Homotopy theory of complete Lie algebras and Lie models of simplicial sets
(2018) Journal of Topology — Vol. 11, n° 3, p. 799-825 (2018)
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- Abstract
- In a previous work, by extending the classical Quillen construction to the non‐simply connected case, we have built a pair of adjoint functors, model and realization, between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow‐up of this work. We show that when X is a finite connected simplicial set, then ⟨𝔏X⟩ coincides with ℚ∞X+, the disjoint union of the Bousfield–Kan completion of X with an external point. We also define a model category structure on cDGL making 𝔏 and ⟨−⟩ a Quillen pair, and we construct an explicit cylinder. In particular, these functors preserve homotopies and weak equivalences and therefore, this gives the basis for developing a cDGL rational homotopy theory for all spaces.
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Urtzi, B., Félix, Y., Aniceto, M., Daniel, T., & et al. (2018). Homotopy theory of complete Lie algebras and Lie models of simplicial sets. Journal of Topology, 11(3), 799-825. https://doi.org/10.1112/topo.12073 (Original work published 2018)