Hilbert $*$-categories: Where limits in analysis and category theory meet
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911305739272192 |
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| author | Di Meglio, Matthew Heunen, Chris |
| author_facet | Di Meglio, Matthew Heunen, Chris |
| contents | This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include categories of Hilbert W*-modules and of unitary group-representations. Hilbert $*$-categories are "analytically" complete in two ways: every bounded increasing sequence of Hermitian endomorphisms has a supremum, and every suitably bounded orthogonal family of parallel morphisms is summable. These "analytic" completeness properties are not assumed outright; rather, they are derived, respectively, from two new universal constructions: codirected $\ell^2$-limits of contractions and $\ell^2$-products. In turn, these are built from directed colimits in the wide subcategory of isometries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_17432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert $*$-categories: Where limits in analysis and category theory meet Di Meglio, Matthew Heunen, Chris Category Theory Functional Analysis Operator Algebras 18M40, 46B15, 46L08, 46M15, 46M40, 06F25 This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include categories of Hilbert W*-modules and of unitary group-representations. Hilbert $*$-categories are "analytically" complete in two ways: every bounded increasing sequence of Hermitian endomorphisms has a supremum, and every suitably bounded orthogonal family of parallel morphisms is summable. These "analytic" completeness properties are not assumed outright; rather, they are derived, respectively, from two new universal constructions: codirected $\ell^2$-limits of contractions and $\ell^2$-products. In turn, these are built from directed colimits in the wide subcategory of isometries. |
| title | Hilbert $*$-categories: Where limits in analysis and category theory meet |
| topic | Category Theory Functional Analysis Operator Algebras 18M40, 46B15, 46L08, 46M15, 46M40, 06F25 |
| url | https://arxiv.org/abs/2505.17432 |