Waldhausen category

In mathematics a Waldhausen category (after Friedhelm Waldhausen) is a category C with a zero object equipped with cofibrations co(C) and weak equivalences we(C), both containing all isomorphisms, both compatible with pushout, and co(C) containing the unique morphisms

$\scriptstyle 0\,\rightarrowtail\, A$

from the zero-object to any object A.

To be more precise about the pushouts, we require when

$\scriptstyle A\, \rightarrowtail\, B$

is a cofibration and

$\scriptstyle A\,\to\, C$

is any map, that we have a push-out

$\scriptstyle B\, \cup_A\, C$

where the map

$\scriptstyle C\, \rightarrowtail\, B\,\cup_A\, C$

is a cofibration:

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.

As examples one may think of exact categories, where the cofibrations are the admissible monomorphisms. Another example is the full subcategory of bifibrant objects in a pointed model categories, that is, the full subcategory consisting of those objects $X$ for which $0 \to X$ is a cofibration and $X \to 0$ is a fibration.

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 chaincomplexes on an exact category $\scriptstyle \mathcal{A}$ The category $\scriptstyle S_n \mathcal{C}$ of functors $\scriptstyle Ar(\Delta ^n)\, \to\, \mathcal{C}$ 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.

References

C. Weibel, Introduction to algebraic K-theory — http://www.math.rutgers.edu/~weibel/Kbook.html
G. Garkusha, Systems of Diagram Categories and K-theory — http://front.math.ucdavis.edu/0401.5062