Strict Rezk completions of models of HoTT and homotopy canonicity

Abstract

We give a new constructive proof of homotopy canonicity for homotopy type theory (HoTT). Canonicity proofs typically involve gluing constructions over the syntax of type theory. We instead use a gluing construction over a "strict Rezk completion" of the syntax of HoTT. The strict Rezk completion is specified and constructed in the topos of cartesian cubical sets. It completes a model of HoTT to an equivalent model satisfying a completeness condition, providing an equivalence between terms of identity types and cubical paths between terms. This generalizes the ordinary Rezk completion of a 1-category.