Homotopy theory of complete Lie algebras and Lie models of simplicial sets
Homotopy theory of complete Lie algebras and Lie models of simplicial sets
复制标题
完全李代数的同伦理论和单纯集李模型
DOI：
10.1112/topo.12073
复制
发表时间：
2016
影响因子：
1.1
通讯作者：
Daniel Tanr'e
中科院分区：
文献类型：
--
作者：
Urtzi Buijs;Yves F'elix;A. Murillo;Daniel Tanr'e
在先前的一项工作中，通过将经典的奎伦构造推广到非单连通情形，我们在单纯集范畴和完备微分分次李代数范畴之间构建了一对伴随函子，即模型函子和实现函子。本文是这项工作的后续。我们证明当\(X\)是一个有限连通单纯集时，\(\langle LX\rangle\)与\(Q^{\infty}X_{+}\)一致，\(Q^{\infty}X_{+}\)是\(X\)的布斯菲尔德 - 坎完备化与一个外点的不交并。我们还在\(cDGL\)上定义了一个模型范畴结构，使得\(L\)和\(\langle - \rangle\)成为一个奎伦对，并且我们构造了一个明确的柱体。特别地，这些函子保持同伦和弱等价，因此，这为针对所有空间发展\(cDGL\)有理同伦理论提供了基础。
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.
影响因子：
1.7
作者：
Lazarev A
通讯作者：
Lazarev A