Abstract
AbstractGreenlees 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.
Highlights
The category of non-equivariant rational spectra is very simple; it is equivalent to the derived category of Q-modules
Greenlees has conjectured that for a compact Lie group G, the category of rational equivariant G-spectra is equivalent to the derived category of an abelian category A(G) [21, Conjecture 6.1]
Let X be a connected finite loop space, there is an equivalence of symmetric monoidal ∞-categories
Summary
The category of non-equivariant rational spectra is very simple; it is equivalent to the derived category of Q-modules. Greenlees has conjectured that for a compact Lie group G, the category of rational equivariant G-spectra is equivalent to the derived category of an abelian category A(G) [21, Conjecture 6.1]. G-spectra, which we denote by SpfGre,Qe and SpcGo,fQree, respectively These categories are equivalent, not by the identity functor. Greenlees and Shipley have given two proofs for the equivalence between free G-spectra and torsion H∗(BG)-modules when G is a connected compact Lie group. Let X be a connected finite loop space, there is an equivalence of symmetric monoidal ∞-categories. We note that there do exist connected finite loop spaces not rationally equivalent to compact Lie groups [1]. The proof proceeds through a series of equivalences of symmetric monoidal stable ∞-categories, as indicated below
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