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The Local Universes Model: An Overlooked Coherence Construction for Dependent Type Theories
Institute for Advanced Study, Princeton, New Jersey, U.S.A..ORCID iD: 0000-0003-1390-2970
Institute for Advanced Study, Princeton, New Jersey, U.S.A..ORCID iD: 0000-0002-3221-4797
2015 (English)In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 16, no 3, 23Article in journal (Refereed) Published
Abstract [en]

We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types.

Precisely, we take as input a “weak model”: a comprehension category, equipped with structure corresponding to the desired logical constructions. We assume throughout that the base category is close to locally Cartesian closed: specifically, that products and certain exponentials exist. Beyond this, we require only that the logical structure should be weakly stable—a pure existence statement, not involving any specific choice of structure, weaker than standard categorical Beck–Chevalley conditions, and holding in the now standard homotopy-theoretic models of type theory.

Given such a comprehension category, we construct an equivalent split one whose logical structure is strictly stable under reindexing. This yields an interpretation of type theory with the chosen constructors.

The model is adapted from Voevodsky’s use of universes for coherence, and at the level of fibrations is a classical construction of Giraud. It may be viewed in terms of local universes or delayed substitutions. 

Place, publisher, year, edition, pages
2015. Vol. 16, no 3, 23
Keyword [en]
dependent type theory, categorical semantics, comprehension categories, coherence theorems
National Category
Algebra and Logic
Research subject
Mathematical Logic
Identifiers
URN: urn:nbn:se:su:diva-128393DOI: 10.1145/2754931OAI: oai:DiVA.org:su-128393DiVA: diva2:914762
Available from: 2016-03-25 Created: 2016-03-25 Last updated: 2016-04-04Bibliographically approved

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Lumsdaine, Peter LeFanuWarren, Michael A.
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