{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:WBD6QWON6VNOKXIERRNIH426Q5","short_pith_number":"pith:WBD6QWON","canonical_record":{"source":{"id":"1601.00239","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-01-03T01:39:30Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"d38e44033e33fd91d69fa49d97ebc4621311fad52a4c6548e92ced80776de0a3","abstract_canon_sha256":"3237d80cee841b4d9075bc44625e11d43c34dbd19361f53cae08654e469d8451"},"schema_version":"1.0"},"canonical_sha256":"b047e859cdf55ae55d048c5a83f35e8774f060b9caf33a56ebcf7f886e1629f7","source":{"kind":"arxiv","id":"1601.00239","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00239","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00239v2","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00239","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"pith_short_12","alias_value":"WBD6QWON6VNO","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"WBD6QWON6VNOKXIE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"WBD6QWON","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:WBD6QWON6VNOKXIERRNIH426Q5","target":"record","payload":{"canonical_record":{"source":{"id":"1601.00239","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-01-03T01:39:30Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"d38e44033e33fd91d69fa49d97ebc4621311fad52a4c6548e92ced80776de0a3","abstract_canon_sha256":"3237d80cee841b4d9075bc44625e11d43c34dbd19361f53cae08654e469d8451"},"schema_version":"1.0"},"canonical_sha256":"b047e859cdf55ae55d048c5a83f35e8774f060b9caf33a56ebcf7f886e1629f7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:31.441818Z","signature_b64":"aklbUYjR5zi44J5HgYObmIPhF6TuWgM4UWhgX+7urJhONmxWCwnktFqTIhu9pSUpt3ZrwD5k+dRdmhinRl7dBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b047e859cdf55ae55d048c5a83f35e8774f060b9caf33a56ebcf7f886e1629f7","last_reissued_at":"2026-05-18T00:38:31.441237Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:31.441237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.00239","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2/r9aHukFMcivKMs2yxuo+R0RLNlVxppPvlupOnaoylvzGtwLt5DI4sD3wELVHId20GCBOUOGKizHRUHQyN0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:44:50.855410Z"},"content_sha256":"19dd17bb160fe975e608dc625fcd4518e0a64ec57047eaa1fbfc7ffb396b600d","schema_version":"1.0","event_id":"sha256:19dd17bb160fe975e608dc625fcd4518e0a64ec57047eaa1fbfc7ffb396b600d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:WBD6QWON6VNOKXIERRNIH426Q5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fixed points of adjoint functors enriched in a quantaloid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"cs.LO","authors_text":"Hongliang Lai, Lili Shen","submitted_at":"2016-01-03T01:39:30Z","abstract_excerpt":"Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00239","kind":"arxiv","version":2},"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":"2026-05-18T00:38:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UYq0SBR75vckszGzhv5pvL13BmHZbYCE2wS8ieaYoLzQodMTL/jOO5J11oVVi0BT3JWhBPVqGzsAsWzPRTsrCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:44:50.855751Z"},"content_sha256":"8ac6eaa53ce580d2efe80e71e731530d9745fb61bcccad1458db261d353765c9","schema_version":"1.0","event_id":"sha256:8ac6eaa53ce580d2efe80e71e731530d9745fb61bcccad1458db261d353765c9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WBD6QWON6VNOKXIERRNIH426Q5/bundle.json","state_url":"https://pith.science/pith/WBD6QWON6VNOKXIERRNIH426Q5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WBD6QWON6VNOKXIERRNIH426Q5/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":"2026-06-20T12:44:50Z","links":{"resolver":"https://pith.science/pith/WBD6QWON6VNOKXIERRNIH426Q5","bundle":"https://pith.science/pith/WBD6QWON6VNOKXIERRNIH426Q5/bundle.json","state":"https://pith.science/pith/WBD6QWON6VNOKXIERRNIH426Q5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WBD6QWON6VNOKXIERRNIH426Q5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WBD6QWON6VNOKXIERRNIH426Q5","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":"3237d80cee841b4d9075bc44625e11d43c34dbd19361f53cae08654e469d8451","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-01-03T01:39:30Z","title_canon_sha256":"d38e44033e33fd91d69fa49d97ebc4621311fad52a4c6548e92ced80776de0a3"},"schema_version":"1.0","source":{"id":"1601.00239","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00239","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00239v2","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00239","created_at":"2026-05-18T00:38:31Z"},{"alias_kind":"pith_short_12","alias_value":"WBD6QWON6VNO","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"WBD6QWON6VNOKXIE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"WBD6QWON","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:8ac6eaa53ce580d2efe80e71e731530d9745fb61bcccad1458db261d353765c9","target":"graph","created_at":"2026-05-18T00:38:31Z","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":"Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it i","authors_text":"Hongliang Lai, Lili Shen","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-01-03T01:39:30Z","title":"Fixed points of adjoint functors enriched in a quantaloid"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00239","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":"sha256:19dd17bb160fe975e608dc625fcd4518e0a64ec57047eaa1fbfc7ffb396b600d","target":"record","created_at":"2026-05-18T00:38:31Z","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":"3237d80cee841b4d9075bc44625e11d43c34dbd19361f53cae08654e469d8451","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2016-01-03T01:39:30Z","title_canon_sha256":"d38e44033e33fd91d69fa49d97ebc4621311fad52a4c6548e92ced80776de0a3"},"schema_version":"1.0","source":{"id":"1601.00239","kind":"arxiv","version":2}},"canonical_sha256":"b047e859cdf55ae55d048c5a83f35e8774f060b9caf33a56ebcf7f886e1629f7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b047e859cdf55ae55d048c5a83f35e8774f060b9caf33a56ebcf7f886e1629f7","first_computed_at":"2026-05-18T00:38:31.441237Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:31.441237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aklbUYjR5zi44J5HgYObmIPhF6TuWgM4UWhgX+7urJhONmxWCwnktFqTIhu9pSUpt3ZrwD5k+dRdmhinRl7dBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:31.441818Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.00239","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19dd17bb160fe975e608dc625fcd4518e0a64ec57047eaa1fbfc7ffb396b600d","sha256:8ac6eaa53ce580d2efe80e71e731530d9745fb61bcccad1458db261d353765c9"],"state_sha256":"3b7ca374a7abb8643b9ec6e131e2f4ed36c4086e751cb14052e7ef9a0aa1d5b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rk8SnH+6jxDj/AfLdi+JKX0Y7k3lt/ctZLnhzfY92XFW6Fy53dwzHXr5kZj3Z+YD9Iv2vFw6uxIYLaYOxA8ECQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T12:44:50.857555Z","bundle_sha256":"c6e6c4ea3eff8568aeba76f7e24273a52402b59e2ee8014e5077846ddaf7f68b"}}