Adding inverses to diagrams II: Invertible homotopy theories are spaces

TL;DR

This paper extends previous work on homotopy theories by establishing that invertible versions of Segal spaces, groupoids, and related structures are all Quillen equivalent to the standard model structure on spaces, confirming their homotopical equivalence.

Contribution

It demonstrates that invertible homotopy theories modeled by Segal groupoids and invertible Segal spaces are Quillen equivalent to classical spaces, generalizing prior results to the invertible case.

Findings

Invertible Segal spaces are Quillen equivalent to spaces.

Model structures on simplicial groupoids and Segal pregroupoids are Quillen equivalent.

Results hold for invertible cases using two different approaches.

Abstract

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or "invertible" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard model structure on the category of spaces. We prove this result using two different approaches to invertible complete Segal spaces and Segal groupoids.