arXiv subjects

Catrin Mair

Publications and source records attributed to Catrin Mair.

4 recordsLinked to original sources

arXiv2026

On Galois categories and condensed contractible schemes

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.

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arXiv2026

Shape theory for condensed anima

We give different perspectives on the notion of shape for condensed anima. We prove that it recovers more classical notions of shape for topological spaces in the cases of all paracompact compactly generated spaces and all locally contractible spaces. These recovering statements imply and extend comparison results on sheaf and condensed cohomology by Clausen-Scholze and Haine. Another homotopy-theoretical direction for condensed anima are their condensed homotopy groups. Connected to this, we give a description of the underlying topological group functor on condensed groups via quasi-topological groups.

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arXiv2025

The condensed homotopy type of a scheme

We study a condensed version of the \'etale homotopy type of a scheme, which refines both the usual \'etale homotopy type of Friedlander-Artin-Mazur and the pro\'etale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a description of the condensed homotopy type using the Galois category of a scheme introduced by Barwick-Glasman-Haine. In the second part, we focus on the fundamental group of the condensed homotopy type in more detail. We show that, unexpectedly, the fundamental group of the condensed homotopy type of the affine line $\mathbf{A}^1_{\mathbf{C}}$ over the complex numbers is nontrivial. Nonetheless, its Noohi completion recovers the pro\'etale fundamental group of Bhatt-Scholze. Moreover, we show that a mild correction, passing to the quasiseparated quotient, fixes most of this group's quirks. Surprisingly, this quotient is often a topological group.

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arXiv2021

Animated Condensed Sets and Their Homotopy Groups

The theory of condensed mathematics by Dustin Clausen and Peter Scholze claims that topological spaces should be replaced by the definition of condensed sets. The main purpose of this paper is to investigate in which way the theory of homotopy groups on topological spaces can be extended to the theory of condensed sets. We show that there exist functors from the category of condensed sets to the category of pro-groups, s.t. restriction to CW-spaces coincides with the ordinary notion of homotopy groups on topological spaces. These functors can be extended to the $\infty$-categories of condensed anima and pro-anima. On our way to these results we will prove that the compact projective objects in the category of condensed sets are given by the extremally disconnected profinite sets.

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