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:
For example, if formula_1 is a cofibration and formula_2 is any map, then there must exist a pushout formula_3, and the natural map formula_4 should be cofibration:
Relations with other notions.
In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms. If "C" has a structure of an exact category, then by defining we("C") to be isomorphisms, co("C") to be admissible monomorphisms, one obtains a structure of a Waldhausen category on "C". Both kinds of structure may be used to define K-theory of "C", using the Q-construction for an exact structure and S-construction for a Waldhausen structure. 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.
S-construction.
The Waldhausen S-construction produces from a Waldhausen category "C" a sequence of Kan complexes formula_5, which forms a spectrum. Let formula_6 denote the loop space of the geometric realization formula_7 of formula_8. Then the group
is the "n"-th "K"-group of "C". Thus, it gives a way to define higher "K"-groups. Another approach for higher "K"-theory is Quillen's Q-construction.
The construction is due to Friedhelm Waldhausen.
biWaldhausen categories.
A category "C" is equipped with bifibrations if it has cofibrations and its opposite category "C"OP has so also. In that case, we denote the fibrations of "C"OP 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 ("C"OP, quot("C"), weOP) are Waldhausen categories.
Waldhausen and biWaldhausen categories are linked with algebraic K-theory. There, many interesting categories are complicial biWaldhausen categories. For example: 
The category formula_10 of bounded chain complexes on an exact category formula_11.
The category formula_12 of functors formula_13 when formula_14 is so.
And given a diagram formula_15, then formula_16 is a nice complicial biWaldhausen category when formula_17 is.