{"paper":{"title":"Implementing the biset category of finite groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The biset category of finite groups is implemented as a tower of standard categorical constructions in the CAP software.","cross_cats":["math.GR"],"primary_cat":"math.CT","authors_text":"Fabian Zickgraf, Marc Talleux, Mohamed Barakat","submitted_at":"2026-04-20T14:42:12Z","abstract_excerpt":"We describe an implementation of the biset category of finite groups as a tower of standard categorical constructions, all of which are implemented in the software projec t CAP for algorithmic category theory. In particular, we describe the composition of bisets as a composition in a Kleisli category of some biadjunction monad. This composition relies on the universal property of the coequalizer completion of a group viewed as a groupoid on one object. Expressing this universal property offers an elegant categorical interpretation of the Schreier-Sims orbit algorithm. Indeed, the implementatio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We describe an implementation of the biset category of finite groups as a tower of standard categorical constructions, all of which are implemented in the software project CAP for algorithmic category theory. In particular, we describe the composition of bisets as a composition in a Kleisli category of some biadjunction monad. This composition relies on the universal property of the coequalizer completion of a group viewed as a groupoid on one object. Implementing this universal property makes nontrivial use of the Schreier-Sims orbit algorithm.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the universal property of the coequalizer completion of a one-object groupoid can be realized algorithmically via the Schreier-Sims orbit algorithm in a manner that correctly implements biset composition inside the existing CAP framework.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Implementation of the biset category of finite groups in CAP as a tower of categorical constructions, with biset composition realized as Kleisli composition of a biadjunction monad using the Schreier-Sims algorithm on coequalizer completions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The biset category of finite groups is implemented as a tower of standard categorical constructions in the CAP software.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b29747137db5d69c02d1a69bc1caf082d15d588af7e988f4a323f80063025ef5"},"source":{"id":"2604.18346","kind":"arxiv","version":2},"verdict":{"id":"ecc07f85-5dba-4d6e-b29c-180b7357d187","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T03:03:17.302731Z","strongest_claim":"We describe an implementation of the biset category of finite groups as a tower of standard categorical constructions, all of which are implemented in the software project CAP for algorithmic category theory. In particular, we describe the composition of bisets as a composition in a Kleisli category of some biadjunction monad. This composition relies on the universal property of the coequalizer completion of a group viewed as a groupoid on one object. Implementing this universal property makes nontrivial use of the Schreier-Sims orbit algorithm.","one_line_summary":"Implementation of the biset category of finite groups in CAP as a tower of categorical constructions, with biset composition realized as Kleisli composition of a biadjunction monad using the Schreier-Sims algorithm on coequalizer completions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the universal property of the coequalizer completion of a one-object groupoid can be realized algorithmically via the Schreier-Sims orbit algorithm in a manner that correctly implements biset composition inside the existing CAP framework.","pith_extraction_headline":"The biset category of finite groups is implemented as a tower of standard categorical constructions in the CAP software."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.18346/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T04:09:26.190554Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b762c3b016ec7611332dab601ad9465deff018b7ce790d17297c73c2c581c938"},"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"}