Homotopy type theory as a language for diagrams of $\infty$-logoses

Abstract: We show that certain diagrams of $\infty$-logoses are reconstructed in homotopy type theory extended with some lex, accessible modalities, which enables us to use plain homotopy type theory to reason about not only a single $\infty$-logos but also a diagram of $\infty$-logoses. This also provides a higher dimensional version of Sterling's synthetic Tait computability -- a type theory for higher dimensional logical relations.