Categorical models for path spaces
Categorical models for path spaces
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DOI：
10.1016/j.aim.2023.108898
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发表时间：
2022-01
影响因子：
1.7
通讯作者：
Emilio Minichiello;M. Rivera;M. Zeinalian
中科院分区：
文献类型：
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作者：
Emilio Minichiello;M. Rivera;M. Zeinalian
我们建立了同伦理论中两种结构的显式比较：同伦相干神经函子的左伴随，也称为刚性函子，和Kan环群函子。这是通过考虑刚性函子的局域化，展开Hinich的构造，并使用Szczarba于1961年最初引入的一系列算子来实现的。得到了简单集路径范畴的几种组合模型。然后，我们传递到链级，并描述路径范畴的模型，现在被认为是在微分梯度（dg）余代数上丰富的范畴，根据一个合适的代数链模型为基础的简单集。这是通过受Lazarev和Holstein的绝对Koszul对偶启发的cobar函子的一个版本来实现的。由此，我们得到了Franz关于简化集链上的扩展cobar构造与它的Kan环群链之间存在天然的dg双代数拟同构的结果的概念解释。
We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of the homotopy coherent nerve functor, also known as the rigidification functor, and the Kan loop groupoid functor. This is achieved by considering localizations of the rigidification functor, unraveling a construction of Hinich, and using a sequence of operators originally introduced by Szczarba in 1961. As a result, we obtain several combinatorial models for the path category of a simplicial set. We then pass to the chain-level and describe a model for the path category, now considered as a category enriched over differential graded (dg) coalgebras, in terms of a suitable algebraic chain model for the underlying simplicial set. This is achieved through a version of the cobar functor inspired by Lazarev and Holstein's categorical Koszul duality. As a consequence, we obtain a conceptual explanation of a result of Franz stating that there is a natural dg bialgebra quasi-isomorphism from the extended cobar construction on the chains of a reduced simplicial set to the chains on its Kan loop group.