Waldhausen category explained

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 → tail 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 → tail 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(l{A})} of bounded chain complexes on an exact category \scriptstyle l{A} .The category \scriptstyle S _{n} l{C} of functors \scriptstyle \operatorname{Ar}(\Delta ^{n)} \to l{C} when \scriptstyle l{C} is so.And given a diagram \scriptstyle I , then \scriptstyle l{C} ^{I} is a nice complicial biWaldhausen category when \scriptstyle l{C} is.

References

• • C. Weibel, The K-book, an introduction to algebraic K-theory - http://www.math.rutgers.edu/~weibel/Kbook.html
• G. Garkusha, Systems of Diagram Categories and K-theory - https://arxiv.org/abs/math/0401062
• Sagave . S. . 2004. On the algebraic K-theory of model categories . Journal of Pure and Applied Algebra. 190. 1–3. 329–340. 10.1016/j.jpaa.2003.11.002.

See also

• Complete Segal space

External links

• Web site: Waldhausen S-construction. nLab.