{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:BUHHJDGGKIYS6MH7M52ENDQQUC","short_pith_number":"pith:BUHHJDGG","schema_version":"1.0","canonical_sha256":"0d0e748cc652312f30ff6774468e10a09751ecc0c440aa98d5581b69382f4251","source":{"kind":"arxiv","id":"1309.6120","version":2},"attestation_state":"computed","paper":{"title":"The Catalan simplicial set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CT","authors_text":"Mitchell Buckley, Richard Garner, Ross Street, Stephen Lack","submitted_at":"2013-09-24T11:45:15Z","abstract_excerpt":"The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.6120","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-09-24T11:45:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5561382e899cbc032905c2637efdc311576de7485b5b87a04138b1c05075e25e","abstract_canon_sha256":"8bbb562ad1a53957948ae58f109c135d562fd7f56a457db421e0675dd4aa17cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:28.274873Z","signature_b64":"jI3BbN6d21AqAogibduOa6uQ7Lp/vqhM9orQYkvVvN3DAdqquXZX8s9MXcSfucy13Ymq/3wsRK4kytMKaomUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d0e748cc652312f30ff6774468e10a09751ecc0c440aa98d5581b69382f4251","last_reissued_at":"2026-05-17T23:41:28.274084Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:28.274084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Catalan simplicial set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CT","authors_text":"Mitchell Buckley, Richard Garner, Ross Street, Stephen Lack","submitted_at":"2013-09-24T11:45:15Z","abstract_excerpt":"The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6120","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1309.6120","created_at":"2026-05-17T23:41:28.274206+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.6120v2","created_at":"2026-05-17T23:41:28.274206+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6120","created_at":"2026-05-17T23:41:28.274206+00:00"},{"alias_kind":"pith_short_12","alias_value":"BUHHJDGGKIYS","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_16","alias_value":"BUHHJDGGKIYS6MH7","created_at":"2026-05-18T12:27:40.988391+00:00"},{"alias_kind":"pith_short_8","alias_value":"BUHHJDGG","created_at":"2026-05-18T12:27:40.988391+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC","json":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC.json","graph_json":"https://pith.science/api/pith-number/BUHHJDGGKIYS6MH7M52ENDQQUC/graph.json","events_json":"https://pith.science/api/pith-number/BUHHJDGGKIYS6MH7M52ENDQQUC/events.json","paper":"https://pith.science/paper/BUHHJDGG"},"agent_actions":{"view_html":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC","download_json":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC.json","view_paper":"https://pith.science/paper/BUHHJDGG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.6120&json=true","fetch_graph":"https://pith.science/api/pith-number/BUHHJDGGKIYS6MH7M52ENDQQUC/graph.json","fetch_events":"https://pith.science/api/pith-number/BUHHJDGGKIYS6MH7M52ENDQQUC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC/action/storage_attestation","attest_author":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC/action/author_attestation","sign_citation":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC/action/citation_signature","submit_replication":"https://pith.science/pith/BUHHJDGGKIYS6MH7M52ENDQQUC/action/replication_record"}},"created_at":"2026-05-17T23:41:28.274206+00:00","updated_at":"2026-05-17T23:41:28.274206+00:00"}