{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:IVSGOQNY35GSM3N2TAOHPLCWAB","short_pith_number":"pith:IVSGOQNY","schema_version":"1.0","canonical_sha256":"45646741b8df4d266dba981c77ac56004b29601b588cb7b7d9c4941a4afe8f3a","source":{"kind":"arxiv","id":"2205.04890","version":3},"attestation_state":"computed","paper":{"title":"Formal category theory in augmented virtual double categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Seerp Roald Koudenburg","submitted_at":"2022-05-10T13:40:45Z","abstract_excerpt":"In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the classical notions of Kan extension, Yoneda embedding $\\text y_A\\colon A \\to \\hat A$, exact square, total category and 'small' cocompletion; the latter in an appropriate sense. Throughout we compare our formalisations to their corresponding $2$-categorical counterparts. Our approach has several advantages. For instance, the structure of augmented virtual double categories naturally allows us to isolate conditions that ensure small cocompleteness of formal presheaf objects $\\hat"},"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":"2205.04890","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2022-05-10T13:40:45Z","cross_cats_sorted":[],"title_canon_sha256":"4544d402b2b5f7780470cb14784e79b6936f7477549c539bd96e66ebecaa3091","abstract_canon_sha256":"cd0caed74130aafb0d52cd574df8d90c820d75982d5705a0792965ffd910151a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:03:38.734801Z","signature_b64":"Yjw3RIVsdhDJ3sEBzTp2wlWLbMoPSFeI2wr6Db2dqp9631LLf6bq05Jp8QxTOQlUs/42REG5MNHw0t2FojhJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45646741b8df4d266dba981c77ac56004b29601b588cb7b7d9c4941a4afe8f3a","last_reissued_at":"2026-07-05T08:03:38.734304Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:03:38.734304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Formal category theory in augmented virtual double categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Seerp Roald Koudenburg","submitted_at":"2022-05-10T13:40:45Z","abstract_excerpt":"In this article we develop formal category theory within augmented virtual double categories. Notably we formalise the classical notions of Kan extension, Yoneda embedding $\\text y_A\\colon A \\to \\hat A$, exact square, total category and 'small' cocompletion; the latter in an appropriate sense. Throughout we compare our formalisations to their corresponding $2$-categorical counterparts. Our approach has several advantages. For instance, the structure of augmented virtual double categories naturally allows us to isolate conditions that ensure small cocompleteness of formal presheaf objects $\\hat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.04890","kind":"arxiv","version":3},"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/2205.04890/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2205.04890","created_at":"2026-07-05T08:03:38.734373+00:00"},{"alias_kind":"arxiv_version","alias_value":"2205.04890v3","created_at":"2026-07-05T08:03:38.734373+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2205.04890","created_at":"2026-07-05T08:03:38.734373+00:00"},{"alias_kind":"pith_short_12","alias_value":"IVSGOQNY35GS","created_at":"2026-07-05T08:03:38.734373+00:00"},{"alias_kind":"pith_short_16","alias_value":"IVSGOQNY35GSM3N2","created_at":"2026-07-05T08:03:38.734373+00:00"},{"alias_kind":"pith_short_8","alias_value":"IVSGOQNY","created_at":"2026-07-05T08:03:38.734373+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2504.16690","citing_title":"Logic and Concepts in the 2-category of Topoi","ref_index":30,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB","json":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB.json","graph_json":"https://pith.science/api/pith-number/IVSGOQNY35GSM3N2TAOHPLCWAB/graph.json","events_json":"https://pith.science/api/pith-number/IVSGOQNY35GSM3N2TAOHPLCWAB/events.json","paper":"https://pith.science/paper/IVSGOQNY"},"agent_actions":{"view_html":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB","download_json":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB.json","view_paper":"https://pith.science/paper/IVSGOQNY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2205.04890&json=true","fetch_graph":"https://pith.science/api/pith-number/IVSGOQNY35GSM3N2TAOHPLCWAB/graph.json","fetch_events":"https://pith.science/api/pith-number/IVSGOQNY35GSM3N2TAOHPLCWAB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB/action/storage_attestation","attest_author":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB/action/author_attestation","sign_citation":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB/action/citation_signature","submit_replication":"https://pith.science/pith/IVSGOQNY35GSM3N2TAOHPLCWAB/action/replication_record"}},"created_at":"2026-07-05T08:03:38.734373+00:00","updated_at":"2026-07-05T08:03:38.734373+00:00"}