论文标题
严格的代数模型，用于理性参数光谱II
Strict algebraic models for rational parametrised spectra II
论文作者
论文摘要
在本文中，我们扩展了Sullivan的Pl de Rham理论，以获取简单的代数模型，以用于参数光谱的理性同义理论。这简化并补充了基于Quillen的理性同义理论的Arxiv：1910.14608的结果。 根据沙利文（Sullivan）的说法，有限的贝蒂（Betti）数字的nilpotent Space $ x $的合理同型类型完全取决于交换性差分等级的代数$ a $ a $ a $在有理共同体上建模杯子。在本文中，我们将拓扑与代数之间的对应关系扩展到参数式的稳定同型理论：对于与CDGA $ A $相对应的空间$ x $，我们证明，超过$ x $的参数谱的特定合理同拷贝类别与差速器分级$ a $ a $ -modules之间的等效性。虽然不满，但我们认为的理性同拷贝类别包含一类大量的参数谱。我们开发的方法的简单性可以使用差分模块进行参数化的稳定同义理论的直接计算。 为了说明我们方法的有用性，我们构建了拓扑结构的代数翻译的综合词典；为基本变化函数，纤维稳定，参数式后的后切片，纤维粉碎产物和纤维稳定图的复合物提供代数模型。
In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of arXiv:1910.14608, which are based on Quillen's rational homotopy theory. According to Sullivan, the rational homotopy type of a nilpotent space $X$ with finite Betti numbers is completely determined by a commutative differential graded algebra $A$ modelling the cup product on rational cohomology. In this article we extend this correspondence between topology and algebra to parametrised stable homotopy theory: for a space $X$ corresponding to the cdga $A$, we prove an equivalence between specific rational homotopy categories for parametrised spectra over $X$ and for differential graded $A$-modules. While not full, the rational homotopy categories we consider contain a large class of parametrised spectra. The simplicity of the approach that we develop enables direct calculations in parametrised stable homotopy theory using differential graded modules. To illustrate the usefulness of our approach, we build a comprehensive dictionary of algebraic translations of topological constructions; providing algebraic models for base change functors, fibrewise stabilisations, parametrised Postnikov sections, fibrewise smash products, and complexes of fibrewise stable maps.
