The Maurer--Cartan equation is the equation: $$d\gamma+\frac 12[\gamma,\gamma]=0$$ where $\gamma$ represents a degree one element in a differential graded Lie algebra $\mathfrak g^\ast$. Let's denote the set of solutions by $MC(\mathfrak g^\ast)$. I need to have some notion of (higher) homotopies between elements of the set $MC(\mathfrak g^\ast)$. One way of doing this would be to define a simplicial set $\mathsf{MC}(\mathfrak g^\ast)$ whose zero simplices are the set $MC(\mathfrak g^\ast)$. I have indeed seen the phrase "simplicial set of solutions to the Maurer--Cartan equation" in papers. Is there a standard construction of this simplicial set? If so, how should I think about it, and what are some good references?

In fact, it seems that the $n$-simplices in $\mathsf{MC}(\mathfrak g^\ast)$ should be $MC(\mathfrak g^\ast\otimes\Omega^\ast(\Delta^n))$ where $\Omega^\ast(\Delta^n)$ is the differential graded algebra of differential forms on the standard simplex $\Delta^n$. Can I use something smaller (hopefully finite dimensional) instead of $\Omega^\ast(\Delta^n)$? Perhaps just the simplicial cochain complex $C^\ast(\Delta^n)$?