Lie models of simplicial sets and representability of the Quillen functor
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LIE MODELS OF SIMPLICIAL SETS AND REPRESENTABILITY OF THE QUILLEN FUNCTOR
BY
Urtzi Buijs Departamento de Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´ alaga M´ alaga, Ap. 59 29080, Spain e-mail: [email protected] AND
´lix Yves Fe Institut de Math´ematiques et Physique Universit´e Catholique de Louvain-la-Neuve Louvain-la-Neuve, Belgium e-mail: [email protected] AND
Aniceto Murillo Departamento de Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´ alaga M´ alaga, Ap. 59 29080, Spain e-mail: [email protected] AND
´ Daniel Tanre D´epartement de Math´ematiques, UMR-CNRS 8524, Universit´e de Lille 1 Villeneuve d’Ascq Cedex 59655, France e-mail: [email protected]
Received May 29, 2017 and in revised form July 1, 2019
1
´ ´ U. BUIJS, Y. FELIX, A. MURILLO AND D. TANRE
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Isr. J. Math.
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 ∆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 {L• }n≥0 is endowed with a cosimplicial differential graded Lie algebra structure which we use to construct two adjoint functors SimpSet
o
L
/
DGL
h·i
given by hLi• = DGL(L• , L) and L(K) = limK 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 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 ∂a , H0 (L(K), ∂a ) is the Malcev Lie completion of π1 (K). Analogous results are obtained for the realization hLi of any complete DGL.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.
The BCH product and the Lawrence–Sullivan model for the interval . . . . . . . . . . . . . . . . . . . . . . .
3 7
2.
Sequences of compatible models of ∆
. . . . . . . . . . .
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3.
Symmetric models of ∆ and the cosimplicial structure . .
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4.
The realization functor and its adjoint . . . . . . . . . . .
23
5.
The L-model of K . . . . . . . . . . . . . . . . . . . . . .
29
6.
Differential graded Lie coalgebras and the Transfer Theorem 32
7.
The cosimplicial structure via a transfer . . . . . . . . . .
37
8.
Representability of the Quillen realization functor
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9.
The Malcev completion of the fundamental group . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. . . .
Vol. TBD, 2020
LIE MODELS OF SIMPLICIAL SETS
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Introduction In [18], R. Lawrence and D. Sullivan raise the following observation and subsequent general questions: the rational singular chains on a cellular complex are naturally endowed with a structure of cocommutative, coassociative infinity coalgebra and hence, taking the commutators of a “generalized bar constru
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