论文标题
Quillen的模型类别的2分集化
The 2-Localization of a Quillen's model category
论文作者
论文摘要
在[同型代数，Springer LNM 43]中，Quillen介绍了模型类别的概念：一个类别$ \ Mathcal {C} $，其中包含三个杰出的映射类$ \ {\ Mathcal {w} {w}，\，\，\，\，\ Mathcal {f}，\ f}，co \ fim， COFIBRATIONS），并构建本地化$ \ MATHCAL {c} [\ MATHCAL {w}^{ - 1}] $作为$ \ Mathcal {C} $的商，由同型箭头集合$ \ Mathcal $ \ \ m nathCal {c} c}（c}（x，y）的同质性关系确定。我们在这里开发了两类本地化，其中同型通过同型给出了这种2个定位的2个细胞，并且在将连接的组件函数$π_0$应用于2列力化的HOM类别时，可以获得Quillen的本地化。我们的证明不仅是对著名Quillen的概括。我们处理[M.E. Descotte，E.J。 Dubuc，M。Szyld;模型BICATEGOIRE及其同型BICATEGOID，ARXIV：1805.07749（2018）]仅考虑一个单个箭头$σ$。当$σ$是模型类别弱等价的类$ \ Mathcal {w} $时，我们会得到Quillen的结果。
In [Homotopical Algebra, Springer LNM 43] Quillen introduces the notion of a model category: a category $\mathcal{C}$ provided with three distinguished classes of maps $\{\mathcal{W},\, \mathcal{F},\, co\mathcal{F}\}$ (weak equivalences, fibrations, cofibrations), and gives a construction of the localization $\mathcal{C}[\mathcal{W}^{-1}]$ as the quotient of $\mathcal{C}$ by the congruence relation determined by the homotopies on the sets of arrows $\mathcal{C}(X,\,Y)$. We develop here the 2-categorical localization, in which the 2-cells of this 2-localization are given by homotopies, and one can get the Quillen's localization when applying the connected components functor $π_0$ on the hom-categories of the 2-localization. Our proof is not just a generalization of the well-known Quillen's one. We work with definitions of cylinders and homotopies introduced in [M.E. Descotte, E.J. Dubuc, M. Szyld; Model bicategories and their homotopy bicategories, arXiv:1805.07749 (2018)] considering only a single family of arrows $Σ$. When $Σ$ is the class $\mathcal{W}$ of weak equivalences of a model category, we get the Quillen's results.
