{"paper":{"title":"Constructive higher sheaf models with applications to synthetic mathematics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Higher sheaf models of type theory with univalence and higher inductive types can be built in a constructive metatheory.","cross_cats":["math.LO"],"primary_cat":"cs.LO","authors_text":"Christian Sattler, Jonas H\\\"ofer, Thierry Coquand","submitted_at":"2026-05-14T17:37:19Z","abstract_excerpt":"There have recently been several developments in synthetic mathematics using extensions of dependent type theory with univalence and higher inductive types: simplicial homotopy type theory, synthetic algebraic geometry and synthetic Stone duality. We provide a foundation of higher sheaf models of type theory in a constructive metatheory and, in particular, build constructive models of these formal systems."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide a foundation of higher sheaf models of type theory in a constructive metatheory and, in particular, build constructive models of these formal systems.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That higher sheaf models can be constructed constructively for type theories extended with univalence and higher inductive types without requiring additional non-constructive assumptions in the metatheory.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructive higher sheaf models of type theory with univalence and higher inductive types are constructed to underpin synthetic mathematics.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Higher sheaf models of type theory with univalence and higher inductive types can be built in a constructive metatheory.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3090d9a3768f239a1bff20680f332db0e23fa60467a2a4d52b1542f8e260d15b"},"source":{"id":"2605.15126","kind":"arxiv","version":1},"verdict":{"id":"5844fd96-3e32-4131-8628-b812b33a5a11","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:59:00.283425Z","strongest_claim":"We provide a foundation of higher sheaf models of type theory in a constructive metatheory and, in particular, build constructive models of these formal systems.","one_line_summary":"Constructive higher sheaf models of type theory with univalence and higher inductive types are constructed to underpin synthetic mathematics.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That higher sheaf models can be constructed constructively for type theories extended with univalence and higher inductive types without requiring additional non-constructive assumptions in the metatheory.","pith_extraction_headline":"Higher sheaf models of type theory with univalence and higher inductive types can be built in a constructive metatheory."},"references":{"count":41,"sample":[{"doi":"","year":1998,"title":"On Relating Type Theories and Set Theories","work_id":"bd497a35-ac73-409f-8acf-1ef552b04d71","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1017/s0960129523000130","year":2023,"title":"Two-level type theory and applications","work_id":"ce4491ab-9c34-468a-accd-8fca7036ec4e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1963,"title":"M. Artin, A. Grothendieck, and J. L. Verdier. Th \\'e orie de Topos et Cohomologie Etale des Sch \\'e mas. Tome 1: Th\\'eorie des topos, S\\'eminaire de G\\'eom\\'etrie Alg\\'ebrique du Bois-Marie 1963--1964","work_id":"f2323910-7a34-404f-8ed2-91fe8ef9b7db","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1017/s0960129516000268","year":2018,"title":"https://doi.org/10.1017/s0960129516000268","work_id":"3aff4fe4-b3f3-4ab2-8686-c35cf23cb98a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4230/lipics.tlca.2015.92","year":2015,"title":"Non-Constructivity in Kan Simplicial Sets","work_id":"c55ae7b3-c95b-4aba-bcba-5df6c8fa4e0d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":41,"snapshot_sha256":"afc3bc9aedec3f76176f457b0654d7c21139f494cc6ccdec2b7abf9ea35aaf82","internal_anchors":4},"formal_canon":{"evidence_count":2,"snapshot_sha256":"129244352d36af9ea601c717046061f2a26939be0faf95acbbf10c1181660cae"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}