  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 $\text{DGL}$.
