{"paper":{"title":"What can Topology tell us about Logical Complexity?","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"A computable variant of the gamified Katětov order on filters is isomorphic to the Lawvere-Tierney order, linking combinatorial complexity measures to computability in topos theory.","cross_cats":["math.CT"],"primary_cat":"math.LO","authors_text":"Ming Ng, Takayuki Kihara","submitted_at":"2026-05-13T20:12:08Z","abstract_excerpt":"In the 1980s, category theorists introduced the Lawvere-Tierney $(\\leq_{\\mathrm{LT}})$ order in the Effective Topos, known to effectively embed the Turing degrees. Understanding its structure is a longstanding open problem in the area. In particular, there was an informal sense that the $\\leq_{\\mathrm{LT}}$-order reflects certain shifts in combinatorial complexity, but a precise characterisation remained elusive for some time.\n  Recent work by the authors has substantially clarified the picture. In arXiv:2602.08138, the authors introduced a game-theoretic (''gamified'') version of the Kat\\v{e}"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"a computable variant of the gamified Katětov order is isomorphic to the original ≤_LT-order","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the gamified Katětov order, closed under well-founded iterations of Fubini powers, precisely captures the combinatorial complexity shifts already present in the Lawvere-Tierney order.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A computable variant of the gamified Katětov order on filters is isomorphic to the Lawvere-Tierney order, linking combinatorial complexity measures to computability in topos theory.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"798209be55e4d94c2ec44d2d68633834b64c70c9030b2b668cc282c84c9006ba"},"source":{"id":"2605.14086","kind":"arxiv","version":1},"verdict":{"id":"739ad3aa-b671-489e-bf1e-2eb202c5363e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:17:25.368934Z","strongest_claim":"a computable variant of the gamified Katětov order is isomorphic to the original ≤_LT-order","one_line_summary":"A computable variant of the gamified Katětov order on filters is isomorphic to the Lawvere-Tierney order, linking combinatorial complexity measures to computability in topos theory.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the gamified Katětov order, closed under well-founded iterations of Fubini powers, precisely captures the combinatorial complexity shifts already present in the Lawvere-Tierney order.","pith_extraction_headline":""},"references":{"count":145,"sample":[{"doi":"","year":null,"title":"Orderings of ultrafilters , year =","work_id":"6007f3e8-0f52-4770-9308-c2d8b74cf201","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1971,"title":"Partial orders on the types in N , url =","work_id":"4f1a54e2-d720-49b3-9804-412b6cbaaea9","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Takayuki Kihara and Ming Ng , date-added =. The","work_id":"e944591f-7236-4ba0-a32d-1ed5e930c23e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/j.topol.2013.08.007","year":2013,"title":"Filip. On. 2013 , bdsk-url-1 =. doi:10.1016/j.topol.2013.08.007 , journal =","work_id":"bdbd02f6-468a-482d-9e14-3dcb36f7b3e5","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1017/jsl.2024.8","year":2024,"title":"A UNIFIED APPROACH TO","work_id":"596aa850-a8cb-46ec-a3da-044e01ac19eb","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":145,"snapshot_sha256":"622230e33d46d68e8d86b2b58444a54cf797746a1485ffb6204887ecc94fbead","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b4e8b40383b97990675ece57ce0118be2e3e27dc4840325e16c1de6ac6cd7726"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}