On the homotopy groups of symmetric spectra-猫眼课题宝

关于对称谱的同伦群

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2006

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S. Schwede

我们在对称谱的同伦群上构造自然数集的单射自映射幺半群的自然、温和的作用。这种额外的代数结构允许对与同构、半稳定性以及对称谱的朴素同伦群和真实同伦群之间的关系相关的各种现象有概念性和统一的理解。 55P42； 55U35 对称谱是一个易于定义且方便的稳定同伦类别模型，具有良好的粉碎产品。对称环谱首先在代数 K 理论和拓扑 Hochschild 同调的背景下以“球体上的 FSP”的名称出现。 1993年左右，杰夫·史密斯做出了重要的观察，即“球面上的FSP”是关于缔合和交换粉碎乘积的“对称谱”类别中的幺半群，并且他怀疑兼容的模型类别结构，以便人们获得同伦类别“稳定同伦类别（对于对称谱），A1环谱的同伦类别（对于对称环谱），分别是E1环谱的同伦类别（对于交换律）对称环光谱）。 Hovey、Shipley 和 Smith 在 [3] 中制定了各种模型结构的细节。也许对称谱的唯一棘手点是稳定等价不能通过查看稳定同伦群来定义（定义为对称谱中项的不稳定同伦群的经典顺序余极限）。形式上反转同构，即那些引起稳定同伦群同构的态射，会留下太多的同伦类型。相反，霍维、希普利和史密斯引入了一个严格更大的稳定等价类，定义为在所有上同调理论上引发同构的态射。 ‐同构和稳定等价之间的区别之前至少让本作者感到困惑。朴素定义的同伦群和“真正的”同伦群（稳定同伦范畴中球体的态射）之间的精确关系在很大程度上是神秘的（尽管 Shipley 的检测函子 [7，第 3 节] 对此提供了相当多的启示）。在本文中，我们宣传并系统地利用对称谱的（经典）同伦群上的额外代数结构，在作者看来，

We construct a natural, tame action of the monoid of injective self-maps of the set of natural numbers on the homotopy groups of a symmetric spectrum. This extra algebraic structure allows a conceptual and uniform understanding of various phenomena related to ‐isomorphisms, semistability and the relationship between naive and true homotopy groups for symmetric spectra. 55P42; 55U35 Symmetric spectra are an easy-to-define and convenient model for the stable homotopy category with a nice smash product. Symmetric ring spectra first showed up under the name “FSP on spheres” in the context of algebraic K ‐theory and topological Hochschild homology. Around 1993, Jeff Smith made the crucial observation that the “FSPs on spheres” are the monoids in a category of “symmetric spectra” with respect to an associative and commutative smash product, and he suspected compatible model category structures so that one obtains as homotopy categories “the” stable homotopy category (for symmetric spectra), the homotopy category of A1 ring spectra (for symmetric ring spectra), respectively the homotopy category of E1 ring spectra (for commutative symmetric ring spectra). The details of various model structures were worked out by Hovey, Shipley and Smith in [3]. Maybe the only tricky point with symmetric spectra is that the stable equivalences can not be defined by looking at stable homotopy groups (defined as the classical sequential colimit of the unstable homotopy groups of the terms in a symmetric spectrum). Formally inverting the ‐isomorphisms, ie, those morphisms which induce isomorphisms of stable homotopy groups, leaves too many homotopy types. Instead, Hovey, Shipley and Smith introduce a strictly larger class of stable equivalences, defined as the morphisms which induce isomorphisms on all cohomology theories. The difference between ‐isomorphisms and stable equivalences has previously confused at least the present author. The precise relationship between the naively defined homotopy groups and the “true” homotopy groups (morphisms from sphere in the stable homotopy category) has largely been mysterious (although Shipley’s detection functor [7, Section 3] sheds considerable light on this). In this paper we advertise and systematically exploit extra algebraic structure on the (classical) homotopy groups of a symmetric spectrum which, in the authors opinion,

On the homotopy groups of symmetric spectra

关于对称谱的同伦群

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