论文标题
$ \ Mathbb {e} _ \ infty $ oprads和$ p $ - ad-adic稳定同义理论理论的稳定
Stabilizations of $\mathbb{E}_\infty$ Operads and $p$-Adic Stable Homotopy Theory
论文作者
论文摘要
我们研究差分级的阶级作业和$ p $ - 亚种稳定的同型理论。我们首先构建了一个新的差异分级的阶级，我们称之为稳定的Operads。从某种意义上说，这些作业是$ \ mathbb {e} _ \ infty $ oprads的稳定。例如，我们构建了一个稳定的Barratt-Eccles Operad。我们在这些稳定的作业上开发了代数的同义理论，以及对这些稳定作业的代数的（CO）同源性操作的理论。我们注意到这些作业的有趣特性，例如，在每个贫乏中，它们具有（几乎）琐碎的同源性，而等效地，这些同源性总和到一定的广义steenrod代数的一定完成，也是高度不大的。我们还通过表明与这些作业相关的单子相干或$ \ infty $ - ，Sense Sense的形容词“稳定”是合理的。然后，我们将稳定作业的应用程序应用于$ p $ - 亚种稳定的同型理论。众所周知，在空间上的Cochains在$ \ mathbb {e} _ \ infty $ oprads上产生代数的示例。我们表明，在稳定的情况下，科雪郡的光谱收益在我们稳定的作战中产生代数的例子。此外，曼德尔（Mandell）的结果说，赋予了$ \ mathbb {e} _ \ infty $代数结构，空间上的Cochains提供了代数型号的$ p $ addic同拷贝类型。我们表明，光谱Cochains赋予了由我们稳定的作业编码的代数结构，为$ p $ - 亚种稳定同型类型的代数模型提供了代数模型。
We study differential graded operads and $p$-adic stable homotopy theory. We first construct a new class of differential graded operads, which we call the stable operads. These operads are, in a particular sense, stabilizations of $\mathbb{E}_\infty$ operads. For example, we construct a stable Barratt-Eccles operad. We develop a homotopy theory of algebras over these stable operads and a theory of (co)homology operations for algebras over these stable operads. We note interesting properties of these operads, such as that, non-equivariantly, in each arity, they have (almost) trivial homology, whereas, equivariantly, these homologies sum to a certain completion of the generalized Steenrod algebra and so are highly non-trivial. We also justify the adjective "stable" by showing that, among other things, the monads associated to these operads are additive in the homotopy coherent, or $\infty$-, sense. We then provide an application of our stable operads to $p$-adic stable homotopy theory. It is well-known that cochains on spaces yield examples of algebras over $\mathbb{E}_\infty$ operads. We show that in the stable case, cochains on spectra yield examples of algebras over our stable operads. Moreover, a result of Mandell says that, endowed with the $\mathbb{E}_\infty$ algebraic structure, cochains on spaces provide algebraic models of $p$-adic homotopy types. We show that, endowed with the algebraic structure encoded by our stable operads, spectral cochains provide algebraic models for $p$-adic stable homotopy types.
