A Waldhausen category is a homotopical category equipped with a bit of extra structure that allows us to consider it as a presentation (via simplicial localization) of an (infinity,1)-category such that the extra structure allows us to conveniently compute the K-theory Grothendieck group of .
Notably a Waldhausen category provides the notion of cofibration sequences, which are crucial structures controlling . Dual to the discussion at homotopy limit and homotopy pullback, ordinary pushouts in Waldhausen categories of the form
with a special morphism called a Waldhausen cofibration compute homotopy pushouts and hence coexact sequences in the corresponding stable (infinity,1)-category.
Waldhausen in his work in K-theory introduced the notion of a category with cofibrations and weak equivalences, nowadays known as Waldhausen category. As the original name suggests, this is a category with zero object , equipped with a choice of two classes of maps of the cofibrations and of weak equivalences such that
(C1) all isomorphisms are cofibrations
(C2) there is a zero object and for any object the unique morphism is a cofibration
(C3) if is a cofibration and any morphism then the pushout is a cofibration
(W1) all isomorphisms are weak equivalences
(W2) weak equivalences are closed under composition (make a subcategory)
(W3) “glueing for weak equivalences”: Given any commutative diagram of the form
in which the vertical arrows are weak equivalences and right horizontal maps cofibrations, the induced map is a weak equivalence.
The axioms imply that for any cofibration there is a cofibration sequence where is the choice of the cokernel .
Given a Waldhausen category whose weak equivalence classes from a set, one defines as an abelian group whose elements are the weak equivalence classes modulo the relation for any cofibration sequence .
Waldhausen then devises the so called S-construction from Waldhausen categories to simplicial categories with cofibrations and weak equivalences (hence one can iterate the construction producing multisimplicial categories).
The K-theory space? of a Waldhausen construction is given by formula , where is the loop space functor, is the simplicial nerve, w.e. takes the (simplicial) subcategory of weak equivalence and . This construction can be improved (using iterated Waldhausen S-construction) to the K-theory -spectrum of ; the K-theory space will be just the one-space of the K-theory spectrum.
Then the K-groups are the homotopy groups of the K-theory space.
a cofibration is a chain morphism that is a monomorphism in in each degree .
weak equivalences are the morphisms that are quasi-isomorphisms when regarded as morphisms in ;
the cofibrations are the degreewise admissible morphisms, i.e. those morphisms such that the pushout computed in the ambient abelian category is in .