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Homotopy Type Theory as Internal Languages of Diagrams of ∞-Logoses

Author Taichi Uemura

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Author Details

Taichi Uemura
  • Stockholm University, Sweden

Funding

  • Uemura, Taichi: Supported by KAW Grant "Type Theory for Mathematics and Computer Science" investigated by Thierry Coquand and Peter LeFanu Lumsdaine.

Acknowledgements

The author thanks Jonathan Sterling for useful conversations on the current work. The author also thanks anonymous referees for corrections, comments, and suggestions.

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Taichi Uemura. Homotopy Type Theory as Internal Languages of Diagrams of ∞-Logoses. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSCD.2023.5

Abstract

We show that certain diagrams of ∞-logoses are reconstructed in internal languages of their oplax limits via lex, accessible modalities, which enables us to use plain homotopy type theory to reason about not only a single ∞-logos but also a diagram of ∞-logoses. This also provides a higher dimensional version of Sterling’s synthetic Tait computability - a type theory for higher dimensional logical relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Denotational semantics
  • Theory of computation → Categorical semantics
Keywords
  • Homotopy type theory
  • ∞-logos
  • ∞-topos
  • oplax limit
  • Artin gluing
  • modality
  • synthetic Tait computability
  • logical relation

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