Lie models of simplicial sets and representability of the Quillen functor

Abstract

Extending the model of the interval, we explicitly define for each $n\ge 0$ a free complete differential graded Lie algebra $\mathfrak{L}_n$ generated by the simplices of $\Delta^n$, with desuspended degrees, in which the vertices are Maurer-Cartan elements and the differential extends the simplicial chain complex of the standard $n$-simplex. The family $\{\mathfrak{L}_\bullet\}_{n\ge 0}$ is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct a pair of adjoint functors between the categories of simplicial sets and complete differential graded Lie algebras given by $\langle L\rangle_\bullet=\text{ DGL} (\mathfrak{L}_\bullet,L)$ and $ \mathfrak{L}(K)=\varinjlim_K\mathfrak{L}_{\bullet} $. This new tools let us extend Quillen rational homotopy theory approach to any simplicial set $K$ whose path components are non necessarily simply connected. We prove that $\mathfrak{L} (K)$ contains a model of each component of $K$. When $K$ is a $1$-connected finite simplicial complex, the Quillen model of $K$ can be extracted from $\mathfrak{L} (K)$. When $K$ is connected then, for a perturbed differential $\partial_a$, $H_0(\mathfrak{L} (K),\partial_a)$ is the Malcev Lie completion of $\pi_1(K)$. Analogous results are obtained for the realization $\langle L\rangle$ of any complete $\text{DGL}$.