Six model structures for DG-modules over DGAs: model category theory in homological action
Six model structures for DG-modules over DGAs: model category theory in homological action
复制标题
DGA 上的 DG 模块的六种模型结构：同调作用中的模型范畴论
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
E. Riehl
中科院分区：
文献类型：
--
作者：
T. Barthel;Jon P. May;E. Riehl
在第一部分中，我们刻画了交换环R上的可微分次代数A上的六个投射型模型结构。当R是一个环R时，这六个模型结构坍塌为三个，至少在民间传说中是众所周知的，但在一般情况下，新的相对和混合模型结构是常用模型结构的有趣替代。其中一些模型结构的构建需要小对象论点的两个新变体，一个是丰富的，一个是代数的，我们对它们进行了更一般的描述。在第二部分中，我们给出了这些模型范畴中的各种理论和计算上的共生逼近。经典的杆件结构在相对模型结构中给出了一致的近似，但通常不是在通常的模型结构中。在通常的模型结构中，将Cobrant逼近从同调代数上的同调模级提升到DG-代数上的DG-模级有两种截然不同的方法。这一新理论使早期基于其中一种结构的显式计算具有模型理论意义。我们遇到的一个新现象是具有不同组合结构的同构Cobrant近似，使得在一个化身中证明的东西在另一个化身中不容易被证明。
In Part 1, we describe six projective-type model structures on the category of dierential graded modules over a dierential graded algebra A over a commutative ring R. When R is a eld, the six collapse to three and are well-known, at least to folklore, but in the general case the new relative and mixed model structures oer interesting alterna- tives to the model structures in common use. The construction of some of these model structures requires two new variants of the small object argument, an enriched and an algebraic one, and we describe these more generally. In Part 2, we present a variety of theoretical and calculational co- brant approximations in these model categories. The classical bar con- struction gives cobrant approximations in the relative model structure, but generally not in the usual one. In the usual model structure, there are two quite dierent ways to lift cobrant approximations from the level of homology modules over homology algebras, where they are classi- cal projective resolutions, to the level of DG-modules over DG-algebras. The new theory makes model theoretic sense of earlier explicit calcu- lations based on one of these constructions. A novel phenomenon we encounter is isomorphic cobrant approximations with dierent combi- natorial structure such that things proven in one avatar are not readily proven in the other.