The Pro-\'Etale Topos as a Category of Pyknotic Presheaves

Sebastian Wolf

The Pro-\'Etale Topos as a Category of Pyknotic Presheaves

Algebraic Geometry 2022-12-23 v3

Authors: Sebastian Wolf

Abstract

Let $X$ be a coherent scheme and let $\operatorname{Gal}(X)$ denote the Galois category of $X$ , as introduced by Barwick, Glasman and Haine. In this paper we prove that the hypercomplete pro-\'etale $\infty$ -topos $X_{\operatorname{proet}}$ of $X$ , introduced by Bhatt and Scholze, is equivalent to the category of continuous representations of $\operatorname{Gal}(X)$ with values in the $\infty$ -category of pyknotic spaces.

Keywords

pro-étale topos coherent scheme galois category pyknotic spaces continuous representations ∞-category

Cite

@article{arxiv.2012.10502,
  title  = {The Pro-\'Etale Topos as a Category of Pyknotic Presheaves},
  author = {Sebastian Wolf},
  journal= {arXiv preprint arXiv:2012.10502},
  year   = {2022}
}

Comments

Final version, to appear in Documenta Mathematica. Theorem 3.11 now explicitly describes the functors in both directions; many other minor changes to improve readability

The Pro-\'Etale Topos as a Category of Pyknotic Presheaves

Abstract

Keywords

Cite

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