Thomas, Sebastian On the 3-arrow calculus for homotopy categories. (English) Zbl 1218.18011 Homology Homotopy Appl. 13, No. 1, 89-119 (2011). The author constructs a class of categories, called uni-fractionable categories, which have good localisations in the following sense: every morphism in the lcoalisation is represented by a zig-zag of length \(3\); two zig-zags represent the same morphism if and only if they can be embedded in a suitable \(3\times 3\) diagram. The axioms for a uni-fractionable category require the existence of certain factorisations, but they do not require these factorisations to be functorial. The result can be applied to arbitrary Quillen model categories. Reviewer: Richard John Steiner (Glasgow) MSC: 18G55 Nonabelian homotopical algebra (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18E35 Localization of categories, calculus of fractions Keywords:homotopy category; localisation of categories; 3-arrow calculus; derived category; Quillen model category PDF BibTeX XML Cite \textit{S. Thomas}, Homology Homotopy Appl. 13, No. 1, 89--119 (2011; Zbl 1218.18011) Full Text: DOI arXiv Link OpenURL