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On Postnikov completeness for replete topoi

Shubhodip Mondal, Emanuel Reinecke

TL;DR

The paper addresses whether Postnikov towers converge in the hypercomplete ∞-topos of sheaves on a replete topos, without finiteness hypotheses. It introduces multiplicative presheaves and establishes Milnor sequences in this setting by proving sheafification commutes with inverse limits and countable products for such presheaves in replete topoi. Using these tools, it proves that for any replete topos $X$, the hypercomplete ∞-topos $ ext{Shv}_{\infty}(\u001bX\u001b)^{\wedge}$ is Postnikov complete, and deduces corollaries including left completeness of derived categories $D(\mathcal{X},R)$ and a general affine-stacks descent result. The results extend Toën’s affine-stack work and apply to naturally occurring topoi like fpqc, v-, and quasisyntomic topologies, while also highlighting that hypercompleteness alone does not imply Postnikov completeness in general.

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 Toën on affine stacks.

On Postnikov completeness for replete topoi

TL;DR

The paper addresses whether Postnikov towers converge in the hypercomplete ∞-topos of sheaves on a replete topos, without finiteness hypotheses. It introduces multiplicative presheaves and establishes Milnor sequences in this setting by proving sheafification commutes with inverse limits and countable products for such presheaves in replete topoi. Using these tools, it proves that for any replete topos , the hypercomplete ∞-topos is Postnikov complete, and deduces corollaries including left completeness of derived categories and a general affine-stacks descent result. The results extend Toën’s affine-stack work and apply to naturally occurring topoi like fpqc, v-, and quasisyntomic topologies, while also highlighting that hypercompleteness alone does not imply Postnikov completeness in general.

Abstract

We show that the hypercomplete -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 Toën on affine stacks.
Paper Structure (5 sections, 18 theorems, 29 equations)

This paper contains 5 sections, 18 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Multiplicative presheaves on Grothendieck sites
  3. Postnikov towers in replete topoi
  4. Preliminaries on oo-topoi
  5. The question of Bhatt--Scholze

Key Result

Theorem A

Let $\mathcal{X}$ be a replete topos. Then the hypercomplete $\infty$-topos $\mathop{\mathrm{Shv}}\nolimits_{\infty}(\mathcal{X})^{\wedge}$ is Postnikov complete.

Theorems & Definitions (63)

  • Definition 1.1: proet
  • Theorem A
  • Remark 1.3
  • Remark 1.4
  • Example 1.5
  • Example 1.6: proet
  • Corollary 1.7: \ref{['lastcor']}
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 53 more