Mayer-Vietoris sequences in stable derivators

TL;DR

This paper explores the properties of stable derivators, demonstrating the existence of Mayer-Vietoris sequences, characterizing homotopy exact squares, and establishing a criterion for stability based on the suspension functor.

Contribution

It introduces new characterizations of stable derivators, including a detection method for colimiting diagrams and a stability criterion via the suspension functor.

Findings

Stable derivators admit Mayer-Vietoris sequences from cocartesian squares.

Homotopy exact squares are characterized within derivators.

A derivator is stable iff its suspension functor is an equivalence.

Abstract

We show that stable derivators, like stable model categories, admit Mayer-Vietoris sequences arising from cocartesian squares. Along the way we characterize homotopy exact squares, and give a detection result for colimiting diagrams in derivators. As an application, we show that a derivator is stable if and only if its suspension functor is an equivalence.