{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:K4UCAQ76CHUV2X3BDVREXIMOYU","short_pith_number":"pith:K4UCAQ76","schema_version":"1.0","canonical_sha256":"57282043fe11e95d5f611d624ba18ec536f3e9344a014b77cae02362c09d4649","source":{"kind":"arxiv","id":"1904.07004","version":2},"attestation_state":"computed","paper":{"title":"All $(\\infty,1)$-toposes have strict univalent universes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Michael Shulman","submitted_at":"2019-04-15T12:45:00Z","abstract_excerpt":"We prove the conjecture that any Grothendieck $(\\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\\infty,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory."},"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":"1904.07004","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-04-15T12:45:00Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"8620ee5a275de66a4de4cbbb729b4d07a6db4694b3de23b32070d56555e4e283","abstract_canon_sha256":"c8d0594cb8fe8d0044c42bd0b2c31673ca02eba873c0e621afe454da1434828c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:39.210304Z","signature_b64":"cVuJShXZ7uMDzcCiOCZ+LiffbAWlhwo0CV5DMKggXaQ5yjkEoTIJ/d16BAuRedPAcc1TrKDtgs4KF2kIxY0HCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57282043fe11e95d5f611d624ba18ec536f3e9344a014b77cae02362c09d4649","last_reissued_at":"2026-05-17T23:47:39.209918Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:39.209918Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"All $(\\infty,1)$-toposes have strict univalent universes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Michael Shulman","submitted_at":"2019-04-15T12:45:00Z","abstract_excerpt":"We prove the conjecture that any Grothendieck $(\\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\\infty,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.07004","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":"1904.07004","created_at":"2026-05-17T23:47:39.209978+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.07004v2","created_at":"2026-05-17T23:47:39.209978+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.07004","created_at":"2026-05-17T23:47:39.209978+00:00"},{"alias_kind":"pith_short_12","alias_value":"K4UCAQ76CHUV","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"K4UCAQ76CHUV2X3B","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"K4UCAQ76","created_at":"2026-05-18T12:33:21.387695+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":5,"internal_anchor_count":4,"sample":[{"citing_arxiv_id":"2605.15126","citing_title":"Constructive higher sheaf models with applications to synthetic mathematics","ref_index":38,"is_internal_anchor":true},{"citing_arxiv_id":"2505.22931","citing_title":"From Copying to Corelations via Ancestry Partitions","ref_index":15,"is_internal_anchor":true},{"citing_arxiv_id":"2512.18891","citing_title":"Elementary $\\infty$-toposes from type theory","ref_index":7,"is_internal_anchor":true},{"citing_arxiv_id":"2605.15126","citing_title":"Constructive higher sheaf models with applications to synthetic mathematics","ref_index":38,"is_internal_anchor":true},{"citing_arxiv_id":"2605.00812","citing_title":"Univalence without function extensionality","ref_index":39,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU","json":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU.json","graph_json":"https://pith.science/api/pith-number/K4UCAQ76CHUV2X3BDVREXIMOYU/graph.json","events_json":"https://pith.science/api/pith-number/K4UCAQ76CHUV2X3BDVREXIMOYU/events.json","paper":"https://pith.science/paper/K4UCAQ76"},"agent_actions":{"view_html":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU","download_json":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU.json","view_paper":"https://pith.science/paper/K4UCAQ76","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.07004&json=true","fetch_graph":"https://pith.science/api/pith-number/K4UCAQ76CHUV2X3BDVREXIMOYU/graph.json","fetch_events":"https://pith.science/api/pith-number/K4UCAQ76CHUV2X3BDVREXIMOYU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU/action/storage_attestation","attest_author":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU/action/author_attestation","sign_citation":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU/action/citation_signature","submit_replication":"https://pith.science/pith/K4UCAQ76CHUV2X3BDVREXIMOYU/action/replication_record"}},"created_at":"2026-05-17T23:47:39.209978+00:00","updated_at":"2026-05-17T23:47:39.209978+00:00"}