论文标题
路径空间的分类模型
Categorical models for path spaces
论文作者
论文摘要
我们在同型理论中建立了两个结构之间的明确比较：同型相干函数的左伴，也称为刚化函数和KAN Loop groupoid fuctor。这是通过考虑刚化函数的定位，阐明hinich的结构，并使用Szczarba最初在1961年引入的一系列运算符来实现。结果，我们获得了几种简单集合路径类别的组合模型。然后，我们将路径类别的模型传递到链级，该模型现在被认为是在差异分级（DG）煤层上富集的类别，该类别用适合基础简单集合的合适代数链模型而言。这是通过Lazarev和Holstein的分类Koszul二元性启发的Cobar Foundor版本来实现的。结果，我们获得了弗朗兹（Franz）的结果的概念解释，表明从扩展的cobar构造中有天然的dg bialgebra准同态，这些cobar构造在其kan loop群体上的链条减少到链条的链条上。
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.
