论文标题
在三种不同的弱等效概念下，链条的简单界面
The simplicial coalgebra of chains under three different notions of weak equivalence
论文作者
论文摘要
我们研究了链条的简单界面，相对于三个弱等效概念的简单集。为此，我们将三个模型结构构建在任何交换环的简单式集合的类别上。弱等效性由以下方式给出：（1）分类等价的R线性化版本，（2）诱导基本群体的同构和（2）对基本组和R-HOLOMOLOGY等价的同构图和（3））R-Homology等价式。类似地，对于任何字段F，我们在连接的简单共同辅助F-Coalgebras的类别上构建了三个模型结构。在这种情况下的弱等价是（1'）映射，在应用cobar函数后诱导DG代数的准同构，（2'）映射诱导DG代数的准代数形态，在应用了Cobar Functor的局部化版本后，以及（3''''''''''''）quasi-Isomorphismss。在（3） - （3'）背景下的Goers先前工作的基础上，我们证明，当F封闭F时，链的简单f-Coalgebra定义了每对模型类别的同型全且忠实的左quillen fuctor。更普遍的是，当F是一个完美的领域时，我们会根据适当的同型固定点相对于F的绝对Galois组比较三对模型类别。
We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring R. The weak equivalences are given by: (1) an R-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an R-homology equivalence between universal covers, and (3) R-homology equivalences. Analogously, for any field F, we construct three model structures on the category of connected simplicial cocommutative F-coalgebras. The weak equivalences in this context are (1') maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2') maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3') quasi-isomorphisms. Building on previous work of Goerss in the context of (3)-(3'), we prove that, when F is algebraically closed, the simplicial F-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each pair of model categories. More generally, when F is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of F.
