Models for dagger $(\infty ,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence

Abstract

Building on the dagger Bergner and dagger Joyal models constructed in Part I, we develop simplicial-space models for dagger $(\infty ,1)$-categories. We first construct a homotopically constant resolution model and a model structure for unitarily complete dagger Segal spaces. A change-of-index adjunction compares the two presentations of dagger simplicial spaces. We then prove a dagger analogue of the Joyal--Tierney Quillen equivalence. Together with Part I, these results give a chain of Quillen equivalences between the dagger Bergner, dagger Joyal, and dagger Rezk models.

arXiv comment41pages. Comments welcome!