Academic Journal
The Biequivalence of Path Categories and Axiomatic Martin-Löf Type Theories
| Title: | The Biequivalence of Path Categories and Axiomatic Martin-Löf Type Theories |
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| Authors: | Otten, Daniël, Spadetto, Matteo |
| Contributors: | Daniël Otten and Matteo Spadetto |
| Publisher Information: | Schloss Dagstuhl – Leibniz-Zentrum für Informatik |
| Publication Year: | 2026 |
| Collection: | DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics ) |
| Subject Terms: | Axiomatic type theory, cubical type theory, propositional equality, biequivalence, display map categories, path categories, homotopy theory, coherence |
| Description: | The semantics of extensional type theory has an elegant categorical description: models of extensional =-types, 𝟙-types, and Σ-types are biequivalent to finitely complete categories, while adding Π-types yields locally Cartesian closed categories. We establish parallel results for axiomatic type theory, which includes systems like cubical type theory, where the computation rule of the =-types only holds as a propositional axiom instead of a definitional reduction. In particular, we prove that models of axiomatic =-types, and standard 𝟙- and Σ-types are biequivalent to certain path categories, while adding axiomatic Π-types yields dependent homotopy exponents. This biequivalence simplifies axiomatic =-types, which are more intricate than extensional ones since they permit higher dimensional structure. Specifically, path categories use a primitive notion of equivalence instead of a direct reproduction of the syntactic elimination rules and computation axioms. We apply our correspondence to prove a coherence theorem: we show that these weak homotopical models can be turned into equivalent strict models of axiomatic type theory. In addition, we introduce a more modular notion, that of a display map path category, which only models axiomatic =-types by default, while leaving room to add other axiomatic type formers such as 𝟙-, Σ-, and Π-types. |
| Document Type: | article in journal/newspaper conference object |
| File Description: | application/pdf |
| Language: | English |
| Relation: | Is Part Of LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026); https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.38 |
| DOI: | 10.4230/LIPIcs.CSL.2026.38 |
| Availability: | https://doi.org/10.4230/LIPIcs.CSL.2026.38 https://nbn-resolving.org/urn:nbn:de:0030-drops-254633 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.38 |
| Rights: | https://creativecommons.org/licenses/by/4.0/legalcode |
| Accession Number: | edsbas.E62618AE |
| Database: | BASE |
| DOI: | 10.4230/LIPIcs.CSL.2026.38 |
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