{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BUHHJDGGKIYS6MH7M52ENDQQUC","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":"8bbb562ad1a53957948ae58f109c135d562fd7f56a457db421e0675dd4aa17cd","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-09-24T11:45:15Z","title_canon_sha256":"5561382e899cbc032905c2637efdc311576de7485b5b87a04138b1c05075e25e"},"schema_version":"1.0","source":{"id":"1309.6120","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6120","created_at":"2026-05-17T23:41:28Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6120v2","created_at":"2026-05-17T23:41:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6120","created_at":"2026-05-17T23:41:28Z"},{"alias_kind":"pith_short_12","alias_value":"BUHHJDGGKIYS","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BUHHJDGGKIYS6MH7","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BUHHJDGG","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:85392dace70acf0dfb47152bdf4d57cb16ce42ee7ba7b01243fcb7006afde548","target":"graph","created_at":"2026-05-17T23:41:28Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"Mitchell Buckley, Richard Garner, Ross Street, Stephen Lack","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-09-24T11:45:15Z","title":"The Catalan simplicial set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6120","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3faaa3d1bfb187cdac0fbad7f75c0262d07eb4e7d6658f23d1470f2746685946","target":"record","created_at":"2026-05-17T23:41:28Z","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":"8bbb562ad1a53957948ae58f109c135d562fd7f56a457db421e0675dd4aa17cd","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-09-24T11:45:15Z","title_canon_sha256":"5561382e899cbc032905c2637efdc311576de7485b5b87a04138b1c05075e25e"},"schema_version":"1.0","source":{"id":"1309.6120","kind":"arxiv","version":2}},"canonical_sha256":"0d0e748cc652312f30ff6774468e10a09751ecc0c440aa98d5581b69382f4251","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d0e748cc652312f30ff6774468e10a09751ecc0c440aa98d5581b69382f4251","first_computed_at":"2026-05-17T23:41:28.274084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:28.274084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jI3BbN6d21AqAogibduOa6uQ7Lp/vqhM9orQYkvVvN3DAdqquXZX8s9MXcSfucy13Ymq/3wsRK4kytMKaomUAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:28.274873Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.6120","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3faaa3d1bfb187cdac0fbad7f75c0262d07eb4e7d6658f23d1470f2746685946","sha256:85392dace70acf0dfb47152bdf4d57cb16ce42ee7ba7b01243fcb7006afde548"],"state_sha256":"ae3204010dc01ab1872923fd96a80ae328d3ef13544bf78946c445d7a3a391e4"}