{"paper":{"title":"Elementary $\\infty$-toposes from type theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Every categorical model of dependent type theory with univalent universes becomes an elementary ∞-topos after ∞-localization.","cross_cats":["math.AT","math.LO"],"primary_cat":"math.CT","authors_text":"Dani\\\"el Apol, Maximilian Petrowitsch","submitted_at":"2025-12-21T21:20:13Z","abstract_excerpt":"We prove that every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its $\\infty$-localisation an elementary $\\infty$-topos, that is, a finitely complete, locally cartesian closed $\\infty$-category with enough univalent universal morphisms. We also show that elementary $\\infty$-toposes have small subobject classifiers. To achieve this, we extend Joyal's theory of tribes by introducing the notion of a univalent tribe and a univalent fibration in a tribe."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its ∞-localisation an elementary ∞-topos, that is, a finitely complete, locally cartesian closed ∞-category with enough univalent universal morphisms","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The ∞-localization functor applied to a categorical model of the given type theory preserves finite completeness, local cartesian closure, and the existence of enough univalent universal morphisms; this relies on the newly introduced notions of univalent tribe and univalent fibration being correctly defined and interacting properly with the localization.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Categorical models of dependent type theory with sums, products, identity types and univalent universes present elementary ∞-toposes via ∞-localization, which also possess small subobject classifiers.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every categorical model of dependent type theory with univalent universes becomes an elementary ∞-topos after ∞-localization.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4afa9e2bcfb4cc1c84b0b974a5336fcc250bc98a51d46e9cba6d99cdb8f881cb"},"source":{"id":"2512.18891","kind":"arxiv","version":3},"verdict":{"id":"cc3f4a5f-353f-4494-a533-30de16fca64e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T20:44:54.460336Z","strongest_claim":"every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its ∞-localisation an elementary ∞-topos, that is, a finitely complete, locally cartesian closed ∞-category with enough univalent universal morphisms","one_line_summary":"Categorical models of dependent type theory with sums, products, identity types and univalent universes present elementary ∞-toposes via ∞-localization, which also possess small subobject classifiers.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The ∞-localization functor applied to a categorical model of the given type theory preserves finite completeness, local cartesian closure, and the existence of enough univalent universal morphisms; this relies on the newly introduced notions of univalent tribe and univalent fibration being correctly defined and interacting properly with the localization.","pith_extraction_headline":"Every categorical model of dependent type theory with univalent universes becomes an elementary ∞-topos after ∞-localization."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.18891/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":2,"snapshot_sha256":"855b7ab0f2b16a6388b03ff06bb8779562bc9d8c85335e0c069a2cd17ea13425"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}