Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group WGK is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

中文翻译：

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合理的本地系统和连接的有限循环空间

Greenlees 猜想紧李群的有理稳定等变同伦范畴总是有一个代数模型。基于这个思想，我们证明了连通有限环空间上的有理局部系统范畴总是有一个简单的代数模型。当环空间来自一个连通的紧李群时，这恢复了 Pol 和 Williamson 关于有理 cofree 的结果的一个特例G-光谱。更一般地，我们证明如果ķ是紧李群的闭子群G使得 Weyl 群WGķ是连通的，那么某一类有理G-光谱“在ķ”有一个代数模型。例如，当ķ是平凡群，这只是理性cofree的范畴G-spectra，这将恢复上述结果。在整个过程中，我们都非常注意扭转和完整类别的作用。