{"paper":{"title":"Cubes are dense in $(\\infty,\\infty)$-categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Tim Campion","submitted_at":"2022-09-19T23:07:52Z","abstract_excerpt":"We show that the strict 1-category $\\square$ of cubes -- defined to be the full subcategory of strict $\\omega$-categories whose objects are the Gray tensor powers of the arrow category -- are dense in the $(\\infty,1)$-category $\\mathsf{Cat}_\\omega$ of weak $(\\infty,\\infty)$-categories, in both Rezk-complete and incomplete variants. More precisely, we show that Joyal's category $\\Theta$ is contained in the idempotent completion of $\\square$, and in fact that the idempotent completion of $\\square$ is closed under suspensions and wedge sums. This result extends a theorem of Campbell and Maehara i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.09376","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/2209.09376/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"}