{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:YMP4LKXYRJGSZIJ453VMLH2ILF","short_pith_number":"pith:YMP4LKXY","canonical_record":{"source":{"id":"2604.18346","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-20T14:42:12Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6a4570da804ee275be570e8cde7c3a0bb7df57b968f48d8c26964a6461516c07","abstract_canon_sha256":"336e15afc8f78bd3afd9a63a6267e424b186fecd57f8dc7adf48019565aa2e00"},"schema_version":"1.0"},"canonical_sha256":"c31fc5aaf88a4d2ca13ceeeac59f485974d7e97630a1e862b181735d2812c4ed","source":{"kind":"arxiv","id":"2604.18346","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.18346","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"arxiv_version","alias_value":"2604.18346v2","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.18346","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"pith_short_12","alias_value":"YMP4LKXYRJGS","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"pith_short_16","alias_value":"YMP4LKXYRJGSZIJ4","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"pith_short_8","alias_value":"YMP4LKXY","created_at":"2026-05-21T02:05:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:YMP4LKXYRJGSZIJ453VMLH2ILF","target":"record","payload":{"canonical_record":{"source":{"id":"2604.18346","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-20T14:42:12Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6a4570da804ee275be570e8cde7c3a0bb7df57b968f48d8c26964a6461516c07","abstract_canon_sha256":"336e15afc8f78bd3afd9a63a6267e424b186fecd57f8dc7adf48019565aa2e00"},"schema_version":"1.0"},"canonical_sha256":"c31fc5aaf88a4d2ca13ceeeac59f485974d7e97630a1e862b181735d2812c4ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T02:05:03.143598Z","signature_b64":"z+zvMtxrFWL2DjIs1ZOLIXcU4i8sBtpMX47xfR0WQh2bWiXbjuumt/+EMQiXpZTUw7xEw937Y5daZuRUavIYCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c31fc5aaf88a4d2ca13ceeeac59f485974d7e97630a1e862b181735d2812c4ed","last_reissued_at":"2026-05-21T02:05:03.142930Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T02:05:03.142930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.18346","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-21T02:05:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mRADrrhaFOpuGZww/DThPXpWNQ6iVSp14RsKzpsydG4s9V52jLbdIwbGxJUFNDEbFNNDAlVMOsutAZZbew1TAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:26:44.370534Z"},"content_sha256":"f89730632fb57a44e21c94bd12f203b8893e77f76ed0a52af813f866ee56ac83","schema_version":"1.0","event_id":"sha256:f89730632fb57a44e21c94bd12f203b8893e77f76ed0a52af813f866ee56ac83"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:YMP4LKXYRJGSZIJ453VMLH2ILF","target":"graph","payload":{"graph_snapshot":{"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"},"verdict_id":"ecc07f85-5dba-4d6e-b29c-180b7357d187"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-21T02:05:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eaCkmyUPTDO0a59PY3JmpUZkcbWOA640xw5j8feca3OVUq6WkQTxd2LvMrYIly1cOL3a0uS5xI9BhKYGc5nXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T22:26:44.371600Z"},"content_sha256":"a7ff6a27daff7d54360307c203ec99dbef53632f9eee919d9a343ad1ce0f0553","schema_version":"1.0","event_id":"sha256:a7ff6a27daff7d54360307c203ec99dbef53632f9eee919d9a343ad1ce0f0553"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YMP4LKXYRJGSZIJ453VMLH2ILF/bundle.json","state_url":"https://pith.science/pith/YMP4LKXYRJGSZIJ453VMLH2ILF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YMP4LKXYRJGSZIJ453VMLH2ILF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T22:26:44Z","links":{"resolver":"https://pith.science/pith/YMP4LKXYRJGSZIJ453VMLH2ILF","bundle":"https://pith.science/pith/YMP4LKXYRJGSZIJ453VMLH2ILF/bundle.json","state":"https://pith.science/pith/YMP4LKXYRJGSZIJ453VMLH2ILF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YMP4LKXYRJGSZIJ453VMLH2ILF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:YMP4LKXYRJGSZIJ453VMLH2ILF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"336e15afc8f78bd3afd9a63a6267e424b186fecd57f8dc7adf48019565aa2e00","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-20T14:42:12Z","title_canon_sha256":"6a4570da804ee275be570e8cde7c3a0bb7df57b968f48d8c26964a6461516c07"},"schema_version":"1.0","source":{"id":"2604.18346","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.18346","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"arxiv_version","alias_value":"2604.18346v2","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.18346","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"pith_short_12","alias_value":"YMP4LKXYRJGS","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"pith_short_16","alias_value":"YMP4LKXYRJGSZIJ4","created_at":"2026-05-21T02:05:03Z"},{"alias_kind":"pith_short_8","alias_value":"YMP4LKXY","created_at":"2026-05-21T02:05:03Z"}],"graph_snapshots":[{"event_id":"sha256:a7ff6a27daff7d54360307c203ec99dbef53632f9eee919d9a343ad1ce0f0553","target":"graph","created_at":"2026-05-21T02:05:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The biset category of finite groups is implemented as a tower of standard categorical constructions in the CAP software."}],"snapshot_sha256":"b29747137db5d69c02d1a69bc1caf082d15d588af7e988f4a323f80063025ef5"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-20T04:09:26.190554Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.18346/integrity.json","findings":[],"snapshot_sha256":"b762c3b016ec7611332dab601ad9465deff018b7ce790d17297c73c2c581c938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Fabian Zickgraf, Marc Talleux, Mohamed Barakat","cross_cats":["math.GR"],"headline":"The biset category of finite groups is implemented as a tower of standard categorical constructions in the CAP software.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-20T14:42:12Z","title":"Implementing the biset category of finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.18346","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T03:03:17.302731Z","id":"ecc07f85-5dba-4d6e-b29c-180b7357d187","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"The biset category of finite groups is implemented as a tower of standard categorical constructions in the CAP software.","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.","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."}},"verdict_id":"ecc07f85-5dba-4d6e-b29c-180b7357d187"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f89730632fb57a44e21c94bd12f203b8893e77f76ed0a52af813f866ee56ac83","target":"record","created_at":"2026-05-21T02:05:03Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"336e15afc8f78bd3afd9a63a6267e424b186fecd57f8dc7adf48019565aa2e00","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CT","submitted_at":"2026-04-20T14:42:12Z","title_canon_sha256":"6a4570da804ee275be570e8cde7c3a0bb7df57b968f48d8c26964a6461516c07"},"schema_version":"1.0","source":{"id":"2604.18346","kind":"arxiv","version":2}},"canonical_sha256":"c31fc5aaf88a4d2ca13ceeeac59f485974d7e97630a1e862b181735d2812c4ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c31fc5aaf88a4d2ca13ceeeac59f485974d7e97630a1e862b181735d2812c4ed","first_computed_at":"2026-05-21T02:05:03.142930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T02:05:03.142930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z+zvMtxrFWL2DjIs1ZOLIXcU4i8sBtpMX47xfR0WQh2bWiXbjuumt/+EMQiXpZTUw7xEw937Y5daZuRUavIYCQ==","signature_status":"signed_v1","signed_at":"2026-05-21T02:05:03.143598Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.18346","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f89730632fb57a44e21c94bd12f203b8893e77f76ed0a52af813f866ee56ac83","sha256:a7ff6a27daff7d54360307c203ec99dbef53632f9eee919d9a343ad1ce0f0553"],"state_sha256":"71f99913a7968082285f156f84880d61b1675c4dadcd0913452a697e0ff0084f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uuVytpF7sVPCNyKCXpbZ1p2c2GpuqQwZ3hWVqaKjhog/+/alsxaq53/2c6S9MeCgnj95f6g3lJJ28LIyAT5/Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T22:26:44.376197Z","bundle_sha256":"9f294f7bbf313e28623a0b98b9de1d2f22f7249add1e7f05b19a940968efddf7"}}