{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:BD7J6G3XJQUF5T2WX7GORIM5LE","short_pith_number":"pith:BD7J6G3X","canonical_record":{"source":{"id":"1604.01187","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-04-05T09:04:22Z","cross_cats_sorted":["cs.PL","math.CT"],"title_canon_sha256":"6ba97df609f1cf35166bdeb6a0b2842567de40334e90a467fceb230ce0a368ac","abstract_canon_sha256":"b7c315b2f9f9cb9bbddb41b47b2baaa640ce22acb72fec249c590a73b6f4f45e"},"schema_version":"1.0"},"canonical_sha256":"08fe9f1b774c285ecf56bfcce8a19d591419eb47085558c5fdbc98fdb46ec1e5","source":{"kind":"arxiv","id":"1604.01187","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.01187","created_at":"2026-05-18T01:15:33Z"},{"alias_kind":"arxiv_version","alias_value":"1604.01187v1","created_at":"2026-05-18T01:15:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.01187","created_at":"2026-05-18T01:15:33Z"},{"alias_kind":"pith_short_12","alias_value":"BD7J6G3XJQUF","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BD7J6G3XJQUF5T2W","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BD7J6G3X","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:BD7J6G3XJQUF5T2WX7GORIM5LE","target":"record","payload":{"canonical_record":{"source":{"id":"1604.01187","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-04-05T09:04:22Z","cross_cats_sorted":["cs.PL","math.CT"],"title_canon_sha256":"6ba97df609f1cf35166bdeb6a0b2842567de40334e90a467fceb230ce0a368ac","abstract_canon_sha256":"b7c315b2f9f9cb9bbddb41b47b2baaa640ce22acb72fec249c590a73b6f4f45e"},"schema_version":"1.0"},"canonical_sha256":"08fe9f1b774c285ecf56bfcce8a19d591419eb47085558c5fdbc98fdb46ec1e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:33.958498Z","signature_b64":"pzxS5UgVdI4BhvB7feGxllFyK6ZGGmonwQVtO69lPiFOAd3rRyoiMJ70ImLTkZkGpGwe9JB1jORjK3FCNvzyDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08fe9f1b774c285ecf56bfcce8a19d591419eb47085558c5fdbc98fdb46ec1e5","last_reissued_at":"2026-05-18T01:15:33.957916Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:33.957916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.01187","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":"2026-05-18T01:15:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n62K1DOdfRtOBNvCf2KFN0epKTKc7D+EB5M70FIRr1p3r4wkpwKrKwpALKG1jwYvOjYhENrOA1PeWlosHY8BCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T23:04:34.609831Z"},"content_sha256":"114facc0205d199c46b2d075e2e8903ea608ecede07be56a9519dfee4598f5cf","schema_version":"1.0","event_id":"sha256:114facc0205d199c46b2d075e2e8903ea608ecede07be56a9519dfee4598f5cf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:BD7J6G3XJQUF5T2WX7GORIM5LE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Directed Containers as Categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.PL","math.CT"],"primary_cat":"cs.LO","authors_text":"Danel Ahman, Tarmo Uustalu","submitted_at":"2016-04-05T09:04:22Z","abstract_excerpt":"Directed containers make explicit the additional structure of those containers whose set functor interpretation carries a comonad structure. The data and laws of a directed container resemble those of a monoid, while the data and laws of a directed container morphism those of a monoid morphism in the reverse direction. With some reorganization, a directed container is the same as a small category, but a directed container morphism is opcleavage-like. We draw some conclusions for comonads from this observation, considering in particular basic constructions and concepts like the opposite categor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01187","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":"2026-05-18T01:15:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"igccpoHhK/McgRuRfM3HOb3/pJxKPyK9NU/nSTNCsKR8MaXH7L1+Zk0FJ91M2pD7U8tgb1NVAaOsuAUlZHOYAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T23:04:34.610506Z"},"content_sha256":"95023a23b8bd9a5ac3215f8d7fe1dfdb63ddd0ca9268fcf358e90a232f0f8c51","schema_version":"1.0","event_id":"sha256:95023a23b8bd9a5ac3215f8d7fe1dfdb63ddd0ca9268fcf358e90a232f0f8c51"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BD7J6G3XJQUF5T2WX7GORIM5LE/bundle.json","state_url":"https://pith.science/pith/BD7J6G3XJQUF5T2WX7GORIM5LE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BD7J6G3XJQUF5T2WX7GORIM5LE/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":"2026-05-26T23:04:34Z","links":{"resolver":"https://pith.science/pith/BD7J6G3XJQUF5T2WX7GORIM5LE","bundle":"https://pith.science/pith/BD7J6G3XJQUF5T2WX7GORIM5LE/bundle.json","state":"https://pith.science/pith/BD7J6G3XJQUF5T2WX7GORIM5LE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BD7J6G3XJQUF5T2WX7GORIM5LE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BD7J6G3XJQUF5T2WX7GORIM5LE","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":"b7c315b2f9f9cb9bbddb41b47b2baaa640ce22acb72fec249c590a73b6f4f45e","cross_cats_sorted":["cs.PL","math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-04-05T09:04:22Z","title_canon_sha256":"6ba97df609f1cf35166bdeb6a0b2842567de40334e90a467fceb230ce0a368ac"},"schema_version":"1.0","source":{"id":"1604.01187","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.01187","created_at":"2026-05-18T01:15:33Z"},{"alias_kind":"arxiv_version","alias_value":"1604.01187v1","created_at":"2026-05-18T01:15:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.01187","created_at":"2026-05-18T01:15:33Z"},{"alias_kind":"pith_short_12","alias_value":"BD7J6G3XJQUF","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"BD7J6G3XJQUF5T2W","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"BD7J6G3X","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:95023a23b8bd9a5ac3215f8d7fe1dfdb63ddd0ca9268fcf358e90a232f0f8c51","target":"graph","created_at":"2026-05-18T01:15:33Z","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":"Directed containers make explicit the additional structure of those containers whose set functor interpretation carries a comonad structure. The data and laws of a directed container resemble those of a monoid, while the data and laws of a directed container morphism those of a monoid morphism in the reverse direction. With some reorganization, a directed container is the same as a small category, but a directed container morphism is opcleavage-like. We draw some conclusions for comonads from this observation, considering in particular basic constructions and concepts like the opposite categor","authors_text":"Danel Ahman, Tarmo Uustalu","cross_cats":["cs.PL","math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-04-05T09:04:22Z","title":"Directed Containers as Categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01187","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":"sha256:114facc0205d199c46b2d075e2e8903ea608ecede07be56a9519dfee4598f5cf","target":"record","created_at":"2026-05-18T01:15:33Z","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":"b7c315b2f9f9cb9bbddb41b47b2baaa640ce22acb72fec249c590a73b6f4f45e","cross_cats_sorted":["cs.PL","math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-04-05T09:04:22Z","title_canon_sha256":"6ba97df609f1cf35166bdeb6a0b2842567de40334e90a467fceb230ce0a368ac"},"schema_version":"1.0","source":{"id":"1604.01187","kind":"arxiv","version":1}},"canonical_sha256":"08fe9f1b774c285ecf56bfcce8a19d591419eb47085558c5fdbc98fdb46ec1e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"08fe9f1b774c285ecf56bfcce8a19d591419eb47085558c5fdbc98fdb46ec1e5","first_computed_at":"2026-05-18T01:15:33.957916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:33.957916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pzxS5UgVdI4BhvB7feGxllFyK6ZGGmonwQVtO69lPiFOAd3rRyoiMJ70ImLTkZkGpGwe9JB1jORjK3FCNvzyDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:33.958498Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.01187","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:114facc0205d199c46b2d075e2e8903ea608ecede07be56a9519dfee4598f5cf","sha256:95023a23b8bd9a5ac3215f8d7fe1dfdb63ddd0ca9268fcf358e90a232f0f8c51"],"state_sha256":"d55b8494c0425347db048f2aaaf4390e48cec871a1a8d7c9841ee065468cd45a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fUzmLhujzGzDPq5me+Fv95pQ/x3IS+eljN34emPFlCNvDUX9bUK1XlfH5QzfIsvQwgdmYyd0LwkgKasH5zmuAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T23:04:34.614143Z","bundle_sha256":"586cae0e57c78cf1df759c5b45d2d942649ada9bf0f68afa71d297e694b6a7ea"}}