Open Access
Translator Disclaimer
2011 On the 3-arrow calculus for homotopy categories
Sebastian Thomas
Homology Homotopy Appl. 13(1): 89-119 (2011).


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.


Download Citation

Sebastian Thomas. "On the 3-arrow calculus for homotopy categories." Homology Homotopy Appl. 13 (1) 89 - 119, 2011.


Published: 2011
First available in Project Euclid: 29 July 2011

zbMATH: 1218.18011
MathSciNet: MR2481463

Primary: 18E30 , 18E35 , 18G55 , 55U35

Keywords: 3-arrow calculus , derived category , homotopy category , localisation

Rights: Copyright © 2011 International Press of Boston


Vol.13 • No. 1 • 2011
Back to Top