In the category of relative categories the Rezk equivalences are exactly the DK-equivalences

TL;DR

This paper establishes that in the context of relative categories, Rezk equivalences precisely correspond to DK-equivalences, providing a clearer understanding of weak equivalences in this setting.

Contribution

The paper proves that Rezk equivalences in relative categories are exactly the DK-equivalences, clarifying the nature of weak equivalences in the model structure.

Findings

Rezk equivalences are exactly DK-equivalences in relative categories.

Provides an explicit characterization of weak equivalences in the model structure.

Strengthens the connection between simplicial localizations and model structures.

Abstract

In a previous paper we lifted Charles Rezk's complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of "relative categories," and our aim in this successor paper is to obtain a more explicit description of the weak equivalences in this model structure by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.