Joins of closed sublocales are not always a coframe
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912919053139968 |
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| author | Arrieta, Igor |
| author_facet | Arrieta, Igor |
| contents | Given a locale $L$, the collection $\mathsf{S}_c(L)$ of joins of closed sublocales forms a frame--somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of $L$, where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc.
A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $\mathsf{S}_c(L)$ is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale $L$ such that $\mathsf{S}_c(L)$ is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in $\mathsf{S}_c(L)$; we circumvent this by analysing a certain separation property satisfied by $\mathsf{S}_c(L)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Joins of closed sublocales are not always a coframe Arrieta, Igor General Topology Category Theory 18F70, 06D22 Given a locale $L$, the collection $\mathsf{S}_c(L)$ of joins of closed sublocales forms a frame--somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of $L$, where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $\mathsf{S}_c(L)$ is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale $L$ such that $\mathsf{S}_c(L)$ is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in $\mathsf{S}_c(L)$; we circumvent this by analysing a certain separation property satisfied by $\mathsf{S}_c(L)$. |
| title | Joins of closed sublocales are not always a coframe |
| topic | General Topology Category Theory 18F70, 06D22 |
| url | https://arxiv.org/abs/2510.00987 |