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Lie models of simplicial sets and representability of the Quillen functor. (English) Zbl 1460.55011

The authors bring to fruition the program initiated by R. Lawrence and D. Sullivan [Fundam. Math. 225, 229–242 (2014; Zbl 1300.55024)]. They construct a complete DG Lie algebra \( \mathcal{L}(\Delta^n)\) for the standard \(n\)-simplex together with coface and codegeneracy operators such that the family \(\{\mathcal{L}(\Delta^n)\}_{n \geq 0} \) forms a cosimplicial DG Lie algebra. They obtain a complete DG Lie algebra \({\mathcal L}(K)\) for any simplicial complex \(K\) as a consequence. Further, they construct a geometric realization functor for complete DG Lie algebras so that the pair give adjoint functors between the categories of simplicial sets and of complete DG Lie algebras. The results provide a broad extension of Quillen’s framework for the rational homotopy theory of simply connected spaces developed in [D. Quillen, Ann. Math. (2) 90, 205–295 (1969; Zbl 0191.53702)].

MSC:

55P62 Rational homotopy theory
55U10 Simplicial sets and complexes in algebraic topology
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References:

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