Type-theoretic replacement and univalent completion: applications and interpretations

Paper in proceeding, 2026

We study Rijke’s type theoretic axiom of replacement, which closes a type-theoretic universe under certain images, and its cousin the axiom of univalent completions, which postulates an extension of each type family to a univalent family. In a long expository section, we survey applications of these two axioms in the literature, from the construction of truncations to the construction of Eilenberg-MacLane spaces to the interpretation of material set theory, making a case for their value as foundational principles. We then give direct, constructive interpretations of the axioms in cubical sets models of type theory and suggest how they can be understood as higher inductive constructions.