论文标题
富含组合预编码类别的型号2类
A model 2-category of enriched combinatorial premodel categories
论文作者
论文摘要
霍维在他的模型类别中的书中询问模型类别的2类$ \ mathbf {mod} $是否承认了“模型2类别结构”，其弱等价是quillen等价。我们表明$ \ mathbf {mod} $没有撤回，因此无法形成模型2类别。缺乏回调可以追溯到模型类别的弱等价上的三个公理。因此，我们将预编码类别定义为配备两个嵌套弱分解系统的完整且共同的类别。组合预编码类别形成完整且共完成的封闭对称单相2类别$ \ MATHBF {cpm} $，其张量产品代表Quillen Bifunctors。对于单型组合预编码类别$ v $，2类别$ v \ mathbf {cpm} $ $ v $ - 富集的组合前期类别仅是$ v $的模块类别$ \ mathbf {cpm} $。 模型类别的同喻理论取决于弱等价的基本方式，因此它不会直接扩展到一般的预编码类别。我们为满足额外属性的预编码类别开发了一种替代同质理论，该属性可自动适用于模型类别，也适用于富含单型模型类别的预编码类别。特别是，对于单型模型类别$ v $，我们获得了$ v $ premodel类别的quillen等效性概念，该类别扩展了$ v $ - 模型类别的概念。当$ v $是一种可处理的对称单型模型类别时，我们将这些Quillen等价等于薄弱的等价型在$ V \ Mathbf {CPM} $上构建模型2类别结构，并通过调整Szumiło的构造的合作类别纤维类别来构建。
In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have pullbacks and so cannot form a model 2-category. This lack of pullbacks can be traced to the two-out-of-three axiom on the weak equivalences of a model category. Accordingly, we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a complete and cocomplete closed symmetric monoidal 2-category $\mathbf{CPM}$ whose tensor product represents Quillen bifunctors. For a monoidal combinatorial premodel category $V$, the 2-category $V\mathbf{CPM}$ of $V$-enriched combinatorial premodel categories is simply the category of modules over $V$ (viewed as a monoid object of $\mathbf{CPM}$), and therefore inherits the algebraic structure of $\mathbf{CPM}$. The homotopy theory of a model category depends in an essential way on the weak equivalences, so it does not extend directly to a general premodel category. We develop a substitute homotopy theory for premodel categories satisfying an additional property which holds automatically for model categories and also for premodel categories enriched in a monoidal model category. In particular, for a monoidal model category $V$, we obtain a notion of Quillen equivalence of $V$-premodel categories which extends the one for $V$-model categories. When $V$ is a tractable symmetric monoidal model category, we construct a model 2-category structure on $V\mathbf{CPM}$ with these Quillen equivalences as the weak equivalences, by adapting Szumiło's construction of a fibration category of cofibration categories.
