Question

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 $C^\ast(\Delta^n)$?

Answer 1

I thought I'd expand on Fernando's comments and add a little bit. Your instincts are correct, there is in fact a finite dimensional model of $\Omega^{\ast}( \Delta_{n} )$, which are called the polynomial differential forms: $$k[ \Delta_{n} ] = k [ t_0, \ldots , t_n, dt_0, \ldots, dt_n ] / \left( (\sum t_i) - 1, \sum dt_i \right)$$ where $|t_i| = 0, d(t_i) = dt_i.$ Note that we take the free graded commutative algebra, so $dt_i^{2} = 0$ for degree reasons.

Each $k[\Delta_n]$ is a differential graded commutative algebra, and the assignment $$n \mapsto k[\Delta_n]$$ is a simplicial object in the category of differential graded commutative algebras.

In their book "On PL DeRham Theory and Rational Homotopy Type" (Memoirs of the AMS, Number 179), Bousfield and Gugenheim demonstrate that the simplicial sets $cdgA(R, L \otimes k[\Delta_{\bullet}] )$ give a simplicial enrichment of the model category structure on commutative differential graded algebras which behave like a simplicial model category. In "Homological Algebra of Homotopy Algebras," Hinich shows that over a field of characteristic zero, this is true for the model category structure on $O$-algebras for any operad $O$, ie, $dgOA(R, L \otimes k[\Delta_{\bullet}])$ is a simplicial enrichment of the category $dgOA$ which behaves like a simplicial model category.

Now, I don't know how to answer your question for the set $MC(\mathfrak{g}^{\ast})$. However, when you consider $MC(\mathfrak{g}^{\ast} \otimes R)$ for some finite dimensional, nilpotent commutative dgA $R$, we have the identification: $$MC( \mathfrak{g} \otimes R ) = dgLie ( \Omega(R^{\vee}), \mathfrak{g}^{\ast}).$$ where $R^{\vee}$ is the hom-dual commutative differential graded coalgebra to $R$, and $\Omega$ is the cobar construction which carries commutative differential graded coalgebra to quasifree dg Lie algebras. In particular, the right-hand side is precisely the points in the simplicial set $$dgLie(\Omega(R^{\vee}), \mathfrak{g}^{\ast} \otimes k[\Delta^{\bullet}]),$$ and the simplicial set is a Kan-complex because every dg Lie algebra is fibrant, and $\Omega$ takes values in cofibrant dg-Lie algebras.