An alternative to hypercovers

TL;DR

This paper introduces 'atlases', a new class of diagrams in Grothendieck sites, providing a more explicit and universal alternative to hypercovers for studying hyperdescent, and shows how they can generate hypercovers via an indexed nerve construction.

Contribution

The paper proposes atlases as a novel, flexible framework that simplifies and generalizes the construction of hypercovers in the context of hyperdescent.

Findings

Atlases can be used to produce hypercovers through an indexed nerve construction.

Atlases are more explicit and more universal than traditional hypercovers.

Hypersheaves map atlases to limits, facilitating hyperdescent analysis.

Abstract

I introduce a class of diagrams in a Grothendieck site called "atlases" which can be used to study hyperdescent, and show that hypersheaves take atlases to limits using an indexed `nerve' construction that produces hypercovers from atlases. Atlases have the flexibility to be at the same time more explicit and more universal than hypercovers.