Localization theory for derivators

TL;DR

This paper develops a comprehensive localization theory for derivators, extending classical category reflection concepts to the derivator setting, including descriptions via fractions, factorization systems, and monads.

Contribution

It introduces a unified framework for reflections in derivators, generalizing classical category theory results to this higher-level context.

Findings

Reflections for derivators can be characterized as categories of fractions.

Reflective factorization systems are established for derivators.

Categories of algebras for idempotent monads describe derivator reflections.

Abstract

We outline the theory of reflections for prederivators, derivators and stable derivators. In order to parallel the classical theory valid for categories, we outline how reflections can be equivalently described as categories of fractions, reflective factorization systems, and categories of algebras for idempotent monads. This is a further development of the theory of monads and factorization systems for derivators.