- Waldhausen category
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 withpushout , and co("C") containing the unique morphisms:scriptstyle 0, ightarrowtail, A
from the
zero-object to any object "A".To be more precise about the pushouts, we require when
:scriptstyle A, ightarrowtail, B
is a cofibration and
:scriptstyle A, o, C
is any map, that we have a push-out
:scriptstyle B, cup_A, C
where the map
:scriptstyle C, ightarrowtail, 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"), weOP) 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), o, mathcal{C} when scriptstylemathcal{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
Wikimedia Foundation. 2010.