A condensed proof of the pro-\'etale and \'etale exodromy theorems

Abstract

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-\'etale sheaves. Using the condensed perspective from the outset, we give a quick and self-contained proof of the pro-\'etale exodromy theorem. This is used to extract an exodromy theorem for (Postnikov complete) \'etale 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 \'etale 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 $\kappa$-condensed statement for any uncountable cardinal $\kappa$ such that $\kappa > \lvert \mathcal O_X(U) \rvert$ for every affine open $U \subseteq X$.