Lie models of simplicial sets and representability of the Quillen functor
(2020) Israel Journal of Mathematics — Vol. 238, n° 1, p. 313-358 (2020)
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- Abstract
- Extending the model of the interval, we explicitly define for each n ≥ 0 a free complete differential graded Lie algebra Ln 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 { L∙} n≥ is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors Simpset⟷L⟨⋅⟩DGL given by ⟨L⟩∙=DGL(L∙,L) and L(K)=lim→KL∙. 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 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 L(K). When K is connected then, for a perturbed differential ϑa, H(L(K) , ∂ a) is the Malcev Lie completion of π1(K). Analogous results are obtained for the realization 〈L〉 of any complete DGL.
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Buijs, U., Félix, Y., Murillo, A., & Tanré, D. (2020). Lie models of simplicial sets and representability of the Quillen functor. Israel Journal of Mathematics, 238(1), 313-358. https://doi.org/10.1007/s11856-020-2026-8 (Original work published 2020)