Obstructions for Associativity in Stable Homotopy Theory

TL;DR

This paper develops an obstruction theory for $ ext{A}_n$-algebra structures in stable $ ext{infty}$-categories and applies it to show that the spectrum $ ext{S}/4$ admits an $ ext{A}_5$-multiplication.

Contribution

It introduces a new obstruction theory for $ ext{A}_n$-algebras in stable $ ext{infty}$-categories and demonstrates its application to specific spectra.

Findings

Constructed obstruction theory for $ ext{A}_n$-algebras in stable $ ext{infty}$-categories.

Proved $ ext{S}/4$ admits an $ ext{A}_5$-multiplication.

Analyzed properties of the obstruction theory in the context of synthetic spectra.

Abstract

We give a construction of the obstruction theory for $\mathbb{A}_{n}$-algebra structures in stable $\infty$-categories, and give some properties of it. We use this to show that the spectrum $\mathbb{S} / 4$ admits an $\mathbb{A}_5$-multiplication using synthetic spectra.