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. 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 A ↣ B {displaystyle scriptstyle A, ightarrowtail ,B} is a cofibration and A → C {displaystyle scriptstyle A, o ,C} is any map, then there must exist a pushout B ∪ A C {displaystyle scriptstyle B,cup _{A},C} , and the natural map C ↣ B ∪ A C {displaystyle scriptstyle C, ightarrowtail ,B,cup _{A},C} should be cofibration: 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. The Waldhausen S-construction produces from a Waldhausen category C a sequence of Kan complexes S n ( C ) {displaystyle S_{n}(C)} , which forms a spectrum. Let K ( C ) {displaystyle K(C)} denote the loop space of the geometric realization | S ∗ ( C ) | {displaystyle |S_{*}(C)|} of S ∗ ( C ) {displaystyle S_{*}(C)} . 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. A category C is equipped with bifibrations if it has cofibrations and its opposite category COP has so also. 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.