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Greenlees has conjectured that the rational stable equivariant homotopycategory of a compact Lie group always has an algebraic model. Based on thisidea, we show that the category of rational local systems on a connected finiteloop space always has a simple algebraic model. When the loop space arises froma connected compact Lie group, this recovers a special case of a result of Poland Williamson about rational cofree $G$-spectra. More generally, we show thatif $K$ is a closed subgroup of a compact Lie group $G$ such that the Weyl group$W_GK$ 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 isjust the category of rational cofree $G$-spectra, and this recovers theaforementioned result. Throughout, we pay careful attention to the role oftorsion and complete categories.

Rational local systems and connected finite loop spaces