{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:EYPQUH45Z4UYO4RMYQKHLKQ5FM","short_pith_number":"pith:EYPQUH45","canonical_record":{"source":{"id":"2005.01340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-05-04T09:32:52Z","cross_cats_sorted":[],"title_canon_sha256":"85917fed9172d8f136ba3788ac3986be416dc62c838cc628011a621bd76a491b","abstract_canon_sha256":"a6fa9a5c4577ae2e0df87a21a15a3a094e2fedca1d26887ecf35c059ba0dce4b"},"schema_version":"1.0"},"canonical_sha256":"261f0a1f9dcf2987722cc41475aa1d2b15eebd99c6d45a87733b90d42ecd70d7","source":{"kind":"arxiv","id":"2005.01340","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.01340","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"arxiv_version","alias_value":"2005.01340v1","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.01340","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"pith_short_12","alias_value":"EYPQUH45Z4UY","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"pith_short_16","alias_value":"EYPQUH45Z4UYO4RM","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"pith_short_8","alias_value":"EYPQUH45","created_at":"2026-07-05T01:00:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:EYPQUH45Z4UYO4RMYQKHLKQ5FM","target":"record","payload":{"canonical_record":{"source":{"id":"2005.01340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-05-04T09:32:52Z","cross_cats_sorted":[],"title_canon_sha256":"85917fed9172d8f136ba3788ac3986be416dc62c838cc628011a621bd76a491b","abstract_canon_sha256":"a6fa9a5c4577ae2e0df87a21a15a3a094e2fedca1d26887ecf35c059ba0dce4b"},"schema_version":"1.0"},"canonical_sha256":"261f0a1f9dcf2987722cc41475aa1d2b15eebd99c6d45a87733b90d42ecd70d7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:00:06.751337Z","signature_b64":"fRZZevtb1EtktXojgBs9zSIkZY5yDTLbPh5mTXFoShCJUwLZS6/pyzVBqbuGMhdEqizmEIPaFhI+XYhNgvRcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"261f0a1f9dcf2987722cc41475aa1d2b15eebd99c6d45a87733b90d42ecd70d7","last_reissued_at":"2026-07-05T01:00:06.750897Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:00:06.750897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2005.01340","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":"2026-07-05T01:00:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cl2ZMqVgY8f4rIpP//eY/FTrw5NRDnJULlp4+0SEo2Rizp6F/pb8bZ51NbubyZqUuS0gkdv/SgIiSyy+lt3tAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T11:15:58.102921Z"},"content_sha256":"e36c637c09d8d0ae8ef3ff5376ccbba893ab14c47a16c43f05c278be8395eebe","schema_version":"1.0","event_id":"sha256:e36c637c09d8d0ae8ef3ff5376ccbba893ab14c47a16c43f05c278be8395eebe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:EYPQUH45Z4UYO4RMYQKHLKQ5FM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Duoidal categories, measuring comonoids and enrichment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Christina Vasilakopoulou, Ignacio L\\'opez Franco","submitted_at":"2020-05-04T09:32:52Z","abstract_excerpt":"We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with an enrichment in the category of comonoids. The enriched homs are provided by the universal measuring comonoids. We study a number of duoidal structures on categories of graded objects and of species and the associated enriched categories, such as an enrichment of graded (twisted) monoids in graded (twisted) comonoids, as well as two enrichments of symmetri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.01340","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2005.01340/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:00:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QFwMr2lOspV0mk6U+EziiKbsC51SrQ7KHPfTktVI6KbhWzBFaTOPJCco2LaARwOhvfCbwJEjyLROl0B8rjBEAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T11:15:58.103479Z"},"content_sha256":"87e272934ba893bee45a13e8fd109b81d622cd87655bf53836a1fe594b568c63","schema_version":"1.0","event_id":"sha256:87e272934ba893bee45a13e8fd109b81d622cd87655bf53836a1fe594b568c63"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM/bundle.json","state_url":"https://pith.science/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM/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":"2026-08-19T11:15:58Z","links":{"resolver":"https://pith.science/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM","bundle":"https://pith.science/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM/bundle.json","state":"https://pith.science/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EYPQUH45Z4UYO4RMYQKHLKQ5FM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:EYPQUH45Z4UYO4RMYQKHLKQ5FM","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":"a6fa9a5c4577ae2e0df87a21a15a3a094e2fedca1d26887ecf35c059ba0dce4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-05-04T09:32:52Z","title_canon_sha256":"85917fed9172d8f136ba3788ac3986be416dc62c838cc628011a621bd76a491b"},"schema_version":"1.0","source":{"id":"2005.01340","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.01340","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"arxiv_version","alias_value":"2005.01340v1","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.01340","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"pith_short_12","alias_value":"EYPQUH45Z4UY","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"pith_short_16","alias_value":"EYPQUH45Z4UYO4RM","created_at":"2026-07-05T01:00:06Z"},{"alias_kind":"pith_short_8","alias_value":"EYPQUH45","created_at":"2026-07-05T01:00:06Z"}],"graph_snapshots":[{"event_id":"sha256:87e272934ba893bee45a13e8fd109b81d622cd87655bf53836a1fe594b568c63","target":"graph","created_at":"2026-07-05T01:00:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2005.01340/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with an enrichment in the category of comonoids. The enriched homs are provided by the universal measuring comonoids. We study a number of duoidal structures on categories of graded objects and of species and the associated enriched categories, such as an enrichment of graded (twisted) monoids in graded (twisted) comonoids, as well as two enrichments of symmetri","authors_text":"Christina Vasilakopoulou, Ignacio L\\'opez Franco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-05-04T09:32:52Z","title":"Duoidal categories, measuring comonoids and enrichment"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.01340","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:e36c637c09d8d0ae8ef3ff5376ccbba893ab14c47a16c43f05c278be8395eebe","target":"record","created_at":"2026-07-05T01:00:06Z","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":"a6fa9a5c4577ae2e0df87a21a15a3a094e2fedca1d26887ecf35c059ba0dce4b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-05-04T09:32:52Z","title_canon_sha256":"85917fed9172d8f136ba3788ac3986be416dc62c838cc628011a621bd76a491b"},"schema_version":"1.0","source":{"id":"2005.01340","kind":"arxiv","version":1}},"canonical_sha256":"261f0a1f9dcf2987722cc41475aa1d2b15eebd99c6d45a87733b90d42ecd70d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"261f0a1f9dcf2987722cc41475aa1d2b15eebd99c6d45a87733b90d42ecd70d7","first_computed_at":"2026-07-05T01:00:06.750897Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:00:06.750897Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fRZZevtb1EtktXojgBs9zSIkZY5yDTLbPh5mTXFoShCJUwLZS6/pyzVBqbuGMhdEqizmEIPaFhI+XYhNgvRcBg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:00:06.751337Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.01340","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e36c637c09d8d0ae8ef3ff5376ccbba893ab14c47a16c43f05c278be8395eebe","sha256:87e272934ba893bee45a13e8fd109b81d622cd87655bf53836a1fe594b568c63"],"state_sha256":"ca8d90d74ad04319eedc97cc491af0903edb739e3a48920bbabc59638948f5f9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C2O7M/iLNr1jEVv0QHR89pxUZFPmEl2dpHalkjcV7qy8T+XXXDo2lTpKeiy5Z4OjxNaYkh056bI/boGccJxMCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T11:15:58.106894Z","bundle_sha256":"8140bbf1ddec95af81a64f38ab81656b6782dd8e009be3299cb958b985cad645"}}