Adjoining roots in homotopy theory

Abstract

We use a "twisted group algebra" method to constructively adjoin formal radicals $\sqrt[n]{\alpha }$, for $\alpha$ a unit in a commutative ring spectrum or an invertible object in a symmetric monoidal $\infty$-category. We show that this construction is classified by maps from Eilenberg-Mac Lane objects to the unit spectrum, the Picard spectrum, and the Brauer spectrum.

arXiv comment21 pages. Comments welcome

References in corpus (11)

• Higher Topos Theory
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• The Picard group of topological modular forms via descent theory
• Brauer groups and étale cohomology in derived algebraic geometry
• Units of ring spectra and Thom spectra
• A simple universal property of Thom ring spectra
• Brauer groups and Galois cohomology of commutative ring spectra
• Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
• Brauer groups for commutative $S$-algebras
• p-adic J-homomorphisms and a product formula
• Roots of unity in $K(n)$-local rings

Cited by in corpus (2)

• Chromatic Cyclotomic Extensions
• Strict units of commutative ring spectra