Waldhausen category
In mathematics, a Waldhausen category is a category C equipped with some additional data, which makes it possible to construct the K-theory spectrum of C using a so-called S-construction. It's named after Friedhelm Waldhausen, who introduced this notion (under the term category with cofibrations and weak equivalences) to extend the methods of algebraic K-theory to categories not necessarily of algebraic origin, for example the category of topological spaces. Definition Let C be a category, co(C) and we(C) two classes of morphisms in C, called cofibrations and weak equivalences respectively. The triple (C, co(C), we(C)) is called a Waldhausen category if it satisfies the following axioms, motivated by the similar properties for the notions of cofibrations and weak homotopy equivalences of topological spaces: C has a zero object, denoted by 0; isomorphisms are included in both co(C) and we(C); co(C) and we(C) are closed under composition; for each object A ∈ C the unique map 0 → A is a cofibration, i.e. is an element of co(C); co(C) and we(C) are compatible with pushouts in a certain sense. For example, if is a cofibration and is any map, then there must exist a pushout , and the natural map should be