{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:A3FS4OLLOAG4AQ4DE7QHW3XATV","short_pith_number":"pith:A3FS4OLL","canonical_record":{"source":{"id":"1804.03527","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-10T13:43:01Z","cross_cats_sorted":["cs.LO","math.CT","math.QA"],"title_canon_sha256":"9c92c9ce59def27e39e7b22a6a56c18f397bf81218d473e069d34d3e54539c5c","abstract_canon_sha256":"7d15fcf93980ac84a5490259fa296375d7da5f4d55c1234cb3fc3a1768170f7f"},"schema_version":"1.0"},"canonical_sha256":"06cb2e396b700dc0438327e07b6ee09d7988e08c975add05a4bc238c2d458561","source":{"kind":"arxiv","id":"1804.03527","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.03527","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"arxiv_version","alias_value":"1804.03527v4","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03527","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"pith_short_12","alias_value":"A3FS4OLLOAG4","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"pith_short_16","alias_value":"A3FS4OLLOAG4AQ4D","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"pith_short_8","alias_value":"A3FS4OLL","created_at":"2026-07-05T00:37:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:A3FS4OLLOAG4AQ4DE7QHW3XATV","target":"record","payload":{"canonical_record":{"source":{"id":"1804.03527","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-10T13:43:01Z","cross_cats_sorted":["cs.LO","math.CT","math.QA"],"title_canon_sha256":"9c92c9ce59def27e39e7b22a6a56c18f397bf81218d473e069d34d3e54539c5c","abstract_canon_sha256":"7d15fcf93980ac84a5490259fa296375d7da5f4d55c1234cb3fc3a1768170f7f"},"schema_version":"1.0"},"canonical_sha256":"06cb2e396b700dc0438327e07b6ee09d7988e08c975add05a4bc238c2d458561","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:37:25.717556Z","signature_b64":"OSw+0jAsiK+4mXIzrzHRo+dyMM0ZfntsL98ccqhOgsOuC0DbmAVgyEjOd7LzHBbUm7Iu6CLcIdlPXxz90BU7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06cb2e396b700dc0438327e07b6ee09d7988e08c975add05a4bc238c2d458561","last_reissued_at":"2026-07-05T00:37:25.717020Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:37:25.717020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.03527","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:37:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NyEWEUMNuLgY9h3BX6i/tyw/a36CfW4Z1MX3iT/chVFHOcvlZ6fNut35wC0FXQX0+de55Bs0oSmPstK5cVDZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T03:07:54.341933Z"},"content_sha256":"1beeec4d78e1283ac139e1fd08d28edcba44bde8b9b0c6959aa2d5c1114109f0","schema_version":"1.0","event_id":"sha256:1beeec4d78e1283ac139e1fd08d28edcba44bde8b9b0c6959aa2d5c1114109f0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:A3FS4OLLOAG4AQ4DE7QHW3XATV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bimonoidal Structure of Probability Monads","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO","math.CT","math.QA"],"primary_cat":"math.PR","authors_text":"Paolo Perrone, Tobias Fritz","submitted_at":"2018-04-10T13:43:01Z","abstract_excerpt":"We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimono"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03527","kind":"arxiv","version":4},"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/1804.03527/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":"2026-07-05T00:37:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eRDxAcXL9ZqJpBBOCzRZgiV+xMb2epjACD3oPhZqn/1+yz2b2FS9McF2pY/xjoCJp4xVf5JuN/r4rH7+yM7jAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T03:07:54.342618Z"},"content_sha256":"c289c664914d3c2063736084f245f770a9b2b56252597b11647dc1b4277672a6","schema_version":"1.0","event_id":"sha256:c289c664914d3c2063736084f245f770a9b2b56252597b11647dc1b4277672a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV/bundle.json","state_url":"https://pith.science/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV/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":"2026-08-21T03:07:54Z","links":{"resolver":"https://pith.science/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV","bundle":"https://pith.science/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV/bundle.json","state":"https://pith.science/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A3FS4OLLOAG4AQ4DE7QHW3XATV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:A3FS4OLLOAG4AQ4DE7QHW3XATV","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":"7d15fcf93980ac84a5490259fa296375d7da5f4d55c1234cb3fc3a1768170f7f","cross_cats_sorted":["cs.LO","math.CT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-10T13:43:01Z","title_canon_sha256":"9c92c9ce59def27e39e7b22a6a56c18f397bf81218d473e069d34d3e54539c5c"},"schema_version":"1.0","source":{"id":"1804.03527","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.03527","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"arxiv_version","alias_value":"1804.03527v4","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.03527","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"pith_short_12","alias_value":"A3FS4OLLOAG4","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"pith_short_16","alias_value":"A3FS4OLLOAG4AQ4D","created_at":"2026-07-05T00:37:25Z"},{"alias_kind":"pith_short_8","alias_value":"A3FS4OLL","created_at":"2026-07-05T00:37:25Z"}],"graph_snapshots":[{"event_id":"sha256:c289c664914d3c2063736084f245f770a9b2b56252597b11647dc1b4277672a6","target":"graph","created_at":"2026-07-05T00:37:25Z","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/1804.03527/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimono","authors_text":"Paolo Perrone, Tobias Fritz","cross_cats":["cs.LO","math.CT","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-10T13:43:01Z","title":"Bimonoidal Structure of Probability Monads"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.03527","kind":"arxiv","version":4},"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":"sha256:1beeec4d78e1283ac139e1fd08d28edcba44bde8b9b0c6959aa2d5c1114109f0","target":"record","created_at":"2026-07-05T00:37:25Z","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":"7d15fcf93980ac84a5490259fa296375d7da5f4d55c1234cb3fc3a1768170f7f","cross_cats_sorted":["cs.LO","math.CT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-10T13:43:01Z","title_canon_sha256":"9c92c9ce59def27e39e7b22a6a56c18f397bf81218d473e069d34d3e54539c5c"},"schema_version":"1.0","source":{"id":"1804.03527","kind":"arxiv","version":4}},"canonical_sha256":"06cb2e396b700dc0438327e07b6ee09d7988e08c975add05a4bc238c2d458561","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06cb2e396b700dc0438327e07b6ee09d7988e08c975add05a4bc238c2d458561","first_computed_at":"2026-07-05T00:37:25.717020Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:37:25.717020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OSw+0jAsiK+4mXIzrzHRo+dyMM0ZfntsL98ccqhOgsOuC0DbmAVgyEjOd7LzHBbUm7Iu6CLcIdlPXxz90BU7DA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:37:25.717556Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.03527","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1beeec4d78e1283ac139e1fd08d28edcba44bde8b9b0c6959aa2d5c1114109f0","sha256:c289c664914d3c2063736084f245f770a9b2b56252597b11647dc1b4277672a6"],"state_sha256":"b94a6f8abc90d51e74ab417225afdac6eb35ead3982f3c0721732dddf75b5aa2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z/nGkiKK8J8KsXUtyEoyiEt55T9KEIJRgUqq8KnPShw+Tr4p+J5dlKmPgKxmRh7+ZvMIAM+MsIjvxJEMJVriDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T03:07:54.353606Z","bundle_sha256":"ad4f486de43a5903bcf5bd17e6ab2b030136a2ce8b65eb101111a6896d10a965"}}