{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:PXTCC2E4R4BI3FI4FXEQ4K73GY","short_pith_number":"pith:PXTCC2E4","canonical_record":{"source":{"id":"1302.4488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-02-19T00:03:14Z","cross_cats_sorted":["math.AT","math.CO","math.QA"],"title_canon_sha256":"c71992f4a5c4252c18e4760070f34cde75d47ed65ae3af530627328d295eae88","abstract_canon_sha256":"dfc045a1f2a46ac6c270b10dcc5414399ec5e97feb7c7d64d208e186e976c9b3"},"schema_version":"1.0"},"canonical_sha256":"7de621689c8f028d951c2dc90e2bfb3625f2ea658684913d0c1aae210eb66bce","source":{"kind":"arxiv","id":"1302.4488","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.4488","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"1302.4488v1","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4488","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"PXTCC2E4R4BI","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"PXTCC2E4R4BI3FI4","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"PXTCC2E4","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:PXTCC2E4R4BI3FI4FXEQ4K73GY","target":"record","payload":{"canonical_record":{"source":{"id":"1302.4488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-02-19T00:03:14Z","cross_cats_sorted":["math.AT","math.CO","math.QA"],"title_canon_sha256":"c71992f4a5c4252c18e4760070f34cde75d47ed65ae3af530627328d295eae88","abstract_canon_sha256":"dfc045a1f2a46ac6c270b10dcc5414399ec5e97feb7c7d64d208e186e976c9b3"},"schema_version":"1.0"},"canonical_sha256":"7de621689c8f028d951c2dc90e2bfb3625f2ea658684913d0c1aae210eb66bce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:34.309568Z","signature_b64":"k1CaT02y5NDzqy21VGge3TmfwhRcPoYYbTJtvwfzCUw4pIGCbZiSSoC5zm+719Tp/wh6caboP02N9A2A6ldrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7de621689c8f028d951c2dc90e2bfb3625f2ea658684913d0c1aae210eb66bce","last_reissued_at":"2026-05-18T02:51:34.309070Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:34.309070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.4488","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kJq6PfRtN5CKdnINGbtaDOs5OldFbgJ5LDD9iZ3p4QEDUmUHGEmKG4qa4FClWelXLLico19VYP729RIqvn2MDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T23:17:38.462429Z"},"content_sha256":"1c02d864e15ec3e9aa4dea3f8e197bdb514400b79acc70814b25231b8e78c56d","schema_version":"1.0","event_id":"sha256:1c02d864e15ec3e9aa4dea3f8e197bdb514400b79acc70814b25231b8e78c56d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:PXTCC2E4R4BI3FI4FXEQ4K73GY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Triangulations, orientals, and skew monoidal categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.CO","math.QA"],"primary_cat":"math.CT","authors_text":"Ross Street, Stephen Lack","submitted_at":"2013-02-19T00:03:14Z","abstract_excerpt":"A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As the objects of Fsk are meaningfully bracketed words in the skew unit I and the generating object X, it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them. This leads us to relationships between triangulations of polygons, the Tamari lattice, left and right bracketing functions, and the orientals. A consequence of ou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4488","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f/rm3rdgJfsCs/oN7wzwLDMCPR00eRd8PwUVwyF89pKGWxqrPIKvKStDk7LJ2JEo95d3qgy0uf46onqdkLWZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T23:17:38.462815Z"},"content_sha256":"bc1428175f8ebcbb07f87b211a6855be51217533021731249c93a90fe84e70f7","schema_version":"1.0","event_id":"sha256:bc1428175f8ebcbb07f87b211a6855be51217533021731249c93a90fe84e70f7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY/bundle.json","state_url":"https://pith.science/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY/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":"2026-06-03T23:17:38Z","links":{"resolver":"https://pith.science/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY","bundle":"https://pith.science/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY/bundle.json","state":"https://pith.science/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PXTCC2E4R4BI3FI4FXEQ4K73GY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PXTCC2E4R4BI3FI4FXEQ4K73GY","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":"dfc045a1f2a46ac6c270b10dcc5414399ec5e97feb7c7d64d208e186e976c9b3","cross_cats_sorted":["math.AT","math.CO","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-02-19T00:03:14Z","title_canon_sha256":"c71992f4a5c4252c18e4760070f34cde75d47ed65ae3af530627328d295eae88"},"schema_version":"1.0","source":{"id":"1302.4488","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.4488","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"1302.4488v1","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4488","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"PXTCC2E4R4BI","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"PXTCC2E4R4BI3FI4","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"PXTCC2E4","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:bc1428175f8ebcbb07f87b211a6855be51217533021731249c93a90fe84e70f7","target":"graph","created_at":"2026-05-18T02:51:34Z","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":"A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As the objects of Fsk are meaningfully bracketed words in the skew unit I and the generating object X, it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them. This leads us to relationships between triangulations of polygons, the Tamari lattice, left and right bracketing functions, and the orientals. A consequence of ou","authors_text":"Ross Street, Stephen Lack","cross_cats":["math.AT","math.CO","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-02-19T00:03:14Z","title":"Triangulations, orientals, and skew monoidal categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4488","kind":"arxiv","version":1},"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":"sha256:1c02d864e15ec3e9aa4dea3f8e197bdb514400b79acc70814b25231b8e78c56d","target":"record","created_at":"2026-05-18T02:51:34Z","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":"dfc045a1f2a46ac6c270b10dcc5414399ec5e97feb7c7d64d208e186e976c9b3","cross_cats_sorted":["math.AT","math.CO","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2013-02-19T00:03:14Z","title_canon_sha256":"c71992f4a5c4252c18e4760070f34cde75d47ed65ae3af530627328d295eae88"},"schema_version":"1.0","source":{"id":"1302.4488","kind":"arxiv","version":1}},"canonical_sha256":"7de621689c8f028d951c2dc90e2bfb3625f2ea658684913d0c1aae210eb66bce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7de621689c8f028d951c2dc90e2bfb3625f2ea658684913d0c1aae210eb66bce","first_computed_at":"2026-05-18T02:51:34.309070Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:34.309070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k1CaT02y5NDzqy21VGge3TmfwhRcPoYYbTJtvwfzCUw4pIGCbZiSSoC5zm+719Tp/wh6caboP02N9A2A6ldrCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:34.309568Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.4488","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c02d864e15ec3e9aa4dea3f8e197bdb514400b79acc70814b25231b8e78c56d","sha256:bc1428175f8ebcbb07f87b211a6855be51217533021731249c93a90fe84e70f7"],"state_sha256":"921b0f30070e6c79cbc54267d39055b222834d0432618bb58500513e6dd752c3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2yX3UvHAK+AweBK5NvpXWIfso1d8zKmd75GzLbZL41BdjbGdijmNHzMGK00/ofhpRJUUeIwdeYbOdQgoD0C2DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T23:17:38.464940Z","bundle_sha256":"016eec3adb796a437972483b271bad251cb58718621ac06b8918dc58ed5edb44"}}