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Extending the model of the interval, we explicitly define for each n ≥ 0 a free complete differential graded Lie algebra $$\mathfrak{L}_n$$ L n generated by the simplices of Δ 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 $$\{\mathfrak{L{_\bullet}}\}_{n\geq0}$$ { L ∙ } n ≥ 0 is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors $$Simpset\underset{{\left\langle \cdot \right\rangle }}{\overset{\mathfrak{L}}{\longleftrightarrow}}DGL$$ S i m p s e t ⟷ L ⟨ ⋅ ⟩ D G L given by $$\langle{L}\rangle_\bullet={\rm{DGL}}(\mathfrak{L}_\bullet,L)$$ ⟨ L ⟩ ∙ = D G L ( L ∙ , L ) and $$\mathfrak{L}(K)=\lim_{\rightarrow K}\mathfrak{L_\bullet}$$ L ( K ) = lim → K L ∙ . This new tool lets us extend the Quillen rational homotopy theory approach to any simplicial set K whose path components are not necessarily simply connected. We prove that $$\mathfrak{L}(K)$$ 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 $$\mathfrak{L}(K)$$ L ( K ) . When K is connected then, for a perturbed differential ϑ a , $$H_0(\mathfrak{L}(K),\partial_a)$$ H 0 ( L ( K ) , ∂ a ) is the Malcev Lie completion of π 1 ( K ). Analogous results are obtained for the realization 〈 L 〉 of any complete DGL.

Lie models of simplicial sets and representability of the Quillen functor