Homology, Homotopy and Applications

On the 3-arrow calculus for homotopy categories

Sebastian Thomas

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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.

Article information

Homology Homotopy Appl., Volume 13, Number 1 (2011), 89-119.

First available in Project Euclid: 29 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18E35: Localization of categories 18G55: Homotopical algebra 18E30: Derived categories, triangulated categories 55U35: Abstract and axiomatic homotopy theory

Localisation 3-arrow calculus homotopy category derived category


Thomas, Sebastian. On the 3-arrow calculus for homotopy categories. Homology Homotopy Appl. 13 (2011), no. 1, 89--119. https://projecteuclid.org/euclid.hha/1311953348

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