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.

arXiv comment34 pages. Comments welcome!