Model Categories of Diagram Spectra
Model Categories of Diagram Spectra
复制标题
图谱的模型类别
DOI：
10.1112/s0024611501012692
复制
发表时间：
2001
影响因子：
1.8
通讯作者：
B. Shipley
中科院分区：
文献类型：
--
作者：
Michael A. Mandell;Jon P. May;S. Schwede;B. Shipley
在基空间范畴T中，给出了图空间和图谱的基本理论。这些函子D →T对于一个适当的小拓扑范畴D。当D是对称monoidal时，有一个smash积使D空间范畴具有对称monoidal结构。例子包括经典定义的前谱、杰夫·史密斯定义的对称谱、正交谱、在域范畴中用正交群代替对称群的对称谱的无坐标模拟、格雷姆·西格尔定义的Γ-空间、W-空间、在域范畴中用有限CW复形代替有限集合的Γ-空间的模拟。我们构建和比较这些类别的模型结构。由于Γ-空间总是连通的，这些范畴和它们的单形类似物是奎伦等价的，它们的同伦范畴等价于经典的稳定同伦范畴。这些范畴中的幺半群是（严格）环谱。通常环谱、环谱上的模谱和交换环谱的子范畴也是模型范畴。当这成立时，环和模谱的各自范畴是奎伦等价的，因此有等价的同伦范畴。这允许在应用程序中互换使用这些类别。
Working in the category T of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functorsD→T for a suitable small topological categoryD. WhenD is symmetric monoidal, there is a smash product that gives the category of D‐spaces a symmetric monoidal structure. Examples include prespectra, as defined classically, symmetric spectra, as defined by Jeff Smith, orthogonal spectra, a coordinate‐free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category, Γ‐spaces, as defined by Graeme Segal, W‐spaces, an analogue of Γ‐spaces with finite sets replaced by finite CW complexes in the domain category. We construct and compare model structures on these categories. With the caveat that Γ‐spaces are always connective, these categories, and their simplicial analogues, are Quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category. Monoids in these categories are (strict) ring spectra. Often the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. When this holds, the respective categories of ring and module spectra are Quillen equivalent and thus have equivalent homotopy categories. This allows interchangeable use of these categories in applications.