A category of quantum posets
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866908832355057664 |
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| author | Kornell, Andre Lindenhovius, Bert Mislove, Michael |
| author_facet | Kornell, Andre Lindenhovius, Bert Mislove, Michael |
| contents | We investigate a category of quantum posets that generalizes the category of posets and monotone functions. Up to equivalence, its objects are hereditarily atomic von Neumann algebras equipped with quantum partial orders in Weaver's sense. We show that this category is complete, cocomplete and symmetric monoidal closed. As a consequence, any discrete quantum family of maps in Sołtan's sense from a discrete quantum space to a partially ordered set is canonically equipped with quantum preorder in Weaver's sense. In particular, the quantum power set of a quantum set is so ordered. As an application, we show that each quantum poset embeds into its quantum power set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_11184 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A category of quantum posets Kornell, Andre Lindenhovius, Bert Mislove, Michael Operator Algebras Mathematical Physics Category Theory 46L89 (Primary) 06A75, 68Q55 (Secondary) We investigate a category of quantum posets that generalizes the category of posets and monotone functions. Up to equivalence, its objects are hereditarily atomic von Neumann algebras equipped with quantum partial orders in Weaver's sense. We show that this category is complete, cocomplete and symmetric monoidal closed. As a consequence, any discrete quantum family of maps in Sołtan's sense from a discrete quantum space to a partially ordered set is canonically equipped with quantum preorder in Weaver's sense. In particular, the quantum power set of a quantum set is so ordered. As an application, we show that each quantum poset embeds into its quantum power set. |
| title | A category of quantum posets |
| topic | Operator Algebras Mathematical Physics Category Theory 46L89 (Primary) 06A75, 68Q55 (Secondary) |
| url | https://arxiv.org/abs/2101.11184 |