Denotational semantics for guarded dependent type theory
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We present a new model of Guarded Dependent Type Theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with, and reason about coinductive types. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion, and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves orthogonal to the object of clocks. In the case of dependent types, this translates to a unique lifting condition similar to the one found in homotopy theoretic models of type theory. Since the universes defined by the standard Hofmann-Streicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. A large and technical part of the paper is devoted to showing that these can be constructed in such a way that inclusions between universes induced by inclusions of clock variable contexts commute on the nose with type operations on the universes.
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