Symmetric monoidal categories of conveniently-constructible Banach bundles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909232221126656 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | We show that a continuously-normed Banach bundle $\mathcal{E}$ over a compact Hausdorff space $X$ whose space of sections is algebraically finitely-generated (f.g.) over $C(X)$ is locally trivial (and hence the section space is projective f.g over $C(X)$); this answers a question of I. Gogić. As a preliminary we also provide sufficient conditions for a quotient bundle to be continuous phrased in terms of the Vietoris continuity of the unit-ball maps attached to the bundles. Related results include (a) the fact that the category of topologically f.g. continuous Banach bundles over $X$ is symmetric monoidal under the (fiber-wise-maximal) tensor product, (b) the full faithfulness of the global-section functor from topologically f.g. continuous bundles to $C(X)$-modules and (c) the consequent identification of the algebraically f.g. bundles as precisely the rigid objects in the aforementioned symmetric monoidal category. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_11221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric monoidal categories of conveniently-constructible Banach bundles Chirvasitu, Alexandru Functional Analysis Category Theory General Topology Operator Algebras Rings and Algebras 46H25, 18M05, 13C10, 46J10, 46E25, 46M20, 13C11, 18A30, 18D15, 18D20 We show that a continuously-normed Banach bundle $\mathcal{E}$ over a compact Hausdorff space $X$ whose space of sections is algebraically finitely-generated (f.g.) over $C(X)$ is locally trivial (and hence the section space is projective f.g over $C(X)$); this answers a question of I. Gogić. As a preliminary we also provide sufficient conditions for a quotient bundle to be continuous phrased in terms of the Vietoris continuity of the unit-ball maps attached to the bundles. Related results include (a) the fact that the category of topologically f.g. continuous Banach bundles over $X$ is symmetric monoidal under the (fiber-wise-maximal) tensor product, (b) the full faithfulness of the global-section functor from topologically f.g. continuous bundles to $C(X)$-modules and (c) the consequent identification of the algebraically f.g. bundles as precisely the rigid objects in the aforementioned symmetric monoidal category. |
| title | Symmetric monoidal categories of conveniently-constructible Banach bundles |
| topic | Functional Analysis Category Theory General Topology Operator Algebras Rings and Algebras 46H25, 18M05, 13C10, 46J10, 46E25, 46M20, 13C11, 18A30, 18D15, 18D20 |
| url | https://arxiv.org/abs/2406.11221 |