Bousfield-Segal spaces

TL;DR

This paper investigates Bousfield-Segal spaces, showing they are equivalent to Segal spaces with invertible arrows, and establishes their role as models for ∞-groupoids and Homotopy Type Theory.

Contribution

It demonstrates that Bousfield-Segal spaces can be derived from Segal spaces via localization and colocalization, confirming their equivalence and applications.

Findings

Bousfield-Segal spaces are Segal spaces with invertible arrows.

Complete Bousfield-Segal spaces are homotopically constant Segal spaces.

The model structure for Bousfield-Segal spaces models ∞-groupoids and Homotopy Type Theory.

Abstract

This paper is a study of Bousfield-Segal spaces, a notion introduced by Julie Bergner drawing on ideas about Eilenberg-Mac Lane objects due to Bousfield. In analogy to Rezk's Segal spaces, they are defined in such a way that Bousfield-Segal spaces naturally come equipped with a homotopy-coherent fraction operation in place of a composition. In this paper we show that Bergner's model structure for Bousfield-Segal spaces in fact can be obtained from the model structure for Segal spaces both as a localization and a colocalization. We thereby prove that Bousfield-Segal spaces really are Segal spaces, and that they characterize exactly those with invertible arrows. We note that the complete Bousfield-Segal spaces are precisely the homotopically constant Segal spaces, and deduce that the associated model structure yields a model for both $\infty$-groupoids and Homotopy Type Theory.