Explicit Square Zero Obstruction Theory

TL;DR

This paper develops a square zero obstruction theory for modules over _1-algebras in stable monoidal infinity-categories, providing explicit descriptions and clarifying subtle non-connective aspects.

Contribution

It introduces a new explicit square zero obstruction theory for modules over _1-algebras in stable monoidal infinity-categories, with detailed descriptions of obstruction elements.

Findings

Explicit description of the obstruction element as a homotopy class

Clarification of subtle features in the non-connective setting

Development of a general framework applicable to arbitrary stable monoidal infinity-categories

Abstract

We develop square zero obstruction theory for modules over $\mathbb{E}_1$-algebras in an arbitrary stable (presentably) monoidal $\infty$-category. We explicitly describe the obstruction element as the homotopy class of a canonically constructed map. Our approach clarifies some subtle features of the non-connective setting.