The theory of half derivators

TL;DR

This paper reviews derivator theory, introduces new classes called half derivators, and explores their properties and applications in K-theory, extending classical results to broader contexts.

Contribution

It defines and analyzes half derivators, extending derivator theory and establishing their relevance in K-theory beyond traditional frameworks.

Findings

Many classical homotopical arguments hold for half derivators.

Introduces the maximal domain for derivator K-theory.

Extends Waldhausen K-theory to derivators.

Abstract

We review the theory of derivators from the ground up, defining new classes of derivators which were originally motivated by derivator K-theory. We prove that many old arguments that relied on homotopical bicompleteness hold also for one-sided half derivators on arbitrary diagram categories. We end by defining the maximal domain for a K-theory of derivators generalising Waldhausen K-theory.