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A symmetric monoidal fracture square

Niko Naumann, Luca Pol, Maxime Ramzi

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.

A symmetric monoidal fracture square

Abstract

Given a symmetric monoidal stable -category which is rigidly-compactly generated and a set of compact objects of , one can form the subcategories of -complete and -local objects. The goal of this paper is to explain how to recover from its -local and -complete subcategories while retaining the symmetric monoidal structure. Specializing to the case where is the -category of -spectra for a finite group , 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.

Paper Structure

This paper contains 8 sections, 32 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Dwyer-Kan Localizations
  3. Atomic objects and internal left adjoints
  4. Local duality
  5. A special case
  6. The fracture square
  7. Equivariant preliminaries
  8. Isotropy separation with coefficients

Key Result

Theorem 1

Let $\mathop{\mathrm{\mathscr{C}}}\nolimits\in\mathop{\mathrm{CAlg}}\nolimits(\mathrm{Pr}^{L}_{\mathrm{st}})$ be rigidly-compactly generated and $\mathcal{K}\subset \mathop{\mathrm{\mathscr{C}}}\nolimits^\omega$. Then the square (mega-square-intro) is a pullback in $\mathop{\mathrm{CAlg}}\nolimits(\

Theorems & Definitions (87)

  • Theorem : \ref{['thm-pullback']}
  • Theorem : \ref{['cor:isotropy_separation-compact']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 77 more