  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 lifting
condition similar to the one found in homotopy theoretic models of type theory,
but here with an additional requirement of uniqueness of lifts. 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. We show how to model these universes in such a way that
inclusions of clock contexts give rise to inclusions of universes commuting
with type operations on the nose.
