On Galois categories and condensed contractible schemes

TL;DR

This paper explores the condensed Galois category of schemes, connecting it to ultracategories, classifying schemes with initial or terminal objects, and computing the condensed fundamental group for Dedekind domains.

Contribution

It extends the theory of condensed Galois categories, relates them to ultracategories, and provides new classifications and computations for schemes like Spec(Z).

Findings

Condensed homotopy type of certain schemes is trivial.

Spec(Z) has a non-trivial condensed fundamental group.

Classification of schemes with initial or terminal Galois objects.

Abstract

We extend the study of the condensed Galois category of a scheme introduced by Barwick, Glasman and Haine in their work on Exodromy. We elaborate its connection to Lurie's work on Ultracategories and provide a description in terms of w-contractible rings. We give a classification of schemes whose Galois category has an initial, respectively, a terminal object. This implies the condensed homotopy type of the scheme, which was studied in more detail in [arXiv:2510.07443v1], to be trivial. Furthermore, we compute a formula for the (underlying group of the) condensed fundamental group of a general Dedekind domain and show that it is non-trivial for the spectrum of the integers Spec(Z).This means that Spec(Z) is not condensed contractible.