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

Author Taichi Uemura

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Taichi Uemura
  • Stockholm University, Sweden


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)


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
  • Homotopy type theory
  • ∞-logos
  • ∞-topos
  • oplax limit
  • Artin gluing
  • modality
  • synthetic Tait computability
  • logical relation


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