论文标题
虚弱的Unital DG类别类别的封闭模型结构，II
A closed model structure on the category of weakly unital dg categories, II
论文作者
论文摘要
在本文之后的本文中，这是我们之前的论文[ps]（但可以独立阅读），我们继续研究类别$ \ mathrm {cat} _ {\ mathrm {\ mathrm {dgwu}}}}（\ bbbk）的封闭模型结构（ $ \ bbbk $。在[PS]中，我们在弱的Unital DG类别的类别上构建了一个封闭的模型结构，对弱的Unital DG类别施加了技术条件，称$ \ m atrm {id} _x \ cdot \ cdot \ cdot \ cdot \ mathrm {id} _x = \ sathrm {id} _x} _x _x _x _x $ for nony object obboce $ x $ $ x $ $ x $。尽管这种情况使我们进行了极大的简化，但它是多余的，必须删除。 在这里，我们摆脱了这种情况，并提供了完全一般性的封闭模型结构。新的封闭模型类别也是共同生成的，事实证明，它相当于封闭的模型类别$ \ mathrm {cat} _ \ mathrm {dg}（\ bbbk）$（属于$ \ bbbk $）的dg类别（\ bbbk）$。放下条件$ \ mathrm {id} _x^2 = \ mathrm {id} _x $使封闭的模型结构的构造更远离loc.cit。，并且需要新的结构。其中之一是WU DG类别的三角式船体，这反过来也被证明是Wu DG类别。 自然出现的虚弱的Unital DG类别的一个例子是DG类别的条形分辨率。我们向本文提供了严格的Unital DG类别的经典棒ob-cobar解决方案的改进（附录B）。可以将类似的结构应用于$ \ mathrm {cat} _ \ mathrm {dgwu}（\ bbbk）$中的cofibrant分辨率。
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $\mathrm{Cat}_{\mathrm{dgwu}}(\Bbbk)$ of small weakly unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field $\Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $\mathrm{id}_x\cdot \mathrm{id}_x=\mathrm{id}_x$ for any object $x$. Although this condition led us to a great simplification, it was redundant and had to be dropped. Here we get rid of this condition, and provide a closed model structure in full generality. The new closed model category is as well cofibrantly generated, and it is proven to be Quillen equivalent to the closed model category $\mathrm{Cat}_\mathrm{dg}(\Bbbk)$ of (strictly unital) dg categories over $\Bbbk$, given by Tabuada [Tab1]. Dropping the condition $\mathrm{id}_x^2=\mathrm{id}_x$ makes the construction of the closed model structure more distant from loc.cit., and requires new constructions. One of them is a pre-triangulated hull of a wu dg category, which in turn is shown to be a wu dg category as well. One example of a weakly unital dg category which naturally appears is the bar-cobar resolution of a dg category. We supply this paper with a refinement of the classical bar-cobar resolution of a unital dg category which is strictly unital (appendix B). A similar construction can be applied to constructing a cofibrant resolution in $\mathrm{Cat}_\mathrm{dgwu}(\Bbbk)$.
