Ticking clocks as dependent right adjoints: Denotational semantics for clocked type theory

Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract form of step-indexing. CloTT has previously been shown to enjoy a number of syntactic properties including strong normalisation, canonicity and decidability of the equational theory. In this paper we present a denotational semantics for CloTT useful, e.g., for studying future extensions of CloTT with constructions such as path types. The main challenge for constructing this model is to model the notion of ticks on a clock used in CloTT for coinductive reasoning about coinductive types. We build on a category previously used to model guarded recursion with multiple clocks. In this category there is an object of clocks but no object of ticks, and so tick-assumptions in a context can not be modelled using standard tools. Instead we model ticks using dependent right adjoint functors, a generalisation of the category theoretic notion of adjunction to the setting of categories with families. Dependent right adjoints are known to model Fitch-style modal types, but in the case of CloTT, the modal operators constitute a family indexed internally in the type theory by clocks. We model this family using a dependent right adjoint on the slice category over the object of clocks. Finally we show how to model the tick constant of CloTT using a semantic substitution. This work improves on a previous model by the first two named authors which not only had a flaw but was also considerably more complicated.