A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types

We describe the categorical semantics for a simply typed variant and simplified dependently of Cocon , contextual modal type theory where box modality mediates between weak function space that is used to represent higher-order abstract syntax (HOAS) trees strong describes (recursive) computations about them. What makes different from standard theories presence first-class contexts objects are closed with respect given context assumptions. Following M. Hofmann’s work, we use presheaf model characterise HOAS trees. Surprisingly, this already provides necessary structure also . In particular, can capture using comonad ♭ restricts presheaves their elements. This gives simple semantic characterisation invariants types (e.g. substitution invariance) identifies as type-theoretic models. further extend dependent categories families show fragment without recursor in Fitch-style presented by Birkedal et al.