On the 3-arrow calculus for homotopy categories. (English) Zbl 1218.18011

Thomas, Sebastian

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