A constructive model of directed univalence in bicubical sets
Directed type theory is an analogue of homotopy type theory where types represent ∞-categories, generalizing groupoids. A bisimplicial approach to directed type theory, developed by Riehl and Shulman, is based on equipping each type with both a notion of path and a separate notion of directed morphism. In this setting, a directed analogue of the univalence axiom asserts that there is a universe of covariant discrete fibrations whose directed morphisms correspond to functions—a higher-categorical analogue of the category of sets and functions. In this talk, I’ll present a constructive model of a directed type theory with directed univalence in bicubical, rather than bisimplicial, sets. We formalize much of this model using Agda as the internal language of a 1-topos, following Orton and Pitts. First, building on the cubical techniques used to give computational models of homotopy type theory, we show that there is a universe of covariant discrete fibrations, with a partial directed univalence principle asserting that functions are a retract of morphisms in this universe. To complete this retraction into an equivalence, we refine the universe of covariant fibrations using the constructive sheaf models by Coquand and Ruch. (Joint work with Dan Licata)This talk is hosted by Western University, Ontario. Link to the talk site.