{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:RKGA3AXXCUO3LIBVV3GMCHGTHE","short_pith_number":"pith:RKGA3AXX","schema_version":"1.0","canonical_sha256":"8a8c0d82f7151db5a035aeccc11cd339311141c3b6b422fcda3adc0b7325e0f4","source":{"kind":"arxiv","id":"1809.00738","version":2},"attestation_state":"computed","paper":{"title":"Categories of Optics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Mitchell Riley","submitted_at":"2018-09-03T22:30:12Z","abstract_excerpt":"Bidirectional data accessors such as lenses, prisms and traversals are all instances of the same general 'optic' construction. We give a careful account of this construction and show that it extends to a functor from the category of symmetric monoidal categories to itself. We also show that this construction enjoys a universal property: it freely adds counit morphisms to a symmetric monoidal category. Missing in the folklore is a general definition of 'lawfulness' that applies directly to any optic category. We provide such a definition and show that it is equivalent to the folklore profunctor"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.00738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2018-09-03T22:30:12Z","cross_cats_sorted":[],"title_canon_sha256":"b99e28c77844ce83c4c2de31de1d488ce05332fcec8a8255e274b8f1812a24db","abstract_canon_sha256":"7b4cac1800a7926a088f0852cbbb2d02c8251141337c5a7ae4fd85fd6221003b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:17.989334Z","signature_b64":"jEDi3sRrE2yi+2OEtSp1Eu+dUCgeIiH6FZqtAsz7d4G6GsLJI1JYtwlwhWJ6z51AKUm/1elfvOe/xxNDRelYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a8c0d82f7151db5a035aeccc11cd339311141c3b6b422fcda3adc0b7325e0f4","last_reissued_at":"2026-05-18T00:06:17.988937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:17.988937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Categories of Optics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Mitchell Riley","submitted_at":"2018-09-03T22:30:12Z","abstract_excerpt":"Bidirectional data accessors such as lenses, prisms and traversals are all instances of the same general 'optic' construction. We give a careful account of this construction and show that it extends to a functor from the category of symmetric monoidal categories to itself. We also show that this construction enjoys a universal property: it freely adds counit morphisms to a symmetric monoidal category. Missing in the folklore is a general definition of 'lawfulness' that applies directly to any optic category. We provide such a definition and show that it is equivalent to the folklore profunctor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00738","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.00738","created_at":"2026-05-18T00:06:17.989004+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.00738v2","created_at":"2026-05-18T00:06:17.989004+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00738","created_at":"2026-05-18T00:06:17.989004+00:00"},{"alias_kind":"pith_short_12","alias_value":"RKGA3AXXCUO3","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"RKGA3AXXCUO3LIBV","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"RKGA3AXX","created_at":"2026-05-18T12:32:50.500415+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2207.09180","citing_title":"Polycategorical Constructions for Unitary Supermaps of Arbitrary Dimension","ref_index":7,"is_internal_anchor":true},{"citing_arxiv_id":"2602.23865","citing_title":"Supermaps on generalised theories","ref_index":56,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE","json":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE.json","graph_json":"https://pith.science/api/pith-number/RKGA3AXXCUO3LIBVV3GMCHGTHE/graph.json","events_json":"https://pith.science/api/pith-number/RKGA3AXXCUO3LIBVV3GMCHGTHE/events.json","paper":"https://pith.science/paper/RKGA3AXX"},"agent_actions":{"view_html":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE","download_json":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE.json","view_paper":"https://pith.science/paper/RKGA3AXX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.00738&json=true","fetch_graph":"https://pith.science/api/pith-number/RKGA3AXXCUO3LIBVV3GMCHGTHE/graph.json","fetch_events":"https://pith.science/api/pith-number/RKGA3AXXCUO3LIBVV3GMCHGTHE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE/action/storage_attestation","attest_author":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE/action/author_attestation","sign_citation":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE/action/citation_signature","submit_replication":"https://pith.science/pith/RKGA3AXXCUO3LIBVV3GMCHGTHE/action/replication_record"}},"created_at":"2026-05-18T00:06:17.989004+00:00","updated_at":"2026-05-18T00:06:17.989004+00:00"}