论文标题
链复合物的模型结构
The model structure for chain complexes
论文作者
论文摘要
令$ \ text {ch} $为Abelian组的（可能是无限的）连锁络合物的类别。在本说明中，我们通过$ \ text {ch} $上的标准Quillen模型结构构建了与标准Qualt的方法不同。从本质上讲，我们为阿贝利亚人群使用功能性的两阶段投影分辨率，并直接从中构建一切。这具有非常具体，明确和功能的优势。它不依赖小对象的参数，也不依赖于任何明确使用转夹诱导。另一方面，它不是那么概念上，它确实使用了一个免费的Abelian群体的子组是免费的，因此除了$ \ Mathbb {Z} $之外，它不会推广到许多其他戒指。我们对这种方法没有任何巨大的技术好处，但这似乎是一种有趣的选择，并且在教学上可能有用。
Let $\text{Ch}$ be the category of (possibly unbounded) chain complexes of abelian groups. In this note we construct the standard Quillen model structure on $\text{Ch}$, by a method that is somewhat different from the standard one. Essentially, we use a functorial two-stage projective resolution for abelian groups, and build everything directly from that. This has the advantage of being very concrete, explicit and functorial. It does not rely on the small object argument, or make any explicit use of transfinite induction. On the other hand, it is not so conceptual, and it does use the fact that subgroups of free abelian groups are free, so it does not generalise to many rings other than $\mathbb{Z}$. We do not claim any great technical benefit for this approach, but it seems like an interesting alternative, and may be pedagogically useful.
