# Adelic models of tensor-triangulated categories

## TL;DR

This paper demonstrates that certain well-structured monoidal model categories can be represented as adelic models constructed from module categories over completed rings, unifying various examples across algebra and topology.

## Contribution

It introduces a general framework for representing Noetherian, finite dimensional, stable monoidal model categories as adelic models, extending known cases in algebra and homotopy theory.

## Key findings

- Unified adelic modeling approach for various categories
- Connections between algebraic and topological models
- Extension of existing models to a broader class

## Abstract

We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, for chromatic homotopy theory this is based on the chromatic fracture square, and for rational torus-equivariant homotopy theory this is the model of Greenlees-Shipley arXiv:1101.2511.