Homotopy type theory as internal languages of diagrams of ∞-logoses

12/05/2022

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by   Taichi Uemura, et al.

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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. To prove the main result, we establish a precise correspondence between the lex, accessible localizations of an ∞-logos and the lex, accessible modalities in the internal language of the ∞-logos. To do this, we also partly develop the Kripke-Joyal semantics of homotopy type theory in ∞-logoses.