A symmetric monoidal fracture square

TL;DR

This paper develops a method to reconstruct a symmetric monoidal stable $mbda$-category from its local and complete subcategories, extending isotropy separation to a monoidal context, with applications to $G$-spectra.

Contribution

It introduces a symmetric monoidal fracture square framework to recover categories from local and complete parts, generalizing previous isotropy separation results.

Findings

Provides a symmetric monoidal reconstruction technique.

Extends isotropy separation to a monoidal setting.

Applicable to categories of $G$-spectra for finite groups.

Abstract

Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ which is rigidly-compactly generated and a set of compact objects $\mathcal{K}$ of $\mathcal{C}$, one can form the subcategories of $\mathcal{K}$-complete and $\mathcal{K}$-local objects. The goal of this paper is to explain how to recover $\mathcal{C}$ from its $\mathcal{K}$-local and $\mathcal{K}$-complete subcategories while retaining the symmetric monoidal structure. Specializing to the case where $\mathcal{C}$ is the $\infty$-category of $G$-spectra for a finite group $G$, our result can be viewed as a symmetric monoidal variant of the isotropy separation decomposition, a version of which appeared previously in work of Krause.