Lie models of simplicial sets and representability of the Quillen functor

Abstract

Extending the model of the interval, we explicitly define for each n ≥ 0 a free complete differential graded Lie algebra $\mathfrak{L}_n$ generated by the simplices of Δⁿ, 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 $\{\mathfrak{L{_\bullet}}\}_{n\geq0}$ is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors

$$Simpset\underset{{\left\langle \cdot \right\rangle }}{\overset{\mathfrak{L}}{\longleftrightarrow}}DGL$$

given by $\langle{L}\rangle_\bullet={\rm{DGL}}(\mathfrak{L}_\bullet,L)$ and $\mathfrak{L}(K)=\lim_{\rightarrow K}\mathfrak{L_\bullet}$. 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 $\mathfrak{L}(K)$ 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 ϑ_a, $H_0(\mathfrak{L}(K),\partial_a)$ is the Malcev Lie completion of π₁(K). Analogous results are obtained for the realization 〈L〉 of any complete DGL.