{"paper":{"title":"Complete W*-categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.OA","authors_text":"Andr\\'e Henriques, David Penneys, Nivedita","submitted_at":"2024-11-03T20:36:37Z","abstract_excerpt":"We study $\\mathrm{W}^*$-categories, and explain the ways in which complete $\\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\\langle\\,\\,,\\,\\rangle_{\\mathrm{Hilb}}\\,:\\,\\overline C\\times C\\,\\to\\, \\mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\\mathrm{W}^*$-categories there is an antilinear equivalence $$\\dagger:\\mathrm{Func}(C,D) \\leftrightarrow \\mathrm{Func}(D,C)$$ characterised by $\\langle c,F^\\dagger(d)\\rangle_{\\mathrm{Hilb}} \\simeq \\langle F(c),d\\rangle_{\\mathrm{Hilb}}$, for $c\\in C$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.01678","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.01678/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}