Cubical homotopy theory: a beginning-猫眼课题宝

三次同伦理论：一个开始

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2002

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J. Jardine

本文展示了立方集范畴的封闭模型结构，并表明所得到的同伦范畴等价于拓扑空间的普通同伦范畴。主要结果是定理19，它给出了模型结构，定理29和推论30一起暗示了同伦范畴的等价性。该理论的共纤维和弱等价性是人们所期望的，即分别引起拓扑空间的弱等价性的水平包含和映射。一旦人们摆脱了纤维化应该通过与 Kan 纤维化进行类比来定义的先入之见，那么封闭模型结构就相对容易推导。纤维化被定义为对于所有微不足道的共纤维化具有正确的提升特性的图。封闭模型公理的验证本质上是形式化的，并且在此处显示（另请参见[4]），这是来自与可数复合体的有界共纤维化条件有关的定位理论的标准技巧的结果。立方复形的同伦范畴与普通同伦范畴的等价性要有趣得多，并且是从立方奇异函子满足非交换意义上的切除的断言得出的。模型有一个基础类别，即框类别，它用于定义三次集，就像序数类别定义单纯集一样。这意味着立方集 X 被定义为盒子类别上的逆变函子 X : op → Set，取集合类别中的值。盒子范畴及其基本属性是本文第一部分的主题，而立方集的第一个属性在第二部分中描述。封闭模型结构在第3节中推导，并作为定理19出现。立方集和拓扑空间（或单纯集）的同伦范畴等价的断言涉及本文的最后三节。需要一位优秀的细分操作员。 n 立方体肯定存在明显的细分，这只是 n 立方体重心细分的产物

This paper displays a closed model structure for the category of cubical sets and shows that the resulting homotopy category is equivalent to the ordinary homotopy category for topological spaces. The main results are Theorem 19, which gives the model structure, and Theorem 29 and Corollary 30 which together imply the equivalence of homotopy categories. The cofibrations and weak equivalences for the theory are what one might expect, namely levelwise inclusions and maps which induce weak equivalences of topological spaces respectively. The closed model structure is relatively easy to derive, once one gets away from the preconception that fibrations should be defined by analogy with Kan fibrations. A fibration is defined to be a map which has the right lifting property with respect to all trivial cofibrations. The verification of the closed model axioms is essentially formal, and is displayed here (see also [4]) as a consequence of standard tricks from localization theory having to do with a bounded cofibration condition for countable complexes. The equivalence of the homotopy category of cubical complexes with the ordinary homotopy category is much more interesting, and follows from the assertion that the cubical singular functor satisfies excision in a non-abelian sense. There is an underlying category of models, namely the box category , which is used to define cubical sets in the same way that the category of ordinal numbers defines simplicial sets. This means that a cubical set X is defined as a contravariant functor X : op → Set on the box category, taking values in the category of sets. The box category and its basic properties are the subject of the first section of this paper, while the first properties of cubical sets are described in the second section. The closed model structure is derived in Section 3, and appears as Theorem 19. The assertion that the homotopy categories of cubical sets and topological spaces (or simplicial sets) are equivalent involves the final three sections of this paper. One needs a good subdivision operator. There is certainly an obvious subdivision of an n-cube, which is just a product of barycentric subdivisions of

Cubical homotopy theory: a beginning

三次同伦理论：一个开始

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