nLab Waldhausen category

Idea

A Waldhausen category $C'$ 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 $C$ such that the extra structure allows us to conveniently compute the K-theory Grothendieck group $\mathbf{K}(C)$ of $C$.

Notably a Waldhausen category provides the notion of cofibration sequences, which are crucial structures controlling $\mathbf{K}(C)$. Dual to the discussion at homotopy limit and homotopy pullback, ordinary pushouts in Waldhausen categories of the form

$$\array{ A &\hookrightarrow& B \\ \downarrow && \downarrow \\ 0 &\to& B//A }$$

with $A \hookrightarrow B$ a special morphism called a Waldhausen cofibration compute homotopy pushouts and hence coexact sequences in the corresponding stable (infinity,1)-category.

Using this, the Waldhausen S-construction on $C'$ is an algorithm for computing the K-theory spectrum of $C$.

Definition

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 $C$ with zero object $0$, equipped with a choice of two classes of maps $\mathrm{cof}$ of the cofibrations and $w.e.$ of weak equivalences such that

• (C1) all isomorphisms are cofibrations

• (C2) there is a zero object $0$ and for any object $a$ the unique morphism $0\to a$ is a cofibration

• (C3) if $a\hookrightarrow b$ is a cofibration and $a\to c$ any morphism then the pushout $c\to b\cup_a c$ 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

  $$\array{ D &\leftarrow& A &\hookrightarrow &B\\ \downarrow^\sim&& \downarrow^\sim &&\downarrow^\sim\\ D' &\leftarrow &A' &\hookrightarrow &B' }$$

  in which the vertical arrows are weak equivalences and right horizontal maps cofibrations, the induced map $B\cup_A D\hookrightarrow B'\cup_{A'} D'$ is a weak equivalence.

The axioms imply that for any cofibration $A\hookrightarrow B$ there is a cofibration sequence $A\hookrightarrow B\to B/A$ where $B/A$ is the choice of the cokernel $B\cup_A 0$.

Given a Waldhausen category $C$ whose weak equivalence classes from a set, one defines $K_0(C)$ as an abelian group whose elements are the weak equivalence classes modulo the relation $[A]+[B/A]=[B]$ for any cofibration sequence $A\hookrightarrow B\to B/A$.

Waldhausen then devises the so called S-construction $C\mapsto S_\bullet C$ 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 $\Omega\mathrm{hocolim}_{\Delta^{\mathrm{op}}}([n]\mapsto N_\bullet(w.e.(S_n C)))$, where $\Omega$ is the loop space functor, $N$ is the simplicial nerve, w.e. takes the (simplicial) subcategory of weak equivalence and $[n]\in\Delta$. This construction can be improved (using iterated Waldhausen S-construction) to the K-theory $\Omega$-spectrum of $C$; 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.

Remarks

• The axioms of a Waldhausen category $C$ are very similar to the axioms of a category of fibrant objects on the opposite category $C^{op}$ in which the initial object is also terminal. One difference is that the weak equivalences in a Waldhausen category are not required to satisfy 2-out-of-3. For example, Waldhausen gives an example of a Waldhausen category where the weak equivalences are simple homotopy equivalences?. Another difference is in axiom W3, whose analog in a category of fibrant objects is the axioms that every object has a path object. It still follows that one has fibration sequences in a category of fibrant objects.

Examples

Waldhausen category of a small abelian category

For $C$ a small abelian category the category of bounded chain complexes $Ch^b(C)$ becomes a Waldhausen category by taking

• a weak equivalence is a quasi-isomorphism of chain complexes;

• a cofibration $f : A_\bullet \to X_\bullet$ is a chain morphism that is a monomorphism in $C$ in each degree $f_n : A_n \to X_n$.

Waldhausen category of a small exact category

For $C$ just a Quillen exact category with ambient abelian category $\hat C$ there is an analogous, slightly more sophisticated construction of a Waldhausen category structure on $Ch^b(C)$:

• weak equivalences are the morphisms that are quasi-isomorphisms when regarded as morphisms in $\hat C$;

• the cofibrations are the degreewise admissible morphisms, i.e. those morphisms $A \to X$ such that the pushout $A \to X \to A/X$ computed in the ambient abelian category $\hat C$ is in $C$.

Related concepts

• category with weak equivalences

• category of fibrant objects, category of cofibrant objects

• model category

References

Waldhausen categories are discussed with an eye towards their application in the computation of Grothendieck groups in chapter 2 of

• Charles Weibel, The K-book: An introduction to algebraic K-theory (web)

Section 1 of

• R. W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, 1990, 247-435.(pdf)