(1)=S_1$. It is not completely clear at first that this is an equivalence relation, but actually it is. Indeed much more is true and the homotopy equivalence just described is just the tip of the iceeberg. To see this, rewrite $g[t,dt]$ as $g\otimes\Omega^1$, where $\Omega^1$ is the differential graded commutative algebra of polynomial differential forms on the (algebraic) 1simplex. Then one sees this is the beginning of a simplicial dgla $g\otimes\Omega^\bullet$, and taking MaurerCartan elements produces a simplicial set $MC(g\otimes \Omega^\bullet)$. This simplicial set turns out to be a Kan complex and the fact that the homotopy relation between solution of the MaurerCartan equation on $g$ is an equivalence relation is precisely the 'horn filling' property of this Kan complex. A good reference is Getzler's [math/0404003] Lie theory for nilpotent Linfinity algebras (in formal deformation theory one produces a nilpotent dgla out of an arbitrary one by tensoring it with the maximal ideal $m_A$ of a local Artin algebra $A$) 

