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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 1

### On the 3-arrow calculus for homotopy categories

Pages: 89 – 119

DOI: http://dx.doi.org/10.4310/HHA.2011.v13.n1.a5

#### Author

#### Abstract

We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. Applications include the localisation of an arbitrary Quillen model category with respect to its weak equivalences as well as the localisation of its full subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy category in all four cases. In contrast to the approach of Dwyer, Hirschhorn, Kan and Smith, the Quillen model category under consideration does not need to admit functorial factorisations.

#### Keywords

localisation; 3-arrow calculus; homotopy category; derived category

#### 2010 Mathematics Subject Classification

18E30, 18E35, 18G55, 55U35