Waldhausen category


Homotopy theory

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Basic facts




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

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

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

with ABA \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 CC' is an algorithm for computing the K-theory spectrum of CC.


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 CC with zero object 00, equipped with a choice of two classes of maps cof\mathrm{cof} of the cofibrations and w.e.w.e. of weak equivalences such that

  • (C1) all isomorphisms are cofibrations

  • (C2) there is a zero object 00 and for any object aa the unique morphism OaO\to a is a cofibration

  • (C3) if aba\hookrightarrow b is a cofibration and aca\to c any morphism then the pushout cb acc\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

    D A B D A B\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 ADB ADB\cup_A D\hookrightarrow B'\cup_{A'} D' is a weak equivalence.

The axioms imply that for any cofibration ABA\hookrightarrow B there is a cofibration sequence ABB/AA\hookrightarrow B\to B/A where B/AB/A is the choice of the cokernel B A0B\cup_A 0.

Given a Waldhausen category CC whose weak equivalence classes from a set, one defines K 0(C)K_0(C) as an abelian group whose elements are the weak equivalence classes modulo the relation [A]+[B/A]=[B][A]+[B/A]=[B] for any cofibration sequence ABB/AA\hookrightarrow B\to B/A.

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



Waldhausen category of a small abelian category

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

Waldhausen category of a small exact category

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

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

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


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

Section 1 of

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

Revised on September 6, 2015 01:40:50 by Tim Porter (