{"paper":{"title":"On the equivalence between the existence of $n$-kernels and $n$-cokernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.","cross_cats":["math.RA","math.RT"],"primary_cat":"math.CT","authors_text":"Vitor Gulisz, Wolfgang Rump","submitted_at":"2025-10-27T14:20:01Z","abstract_excerpt":"We give an elementary proof of the statement that if an idempotent complete preadditive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If an idempotent complete preadditive category has weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, where n is a nonnegative integer.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The category under consideration is idempotent complete and preadditive and already possesses weak kernels and weak cokernels (as stated in the main theorem).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist if and only if n-cokernels exist.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ac0de6feac25256130574af315cf3be3267e92e54e532c47861d3e4df627fe66"},"source":{"id":"2510.23369","kind":"arxiv","version":2},"verdict":{"id":"54be0160-6d03-4ab7-8863-f65b306216a7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T03:50:37.119395Z","strongest_claim":"If an idempotent complete preadditive category has weak kernels and weak cokernels, then it has n-kernels if and only if it has n-cokernels, where n is a nonnegative integer.","one_line_summary":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist if and only if n-cokernels exist.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The category under consideration is idempotent complete and preadditive and already possesses weak kernels and weak cokernels (as stated in the main theorem).","pith_extraction_headline":"In idempotent complete preadditive categories with weak kernels and weak cokernels, n-kernels exist exactly when n-cokernels do."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.23369/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":1,"snapshot_sha256":"f86edd0fdd926385945914e3d7f0cb267d8301f6461086caeb9b8e667709b3b9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}