S-construction

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
:$\pi_n K(C) = \pi_ , S_*(C),$
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 $\scriptstyle C^b(\mathcal)$ of bounded chain complexes on an exact category $\scriptstyle \mathcal$.
The category $\scriptstyle S_n \mathcal$ of functors $\scriptstyle \operatorname(\Delta ^n)\, \to\, \mathcal$ when $\scriptstyle\mathcal$ is so.
And given a diagram $\scriptstyle I$, then $\scriptstyle \mathcal^I$ is a nice complicial biWaldhausen category when $\scriptstyle \mathcal$ 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
*
*

See also

*Complete Segal space

External links

*{{cite web, url=https://ncatlab.org/nlab/show/Waldhausen+S-construction, title=Waldhausen S-construction, website=nLab

Category theory
Algebraic K-theory