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

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.

### MSC:

 18G55 Nonabelian homotopical algebra (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18E35 Localization of categories, calculus of fractions
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