Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Bousfield–Segal spaces

Pages: 217 – 243

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a12


Raffael Stenzel (Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic)


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.


homotopy, simplicial space, $\infty$-groupoid

2010 Mathematics Subject Classification

18G55, 55P20, 55P60

Received 19 November 2019

Received revised 24 November 2020

Accepted 11 March 2021

Published 6 April 2022