Revisiting the canonicity of canonical triangulations

TL;DR

This paper investigates the relationship between stable derivators and their canonical triangulations, demonstrating how morphisms induce exact functors and establishing uniqueness results through a detailed 2-categorical analysis.

Contribution

It shows that exact morphisms of stable derivators induce exact functors of canonical triangulations and provides a uniqueness statement, refining the understanding of their interaction.

Findings

Exact morphisms induce exact functors of triangulations

Canonical triangulations are unique up to 2-categorical refinement

Analysis of morphisms' interaction with limits, colimits, and Kan extensions

Abstract

Stable derivators provide an enhancement of triangulated categories as is indicated by the existence of canonical triangulations. In this paper we show that exact morphisms of stable derivators induce exact functors of canonical triangulations, and similarly for arbitrary natural transformations. This 2-categorical refinement also provides a uniqueness statement concerning canonical triangulations. These results rely on a more careful study of morphisms of derivators and this study is of independent interest. We analyze the interaction of morphisms of derivators with limits, colimits, and Kan extensions, including a discussion of invariance and closure properties of the class of Kan extensions preserved by a fixed morphism.