Derivators, pointed derivators, and stable derivators

TL;DR

This paper advances the theory of derivators by establishing that stable derivators naturally form triangulated categories and that their functors are exact, simplifying axioms and providing new proofs.

Contribution

It introduces a canonical triangulated structure on stable derivators, simplifies axioms of pointed derivators, and offers a new proof linking model categories to derivators.

Findings

Stable derivators can be endowed with a triangulated structure.

Functors in stable derivators are exact with respect to these structures.

A combinatorial model category has an underlying derivator.

Abstract

We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category. Moreover, the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures. Along the way, we give a simplification of the axioms of a pointed derivator and a reformulation of the base change axiom in terms of Grothendieck (op)fibration. Furthermore, we have a new proof that a combinatorial model category has an underlying derivator.