On the symmetry behind duality
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908464366747648 |
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| author | Abbadini, Marco Jung, Achim |
| author_facet | Abbadini, Marco Jung, Achim |
| contents | Open sets and compact saturated sets enjoy a perfect formal symmetry, at least for classes of spaces such as Stone spaces or spectral spaces. For larger classes of spaces, a perfect symmetry may not be available, although strong signs of it may remain. These signs appear especially in the classes of spaces involved in Stone-like dualities (such as sober spaces).
In this article, we introduce a framework with a perfect symmetry between open sets and compact saturated sets, and which includes sober spaces. Our main result is an extension of the duality between sober spaces and spatial frames to a duality between two categories, each equipped with a self-duality. On the spatial side, the self-duality extends de Groot self-duality for stably compact spaces, which swaps open sets with complements of compact saturated sets; this self-duality is made possible using structures reminiscent of bitopological spaces. On the pointfree side, the self-duality is an extension of Lawson self-duality for continuous domains and is achieved via a framework analogous to that of d-frames.
We show how to derive from our main result the well-known duality between locally compact sober spaces and locally compact frames. In doing so, we provide a presentation of continuous domains in a manner akin to d-frames. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the symmetry behind duality Abbadini, Marco Jung, Achim Logic Primary: 54E55. Secondary: 06D50, 06D22, 06E15, 06F30, 06B35, 06D05 Open sets and compact saturated sets enjoy a perfect formal symmetry, at least for classes of spaces such as Stone spaces or spectral spaces. For larger classes of spaces, a perfect symmetry may not be available, although strong signs of it may remain. These signs appear especially in the classes of spaces involved in Stone-like dualities (such as sober spaces). In this article, we introduce a framework with a perfect symmetry between open sets and compact saturated sets, and which includes sober spaces. Our main result is an extension of the duality between sober spaces and spatial frames to a duality between two categories, each equipped with a self-duality. On the spatial side, the self-duality extends de Groot self-duality for stably compact spaces, which swaps open sets with complements of compact saturated sets; this self-duality is made possible using structures reminiscent of bitopological spaces. On the pointfree side, the self-duality is an extension of Lawson self-duality for continuous domains and is achieved via a framework analogous to that of d-frames. We show how to derive from our main result the well-known duality between locally compact sober spaces and locally compact frames. In doing so, we provide a presentation of continuous domains in a manner akin to d-frames. |
| title | On the symmetry behind duality |
| topic | Logic Primary: 54E55. Secondary: 06D50, 06D22, 06E15, 06F30, 06B35, 06D05 |
| url | https://arxiv.org/abs/2507.18245 |