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.
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: For example, if
should be cofibration: In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms.
An important fact is that the resulting K-theory spaces are homotopy equivalent.
If C is a model category with a zero object, then the full subcategory of cofibrant objects in C may be given a Waldhausen structure.
denote the loop space of the geometric realization
Another approach for higher K-theory is Quillen's Q-construction.
The construction is due to Friedhelm Waldhausen.
In that case, we denote the fibrations of COP by quot(C).
In that case, C is a biWaldhausen category if C has bifibrations and weak equivalences such that both (C, co(C), we) and (COP, quot(C), weOP) are Waldhausen categories.
Waldhausen and biWaldhausen categories are linked with algebraic K-theory.
of bounded chain complexes on an exact category
is a nice complicial biWaldhausen category when