Towards 2-derivators for formal $\infty$-category theory

TL;DR

This paper proposes axioms for a 2-dimensional extension of derivators, called 2-derivators, to better understand homotopy 2-categories in higher category theory, and shows their applicability and stability across models.

Contribution

It introduces a set of axioms for 2-derivators, a higher-dimensional analog of derivators, and demonstrates their validity in various models and their preservation under shifts.

Findings

Axioms for 2-derivators are satisfied in multiple models.

2-derivators are stable under a shift operation.

The framework extends homotopy 1-categories to homotopy 2-categories.

Abstract

Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or $(\infty, 1)$-category retains only limited information, the structured collection of homotopy 1-categories of diagram categories often suffices for many homotopical purposes. In this paper, we introduce a set of axioms for a 2-dimensional analog of derivators: a refinement of the homotopy 2-category of an enriched model category or $(\infty, 2)$-category into a coherent system of homotopy 2-categories of higher categories of diagrams. We show that these axioms are satisfied in a variety of models, including standard ones related to $(\infty, 1)$-category theory. Moreover, we prove that the axioms are preserved under a certain shift operation.