Condensed mathematics through compactological spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909965521780736 |
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| author | Böhnlein, Franziska Bruske, Benjamin Wegner, Sven-Ake |
| author_facet | Böhnlein, Franziska Bruske, Benjamin Wegner, Sven-Ake |
| contents | In their 2022 lecture notes on condensed sets, Clausen and Scholze mentioned in a remark that the important subclass of quasiseparated condensed sets is equivalent to the category of so-called compactological spaces defined by Waelbroeck in the 1960s. In this paper we survey the latter category in detail, we give a rigorous proof of Clausen and Scholze's claim, and we establish that condensed sets are a formal categorical completion of Waelbroeck's compactological spaces. The latter answers a question asked by Hanson in 2023 and permits the interpretation of compactological sets as an 'elementary' approach to condensed mathematics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_14612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Condensed mathematics through compactological spaces Böhnlein, Franziska Bruske, Benjamin Wegner, Sven-Ake Functional Analysis Algebraic Topology Category Theory Logic 46M15, 46A17, 18F60, 18A25, 54B30, 54E99, 55U40 In their 2022 lecture notes on condensed sets, Clausen and Scholze mentioned in a remark that the important subclass of quasiseparated condensed sets is equivalent to the category of so-called compactological spaces defined by Waelbroeck in the 1960s. In this paper we survey the latter category in detail, we give a rigorous proof of Clausen and Scholze's claim, and we establish that condensed sets are a formal categorical completion of Waelbroeck's compactological spaces. The latter answers a question asked by Hanson in 2023 and permits the interpretation of compactological sets as an 'elementary' approach to condensed mathematics. |
| title | Condensed mathematics through compactological spaces |
| topic | Functional Analysis Algebraic Topology Category Theory Logic 46M15, 46A17, 18F60, 18A25, 54B30, 54E99, 55U40 |
| url | https://arxiv.org/abs/2512.14612 |