{"paper":{"title":"Sphericalization and the Universal Spherical Adjunction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Any adjunction of stable infinity-categories can be turned into a spherical adjunction by inverting its twist and cotwist functors.","cross_cats":["math.AT","math.RT"],"primary_cat":"math.CT","authors_text":"Fernando Abell\\'an, Jonte G\\\"odicke","submitted_at":"2026-05-14T16:30:52Z","abstract_excerpt":"For every adjunction of stable $\\infty$-categories -- or more generally, in any locally stable $(\\infty,2)$-category -- we give a simple procedure for inverting the twist and cotwist functors associated to this adjunction. As a consequence, we obtain an explicit construction for a left and right adjoint to the inclusion of the $(\\infty,2)$-category of spherical adjunctions of stable $\\infty$-categories into all adjunctions. We utilize these adjoints to give a description of the walking spherical adjunction, a locally stable $(\\infty,2)$-category which classifies spherical adjunctions, and to p"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every adjunction of stable ∞-categories we give a simple procedure for inverting the twist and cotwist functors; this yields an explicit left and right adjoint to the inclusion of the (∞,2)-category of spherical adjunctions into all adjunctions, plus a description of the walking spherical adjunction.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The ambient structure is a locally stable (∞,2)-category in which the given adjunction lives and in which the twist and cotwist functors are well-defined and invertible after the procedure.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A construction inverts twists in adjunctions of stable infinity-categories, producing adjoints to the spherical adjunction inclusion and a walking spherical adjunction that classifies them.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any adjunction of stable infinity-categories can be turned into a spherical adjunction by inverting its twist and cotwist functors.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0ee766817a695e0cb40aa4830286be454bd18c2d7f9e0eafe78a5d76747b84b5"},"source":{"id":"2605.15037","kind":"arxiv","version":1},"verdict":{"id":"433c55e6-55e4-4995-9b74-6177d0b2c7b1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:25:01.211094Z","strongest_claim":"For every adjunction of stable ∞-categories we give a simple procedure for inverting the twist and cotwist functors; this yields an explicit left and right adjoint to the inclusion of the (∞,2)-category of spherical adjunctions into all adjunctions, plus a description of the walking spherical adjunction.","one_line_summary":"A construction inverts twists in adjunctions of stable infinity-categories, producing adjoints to the spherical adjunction inclusion and a walking spherical adjunction that classifies them.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The ambient structure is a locally stable (∞,2)-category in which the given adjunction lives and in which the twist and cotwist functors are well-defined and invertible after the procedure.","pith_extraction_headline":"Any adjunction of stable infinity-categories can be turned into a spherical adjunction by inverting its twist and cotwist functors."},"references":{"count":74,"sample":[{"doi":"","year":null,"title":"Justin Hilburn , title =","work_id":"b9ca0aa8-d7d8-4ced-b90f-f550f7704268","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Fully faithful functors and pushouts of -categories , author=. 2025 , eprint=","work_id":"250917c6-7e6c-45d4-a349-d3632e412106","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Spectral Algebraic Geometry , author=","work_id":"e599d02f-0a57-4ffd-9a47-8ac631efd667","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Free fibrations, lax colimits and","work_id":"297f5006-eaae-4cac-aaea-d38647beaf79","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"( ,2) -Topoi and descent , author=. 2024 , eprint=","work_id":"7c59fe3b-7ffa-40dd-95cb-95ffc3af0357","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":74,"snapshot_sha256":"6d76e2ad92ffe9bff756cdd795c92053cfd41c7ed7e21989b9be42af1ee88843","internal_anchors":2},"formal_canon":{"evidence_count":1,"snapshot_sha256":"c06d1f4d952d989098c6432b86585c08f44c6594899bb3216c2210c925437a0f"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}