aa r X i v : . [ m a t h . K T ] N ov D´EVISSAGE FOR WALDHAUSEN K -THEORY GEORGE RAPTIS
Abstract.
A d´evissage–type theorem in algebraic K -theory is a statementthat identifies the K -theory of a Waldhausen category C in terms of the K -theories of a collection of Waldhausen subcategories of C when a d´evissagecondition about the existence of appropriate finite filtrations is satisfied. Wedistinguish between d´evissage theorems of single type and of multiple type de-pending on the number of Waldhausen subcategories and their properties. Themain representative examples of such theorems are Quillen’s original d´evissagetheorem for abelian categories (single type) and Waldhausen’s theorem onspherical objects for more general Waldhausen categories (multiple type). Inthis paper, we study some general aspects of d´evissage–type theorems andprove a general d´evissage theorem of single type and a general d´evissage the-orem of multiple type. Introduction
The d´evissage theorem of Quillen [5] is a fundamental theorem in algebraic K -theory with many applications. The theorem states that given a full exact inclusionof abelian categories A ֒ → C , where A is closed under subobjects and quotientsin C , and in addition, it satisfies the condition that every object C ∈ C admits afinite filtration: 0 = C ⊆ C ⊆ · · · ⊆ C n − ⊆ C n = C such that C i /C i − ∈ A for i ≥
1, then the induced map K ( A ) ≃ −→ K ( C ) is ahomotopy equivalence. This theorem is an important source of K -equivalenceswhich do not arise from an equivalence between the underlying homotopy theories– in other words, it makes essential use on the “group completion” process thatdefines algebraic K -theory.Waldhausen [9] extended the definition of Quillen K -theory to categories withcofibrations and weak equivalences ( Waldhausen categories ) and generalized manyof Quillen’s fundamental theorems to this more general homotopical context, butthe d´evissage theorem has been a notable exception. The problem of finding asuitable generalization of the d´evissage theorem to Waldhausen K -theory was statedexplicitly by Thomason-Trobaugh [7, 1.11.1] (see also Waldhausen [8, p. 188]).More precisely, the problem asks for a general result in Waldhausen K -theory thatspecializes to Quillen’s d´evissage theorem when applied to the categories of boundedchain complexes.On the other hand, Waldhausen’s theorem on spherical objects [9, Theorem1.7.1] may be considered as a d´evissage–type theorem even though it is not relatedto Quillen’s d´evissage theorem. The theorem states that given a Waldhausen cat-egory C satisfying certain mild technical assumptions and an associated collection A = ( A i ) of Waldhausen subcategories of C consisting of spherical objects in C of dimension i , then under the assumption that the morphisms in C satisfy a multi-ple type version of the d´evissage condition in Quillen’s theorem, it follows that acanonical comparison map: hocolim −−−−−→ (Σ) K ( A i ) ≃ −→ K ( C )is a homotopy equivalence. The motivation for Waldhausen’s theorem and one ofits main applications in [9] had to do with obtaining a description of the algebraic K -theory of a space in terms of matrices and the plus construction in analogy toQuillen’s algebraic K -theory of rings. Waldhausen’s theorem can be consideredas a d´evissage–type result if the latter characterization is understood to refer to astatement that identifies the K -theory of a Waldhausen category C in terms of the K -theories of a collection of Waldhausen subcategories of C under the assumptionof a d´evissage condition about the existence of appropriate finite filtrations.Despite the common d´evissage–type quality of the theorems of Quillen and Wald-hausen, there are also some important differences between these two theorems thatare worth making explicit. Firstly, the obvious one is that in Quillen’s theoremthere is a single subcategory for unscrewing objects in C , whereas in Waldhausen’stheorem, we have for a similar purpose a collection of subcategories A = ( A i ) thatare suitably related. Secondly, in the context of Quillen’s theorem, the subcate-gory A ⊂ C (or rather, Ch b ( A ) ⊂ Ch b ( C )) is closed under (homotopy) pushouts,but this fails for the subcategories A i ⊂ C in Waldhausen’s context. Thirdly, inQuillen’s theorem, the subcategory Ch b ( A ) ⊂ Ch b ( C ) is not homotopically full ingeneral, but this property will typically hold for the subcategories A i ⊂ C in Wald-hausen’s theorem. We will refer to these two cases of d´evissage–type theorems as single type d´evissage and multiple type d´evissage respectively. Other d´evissage–typetheorems include the Gillet–Waldhausen theorem (see [7]), the d´evissage theoremof Blumberg–Mandell [2] and Barwick’s ‘Theorem of the Heart’ [1]. These belongto the category of d´evissage theorems of multiple type in our sense – where addi-tionally we have that K ( A i ) Σ ≃ K ( A i +1 ) for all i .The purpose of this paper is to prove a d´evissage theorem of single type (Theorem5.5) and a d´evissage theorem of multiple type (Theorem 6.9) using a commongeneral method, in the spirit of [9], whose application in each case is distinguishedby arguments specific to each case. Let us next give a general outline of thismethod and describe our main results without going into some of the more technicaldetails. Let C denote a Waldhausen category and let A = ( A i ) i ≥ be a collectionof Waldhausen subcategories of C . We will consider a Waldhausen subcategoryS n C A of S n C which consists of those filtered objects X • , • ∈ S n C : ∗ X , · · · X ,n whose successive cofibers X i − ,i are in A i for i ≥
1. Using Waldhausen’s AdditivityTheorem [9], we may identify the K -theory of this Waldhausen category S n C A asfollows (Proposition 3.2):(1) K (S n C A ) ≃ n Y K ( A i ) . In addition, we will consider the Waldhausen category b S n C A which has the sameunderlying category with cofibrations as S n C A , but the weak equivalences are de-tected by the underlying total object X ,n . We may stabilize these Waldhausen ´EVISSAGE 3 categories and obtain S ∞ C A and b S ∞ C A , respectively. The Waldhausen category b S ∞ C A represents the homotopy theory of objects in C equipped with a bounded fil-tration by objects whose successive cofibers are in A . Under some general technicalassumptions, there is a homotopy fiber sequence:(2) K (S w ∞ C A ) → K (S ∞ C A ) → K ( b S ∞ C A )where S w ∞ C A ⊂ S ∞ C A is the full Waldhausen subcategory of those objects whichare weakly trivial in b S ∞ C A (see Proposition 3.7). We introduce an additional tech-nical admissibility assumption which essentially identifies the K -theory of S w ∞ C A (we refer to Section 4 for the single type case and to Subsection 6.1 for the mul-tiple type case). Furthermore there is a forgetful exact functor ev ∞ : b S ∞ C A → C that forgets the filtration and evaluates at the underlying total object. Under theassumption of a d´evissage condition on ( C , A ) for morphisms in C , we prove thatthe exact functor ev ∞ induces a homotopy equivalence:(3) K ( b S ∞ C A ) ≃ −→ K ( C ) . (See Proposition 5.4 for the single type case and Proposition 6.8 for the multipletype case.) Combining the above, we deduce under certain assumptions an identi-fication of K ( C ) in terms of K ( A i ), for i ≥
1, as in the theorems of Quillen andWaldhausen respectively (see Theorem 5.5 for the single type case and Theorem 6.9for the multiple type case). We show that Quillen’s d´evissage theorem satisfies theassumptions of our d´evissage theorem of single type, but only if we assume
Quillen’stheorem in general – so we do not obtain an independent proof of Quillen’s theo-rem (see Subsection 5.4). On the other hand, we verify that Waldhausen’s theoremon spherical objects fits in the abstract formulation of our d´evissage theorem ofmultiple type (see Section 6).
Organization of the paper.
In Section 2, we recall some background materialabout Waldhausen categories and some of the fundamental theorems of Waldhausen K -theory. In Section 3, we introduce the Waldhausen categories S n C A and b S n C A and prove the homotopy equivalence (1) and the homotopy fiber sequence (2). InSection 4, we restrict to the single type case and discuss the somewhat mysteriousadmissibility assumption and some examples of classes of admissible Waldhausenpairs ( C , A ). In Section 5, we define the d´evissage condition and prove the ho-motopy equivalence (3) in the single type case. As a consequence, we then deducethe d´evissage theorem of single type (Theorem 5.5). In addition, we discuss howthe d´evissage condition arises from the existence of a d´evissage functor (Subsection5.3). We also discuss the relation of the single type d´evissage theorem to Quillen’sd´evissage theorem (Subsection 5.4) and to the Additivity Theorem (Example 5.10).In Section 6, we first introduce an abstract version of the context of Waldhausen’stheorem on spherical objects (Subsection 6.1). Then we prove the homotopy equiva-lence (3) in the multiple type case assuming an appropriate multiple type version ofthe d´evissage condition (Subsection 6.2). Finally in Subsection 6.3, we deduce thed´evissage theorem of multiple type (Theorem 6.9) and discuss a couple of examples. Acknowledgements.
The author is grateful for the support and the hospitality ofthe Hausdorff Institute for Mathematics in Bonn during the program “Topology”– where a first outline of this work was worked out – and for the support andthe hospitality of the Isaac Newton Institute in Cambridge during the research
G. RAPTIS programme “Homotopy harnessing higher structures” – where a final version ofthis work was first completed. The author was also partially supported by
SFB1085 — Higher Invariants (University of Regensburg) funded by the DFG.2.
Recollections
Waldhausen categories.
In this section we fix some terminology and recallsome of the fundamental theorems of Waldhausen K -theory that will be needed inthe proofs of our main results.A Waldhausen category is a small category C with cofibrations co C and weakequivalences w C in the sense of [9]. We recall that C has a zero object ∗ andco C is a subcategory which contains the isomorphisms, the morphisms from ∗ ,and it is closed under pushouts. In addition, w C is a subcategory which containsthe isomorphisms and the gluing axiom holds [9]. Examples include the exactcategories (in the sense of Quillen [5]) and appropriate full subcategories of cofibrantobjects in pointed model categories. Cofibrations will be indicated by and weakequivalences by ∼ → .We will also need to consider Waldhausen categories which have the followingadditional properties. Definition 2.1.
Let C be a Waldhausen category.(a) We say that C has the if given composable morphisms f and g in C , then the three morphisms f , g , and gf are weak equivalenceswhenever any two of them are.(b) We say that C admits factorizations if every morphism can be written asthe composition of a cofibration followed by a weak equivalence.(c) C is called good if it has the 2-out-of-3 property and admits factorizations. Remark 2.2.
A good Waldhausen category is called derivable in [4].A functor between Waldhausen categories F : C → C ′ is called exact if itpreserves all the relevant structure (zero object, cofibrations, weak equivalences,and pushouts along a cofibration).2.2. Waldhausen subcategories.
Let C be a Waldhausen category. A subcat-egory A ⊂ C is called a Waldhausen subcategory if A becomes a Waldhausencategory when equipped with following classes of cofibrations and weak equiva-lences:(i) a morphism in A is in co A if it is in co C and it has a cofiber in A ,(ii) a morphism in A is in w A if it is in w C ,and in addition, the inclusion functor A ֒ → C is exact. In other words, the zeroobject ∗ ∈ C is also a zero object in A , co A is a subcategory of A , and the pushoutin C of a diagram in A along a cofibration (in C with cofiber in A ) defines also apushout in A . Note that we do not assume that A ⊂ C is full in general. A pair( C , A ) where C is a Waldhausen category and A ⊂ C a Waldhausen subcategorywill be called a Waldhausen pair .Clearly a Waldhausen subcategory A ⊂ C has the 2-out-of-3 property if C does.On the other hand, one needs additional assumptions on ( C , A ) in general for theexistence of factorizations on A . ´EVISSAGE 5 Waldhausen K -theory. Waldhausen’s S • -construction associates to eachWaldhausen category C a simplicial object S • C of Waldhausen categories. Fol-lowing [9], let Ar[ n ] denote the category of morphisms and commutative squares inthe poset [ n ], n ≥
0. S n C is the full subcategory of the category of functors X : Ar[ n ] → C that is spanned by the functors X such that:(i) X ( i → i ) is the zero object ∗ ∈ C ,(ii) for every 0 ≤ i ≤ j ≤ k ≤ l ≤ n , the square X ( i → k ) / / (cid:15) (cid:15) X ( i → l ) (cid:15) (cid:15) X ( j → k ) / / X ( j → l )is a pushout where the horizontal morphisms are cofibrations in C .We will abbreviate X ( i → j ) to X i,j . Such a functor is determined up to isomor-phism by the filtered object ∗ = X , X , X , · · · X ,n since any other value of X is either the zero object or a cofiber of a morphismfrom this filtration. We will sometimes refer to such functors ( X • , • ) as staircase diagrams. An object in S C is given by a cofiber sequence ( X , X , ։ X , ).The category S n C is naturally a Waldhausen category where the weak equiv-alences are defined pointwise (see [9]). The simplicial operators are defined byprecomposition and induce exact functors of Waldhausen categories. Moreover, C S • C is a functor from the category of Waldhausen categories and exact func-tors to simplicial objects in this category. We note that if C is good, then so isS n C for any n ≥
0, see [6, A.9], [4].The restriction to the subcategory of weak equivalences w S • C yields a simplicialobject in the category of small categories. The Waldhausen K -theory of C is theloop space of the geometric realization of the bisimplicial set associated to thissimplicial category: K ( C ) := Ω | N • w S • C | . The Additivity Theorem.
We recall the statement of Waldhausen’s funda-mental Additivity Theorem (see [9, 1.3-1.4]).
Theorem 2.3 (Additivity Theorem) . Let C be a Waldhausen category. Then theexact functor ( d , d ) : S C → C × C , ( A C ։ B ) ( A, B ) , induces a homotopy equivalence K (S C ) ≃ −→ K ( C ) × K ( C ) . We recall also a different version of the Additivity Theorem that will be usedin later sections (see [9, Proposition 1.3.2]). Given a Waldhausen category C andWaldhausen subcategories A , B ⊂ C , we denote by E ( A , C , B ) the Waldhausensubcategory of S C whose objects are cofiber sequences A C ։ B, A ∈ A , B ∈ B , G. RAPTIS and morphisms are morphisms of cofiber sequences A / / / / f (cid:15) (cid:15) C / / / / h (cid:15) (cid:15) B g (cid:15) (cid:15) A ′ / / / / C ′ / / / / B ′ where f is in A and g is in B . Theorem 2.4 (Additivity Theorem, Version 2) . Let C be a Waldhausen categoryand let A , B ⊂ C be Waldhausen subcategories. Then the exact functor E ( A , C , B ) → A × B , ( A C ։ B ) ( A, B ) induces a homotopy equivalence K ( E ( A , C , B )) ≃ −→ K ( A ) × K ( B ) . Remark 2.5.
Theorem 2.4 holds more generally in the case where the inclusionfunctors A ֒ → C and B ֒ → C are simply exact functors of Waldhausen categories.2.5. The Approximation Theorem.
The Approximation Theorem provides auseful method for showing that an exact functor F : C → C ′ induces a homotopyequivalence in K -theory. The original formulation of Waldhausen [9, Theorem1.6.7] stated two approximation properties for F as criteria for a K -equivalence.Starting with the seminal work of Thomason-Trobaugh [7], later treatments of thetheorem studied these properties from the viewpoint of homotopical algebra andconnected the theorem with the invariance of K -theory under derived equivalencesor equivalences of homotopy theories (see [3] and [4]). The following version of theApproximation Theorem is due to Cisinski [4, Proposition 2.14]. Theorem 2.6 (Approximation Theorem) . Let F : C → C ′ be an exact functorbetween good Waldausen categories. Suppose that F satisfies the following twoapproximation properties: (App1) A morphism f in C is a weak equivalence if and only if F ( f ) is a weakequivalence in C ′ . (App2) For every morphism f : F ( X ) → Y in C ′ , there is a weak equivalence j : Y → Y ′ in C ′ , a morphism f ′ : X → X ′ in C and a weak equivalence q : F ( X ′ ) → Y ′ such that the square in C ′ F ( X ) f / / F ( f ′ ) (cid:15) (cid:15) Y ∼ j (cid:15) (cid:15) F ( X ′ ) ∼ q / / Y ′ commutes.Then the induced map K ( F ) : K ( C ) ≃ → K ( C ′ ) is a homotopy equivalence. The Fibration Theorem.
The Fibration Theorem provides a way of obtain-ing long exact sequences of K -groups similar to Quillen’s Localization Theorem forthe K -theory of abelian categories. It relates the K -theories of two different Wald-hausen category structures on the same underlying category with cofibrations. Thefollowing version is a slight generalization of Waldhausen’s original formulation [9,Theorem 1.6.4]. ´EVISSAGE 7 Theorem 2.7 (Fibration Theorem) . Let C v = ( C , co C , v C ) and C w = ( C , co C , w C ) be Waldhausen categories that have the same underlying categories, the same cofi-brations and v C ⊂ w C . Suppose that ( C , co C , w C ) is good. Let C w be the fullsubcategory of C spanned by the objects X ∈ C such that ∗ → X is in w C .Then C w = ( C w , co C ∩ C w , v C ∩ C w ) is a full Waldhausen subcategory of ( C , co C , v C ) and the sequence of exact inclusion functors C w → C v → C w induces a homotopy fiber sequence K ( C w ) → K ( C v ) → K ( C w ) . (The null-homotopy of the composition is the canonical one given by the naturalweak w -equivalence from the terminal object.)Proof. This is essentially [9, Theorem 1.6.4]. The replacement of Waldhausen’scylinder axiom with the existence of (non-functorial) factorizations is worked outin [6, Theorem A.3]. The assumption in [9, Theorem 1.6.4] that C w satisfies theextension axiom may be omitted by making a small modification in the proof thatinvolves an additional application of the Additivity Theorem (see below).[ Addendum : We recall that the proof of [9, Theorem 1.6.4] uses the extensionaxiom in order to identify vw S • C with v S • F • ( C , C w ). But the inclusion vw S • C ⊂ v S • F • ( C , C w ) is always a weak equivalence because we have weak equivalences foreach n ≥ v S • C × v S • S n C w ≃ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ ≃ u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ v S • w n C = vw n S • C / / v S • F n ( C , C w )by the Additivity Theorem. Here w n C denotes the Waldhausen subcategory ofdiagrams in C c ∼ c ∼ · · · ∼ c n with the usual cofibrations and where the ( v -)weak equivalences are defined point-wise. This has Waldhausen subcategories which are identified with C , embeddedas constant diagrams, and S n C w , embedded as diagrams with c = ∗ , and there isan equivalence of categories between w n C and E ( C , w n C , S n C w ) . ] (cid:3) The category of A -filtered objects in C The Waldhausen category S n C A . Let C be a Waldhausen category. Con-sider the functor [ n + 1] → [ n ] which sends i i , if i ≤ n , and n + 1 n . Thisdefines a functor Ar[ n + 1] → Ar[ n ]which induces an exact inclusion functor of Waldhausen categories i n : S n C ֒ → S n +1 C . G. RAPTIS
The functor i n simply repeats the last column of a staircase diagram – it is the n -thdegeneracy map. We defineS ∞ C : = colim { C i → S C i → · · · i n → S n +1 C i n +1 −→ · · · } . The subcategory of cofibrations (resp. weak equivalences) in S ∞ C is defined to bethe colimit of the categories of cofibrations (resp. weak equivalences) in S n C forall n ≥
1. The following proposition is now an easy observation.
Proposition 3.1.
Let C be a Waldhausen category. Then S ∞ C endowed with thesubcategories of cofibrations and weak equivalences from the Waldhausen categories S n C , for n ≥ , is a Waldhausen category. Moreover, if C is good, then so is S ∞ C . Let A = ( A i ) i ≥ be an (ordered) collection of Waldhausen subcategories of C .We emphasize here that repetitions among the A i ’s are allowed. In particular,the case where A i = A for a single Waldhausen subcategory A ⊂ C will be animportant example. We consider the subcategory S n C A of S n C whose objects arethose objects of S n C X • , • : Ar[ n ] → C such that X i − ,i ∈ A i for 1 ≤ i ≤ n . The morphisms in S n C A are morphisms F • , • : X • , • → X ′• , • in S n C such that F i − ,i is in A i for each i ≥
1. This definesa Waldhausen subcategory of S n C . The exact inclusion i n restricts to an exactinclusion between the respective Waldhausen subcategories, i n, A : S n C A ֒ → S n +1 C A . Similarly, we consider the corresponding Waldhausen subcategory of S ∞ C :S ∞ C A : = colim { A i , A −−−→ S C A i , A −−−→ · · · i n, A −−−→ S n +1 C A −→ · · · } . Given a collection of Waldhausen subcategories A = ( A i ) i ≥ of C , we write A [ n ] for the new collection of Waldhausen categories obtained after shifting by n ,i.e., A [ n ] i = A n + i for i ≥ Proposition 3.2.
Let C be a Waldhausen category and let A = ( A i ) i ≥ be acollection of Waldhausen subcategories of C . For each n ≥ , the exact functor q n : S n C A −→ n Y A i , ( X • , • ) ( X , , · · · , X n − ,n ) induces a homotopy equivalence K (S n C A ) ≃ −→ n Y K ( A i ) . Moreover, these induce also a homotopy equivalence K (S ∞ C A ) ≃ colim −−−→ n Q n K ( A i ) . Proof.
There are inclusions of Waldhausen subcategories A ⊂ S n C A , as constantfiltered objects, and s : S n − C A [1] ⊂ S n C A , as filtered objects that begin withthe zero object in A . It is easy to see that there is an equivalence of Waldhausencategories S n C A −→ E ( A , S n C A , S n − C A [1] ) . ´EVISSAGE 9 Then the Additivity Theorem (Theorem 2.4) gives inductively homotopy equiva-lences K (S n C A ) ≃ K ( A ) × K (S n − C A [1] ) ≃ · · · ≃ n Y K ( A i ) . The composite homotopy equivalence is defined by the exact functor(4) ( X • , • ) ( X , , X , , · · · , X n − ,n ) . A homotopy inverse Q n K ( A i ) → K (S n C A ) is given by the exact functor thatincludes those filtered objects which are defined by successive trivial cofiber se-quences: ( A , A , · · · , A n ) ( ∗ A A ∨ A · · · _ ≤ i ≤ n A i ) . Moreover, we have commutative diagrams of exact functors:S n C A (cid:10) (cid:10) i n, A / / S n +1 C A (cid:9) (cid:9) Q n − A I I / / Q n A I I where the bottom functor is the canonical inclusion functor, given on objects by( A , · · · , A n − ) ( A , · · · , A n − , ∗ ). Then the case n = ∞ follows immediatelysince hocolim −−−−−→ n K (S n C A ) ≃ −→ K (S ∞ C A ) . (cid:3) The Waldhausen category b S n C A . Let C be a Waldhausen category andlet A = ( A i ) i ≥ be a collection of Waldhausen subcategories of C . The cate-gories S n C A categories can be endowed with the following weaker type of weakequivalence. Definition 3.3.
A morphism F • , • : ( X • , • ) → ( Y • , • ) in S n C A (resp. in S ∞ C A ) iscalled an eventual weak equivalence if F ,n : X ,n → Y ,n is a weak equivalence in C (resp. for all large enough n ).Every weak equivalence in S n C A , for n ∈ { , , · · · , ∞} , is also an eventual weakequivalence and this new class of weak equivalences defines a new Waldhausenstructure on the underlying category of S n C A (with the same subcategory of cofi-brations). We denote this new Waldhausen category by b S n C A . Note that theinclusion functors b i n, A : b S n C A ֒ → b S n +1 C A are again exact and their colimit is b S ∞ C A . The Waldhausen category b S ∞ C A isexactly the Waldhausen category of filtrations (or “unscrewings”) of objects in C by objects in the (ordered) collection of subcategories A .In order to ensure that the Waldhausen categories S n C A and b S n C A are goodwhen C is, it will be convenient to assume in addition that C admits a cylinderfunctor which satisfies the cylinder axiom in the sense of [9]. This structure equips C with factorizations which are functorial in C [1] and additionally satisfy certainexactness properties. In particular, restricting these functorial factorizations to themorphisms of the form ( X → ∗ ) yields an exact functor (“cone”): C : C → C , X C ( X ) , and also an exact functor (“suspension”):Σ : C → C , X Σ( X ) : = C ( X ) /X. We refer to [9, 1.6] for more details. The assumption about the existence of cylinderfunctors is not necessary in all of our proofs, but it will be convenient for somemain examples of Waldhausen pairs ( C , A ) for which the inclusion A ⊂ C is nothomotopically fully faithful. Definition 3.4.
A Waldhausen pair ( C , A ) is called good if C has the 2-out-of-3 property and admits a cylinder functor which satisfies the cylinder axiom andrestricts to a cylinder functor on A .Note that if ( C , A ) is a good Waldhausen pair, then both C and A are goodWaldhausen categories. We will be also interested in the following type of a Wald-hausen pair, which may fail to be good in general, but it has different stronghomotopical properties. Definition 3.5.
A Waldhausen subcategory A ⊂ C is called replete if it is a fullsubcategory and has the following property: given X ∈ A , then any object Y ∈ C which is weakly equivalent to X is also in A . In this case, we also say that theWaldhausen pair ( C , A ) is replete. Lemma 3.6.
Let C be a good Waldhausen category and let A = ( A i ) i ≥ be acollection of Waldhausen subcategories of C . Suppose that C has a cylinder functorwhich satisfies the cylinder axiom. (i) The Waldhausen categories S n C A and b S n C A have the 2-out-of-3 propertyfor any n ∈ { , , . . . , ∞} . (ii) Suppose that ( C , A i ) is replete for every i ≥ and that Σ : C → C sends A i to A i +1 . Then b S ∞ C A admits factorizations. In particular, b S ∞ C A is agood Waldhausen category. (iii) Let n ≥ . Suppose that ( C , A i ) is a good Waldhausen pair for ≤ i ≤ n .Then S n C A and b S n C A admit (functorial) factorizations. In particular, theWaldhausen categories S n C A and b S n C A are good Waldhausen categories.Proof. (i) is obvious. (ii): Let F • , • : X • , • → Y • , • be a morphism in b S ∞ C A . ByProposition 3.1, there is a factorization of the morphism in S ∞ C F • , • + Id : X • , • ∨ Y • , • → Y • , • into a cofibration followed by a weak equivalence (both in the sense of S ∞ C ) X • , • ∨ Y • , • Z • , • ∼ → Y • , • . Then we define a new object Z ′• , • ∈ S ∞ C by Z ′ ,i := X ,i ∪ X ,i − Z ,i − ∪ Y ,i − Y ,i . Using our assumptions on ( C , A ), it follows that the object Z ′• , • is in S ∞ C A .Moreover, the canonical morphisms ( X ,i → Z ′ ,i → Y ,i ) define the required fac-torization in b S ∞ C A : X • , • Z ′• , • ∼ −→ Y • , • . (iii): the functorial factorizations in C , given by the cylinder functor, producefunctorial factorizations in S n C [9, 1.6.1]. Using the assumptions, these restrict tofunctorial factorizations on the Waldhausen subcategory S n C A , as required. Thefactorizations in S n C A define also factorizations in b S n C A . (cid:3) ´EVISSAGE 11 A homotopy fiber sequence.
For n ∈ { , , · · · , ∞} , let S wn C A denote thefull Waldhausen subcategory of S n C A spanned by those objects X • , • which areweakly equivalent to the zero object in b S n C A , that is, the morphism ∗ → X ,n is aweak equivalence. Clearly, if S n C A is good, then so is S wn C A . The exact inclusionfunctors i n restrict to exact functors i wn, A : S wn C A ֒ → S wn +1 C A whose colimit is S w ∞ C A . For n = 1, S w C A is the Waldhausen subcategory of A which consists of the weakly trivial objects. Proposition 3.7.
Let C be a good Waldhausen category, A = ( A i ) i ≥ a collectionof Waldhausen subcategories of C , and let n ∈ { , , · · · , ∞} . Suppose that b S n C A is a good Waldhausen category (see, e.g., Lemma 3.6). Then the exact functors S wn C A ֒ → S n C A → b S n C A induce a homotopy fiber sequence K (S wn C A ) → K (S n C A ) → K ( b S n C A ) . Proof.
This is a direct application of the Fibration Theorem (Theorem 2.7). (cid:3) Admissible Waldhausen Pairs
Basic definitions and properties.
Let ( C , A ) be a good Waldhausen pair.For the purpose of the devisage theorem in the next section, we will need to consideran additional ad hoc condition on ( C , A ) which states an identification of the K -theory of the Waldhausen category S w ∞ C A . For each n ≥
1, we have an exactfunctor q wn : S wn C A −→ n − Y A , ( X • , • ) ( X , , · · · , X n − ,n − ) . Each functor q wn admits a section given by the exact functor j n : n − Y A → S wn C A , ( A , · · · , A n − ) ( ∗ A · · · n − _ A i n − _ C ( A i )) . As a consequence, the induced map K ( j n ) defines a section of K ( q wn ). It will beconvenient to consider also the exact functor τ n : Q n − A → S wn C A which inducesa section only up to a homotopy equivalence. This is defined on objects by τ n : ( A , · · · , A n − ) ( ∗ A A ∨ CA · · · A n − ∨ n − _ CA i n − _ CA i ) . The composite functor q wn ◦ τ n is given on objects by( A , · · · , A n − ) ( A , A ∨ Σ A , · · · , A n − ∨ Σ A n − )which induces a homotopy equivalence in K -theory. More specifically, the map in K -theory is identified with the homotopy equivalence( π , π − π , · · · , π n − − π n − ) : n − Y K ( A ) → n − Y K ( A )where π i denotes the projection onto the i -th factor and the sum corresponds tothe loop sum. Clearly each of the maps K ( q wn ), K ( j n ), and K ( τ n ), is a homotopyequivalence if any one of them is a homotopy equivalence. The functors { τ n } n ≥ are compatible with respect to n ≥ wn C A i wn, A / / S wn +1 C A Q n − A τ n O O / / Q n A τ n +1 O O where the bottom functor is the canonical inclusion functor, given on objects by( A , · · · , A n − ) ( A , · · · , A n − , ∗ ). As a consequence, we obtain an exact functor τ ∞ : colim −−−→ n n − Y A −→ S w ∞ C A . Definition 4.1.
Let ( C , A ) be a good Waldhausen pair. We say that ( C , A ) is admissible if the exact functor τ ∞ induces a homotopy equivalence: K ( τ ∞ ) : colim −−−→ n n − Y K ( A ) ≃ −→ K (S w ∞ C A ) . Remark 4.2.
The comments above imply that K ( τ n ) is (split) injective on ho-motopy groups. As a consequence, K ( τ ∞ ) is also always injective on homotopygroups. Remark 4.3.
The functors { j n } n ≥ are not compatible with the respect to thenatural inclusion functors i wn, A . We consider the exact functor ξ n : n − Y A → n Y A , ( A , · · · , A n − ) ( A , · · · , A n − , n − _ Σ( A i )) . The map in K -theory induced by ξ n can be identified with the map( π , · · · , π n − , − n − X π i ) : n − Y K ( A ) → n Y K ( A )where π i denotes the projection onto the i -th factor. Using the Additivity Theorem(Theorem 2.4), it can be shown that the induced diagram in K -theory(5) K (S wn C A ) K ( i wn, A ) / / K (S wn +1 C A ) Q n − K ( A ) K ( ξ n ) / / K ( j n ) O O Q n K ( A ) K ( j n +1 ) O O commutes up to a preferred homotopy. Thus, we may obtain also in this way amap as n → ∞ : J ∞ : colim −−−→ ( ξ n ) n − Y K ( A ) −→ K (S w ∞ C A ) . We do not know if the diagram analogous to (5) but where the vertical maps arethe retraction maps K ( q wn ) is also homotopy commutative – this is related to theadmissibility assumption. ´EVISSAGE 13 We do not know if every good Waldhausen pair ( C , A ) is admissible. In thefollowing sections, we will show several classes of examples where the admissibilityassumption is satisfied, using arguments specific to each case.4.2. Criteria for admissiblity.
Let ( C , A ) be a good Waldhausen pair. We letS wn C A denote the Waldhausen category whose category with cofibrations is thesame as that of S wn C A and a morphism F • , • : X • , • → Y • , • is a weak equivalence if F , is a weak equivalence in A . This is again a good Waldhausen category. For n ≥
2, there are exact functors as follows: p n : S wn C A → A , ( X • , • ) X , s : A → S wn C A , A ( ∗ A = A = · · · = A C ( A )) . Note that the composite functor p n ◦ s is the identity. Let iso S wn C A (resp. iso A )denote the associated Waldhausen category whose underlying category with cofi-brations is that of S wn C A (resp. A ), and a morphism F • , • : X • , • → Y • , • is a weakequivalence if F , is an isomorphism (resp. the weak equivalences in iso A arethe isomorphisms). The functors defined above are exact also with respect to theseWaldhausen category structures. Proposition 4.4.
Let ( C , A ) be a good Waldhausen pair. Then ( C , A ) is admis-sible if any of the following conditions holds. (1) The map K ( p n ) : K (S wn C A ) → K ( A ) , ( X • , • ) X , , is a homotopyequivalence for every n ≥ . (2) The map K ( p n ) : K (iso S wn C A ) → K (iso A ) , ( X • , • ) X , , is a homotopy equivalence for every n ≥ .Proof. (1) We proceed by induction on n ≥ K ( q wk ) is a homotopyequivalence for k ≤ n if and only if K ( p k ) is a homotopy equivalence for k ≤ n . The claim is obvious for n ≤
2. For the inductive step, consider the fullWaldhausen subcategory E n ⊂ S wn C A which consists of objects ( X • , • ) such that X , is weakly trivial. Then the Fibration Theorem (Theorem 2.7) implies thatthere is a homotopy fiber sequence K ( E n ) → K (S wn C A ) → K (S wn C A ) . There are exact functors ρ : E n → S wn − C A , ( X • , • ) ( ∗ X , · · · X ,n ) ι : S wn − C A → E n , ( Y • , • ) ( ∗ = ∗ Y , · · · Y ,n − ) . The composition ρ ◦ ι is the identity functor. The composition ι ◦ ρ is weaklyequivalent to the identity functor via the following natural weak equivalence offiltered objects: X , / / / / ∼ (cid:15) (cid:15) X , / / / / ∼ (cid:15) (cid:15) · · · / / / / X ,n ∼ (cid:15) (cid:15) ∗ / / / / X , / / / / · · · / / / / X ,n . Therefore, K ( E n ) ≃ K (S wn − C A ) ≃ Q n − K ( A ), using the inductive assumption.Now we consider the following homotopy commutative diagram: K ( E n ) / / K (S wn C A ) / / K (S wn C A ) Q n − K ( A ) / / ≃ K ( ιj n − ) O O Q n − K ( A ) π / / K ( j n ) O O K ( A ) K ( s ) O O where the rows are homotopy fiber sequences (of infinite loop spaces). The rightsquare is homotopy commutative because the underlying exact functors are natu-rally weakly equivalent. Hence the right vertical map is a homotopy equivalenceif and only if the middle vertical map is a homotopy equivalence. Equivalently,the map K ( p n ) is a homotopy equivalence if and only if K ( j n ) – and thereforealso K ( q wn ) and K ( τ n ) – is a homotopy equivalence. The result then follows byinduction.(2) Let D n ⊂ iso S wn C A be the full Waldhausen subcategory spanned by the objects X • , • such that X , is weakly trivial. Consider the following diagram in K -theory: K ( D n ) / / (cid:15) (cid:15) K (iso S wn C A ) / / (cid:15) (cid:15) K (S wn C A ) (cid:15) (cid:15) K (iso A w ) / / K (iso A ) / / K ( A )where the rows define homotopy fiber sequences (of infinite loop spaces) by theFibration Theorem (Theorem 2.7). The vertical maps are induced by the functor p n : X • , • X , . The map K ( D n ) → K (iso A w ) is a homotopy equivalence withhomotopy inverse induced by the functor A ( ∗ A = · · · = A ). Hence the rightvertical map is a homotopy equivalence if (and only if) the middle vertical map isa homotopy equivalence. The result then follows from (1). (cid:3) Example: Replete Waldhausen pairs.
The next proposition shows thatreplete Waldhausen pairs (Definition 3.5) are examples of admissible Waldhausenpairs.
Proposition 4.5.
Let ( C , A ) be a good Waldhausen pair which is also replete.Then ( C , A ) is admissible.Proof. We will show that the exact functor q wn : S wn C A → Q n − A induces a ho-motopy equivalence in K -theory for any n ≥
1. Let S n − ( C A , Σ − A ) denote thefull Waldhausen subcategory of S n − C A spanned by objects ( X • , • ) such that thesuspension of X ,n − is in A . In detail, this means that there is a factorization X ,n − Z ∼ → ∗ such that the cofiber of the first morphism is in A . (Since A ⊂ C is replete, thisproperty is independent of the choice of the factorization.) The full Waldhausensubcategory S n − ( C A , Σ − A ) has the 2-out-of-3 property and admits factorizationsbecause S n − C A has these properties. There is an exact functor U : S wn C A → S n − ( C A , Σ − A )which simply forgets the last column. We show that U induces a homotopy equiv-alence in K -theory by checking that it has properties (App1) and (App2) of the ´EVISSAGE 15 Approximation Theorem (Theorem 2.6). U obviously satisfies (App1). To verify(App2), let f : U ( X • , • ) → Z • , • be a morphism in S n − ( C A , Σ − A ) and consider a factorization Z ,n − ∪ X ,n − X ,n Y ∼ → ∗ . Then let Z ′• , • be the object of S wn C A that corresponds to the filtered object ∗ Z , Z , · · · Z ,n − Y by making choices of cofibers. There is an obvious morphism f ′ : X • , • → Z ′• , • anda weak equivalence q : U ( Z ′• , • ) = −→ Z • , • such that f = q ◦ U ( f ′ ). Thus, (App2)holds, and therefore U induces a homotopy equivalence in K -theory: K ( U ) : K (cid:0) S wn C A (cid:1) ≃ → K (cid:0) S n − ( C A , Σ − A ) (cid:1) . We may view A as the full Waldhausen subcategory of constant filtered objectsin S n − ( C A , Σ − A ), and S n − ( C A , Σ − A ) as the full Waldhausen subcategory ofthose filtered objects with X , = ∗ . We have an equivalence of categoriesS n − ( C A , Σ − A ) ≃ E ( A , S n − ( C A , Σ − A ) , S n − ( C A , Σ − A )) . Therefore the Additivity Theorem (Theorem 2.4) yields inductively homotopy equiv-alences as follows: K (cid:0) S n − ( C A , Σ − A ) (cid:1) ≃ K ( A ) × K (cid:0) S n − ( C A , Σ − A ) (cid:1) ≃ · · · ≃ n − Y K ( A ) . The composite homotopy equivalence is given by the functor:(6) ( X • , • ) ( X , , X , , · · · , X n − ,n − ) . (cid:3) Remark 4.6.
The proof of Proposition 4.5 works also under some slightly weakerassumptions. It suffices that C is good, A ֒ → C detects cofibrations, and ( C , A )is replete. In other words, under these assumptions, it is not required to assumethat C admits functorial factorizations.4.4. Example: Waldhausen pairs with (WHEP).
This example is inspiredby the assumptions in the d´evissage theorem of D. Yao [10]. Let ( C , A ) be agood Waldhausen pair. We say that ( C , A ) satisfies the weak homotopy extensionproperty (WHEP) if the following property is satisfied: given a cofibration A B in A and X ∈ C such that X ∼ −→ ∗ , then every morphism C ( A ) ∪ A B → X extendsto a morphism C ( B ) → X along the cofibration C ( A ) ∪ A B C ( B ). Proposition 4.7.
Let ( C , A ) be a good Waldhausen pair where A ⊂ C is a fullsubcategory. Suppose that ( C , A ) satisfies the weak homotopy extension property.Then ( C , A ) is admissible.Proof. We show that the condition of Proposition 4.4(1) is satisfied. To this end,we claim that the exact functor s : A → S wn C A induces a homotopy equivalencein K -theory. This is shown by applying the Approximation Theorem (Theorem2.6). The functor s clearly satisfies (App1). For (App2), we consider a morphism F • , • : s ( A ) → X • , • and a factorization in A A i Z q, ∼ −−→ X , . By the weak homotopy extension property, we find an extension of the morphism C ( A ) ∪ A Z → X ,n to a morphism q ′ : C ( Z ) → X ,n . This extension is used todefine a morphism Q • , • : s ( Z ) → X • , • by Z q (cid:15) (cid:15) Z (cid:15) (cid:15) · · · Z (cid:15) (cid:15) / / / / C ( Z ) q ′ (cid:15) (cid:15) X , / / / / X , / / / / · · · / / / / X ,n − / / / / X ,n and this gives the required factorization of F • , • as the composition s ( A ) s ( i ) s ( Z ) Q • , • , ∼ −−−−→ X • , • . (cid:3) Example 4.8.
Let ( C , A ) be a good Waldhausen pair where A ⊂ C is a fullsubcategory. Suppose that there is an exact fully faithful functor ι : C ֒ → M c in thefull subcategory of cofibrant objects of a pointed model category M . In addition,suppose that for every X ∈ C which is weakly trivial, the object ι ( X ) ∈ M is alsofibrant. Then ( C , A ) has the weak homotopy extension property and therefore itis admissible. 5. Single Type D´evissage
Assumptions.
We fix the following notation and assumptions throughout thissection. We let ( C , A ) denote a good Waldhausen pair. We denote also by A thecollection of Waldhausen subcategories ( A = A i ) i ≥ which is constant at A . ByLemma 3.6, it follows that the Waldhausen categories S n C A and b S n C A are goodfor every n ∈ { , , . . . , ∞} .5.1. The d´evissage condition.
There are exact “evaluation” functors ev n : b S n C A → C , ( X • , • ) X ,n . These are compatible with stabilization along the inclusion functors b i n, A , so thereis also an induced exact functor, ev ∞ : b S ∞ C A → C , which sends a staircase diagram representing a bounded A -filtration of an object X ,n , for n large enough, to the object itself. One of the main functions of ad´evissage condition on ( C , A ) is to ensure that the exact functor ev ∞ induces a K -equivalence. Definition 5.1.
We say that ( C , A ) satisfies the d´evissage condition if for everymorphism f : X → Y in C , there is a weak equivalence g : Y ∼ → Y ′ such that thecomposition gf : X → Y ′ admits a factorization X = X X · · · X m − X m ∼ → Y ′ where X i /X i − ∈ A for all i ≥ C , and notjust the objects, admit filtrations whose subquotients are in A – not necessarilyfunctorially. ´EVISSAGE 17 Example 5.2.
Suppose that ( C , A ) satisfies the d´evissage condition. Let ( C , A ′ )be another good Waldhausen pair where Ob A ⊆ Ob A ′ . Then ( C , A ′ ) also satisfiesthe d´evissage condition. Example 5.3.
Suppose that ( C , A ) satisfies the d´evissage condition. Let C ′ be an-other Waldhausen category whose underlying category with cofibrations is ( C , co C )and w C ⊆ w C ′ . Let ( C ′ , A ′ ) denote the good Waldhausen pair which correspondsto the subcategory of C ′ defined by A . Then ( C ′ , A ′ ) also satisfies the d´evissagecondition. Proposition 5.4.
Suppose that ( C , A ) satisfies the d´evissage condition. Then ev ∞ : b S ∞ C A → C induces a homotopy equivalence K ( b S ∞ C A ) ≃ → K ( C ) . Proof.
We claim that the functor ev ∞ : b S ∞ C A → C satisfies the conditions of theApproximation Theorem (Theorem 2.6). (App1) holds by definition. For (App2),let ( X • , • ) be an object in b S ∞ C A and consider a morphism in C f : ev ∞ ( X • , • ) = X ,n → Y. Using the d´evissage condition applied to the morphism f , there is a weak equiva-lence g : Y ∼ → Y ′ and a factorization of the composite gf , X ,n = Z Z · · · Z m ∼ → Y ′ such that the subquotients Z i /Z i − are in A . Consequently, the combination ofthe two filtered objects ( X • , • ′ ) ≤•≤• ′ ≤ n and ( Z • )can be extended, by making choices of cofibers, to a new object ( Z • , • ) in b S ∞ C A .There is a canonical morphism in b S ∞ C A f ′ : ( X • , • ) → ( Z • , • )whose components are either identities or a composition of cofibrations in the fac-torization of the map X ,n → Y ′ considered above. Lastly, the weak equivalence ev ∞ ( Z • , • ) = Z m ∼ → Y ′ fits in a commutative diagram ev ∞ ( X • , • ) f / / ev ∞ ( f ′ ) (cid:15) (cid:15) Y ∼ g (cid:15) (cid:15) ev ∞ ( Z • , • ) ∼ / / Y ′ . This shows that (App2) is satisfied and then the result follows from Theorem 2.6. (cid:3)
The man theorem.
A d´evissage theorem of single type is a statement that forcertain Waldhausen pairs ( C , A ) for which a d´evissage–type condition is satisfied,the induced map K ( A ) ≃ −→ K ( C ) is a homotopy equivalence. Combining ouradmissibility and d´evissage conditions (see Definition 4.1 and Definition 5.1), weobtain our main d´evissage theorem of this type. Theorem 5.5 (Single Type D´evissage) . Let ( C , A ) be an admissible Waldhausenpair which satisfies the d´evissage condition. Then the inclusion A ֒ → C induces ahomotopy equivalence K ( A ) ≃ → K ( C ) . Proof.
Consider the following diagram(7) K (S wn C A ) / / K (S n C A ) ∼ (cid:20) (cid:20) / / K ( b S n C A ) Q n − K ( A ) / / K ( τ n ) O O Q n K ( A ) ∨ / / U U K ( A ) O O The middle vertical maps are homotopy equivalences from Propositions 3.2. Theleft vertical map was defined in Subsection 4.1. The left bottom map is induced bythe exact functor( A , · · · , A n − ) ( A , A ∨ Σ A , · · · , A n − ∨ Σ A n − , Σ A n − ) . The left square commutes up to homotopy by direct inspection.The bottom map Q n K ( A ) → K ( A ) is induced by the coproduct functor. Theright vertical map is the canonical inclusion A = b S C A → b S n C A . The right squareis also homotopy commutative, since the underlying exact functors are naturallyweakly equivalent. Note that the bottom row defines a homotopy fiber sequenceand the top row is a homotopy fiber sequence by Proposition 3.7.Passing to the colimit as n → ∞ , we obtain a homotopy commutative diagram(8) K (S w ∞ C A ) / / K (S ∞ C A ) ∼ (cid:20) (cid:20) / / K ( b S ∞ C A )colim −−−→ n Q n − K ( A ) / / ≃ K ( τ ∞ ) O O colim −−−→ n Q n K ( A ) ∨ / / U U K ( A ) O O whose rows are homotopy fiber sequences (of infinite loop spaces) and K ( τ ∞ ) isa homotopy equivalence because ( C , A ) is admissible by assumption. Hence theright vertical map K ( A ) → K ( b S ∞ C A ) is a homotopy equivalence. (Note that bothof the right horizontal maps are π -surjective.)By Proposition 5.4, the exact functor ev ∞ induces a homotopy equivalence K ( b S ∞ C A ) ≃ −→ K ( C ). Then the result follows because the map in K -theory in-duced by the inclusion A ⊂ C agrees with the composition of homotopy equiva-lences K ( A ) ≃ −→ K ( b S ∞ C A ) ≃ −→ K ( C ) . (cid:3) Remark 5.6.
The proof of Theorem 5.5 shows also the following converse state-ment: if ( C , A ) satisfies the d´evissage condition and the inclusion A ֒ → C inducesa homotopy equivalence K ( A ) ≃ → K ( C ), then the pair ( C , A ) is admissible.5.3. D´evissage functors.
The d´evissage condition for a (good) Waldhausen pair( C , A ) is often a consequence of the existence of a d´evissage functor on C definedas follows. Definition 5.7.
A d´evissage functor for ( C , A ) is a (not necessarily exact!) functor D : C → E ( C , C , A ) , X D ( X ) = ( L ( X ) X ։ A ( X ))where L : C → C and A : C → A are (not necessarily exact!) functors such that:(1) D preserves cofibrations,(2) for each X ∈ C , there is n ≥ L n ( X ) = ∗ . ´EVISSAGE 19 Proposition 5.8.
Let D be a d´evissage functor for ( C , A ) . Then ( C , A ) satisfiesthe d´evissage condition. Moreover, if ( C , A ) is also admissible, then the inclusion A ֒ → C induces a homotopy equivalence K ( A ) ≃ → K ( C ) .Proof. Given a cofibration i : X Y , we consider for n large enough the morphismof filtered objects ∗ = L n +1 ( X ) / / / / L n ( X ) / / / / (cid:15) (cid:15) (cid:15) (cid:15) L n − ( X ) / / / / (cid:15) (cid:15) (cid:15) (cid:15) · · · (cid:15) (cid:15) (cid:15) (cid:15) / / / / L ( X ) / / / / (cid:15) (cid:15) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:15) (cid:15) ∗ = L n +1 ( Y ) / / / / L n ( Y ) / / / / L n − ( Y ) / / / / · · · / / / / L ( Y ) / / / / Y. Then the objects L k ( f ), 1 ≤ k ≤ n , defined by the pushouts L k ( X ) / / / / (cid:15) (cid:15) (cid:15) (cid:15) X (cid:15) (cid:15) (cid:15) (cid:15) L k ( Y ) / / / / L k ( f )yield a factorization of i : X Y , X L n ( f ) L n − ( f ) · · · L ( f ) L ( f ) := Y. Here the morphisms are cofibrations in C because D preserves cofibrations byassumption. The cofiber of L k +1 ( f ) L k ( f ), k ≥
0, is the cofiber of A ( L k ( X )) A ( L k ( Y )) which is in A . Hence the d´evisage condition is satisfied for cofibrations.For an arbitrary morphism f : X → Y , we choose a factorization X i Y ′ ∼ → Y and find a factorization of i as shown above. The second claim is a consequence ofTheorem 5.5. (cid:3) Remark 5.9.
As already suggested in the proof of Proposition 5.8, a d´evissagefunctor D for ( C , A ) can be iterated in order to define an exact functor D ∞ : C → b S ∞ C A which is a section to ev ∞ . If, moreover, the d´evissage functor D is also exact , which is too strong an assumption in general, then D ∞ actually defines anexact functor D ∞ : C → S ∞ C A . If, furthermore, D ∞ sends A to S ∞ A , then it canbe shown that the map K ( A ) ≃ −→ K ( C ) is a homotopy equivalence, with homotopyinverse induced by the composite exact functor: C D ∞ −−→ S ∞ C A colim −−−→ n q n −−−−−−→ colim −−−→ n A n ∨ −→ A . Example 5.10.
Let C be a good Waldhausen category with a cylinder functorwhich satisfies the cylinder axiom. We may consider the good Waldhausen pairthat corresponds to the exact inclusion functor (cf. Theorem 2.3): C × C ֒ → S C , ( X, Y ) ( X X ∨ Y ։ Y ) . There is a d´evissage functor D for (S C , C × C ) which is defined on objects bysending ( A C ։ B ) to the cofiber sequence in E (S C , S C , C × C ): D : ( A C ։ B ) (cid:0) ( A = A ։ ∗ ) ( A C ։ B ) ։ ( ∗ B = B ) (cid:1) . By Proposition 5.8, it follows that (S C , C × C ) satisfies the d´evissage condition.According to the Additivity Theorem (Theorem 2.3), the map K ( C × C ) ≃ −→ K (S C ) is a homotopy equivalence. (In fact, D is actually exact in this case, so the argument in the preceding remark applies.) Then it follows from Remark5.6 that the good Waldhausen pair (S C , C × C ) is admissible – so it satisfies theassumptions of Theorem 5.5. In this indirect way, we may view the statement ofthe Additivity Theorem for C as part of the statement of Theorem 5.5 – whoseproof, of course, made essential use of the Additivity Theorem.5.4. Example: Abelian categories.
Let C be a (small) abelian category. Thismay be regarded as a Waldhausen category in the standard way by defining thecofibrations to be the monomorphisms and the weak equivalences to be the iso-morphisms. We let Ch b ( C ) denote the Waldhausen category of bounded chaincomplexes in C , where the cofibrations are the monomorphisms and the weakequivalences are the quasi-isomorphisms of chain complexes. According to theGillet-Waldhausen theorem [7], the exact inclusion functor C → Ch b ( C ), as chaincomplexes concentrated in degree 0, induces a homotopy equivalence in K -theory(9) K ( C ) ≃ −→ K (Ch b ( C )) . Let
A ⊂ C be a full exact abelian subcategory. The corresponding inclusion ofchain complexes Ch b ( A ) ⊂ Ch b ( C ) defines a Waldhausen subcategory. In addition,(Ch b ( C ) , Ch b ( A )) is a good Waldhausen pair using the standard cylinder objectsfor chain complexes. We emphasize that the exact inclusion Ch b ( A ) ⊂ Ch b ( C ) isfull but not homotopically full in general.We recall Quillen’s d´evissage theorem for the K -theory of abelian categories. Theorem 5.11 (Quillen [5], D´evissage) . Let C be an abelian category and A afull exact abelian subcategory which is closed in C under subobjects and quotients.Suppose that every object C ∈ C admits a finite filtration C ⊆ C ⊆ C ⊆ · · · ⊆ C n − ⊆ C n = C such that C i /C i − ∈ A for all i = 1 , · · · , n . Then the inclusion A ֒ → C induces ahomotopy equivalence K ( A ) ≃ → K ( C ) . The purpose of this subsection is to relate Quillen’s d´evissage theorem to Theo-rem 5.5 when applied to the respective categories of bounded chain complexes. Thed´evissage condition in Quillen’s theorem is seemingly weaker than the d´evissage con-dition of Definition 5.1 – while Theorem 5.11 requires the existence of appropriatefiltrations of objects, the d´evissage condition of Definition 5.1 essentially requiresthe existence of such factorizations for all maps of chain complexes. However, itturns out that the d´evissage condition for (Ch b ( C ) , Ch b ( A )) is satisfied under theassumptions of Quillen’s theorem. Lemma 5.12.
Let
A ⊂ C be as in Theorem 5.11. Then for every C ∈ C andsubobject C ′ ⊆ C , there is a filtration C ′ = C ′ ⊆ C ′ ⊆ C ′ ⊆ · · · ⊆ C ′ n − ⊆ C ′ n = C such that C ′ i /C ′ i − ∈ A for all i = 1 , · · · , n . ´EVISSAGE 21 Proof.
There is a filtration 0 = C ⊆ C ⊆ C ⊆ · · · ⊆ C n − ⊆ C n = C such that C i /C i − ∈ A for i ≥
1. For i ≥
0, we consider the pushouts of subobjects of C : C ′ ∩ C i / / (cid:15) (cid:15) C i (cid:15) (cid:15) C ′ / / C ′ i . Thus, there is a filtration C ′ ⊆ C ′ ⊆ C ′ ⊆ · · · ⊆ C ′ n − ⊆ C ′ n = C and the cokernels C ′ i /C ′ i − are quotients of C i /C i − , hence they are again in A . (cid:3) Proposition 5.13.
Let
A ⊂ C be as in Theorem 5.11. Then (Ch b ( C ) , Ch b ( A )) satisfies the d´evissage condition.Proof. Consider an object in Ch b ( C ) C • = ( · · · → → C n → C n − → · · · → C → C → → · · · )and a subobject C ′• ֒ → C • . First, by Lemma 5.12, we may suppose that there is afiltration of C ′ n ⊆ C n : C ′ n = C ,n ⊆ C ,n ⊆ · · · ⊆ C m − ,n ⊆ C m,n ⊆ C n whose successive subquotients lie in A . Set X ,n − := C ′ n − . For i = 1 , · · · , m ,define X i,n − inductively by pushout squares C i − ,n / / (cid:15) (cid:15) C i,n (cid:15) (cid:15) X i − ,n − / / X i,n − . (Note that C ,n = C ′ n ∂ −→ C ′ n − = X ,n − .) We obtain in this way a factorizationof the inclusion C ′ n − ⊆ C n − as follows: C ′ n − = X ,n − ⊆ X ,n − ⊆ · · · ⊆ X m − ,n − ⊆ X m,n − → C n − which has the required property except possibly at the last stage. Let C i,n − denotethe image of X i,n − in C n − . Then there is a filtration of C ′ n − ⊆ C n − (10) C ′ n − = C ,n − ⊆ C ,n − ⊆ · · · ⊆ C m − ,n − ⊆ C m,n − ⊆ C n − whose successive subquotients, except possibly for the last one, are in A , since A is closed under taking quotients in C . By Lemma 5.12, there is a further filtration(11) C m,n − = C ,n − ⊆ C ,n − ⊆ · · · ⊆ C m ′ ,n − = C n − whose successive subquotients are in A . Joining these two filtrations (10) and (11),we obtain a combined filtration of C ′ n − ⊆ C n − with the required property thatits subquotients are in A . Together with the original filtration of C ′ n ⊆ C n , thisdefines a filtration for the inclusion( C ′ n ∂ → C ′ n − ) ֒ → ( C n ∂ → C n − ) . Repeating this process inductively on the length of the chain complex, we obtaina filtration for the inclusion C ′• ⊆ C • as required. This shows that the d´evissagecondition is satisfied for cofibrations between chain complexes. For an arbitrarychain map f : C ′• → C • , we choose a factorization C ′• i C ′′• ∼ → C • and apply theconstruction above to the cofibration i : C ′• ⊆ C ′′• . (cid:3) Remark 5.14.
Let Ch b ( C ) ac (resp. (Ch b ( A ) ac ) denote the full Waldhausen subca-teogry which is spanned by the acyclic chain complexes. Then the Waldhausen pair(Ch b ( C ) ac , Ch b ( A ) ac ) is again good. In addition, the factorizations constructed inProposition 5.13 apply also to the Waldhausen pair (Ch b ( C ) ac , Ch b ( A ) ac ). Hencethis Waldhausen pair satisfies the d´evissage condition too.Let A ⊂ C be as in Theorem 5.11. Applying Theorem 5.11 and the Gillet-Waldhausen homotopy equivalence (9), we conclude that the exact inclusion functorCh b ( A ) ֒ → Ch b ( C ) induces a homotopy equivalence: K (Ch b ( A )) ≃ −→ K (Ch b ( C )) . As a consequence, it follows from Proposition 5.13 and Remark 5.6 that the goodWaldhausen pair (Ch b ( C ) , Ch b ( A )) is admissible – so it satisfies the assumptions ofTheorem 5.5. It would clearly also be desirable to have a proof of the admissibiltyof (Ch b ( C ) , Ch b ( A )) which is independent of Theorem 5.11.6. Multiple Type D´evissage
Assumptions.
We fix the following notation and assumptions throughout thissection. Let C be a good Waldhausen category equipped with a cylinder functorwhich satisfies the cylinder axiom. Let A = ( A i ) i ≥ be a collection of Waldhausensubcategories such that for every i ≥ C , A i ) is a replete Waldhausen pair (Definition 3.5),(b) the Waldhausen subcategory A i is closed under extensions in C , i.e. givena cofiber sequence A X ։ B in C with A, B ∈ A i , then X ∈ A i ,(c) the suspension functor Σ : C → C sends A i to A i +1 .We note that the Waldhausen category b S ∞ C A is good by Lemma 3.6.6.1. Admissibility.
As in the case of d´evissage of single type, the purpose of anadmissibility assumption on ( C , A ) is to identify the K -theory of the Waldhausencategory S w ∞ C A . In the multiple type case, the relevant admissibility assumptionis much stronger, however, at the same time, it is significantly easier to verify inmany examples of interest. Definition 6.1.
We say that ( C , A ) is admissible if for every X • , • ∈ S w ∞ C A , wehave that X ,k ∈ A k for every k ≥ Example 6.2.
Let C be a Waldhausen category associated with a homology theoryin the sense of [9, 1.7], and let A i denote the Waldhausen subcategory of sphericalobjects of dimension i −
1. Then ( C , A = ( A i ) i ≥ ) is admissible by [9, Lemma1.7.4].For each n ≥
1, we have an exact functor (cf. Subsection 4.1): q wn : S wn C A → A × · · · × A n − , X • , • ( X , , · · · , X n − ,n − ) . Assuming that ( C , A ) is admissible, we also have a well-defined exact functor: p n : S wn C A → A × · · · × A n − , X • , • ( X , , · · · , X ,n − ) . ´EVISSAGE 23 Similarly to Subsection 4.1, there are also exact functors τ n : Q n − A i → S wn C A for n ≥
1. In detail, τ n is defined on objects by τ n : ( A , · · · , A n − ) ( ∗ A A ∨ CA · · · A n − ∨ n − _ CA i n − _ CA i ) . (Note that this functor is well defined because of assumption (c) on ( C , A ).) More-over, the following diagram of exact functors commutes:S wn C A i wn, A / / S wn +1 C A Q n − A iτ n O O / / Q n A iτ n +1 O O where the bottom functor is the canonical inclusion functor, given on objects by( A , · · · , A n − ) ( A , · · · , A n − , ∗ ). As a consequence, we also obtain an exactfunctor τ ∞ : colim −−−→ n Q n − A i −→ S w ∞ C A . The composite functor p n ◦ τ n is weakly equivalent to the identity functor. Thecomposite functor q wn ◦ τ n is given on objects by( A , · · · , A n − ) ( A , A ∨ Σ A , · · · , A n − ∨ Σ A n − )and it induces a homotopy equivalence in K -theory. More specifically, the map in K -theory is identified with the homotopy equivalence:( π , π + Σ ◦ π , · · · , π n − + Σ ◦ π n − ) : n − Y K ( A i ) → n − Y K ( A i )where π i denotes the projection onto the i -th factor, the sum corresponds to theloop sum, and Σ : K ( A i ) → K ( A i +1 ) denotes here, by a slight abuse of notation, themap that is induced by the suspension functor on C . Each one of the maps K ( p n ), K ( q wn ) and K ( τ n ) is a homotopy equivalence if any one of them is a homotopyequivalence. Proposition 6.3.
Suppose that ( C , A ) be admissible. Then the map K ( p n ) : K (S wn C A ) → K ( A ) × · · · × K ( A n − ) is a homotopy equivalence.Proof. The proof is essentially the same as the proof of [9, Lemma 1.7.3] and willbe omitted. (cid:3)
The d´evissage condition.
First we define an abstract notion of connectivityin C with respect to A . This definition is inspired by the Hypothesis in [9, 1.7, p.361]. A morphism f : X → Y in C is k–connected (with respect to A ) , k ≥ −
1, ifthere is a weak equivalence g : Y ∼ −→ Y ′ and a factorization of gf : X → Y ′ X = X X X · · · X m ∼ −→ Y ′ such that X i /X i − ∈ A k + i +1 for every i ≥ Definition 6.4.
We say that ( C , A ) has the cancellation property if the followingholds: for any k ≥ − X f −→ Y g −→ Z in C suchthat f and gf are k -connected, then g is also k -connected. Example 6.5.
The meaning of the cancellation property may be unclear directlyfrom the abstract definition of A -connectivity, but it can be easily verified in manyexamples of interest in which A -connectivity corresponds to a standard notion ofconnectivity. For example, the cancellation property is obviously satisfied in thecontext of Waldhausen’s theorem on spherical objects in [9, 1.7, pp. 360–361].As in the case of d´evissage of single type, the main condition on ( C , A ) that weare interested in is the following analogue of Definition 5.1. Definition 6.6.
We say that ( C , A ) satisfies the d´evissage condition if for everymorphism f : X → Y in C , there is a weak equivalence g : Y ∼ → Y ′ such that thecomposition gf : X → Y ′ admits a factorization X = X X · · · X m − X m ∼ → Y ′ where X i /X i − ∈ A i for every i ≥ Remark 6.7.
The d´evissage condition for ( C , A ) exactly says that every morphismin C is (–1)-connected. We emphasize that the factorizations in Definition 6.6 arenot required to be functorial.Similarly to the single type case, there are exact functors ev n : b S n C A → C for n ≥
1, given by ( X • , • ) X ,n , which are compatible with respect to stabilizationalong the inclusion functors b i n, A . Thus we obtain an induced exact functor ev ∞ : b S ∞ C A → C which sends a staircase diagram representing a bounded A -filtration of an object X ,n , for n large enough, to the object itself. As in the single type case, the mainfunction of the d´evissage condition will be to ensure that the functor ev ∞ is a K -equivalence. Proposition 6.8.
Suppose that ( C , A ) satisfies the d´evissage condition and hasthe cancellation property. Then the functor ev ∞ : b S ∞ C A → C induces a homotopyequivalence K ( b S ∞ C A ) ≃ → K ( C ) .Proof. The proof is similar to [9, Lemma 1.7.2]. We show that the exact func-tor ev ∞ : b S ∞ C A → C satisfies the assumptions of the Approximation Theorem(Theorem 2.6). (App1) holds by definition. For (App2), we consider a morphism ev ∞ ( X • , • ) = X ,n → Y where X • , • ∈ b S n C A and proceed by induction on n ≥ b S C A = S C = {∗} ). (App2) holds for n = 0 by the d´evissage conditionapplied to the morphism ∗ = X , → Y . Suppose by induction that (App2) holdsfor morphisms ev ∞ ( X • , • ) → Y where X • , • ∈ b S n − C A . Now consider a morphismas follows (we ignore the cofibers for simplicity):( ∗ X , · · · X ,n ) ∈ b S n C A , f : ev ∞ ( X • , • ) = X ,n → Y. By induction, there is an object X ′• , • ∈ b S n − C A and a morphism in b S k C A , k ≥ n − ∗ / / / / X , / / / / (cid:15) (cid:15) · · · / / / / X ,n − (cid:15) (cid:15) · · · (cid:15) (cid:15) X ,n − (cid:15) (cid:15) ∗ / / / / X ′ , / / / / · · · / / / / X ′ ,n − / / / / · · · / / / / X ′ ,k ´EVISSAGE 25 together with weak equivalences g : Y ∼ −→ Y ′ and f ′ : X ′ ,k ∼ −→ Y ′ such that thefollowing square in C commutes(12) X ,n − / / / / (cid:15) (cid:15) X ,n f / / Y ∼ g (cid:15) (cid:15) X ′ ,n − (cid:15) (cid:15) X ′ ,k ∼ f ′ / / Y ′ . Then, by definition, the composition X ′ ,ℓ → X ′ ,k ∼ −→ Y ′ , 1 ≤ ℓ ≤ k , is ( ℓ − X ′ ,n − → X ′ ,n − ∪ X ,n − X ,n is ( n − A n . It follows by the cancellationproperty that the morphism induced by (12): u : X ′ ,n − ∪ X ,n − X ,n → Y ′ is also ( n − g ′ : Y ′ ∼ −→ Y ′′ anda factorization of g ′ uX ′ ,n − ∪ X ,n − X ,n = Z Z · · · Z m ∼ → Y ′′ where Z i /Z i − ∈ A n + i − for every i ≥
1. Note that there is a cofiber sequence X n − ,n = X ,n /X ,n − Z /X ′ ,n − ։ Z / ( X ′ ,n − ∪ X ,n − X ,n ) . Since A n is closed under extensions in C , it follows that Z /X ′ ,n − ∈ A n . Thus,we obtain a morphism in b S ∞ C A represented by the diagram: ∗ / / / / X , / / / / (cid:15) (cid:15) · · · / / / / X ,n − (cid:15) (cid:15) / / / / X ,n (cid:15) (cid:15) · · · X ,n (cid:15) (cid:15) ∗ / / / / X ′ , / / / / · · · / / / / X ′ ,n − / / / / Z / / / / · · · / / / / Z m together with a commutative square in C : X ,n f / / (cid:15) (cid:15) Y ∼ g (cid:15) (cid:15) Y ′∼ g ′ (cid:15) (cid:15) Z m ∼ / / Y ′′ . This completes the inductive proof that ev ∞ satisfies (App2). Then the resultfollows as an application of Theorem 2.6. (cid:3) The main theorem.
We recall that the suspension functor Σ : C → C re-stricts to exact functors Σ : A i → A i +1 , i ≥
1, by assumption. Thus the collectionof maps K ( A i ) → K ( C ), i ≥
1, induce canonically a map:(13) hocolim −−−−−→ (Σ) K ( A i ) −→ hocolim −−−−−→ (Σ) K ( C ) ≃ K ( C )where the last homotopy equivalence holds because Σ induces a homotopy equiv-alence in K -theory. A d´evissage theorem of multiple type is a statement that forcertain ( C , A ) for which a d´evissage–type condition is satisfied, the map (13) is ahomotopy equivalence. The following result is an abstract version and a general-ization of Waldhausen’s theorem on spherical objects in [9, 1.7]. Theorem 6.9 (Multiple Type D´evissage) . Let ( C , A ) be admissible and supposethat it has the cancellation property and satisfies the d´evissage condition. Then thecanonical map hocolim −−−−−→ (Σ) K ( A i ) ≃ −→ hocolim −−−−−→ (Σ) K ( C ) ≃ K ( C ) is a homotopy equivalence.Proof. Consider the following diagram(14) K (S wn C A ) / / K (S n C A ) / / K ( b S n C A ) Q n − K ( A i ) / / K ( τ n ) ≃ O O Q n K ( A i ) π n / / K ( τ ′ n ) ≃ O O K ( A n ) O O The left vertical map was defined in Subsection 6.1. It is a homotopy equivalenceby Proposition 6.3.The middle map is induced by an exact functor τ ′ n : Q n A i → S n C A , a variationof the functor τ n , given on object by( A , A , · · · , A n ) ( ∗ A A ∨ CA · · · A n ∨ n − _ CA i ) . After we take the composition of τ ′ n with the functor q n : S n C A → Q n A i (seeProposition 3.2), we obtain the exact functor n Y A i → n Y A i , ( A , · · · , A n ) ( A , A ∨ Σ A , · · · , A n ∨ Σ A n − ) . This last functor induces a homotopy equivalence in K -theory. Since K ( q n ) is alsoa homotopy equivalence by Proposition 3.2, it follows that the middle vertical map K ( τ ′ n ) in (14) is also a homotopy equivalence. The left bottom map is induced bythe inclusion functor( A , · · · , A n − ) ( A , A , · · · , A n − , ∗ ) . Then the left square in (14) commutes by direct inspection.The bottom map π n : Q n K ( A i ) → K ( A n ) is the projection onto the last factor.The right vertical map is induced by the exact inclusion functor A n → b S n C A whichis given on objects by A ( ∗ = · · · = ∗ A ) . ´EVISSAGE 27 The right square in (14) is homotopy commutative, since the underlying exactfunctors are naturally weakly equivalent. Note that the bottom row in (14) definesa homotopy fiber sequence.Let σ : S n C A → S n +1 C A be the exact functor that is given on objects by( ∗ X , · · · X ,n ) ( ∗ = ∗ Σ X , · · · Σ X ,n )and let σ ′ : Q n A i → Q n +11 A i be the exact functor given by ( A , · · · , A n ) ( ∗ , Σ A , · · · , Σ A n ). The maps in Diagram 14 are compatible with the functors σ and σ ′ . Passing in (14) to the homotopy colimits as n → ∞ along these stabilizationfunctors, we obtain a homotopy commutative diagram:hocolim −−−−−→ ( σ ) K (S wn C A ) / / hocolim −−−−−→ ( σ ) K (S n C A ) / / hocolim −−−−−→ ( σ ) K ( b S n C A )hocolim −−−−−→ ( σ ′ ) Q n − K ( A i ) / / ≃ O O hocolim −−−−−→ ( σ ′ ) Q n K ( A i ) / / ≃ O O hocolim −−−−−→ (Σ) K ( A n ) . O O Note that the bottom row defines a homotopy fiber sequence. An application ofthe Additivity Theorem (Theorem 2.4) shows that the map K ( σ ) : K ( b S n C A ) → K ( b S n +1 C A ) agrees with the stabilization map K ( b i n, A ) up to sign – this uses thedefinition of the suspension functor in b S ∞ C A , see Lemma 3.6. We consider the mapfrom the top row in the diagram to the respective sequence of maps for n = ∞ :hocolim −−−−−→ (Σ) K (S w ∞ C A ) / / hocolim −−−−−→ (Σ) K (S ∞ C A ) / / hocolim −−−−−→ (Σ) K ( b S ∞ C A )hocolim −−−−−→ ( σ ) K (S wn C A ) / / O O hocolim −−−−−→ ( σ ) K (S n C A ) / / O O hocolim −−−−−→ ( σ ) K ( b S n C A ) ≃ O O where Σ here denotes the exact functor which defines the suspension functor on b S ∞ C A – this restricts to the functor σ defined above. The previous remarks implythat the right vertical map is a homotopy equivalence as indicated in the diagram.The top row is a homotopy fiber sequence (of infinite loop spaces) by Proposition3.7. In addition, it can be verified that the left square is a homotopy pullbackusing the identifications in Diagram 14 to determine its horizontal (co)fibers. As aconsequence, the bottom row of the last diagram is also a homotopy fiber sequence.Returning now to the previous diagram above, it follows that the right verticalmap, hocolim −−−−−→ (Σ) K ( A n ) → hocolim −−−−−→ ( σ ) K ( b S n C A ) , is a homotopy equivalence. Next we consider the commutative diagrams K ( A n ) Σ (cid:15) (cid:15) / / K ( b S n C A ) K ( σ ) (cid:15) (cid:15) K ( ev n ) / / K ( C ) Σ (cid:15) (cid:15) K ( A n +1 ) / / K ( b S n +1 C A ) K ( ev n +1 ) / / K ( C )and passing to the homotopy colimit as n → ∞ , we obtain the following maps:(15) hocolim −−−−−→ (Σ) K ( A n ) ≃ −→ hocolim −−−−−→ ( σ ) K ( b S n C A ) → hocolim −−−−−→ (Σ) K ( C ) ≃ K ( C ) The last map hocolim −−−−−→ ( σ ) K ( b S n C A ) → K ( C ) can be identified with K ( ev ∞ ) andtherefore it is a homotopy equivalence by Proposition 6.8. Then the compositemap (15) is a homotopy equivalence, as required. (cid:3) Example 6.10.
Let ( C , A ) be as in Example 6.2. Under the assumption of [9,Hypothesis, p. 361], ( C , A ) has the cancellation property and satisfies the d´evissagecondition. Thus we recover Waldhausen’s theorem on spherical objects [9, Thorem1.7.1] as a special case of Theorem 6.9. Example 6.11.
Some important examples of d´evissage-type theorems of multipletype involve collections of Waldhausen subcategories A = ( A i ) for i ∈ Z . These ex-amples include the Gillet-Waldhausen theorem [7] (for abelian categories, at least),the d´evissage theorem of Blumberg–Mandell [2], and the ‘Theorem of the Heart’due to Barwick [1]. In each of these cases, A i is the full Waldhausen subcategoryof objects which are (up to weak equivalence) concentrated in degree i (definedappropriately in each context), and the maps Σ : K ( A i ) ≃ −→ K ( A i +1 ) are homotopyequivalences. In order to relate Theorem 6.9 with these results, one may first con-sider applying Theorem 6.9 to the full Waldhausen subcategory C ≥ i which consistsof the objects which are concentrated in degrees ≥ i , in order to obtain a homo-topy equivalence K ( A i ) ≃ K ( C ≥ i ). The exact inclusion functors C ≥ i +1 → C ≥ i , i ∈ Z , will typically induce homotopy equivalences in K -theory by the AdditivityTheorem – the inclusion functor is related to an equivalence of homotopy theories C ≥ i +1 ≃ C ≥ i by a shift in C ≥ i . As a consequence, in these examples, we have that K ( C ≥ i ) ≃ K ( C ). See also [1, 2]. References [1] Clark Barwick,
On exact ∞ -categories and the Theorem of the Heart , Compos. Math. 151(2015), no. 11, 2160–2186.[2] Andrew J. Blumberg; Michael A. Mandell, The localization sequence for the algebraic K-theory of topological K-theory , Acta Math. 200 (2008), no. 2, 155–179.[3] Andrew J. Blumberg; Michael A. Mandell,
Algebraic K-theory and abstract homotopy theory ,Adv. Math. 226 (2011), no. 4, 3760–3812.[4] Denis-Charles Cisinski,
Invariance de la K-th´eorie par ´equivalences d´eriv´ees , J. K-Theory 6(2010), no. 3, 505–546.[5] Daniel Quillen,
Higher algebraic K-theory. I. , Algebraic K-theory, I: Higher K-theories (Proc.Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147, Lecture Notes in Math.341, Springer, Berlin 1973.[6] Marco Schlichting,
Negative K -theory of derived categories , Math. Z. 253 (2006), no. 1, 97–134.[7] R. W. Thomason; Thomas Trobaugh, Higher algebraic K-theory of schemes and of derivedcategories , The Grothendieck Festschrift, Vol. III, pp. 247–435, Progr. Math., 88, BirkhuserBoston, Boston, MA, 1990.[8] Friedhelm Waldhausen,
Algebraic K-theory of spaces, localization, and the chromatic filtra-tion of stable homotopy , Algebraic topology (Aarhus, 1982), pp. 173–195, Lecture Notes inMath. 1051, Springer, Berlin, 1984.[9] Friedhelm Waldhausen,
Algebraic K-theory of spaces , Algebraic and geometric topology (NewBrunswick, N.J., 1983), pp. 318–419, Lecture Notes in Math. 1126, Springer, Berlin, 1985.[10] Dongyuan Yao,
A devissage theorem in Waldhausen K-theory , J. Algebra 176 (1995), no. 3,755–761.
G. RaptisFakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany
E-mail address ::