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ual isomorphisms. And even in category theory, often we do not have to distinguish two categories as long as they are equivalent. These are only three exa mples for the fairly common situation that we start with a pair (C, W ) consisting of a category C and a class W of so-called weak equivalences, a class of morphisms which we would like to treat as isomorphisms. In such situations, functorial constructions are only ‘meaningful’ if they preserve weak equivalences. The search for convenient languag es to study such situatio ns has already quite some his tory and various different approaches have been cons id- ered. This includes triangulated categories, model categories, derivato