THE UNIQUENESS OF INFINITE LOOP SPACE MACHINES
THE UNIQUENESS OF INFINITE LOOP SPACE MACHINES
复制标题
无限循环空间机器的独特性
DOI：
10.1016/0040-9383(78)90026-5
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发表时间：
1978
期刊：
影响因子：
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通讯作者：
R. Thomason
中科院分区：
文献类型：
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作者：
Jon P. May;R. Thomason
无限循环空间机是一种用更简单的空间级数据构造谱的函子。有许多这样的机器是已知的[14,8,151]。他们接受的数据有些不同。更糟糕的是，它们是由如此广泛不同的拓扑结构给出的，以至于当提供相同的数据时，它们会改变我们的等效光谱，这一点远不明显。本文的目的是证明所有满足某些合理性质的机器实际上都能产生等效谱。Segal的机器[15]满足了这些属性，但需要使用比文献中其他机器所能接受的更通用的输入数据。我们推广May的机器[8,9]，使其在必要的一般性中工作并满足必要的性质。因此，May和Segal机器是等价的。这个证明将说明其他机器的相应推广所涉及的内容，我们非常有信心，详尽的逐例验证将得出结论，即实际上只有一个无限循环空间机器。为了避免给人留下错误的印象，我们赶紧补充说，这并不意味着我们现在可以抛弃所有的明确结构，只留下一个。这些构造的目的是证明定理和进行计算，就像在[11]中概述的那样，这样的应用可能只有一台或另一台机器可以访问。例如，从E，环空间到E，环光谱的过渡，基于E，环光谱的束和纤维理论的分类光谱的构建，以及从E，环光谱到H，环光谱的过渡[l2, 131]是一个计算上强大的思想循环的一部分，它依赖于May机器的特殊几何结构的使用。这里的重点是，虽然现在有了无限循环空间机器的唯一性定理，但乘法无限循环空间工厂的装配线却没有唯一性定理。另一方面，西格尔的机器有一个明显的优势，那就是比其他机器构造起来要简单得多。此外，它将在我们的理论中发挥规范作用。我们不直接比较两台机器，而是将它们分别与Segal的机器进行比较。本文对2006年无限循环空间机的输入数据作了一般性的讨论，并给出了2004年无限循环空间机的实例构造方法。我们在第2节和第3节中证明了唯一性定理，只是我们把关于谱的一个关键结果的证明放在了第1附录中。我们在§§S和§6中给出May的机器的承诺泛化。按照这门学科的传统，还有一个关于联合的附录。在证明我们的新结果的过程中，我们不得不重新发展和系统化无限循环空间理论的基础，我们希望本文可以作为其主要思想和技术的可读来源。第一作者想承认关键的新想法完全归功于第二作者，后者想感谢瓦尔德豪森的一次非常有帮助的谈话。两位作者都希望承认，他们的基本见解来自Fiedorowicz的论文b[6]。
AN INFINITE loop space machine is a functor which constructs spectra out of simpler space level data. There are many such machines known [14, 8,151. They differ somewhat in the data they accept. Worse, they are given by such widely disparate topological constructions that it is far from obvious that they turn our equivalent spectra when fed the same data. The purpose of this paper is to prove that all machines which satisfy certain reasonable properties do in fact turn out equivalent spectra. The properties are satisfied by Segal’s machine [l5], but require use of somewhat more general input data than the other machines in the literature are geared to accept. We generalize May’s machine [8, 9] so that it acts in the requisite generality and satisfies the requisite properties. Thus the May and Segal machines are equivalent. This proof will illustrate what would be involved in the corresponding generalization of other machines, and we are quite confident that an exhaustive case-by-case verification would lead to the conclusion that there is really only one infinite loop space machine.To avoid leaving a wrong impression, we hasten to add that this does not mean we can now discard all but one of the explicit constructions. The purpose of the constructions is to prove theorems and make calculations, of the sort sketched in [ll], and such applications may only be accessible to one or another of the machines. For example, the passage from E, ring spaces to E, ring spectra, the construction of classifying spectra for bundle and fibration theories oriented with respect to an E, ring spectrum, and the passage from E, ring spectra to H, ring spectra [l2, 131 are part of a calculationally powerful circle of ideas which depends on use of the particular geometry of May’s machine. The point here is that while there is now a uniqueness theorem for infinite loop space machines, there is no uniqueness theorem for the assembly lines of multiplicative infinite loop space factories. On the other hand, Segal’s machine has the distinct advantage of being very much simpler to construct than the others. Moreover, it will play a canonical role in our theory. Rather than compare two machines directly, we compare each of them to Segal’s machine. We give a general discussion of the input data of infinite loop space machines in 6 I and give a way to construct examples in 04. We prove the uniqueness theorem in $02 and 3, except that we relegate the proof of a key result about spectra to the first appendix. We give the promised generalization of May’s machine in §§ S and 6. As is traditional in this subject, there is also an appendix about cofibrations. In the course of proving our new results, we have had to redevelop and systematize the foundations of infinite loop space theory, and it is our hope that the present paper can serve as a readable source for its main ideas and techniques. The first author wishes to acknowledge that the key new idea is entirely due to the second author and the latter wants to thank Waldhausen for a very helpful conversation. Both authors wish to acknowledge that the basic insight comes from Fiedorowicz’paper [6].