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

from the zero-object to any object *A*.

To be more precise about the pushouts, we require when

is a cofibration and

is any map, that we have a push-out

where the map

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*), 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 bifibrant objects in a pointed model categories, that is, the full subcategory consisting of those objects for which is a cofibration and 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 of bounded chaincomplexes on an exact category The category of functors when is so. And given a diagram, then is a nice complicial biWaldhausen category when is.

### Famous quotes containing the word category:

“I see no reason for calling my work poetry except that there is no other *category* in which to put it.”

—Marianne Moore (1887–1972)