Higher stacks as diagrams

TL;DR

This paper explores various presentations of the homotopy theory of non-hypercomplete $ abla$-stacks on a classical site, introducing a combinatorial diagram-based approach similar to existing models, and extending derivators to $ abla$-stacks.

Contribution

It provides a new combinatorial diagrammatic description of $ abla$-stacks homotopy theory, connecting classical and $ abla$-stack frameworks, and shows how derivators extend to $ abla$-stacks.

Findings

A diagrammatic presentation for $ abla$-stacks homotopy theory is established.

The approach generalizes classical homotopy theory models by Quillen, Thomason, and Grothendieck.

Derivators over a site extend to $ abla$-stacks under mild conditions.

Abstract

Several possible presentations for the homotopy theory of (non-hypercomplete) $\infty$-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in terms of diagrams in S exists, similar to Cisinski's presentation, based on work of Quillen, Thomason and Grothendieck, of usual homotopy theory by small categories and their smallest (basic) localizer. As an application it is shown that any (local) fibered (a.k.a. algebraic) derivator over S with stable fibers extends to $\infty$-stacks in a well-defined way under mild assumptions.