The Stable Homotopy Category Has a Unique Model at the Prime 2
The Stable Homotopy Category Has a Unique Model at the Prime 2
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稳定同伦范畴在素数 2 处有一个独特的模型
DOI：
10.1006/aima.2001.2009
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发表时间：
2001
影响因子：
1.7
通讯作者：
S. Schwede
中科院分区：
文献类型：
--
作者：
S. Schwede
稳定同伦范畴一直是代数拓扑学家广泛研究的范畴。对于许多应用来说，处理光谱的点集水平模型是方便的，甚至是必要的，而不是处理到同伦，并且计算的结果可以取决于模型的选择。近年来，关于稳定同伦范畴的许多新模型被构造出来。在Quillen[Qui]的意义下具有闭合模型范畴的结构和在此上下文中具有谱范畴t的许多示例特别有用[BF，Rob87，EKMM，HSS，LYD，MMSS]。此外，所有已知的例子都符合‘相同同伦理论’(在技术术语中，人们谈论Quillen等价模型范畴[Hov，Def.1.3.12]。因此，不仅同伦范畴，而且函数空间的高阶信息，如Toda括号、同伦余极限和同伦类型也是重合的。在两个Quillen等价模型范畴中，每个同伦理论问题的答案都是相同的。在一个模型范畴中，一个人可以通过形式上对弱等价类进行倒置来传递到相应的同伦范畴。然而，通过到同伦范畴会丢失信息，并且通常不能从同伦范畴恢复‘同伦理论’，见2.1和2.2的两个例子。本文证明了与一般情形不同，稳定同伦范畴完全决定了稳定同伦理论2-局部。我们证明了一个唯一性定理，即2-局部谱的稳定同伦范畴本质上只存在一个模型范畴结构|稳定同伦范畴在素数2处没有‘奇异’模型。
The stable homotopy category has been extensively studied by algebraic topologists for a long time. For many applications it is convenient or even necessary to work with point set level models of spectra as opposed to working up-to-homotopy, and the outcome of a calculation can depend on the choice of model. In recent years many new models for the stable homotopy category have been constructed. It is especially useful to have the structure of a closed model category in the sense of Quillen [Qui] and many examples of spectra categories t into this context [BF, Rob87, EKMM, HSS, Lyd, MMSS]. Moreover all known examples capture the ‘same homotopy theory’ { in technical terms one speaks of Quillen equivalent model categories [Hov, Def. 1.3.12]. Hence not only the homotopy categories, but also higher order information such as Toda brackets, homotopy colimits and homotopy types of function spaces coincide. In two Quillen equivalent model categories the answer to every homotopy theoretic question comes out the same. In a model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences. However, passage to the homotopy category loses information and in general the ‘homotopy theory’ can not be recovered from the homotopy category, see 2.1 and 2.2 for two examples. In this paper we show that in contrast to the general case, the stable homotopy category completely determines the stable homotopy theory 2-locally. We prove a uniqueness theorem which says that there is essentially only one model category structure underlying the stable homotopy category of 2-local spectra | the stable homotopy category has no ‘exotic’ models at the prime 2.