A sketch for derivators

TL;DR

This paper explores derivators as models of homotopy limit 2-sketches, establishing their homotopy local presentability and characterizing derivators of small presentation as homotopy λ-presentable objects.

Contribution

It introduces a new perspective on derivators as models of homotopy limit 2-sketches and proves their homotopy λ-presentability under certain conditions.

Findings

Derivators can be modeled as homotopy limit 2-sketches.

Derivators of small presentation are homotopy λ-presentable.

The paper discusses homotopy local λ-presentability of the 2-category of derivators.

Abstract

We show first that derivators can be seen as models of a suitable homotopy limit 2-sketch. After discussing homotopy local $\lambda$-presentability of the 2-category of derivators, for some appropriate regular cardinal $\lambda$, as an application we prove that derivators of small presentation are homotopy $\lambda$-presentable objects.