Waldhausen category

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In mathematics a Waldhausen category is a category C 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 it's 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 model categories, though they have much more structure than needed.

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.