From fractions to complete Segal spaces

TL;DR

This paper demonstrates that certain relative categories with a homotopical calculus of fractions have classification diagrams that are Segal spaces after Reedy-fibrant replacement, extending previous results to a broader context.

Contribution

It generalizes existing results by showing that the Rezk classification diagram of these categories is a Segal space, broadening the applicability of classification diagram techniques.

Findings

Classification diagram is a Segal space after Reedy-fibrant replacement.

Generalizes Rezk and Bergner's results to categories with a homotopical calculus.

Extends Barwick and Kan's results to a wider class of categories.

Abstract

We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This generalizes the result of Rezk and Bergner on the classification diagram of a closed model category, as well as the result of Barwick and Kan on the classification diagram of a partial model category.