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

${\displaystyle \scriptstyle 0\,\rightarrowtail \,A}$

from the zero-object to any object A.

To be more precise about the pushouts, we require when

${\displaystyle \scriptstyle A\,\rightarrowtail \,B}$

is a cofibration and

${\displaystyle \scriptstyle A\,\to \,C}$

is any map, that we have a push-out

${\displaystyle \scriptstyle B\,\cup _{A}\,C}$

where the map

${\displaystyle \scriptstyle C\,\rightarrowtail \,B\,\cup _{A}\,C}$

is a cofibration:

A category C is equipped with bifibrations if it has cofibrations and its opposite category 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.

As examples one may think of exact categories, where the cofibrations are the admissible monomorphisms. Another example is the full subcategory of cofibrant objects in a pointed model categories, that is, the full subcategory consisting of those objects ${\displaystyle X}$ for which ${\displaystyle 0\to X}$ is a cofibration. (The bifibrant objects do not in general form a Waldhausen category, as a pushout of fibrant objects need not be fibrant. For more information on this second example see the paper by Sagave in the references)

Waldhausen and biWaldhausen categories are linked with algebraic K-theory. There, many interesting categories are complicial biWaldhausen categories. For example: The category ${\displaystyle \scriptstyle C^{b}({\mathcal {A}})}$ of bounded chaincomplexes on an exact category ${\displaystyle \scriptstyle {\mathcal {A}}}$ The category ${\displaystyle \scriptstyle S_{n}{\mathcal {C}}}$ of functors ${\displaystyle \scriptstyle Ar(\Delta ^{n})\,\to \,{\mathcal {C}}}$ when ${\displaystyle \scriptstyle {\mathcal {C}}}$ is so. And given a diagram ${\displaystyle \scriptstyle I}$, then ${\displaystyle \scriptstyle {\mathcal {C}}^{I}}$ is a nice complicial biWaldhausen category when ${\displaystyle \scriptstyle {\mathcal {C}}}$ is.