The pro-Nisnevich topology

TL;DR

This paper introduces the pro-Nisnevich topology, demonstrating its properties and embedding the Nisnevich $$-topos into it, revealing that it is locally of homotopy dimension zero.

Contribution

It constructs the pro-Nisnevich topology and proves key properties including the embedding of the Nisnevich $$-topos and the local homotopy dimension.

Findings

Nisnevich $$-topos embeds into pro-Nisnevich $$-topos

Pro-Nisnevich $$-topos is locally of homotopy dimension 0

Construction of the pro-Nisnevich topology

Abstract

We construct the pro-Nisnevich topology, an analog of the pro-\'etale topology. We then show that the Nisnevich $\infty$-topos embeds into the pro-Nisnevich $\infty$-topos, and that the pro-Nisnevich $\infty$-topos is locally of homotopy dimension $0$.