The Galois theory of $G$-spectra and the Burnside ring

TL;DR

This paper establishes that the Galois groupoid of G-spectra for a finite group G is algebraic, equivalent to the étale fundamental groupoid of G's Burnside ring, and provides an algorithm for computation.

Contribution

It demonstrates the algebraic nature of the Galois groupoid of G-spectra and introduces an algorithm to compute the fundamental groupoid from the table of marks.

Findings

The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring.

An explicit algorithm for computing the fundamental groupoid from the table of marks.

Numerous examples illustrating the computation and theory.

Abstract

We prove that the Galois groupoid of the category of $G$-spectra for a finite group $G$ is algebraic, i.e. equivalent to the \'etale fundamental groupoid of the Burnside ring of $G$. We implement an algorithm that computes the latter from the table of marks of $G$, and provide numerous examples.