Simplicial functors and stable homotopy theory
Simplicial functors and stable homotopy theory
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单纯函子和稳定同伦理论
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发表时间：
1998
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通讯作者：
M. Lydakis
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作者：
M. Lydakis
构造谱的良好粉碎积问题是代数拓扑学中一个古老而著名的问题。这个问题的意思是：找到一个模型范畴，它与谱的模型范畴是quillen等价的，并且有一个对称的单线积对应于谱的粉碎积。最近找到了两个解，即[EKMM]的s模的粉碎积和[HSS]的对称谱的粉碎积。本文给出了从有限点单纯集到点单纯集的单纯函子范畴的另一个解——SF的粉碎积。在我们描述SF的一些有趣的特殊性质之前，我们总结了本文中包含的一些其他结果：尽管SF似乎不是一个自然的研究对象，特别是对于不太熟悉简单技术的读者来说，它实际上应该被认为是从有限点cw -复形到点拓扑空间的函子的类别，这些函子是指向的（取一点空间到一点空间）和同伦函子（取弱等价到弱等价）。很明显，后一个范畴有一个有趣的同伦理论（它有一个自然的弱等价类），它不能来自一个模型结构（因为平凡的原因，边界和极限不能保持一般的弱等价）。然而，这个同伦理论等价于SF的另一个同伦理论（见第8节，特别是我们构造了一个模型结构）普通（Bousfield-Friedlander）谱也可以被看作是指向简单集的点简单函子（只是现在它们不是在所有有限点简单集上定义的）。
The problem of constructing a nice smash product of spectra is an old and well-known problem of algebraic topology. This problem has come to mean the following: Find a model category, which is Quillen-equivalent to the model category of spectra, and which has a symmetric monoidal product corresponding to the smash product of spectra. Two solutions were found recently, namely the smash product of S-modules of [EKMM], and the smash product of symmetric spectra of [HSS]. Here we present another solution, the smash product of SF, the category of simplicial functors from finite pointed simplicial sets to pointed simplicial sets. Before we describe some interesting special properties of SF, we summarize some other results contained in this paper: Although SF might not seem a natural object to study, especially to a reader not very familiar with simplicial techniques, it should in fact be thought of as the category of functors from finite pointed CW-complexes to pointed topological spaces which are pointed (take one-point spaces to one-point spaces) and homotopy functors (take weak equivalences to weak equivalences). It is clear that this latter category has an interesting homotopy theory (it has a natural class of weak equivalences), which cannot come from a model structure (for the trivial reason that colimits and limits do not preserve weak equivalences in general). However, this homotopy theory is equivalent to another one that SF has (see section 8, especially for which we construct a model structure Ordinary (Bousfield-Friedlander) spectra can also be viewed as pointed simplicial functors into pointed simplicial sets (only now they are not defined on all finite pointed simplicial sets,