Models for dagger $(\infty,1)$-categories III: Univalent flagged dagger $(\infty,1)$-categories and unitarily complete dagger Segal anima

Abstract

We identify the $(\infty,1)$-category presented by the dagger Bergner, dagger Joyal, and dagger Rezk models of Parts I and II with that of univalent flagged dagger $(\infty,1)$-categories. We first construct a realization--nerve adjunction between reversal simplicial anima and Hermitian flags. We prove that its nerve restricts to an equivalence between flagged dagger $(\infty,1)$-categories and dagger Segal anima, under which univalence corresponds to unitary completeness. We then construct the adjunctions for univalentization and Hermitian completion and identify univalentization with unitary completion.