## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 10, Number 2 (2008), 175-193.

### Adding inverses to diagrams II: Invertible homotopy theories are spaces

#### 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 10, Number 2 (2008), 175-193.

**Dates**

First available in Project Euclid: 1 September 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1251811072

**Mathematical Reviews number (MathSciNet)**

MR2475608

**Zentralblatt MATH identifier**

1155.55006

**Subjects**

Primary: 55U35: Abstract and axiomatic homotopy theory 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18E35: Localization of categories

**Keywords**

Homotopy theories simplicial categories simplicial groupoids complete Segal spaces Segal groupoids model categories (∞, 1)-categories and groupoids

#### Citation

Bergner, Julia E. Adding inverses to diagrams II: Invertible homotopy theories are spaces. Homology Homotopy Appl. 10 (2008), no. 2, 175--193. https://projecteuclid.org/euclid.hha/1251811072

#### See also

- See: Julia E. Bergner. Adding inverses to diagrams encoding algebraic structures. Homology Homotopy Appl. Volume 10, Number 2 (2008), 149-174.Project Euclid: euclid.hha/1251811071
- See: Julia E. Bergner. Erratum to "Adding inverses to diagrams encoding algebraic structures" and "Adding inverses to diagrams II: Invertible homotopy theories are spaces". Homology Homotopy Appl. Volume 14, Number 1 (2012), 287-291.Project Euclid: euclid.hha/1355321076