On Postnikov completeness for replete topoi

TL;DR

This paper proves that hypercomplete $ abla$-topoi from replete topoi are Postnikov complete, answering a question by Bhatt and Scholze, and extends To"en's results on affine stacks.

Contribution

It establishes Postnikov completeness for hypercomplete $ abla$-topoi associated with replete topoi, using Milnor sequences, and generalizes To"en's affine stacks result.

Findings

Hypercomplete $ abla$-topoi from replete topoi are Postnikov complete.

Constructs Milnor sequences for sheaves of spaces on replete topoi.

Generalizes To"en's theorem on affine stacks.

Abstract

We show that the hypercomplete $\infty$-topos associated with any replete topos is Postnikov complete, positively answering a question of Bhatt and Scholze; this will be deduced from the Milnor sequences for sheaves of spaces on replete topoi that we construct. As a corollary, we generalize a result of To\"en on affine stacks.