On equivariant homotopy theory for model categories
On equivariant homotopy theory for model categories
复制标题
模型范畴的等变同伦理论
DOI：
10.4310/hha.2016.v18.n2.a10
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Marc Stephan
中科院分区：
文献类型：
--
作者：
Marc Stephan
我们介绍并比较了拓扑或普通Quillen模型范畴中等变同伦理论的两种方法。对于空间的拓扑模型范畴，我们推广了Piacenza的结果，即以固定拓扑群的轨道范畴和空间的轨道范畴为指标的拓扑预伸缩范畴可以被赋予Quillen等价模型范畴结构。在一定条件下，证明了任意余原生成模型范畴和离散群$G$的子群的不动点函子的类似结果。这些条件在许多例子中都成立，尽管不是在链复合体的范畴中，但我们仍然建立并推广了单纯$G集合的归一化链复合体的等变Whitehead定理和WALL定理。
We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group $G$ and the category of $G$-spaces can be endowed with Quillen equivalent model category structures. We prove an analogous result for any cofibrantly generated model category and discrete group $G$, under certain conditions on the fixed point functors of the subgroups of $G$. These conditions hold in many examples, though not in the category of chain complexes, where we nevertheless establish and generalize to collections an equivariant Whitehead Theorem \`{a} la Kropholler and Wall for the normalized chain complexes of simplicial $G$-sets.