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 $\scriptstyle A\,\rightarrowtail \,B$ is a cofibration and $\scriptstyle A\,\to \,C$ is any map, then there must exist a pushout $\scriptstyle B\,\cup _{A}\,C$ , and the natural map $\scriptstyle C\,\rightarrowtail \,B\,\cup _{A}\,C$ 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 $S_{n}(C)$ , which forms a spectrum. Let $K(C)$ denote the loop space of the geometric realization $|S_{*}(C)|$ of $S_{*}(C)$ . Then the group

\pi _{n}K(C)=\pi _{n+1}|S_{*}(C)|

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), we^OP) 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 $\scriptstyle C^{b}({\mathcal {A}})$ of bounded chain complexes on an exact category $\scriptstyle {\mathcal {A}}$ . The category $\scriptstyle S_{n}{\mathcal {C}}$ of functors $\scriptstyle \operatorname {Ar} (\Delta ^{n})\,\to \,{\mathcal {C}}$ when $\scriptstyle {\mathcal {C}}$ is so. And given a diagram $\scriptstyle I$ , then $\scriptstyle {\mathcal {C}}^{I}$ is a nice complicial biWaldhausen category when $\scriptstyle {\mathcal {C}}$ is.