Characterizations of abstract stable homotopy theories

TL;DR

This paper provides new characterizations of stable derivators, linking their stability to the commutation of homotopy finite limits and colimits, and offering alternative reformulations involving zero objects and morphism commutations.

Contribution

It introduces novel criteria for stability of derivators, enhancing understanding of the stabilization process from spaces to spectra.

Findings

Homotopy finite limits and colimits commute in stable derivators

Stability characterized by the existence of a zero object and commutation of partial cone and fiber

Variants for finite Kan extensions also characterize stability

Abstract

In this paper we establish new characterizations of stable derivators, thereby obtaining additional interpretations of the passage from (pointed) topological spaces to spectra and, more generally, of the stabilization. We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, and there are variants for sufficiently finite Kan extensions. As an additional reformulation, a derivator is stable if and only if it admits a zero object and if partial cone and partial fiber morphisms commute on squares.