Toward a Galois theory of the integers over the sphere spectrum

TL;DR

This paper explores a new perspective on the algebraic structure of the integers within the framework of the sphere spectrum, using derived Galois theory and topological concepts.

Contribution

It introduces a Galois-theoretic interpretation of the integers over the sphere spectrum, connecting higher algebra, Thom spectra, and topological field theory.

Findings

Reinterpretation of $H\mathbb{Z}$ as a Thom spectrum

Connection between derived Galois theory and the integers over the sphere spectrum

Proposal of a topological field theory analogy

Abstract

Recent work in higher algebra allows the reinterpretation of a classical description of the Eilenberg-MacLane spectrum $H\mathbb{Z}$ as a Thom spectrum, in terms of a kind of derived Galois theory. This essentially expository talk summarizes some of this work, and suggests an interpretation in terms of configuration spaces and monoidal functors on them, with some analogies to a topological field theory.