A condensed proof of the pro-étale and étale exodromy theorems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911704704614400 |
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| author | de Bruyn, Remy van Dobben |
| author_facet | de Bruyn, Remy van Dobben |
| contents | The exodromy correspondence of Barwick, Glasman, and Haine computes constructible sheaves of spaces on a scheme $X$ as an $\infty$-category of continuous functors from the profinite category $\operatorname{Gal}(X)$. Viewing $\operatorname{Gal}(X)$ instead as a condensed category, this was extended by Wolf to an exodromy correspondence for pro-étale sheaves. Using the condensed perspective from the outset, we give a quick and self-contained proof of the pro-étale exodromy theorem. This is used to extract an exodromy theorem for (Postnikov complete) étale sheaves that does not yet appear in the literature, which is closely related to Lurie's work on ultracategories. Finally, we use this to give a new proof of the constructible étale exodromy correspondence of Barwick, Glasman, and Haine. Without additional effort, our method removes the qcqs hypotheses on the schemes, and gives versions for sheaves with coefficients in more general $\infty$-categories. Finally, we refine the methods to obtain a $κ$-condensed statement whenever $κ> \lvert \mathcal O_X(U) \rvert$ for every affine open $U \subseteq X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22499 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A condensed proof of the pro-étale and étale exodromy theorems de Bruyn, Remy van Dobben Algebraic Geometry Category Theory 14F20 (primary), 14F35, 14F06, 18F10, 18D40, 18N60 The exodromy correspondence of Barwick, Glasman, and Haine computes constructible sheaves of spaces on a scheme $X$ as an $\infty$-category of continuous functors from the profinite category $\operatorname{Gal}(X)$. Viewing $\operatorname{Gal}(X)$ instead as a condensed category, this was extended by Wolf to an exodromy correspondence for pro-étale sheaves. Using the condensed perspective from the outset, we give a quick and self-contained proof of the pro-étale exodromy theorem. This is used to extract an exodromy theorem for (Postnikov complete) étale sheaves that does not yet appear in the literature, which is closely related to Lurie's work on ultracategories. Finally, we use this to give a new proof of the constructible étale exodromy correspondence of Barwick, Glasman, and Haine. Without additional effort, our method removes the qcqs hypotheses on the schemes, and gives versions for sheaves with coefficients in more general $\infty$-categories. Finally, we refine the methods to obtain a $κ$-condensed statement whenever $κ> \lvert \mathcal O_X(U) \rvert$ for every affine open $U \subseteq X$. |
| title | A condensed proof of the pro-étale and étale exodromy theorems |
| topic | Algebraic Geometry Category Theory 14F20 (primary), 14F35, 14F06, 18F10, 18D40, 18N60 |
| url | https://arxiv.org/abs/2605.22499 |