The equivalence between rational G-sheaves and rational G-Mackey functors for profinite G

TL;DR

This paper establishes an equivalence between rational G-Mackey functors and G-sheaves over the space of closed subgroups for profinite groups, extending finite group results and aiding in classifying rational G-spectra.

Contribution

It introduces a new equivalence between rational G-Mackey functors and Weyl-G-sheaves for profinite groups, extending finite group classifications.

Findings

Established an equivalence between rational G-Mackey functors and Weyl-G-sheaves.

Extended classification results from finite to profinite groups.

Facilitated the classification of rational G-spectra for profinite groups.

Abstract

For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed. This extends the classification of rational G-Mackey functors for finite G of Th\'{e}venaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second author's thesis.