{"paper":{"title":"Rigidity of non-negligible objects of moderate growth in braided categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.RT"],"primary_cat":"math.QA","authors_text":"David Penneys, Pavel Etingof","submitted_at":"2024-12-23T16:07:48Z","abstract_excerpt":"Let $k$ be a field, and let $\\mathcal{C}$ be a Cauchy complete $k$-linear braided category with finite dimensional morphism spaces and ${{\\rm End}(\\bf 1)}=k$. We call an indecomposable object $X$ of $\\mathcal C$ non-negligible if there exists $Y\\in \\mathcal{C}$ such that $\\bf 1$ is a direct summand of $Y\\otimes X$. We prove that every non-negligible object $X\\in \\mathcal{C}$ such that ${\\rm dim}{\\rm End}(X^{\\otimes n})<n!$ for some $n$ is automatically rigid. In particular, if $\\mathcal{C}$ is semisimple of moderate growth and weakly rigid, then $\\mathcal{C}$ is rigid. As applications, we simp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.17681","kind":"arxiv","version":3},"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/2412.17681/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"}