aa r X i v : . [ m a t h . K T ] S e p THE K-THEORY OF LEFT POINTED DERIVATORS
IAN COLEY
Abstract.
We build on work of Muro-Raptis in [MR17] and Cisinski-Neeman in[CN08] to prove that the additivity of derivator K-theory holds for a large classof derivators that we call left pointed derivators , which includes all triangulatedderivators. The proof methodology is an adaptation of the combinatorial methodsof Grayson in [Gra11]. As a corollary, we prove that derivator K-theory is aninfinite loop space. Finally, we speculate on the role of derivator K-theory as atrace from the algebraic K-theory of a stable ∞ -category `a la [BGT13]. Contents
1. Introduction 12. Preliminaries 53. Left pointed derivators 114. Derivator K-theory 145. Additivity 176. Further properties 28References 441.
Introduction
Algebraic K-theory is a general tool for understanding complicated mathematicalobjects arising in homotopy theory, algebraic geometry, differential topology, repre-sentation theory, and other fields. Since Quillen’s seminal work [Qui73], the field ofalgebraic K-theory has enjoyed incredible popularity and expansion beyond abelianor exact categories. Waldhausen in [Wal85] set the tone for how K-theory would beconstructed for more and more general objects. A common philosophy is that, if weexpand the class of objects on which K-theory is defined, we should make sure thatour new definitions agree with the old. Waldhausen made sure this was the casewhen he defined his K-theory of categories with cofibrations and weak equivalences.A stumbling block, however, was including triangulated categories into Wald-hausen’s framework. Defined first by Verdier in his doctoral thesis [Ver96] in 1963, triangulated categories are invaluable in the study of homological algebra and ho-motopy theory. The bounded derived category associated to an exact or abeliancategory has a natural triangulated structure. The homotopy categories of bothordinary and G -equivariant spectra are similarly naturally triangulated. In alge-braic geometry, the theory of motives is studied using triangulated categories, as the‘abelian category of mixed motives’ remains a conjecture.Neeman in the 1990’s published a series of papers on the K-theory of triangulatedcategories starting with [Nee97a] and [Nee97b]. There are a number of interestingproperties of his construction, but we do not mention them here due to a fundamentaldefect in triangulated category theory. In the years following Neeman’s publications,various authors proved that a satisfactory functorial construction on triangulatedcategories would never be possible, in the following sense. Starting from an exactcategory, we can take its K-theory via Quillen’s or Waldhausen’s definition, or passto its bounded derived category and take its triangulated K-theory. If triangulatedK-theory were to extend Quillen’s K-theory, these two constructions would give thesame K-groups. In other words, if K Q is Quillen’s K-theory and S is the category ofspaces, can we find a functor K ∆ making the diagram below commute? ExCat D b ( − ) $ $ ■■■■■■■■■ K Q / / S TriCat K ∆ > > ⑥⑥⑥⑥⑥⑥⑥⑥ Schlichting in [Sch02] gives a general argument showing that we should not ex-pect the triangulated category to retain all K-theoretic information. Specifically, heconstructs two Waldhausen categories W and W , arising from abelian categoriesof modules over a commutative ring, such that their (triangulated) homotopy cate-gories are equivalent but their Waldhausen K differs. However, on the triangulatedside, each homotopy category Ho( W i ) appears as a Verdier localization of the samecategory with equivalent localising subcategories. This leads to a contradiction be-tween two desirable properties for K-theory: agreement and localization. The abovediagram can only commute if K ∆ does not satisfy localization.Schlichting’s result points to the need for a richer structure for homotopy theorythan triangulated categories alone. This is not a new idea; from our first homo-logical algebra class, we learn that the cone construction in a triangulated categoryis non-functorial. The slogan ‘unique up to unique isomorphism’, central to howwe approach category theory, abandons us. There are a few different ways to giveourselves more data to work with.In some senses, the best replacement for triangulated categories, especially throughthe lens of algebraic K-theory, are stable ∞ -categories , i.e. the triangulated analogueof higher categories. Recent work of Blumberg-Gepner-Tabuada in [BGT13] proves HE K-THEORY OF LEFT POINTED DERIVATORS 3 that algebraic K-theory is the universal additive invariant of a stable (small) ∞ -category. This extends the origins of K-theory precisely; the Grothendieck group ofa commutative monoid is the universal abelian group such that any additive invarianton the monoid must factor through it. However, there are reasons to mistrust ∞ -categories: the literature is daunting and there are competing (though equivalent)models. Though the theory of ∞ -categories is ideal for universal constructions, itis often difficult to know concretely what has been constructed. In the example ofalgebraic K-theory, there is an ∞ -category of ‘non-commutative motives’ in whichalgebraic K-theory is found – but the rest of the category is quite mysterious (fornow – there is much active work on this topic).There are lower-categorical tools that work well and do not have these drawbacks.An early tool in studying triangulated categories, developed by Quillen in [Qui67]before his work on algebraic K-theory, is that of model categories . A model category isthe data of a category we wish to treat homotopically and extra information allowingus to pass from the rigid structure to the homotopy category. A solution to the non-functoriality of the cone can be solved in such a framework. For nice enough modelcategories M , the category of arrows Ar M inherits a compatible model categorystructure. We can define the cone of a morphism before passing to the homotopycategories, i.e. Ho(Ar M ) → Ho( M ) rather than Ar Ho( M ) → Ho( M ). If we knewonly the category Ho( M ), this first approach will not be possible, so in this sensewe have given ourselves more to work with.Let us interpret this in the triangulated setting. Let A be an abelian category.Then the arrow category Ar A is still abelian, so we can take its bounded derivedcategory D b (Ar A ), which we can think of as homotopy classes of maps of chaincomplexes. Then we can define the cone construction as an exact functor of abeliancategories C b (Ar A ) → C b ( A ) before we invert quasi-isomorphisms. We still have afunctor upon passing to the derived category, and so have a functorial cone construc-tion with a new domain. However, there is a forgetful functor D b (Ar A ) → Ar D b ( A )which takes a homotopy class of a map to a map of homotopy classes. While D b (Ar A )is triangulated, Ar D b ( A ) is not, but this functor can be shown to be full and es-sentially surjective (and almost never faithful). We have constructed a cone functor because we had access to A itself and not just the triangulated category D b ( A ) andthus were able to build an auxiliary diagram category Ar A to fill in the gaps ininformation.This is our slogan: we would like to study not only a triangulated category, buta whole system of triangulated diagram categories. An equivalence of homotopycategories as in [Sch02] does not necessarily give rise to an equivalence of systems,and thus we are able to better distinguish distinct homotopy theories. Grothendieckin [Gro90] coined the term derivator for a system of derived categories, and this is IAN COLEY the framework in which we will work to address questions about the K-theory oftriangulated categories.The theory of derivators was developed initially (under different names) by Hellerin [Hel88], Grothendieck in [Gro90], and (in the triangulated setting) Franke in[Fra96]. In brief, a derivator represents an abstract bicomplete homotopy theory;we attach the adjective triangulated to a derivator when it represents a stable (bi-complete) homotopy theory. The fundamental proof techniques used in the theoryof derivators and the diagrammatic flavor which is unique to this field have beenwell-articulated by Moritz Rahn (n´e Groth) in [Gro13].The K-theory of triangulated derivators was defined by Maltsiniotis in [Mal07]and Garkusha in [Gar06] and [Gar05], and revisited by Muro and Raptis in [MR17].Muro-Raptis proved that the definition of K-theory still makes sense for derivatorswhich are not triangulated, and form a class which we call left pointed derivators .We develop in [Col20] a more robust theory of such half derivators , i.e. ones repre-senting homotopy theories that may not be bicomplete, but still admit many limitsor colimits, in order to answer questions about K-theory in the broadest generality.This is one advantage over the approach of [BGT13]: we are not restricted to stablephenomena.Cisinki and Neeman proved that the K-theory of triangulated derivators satisfiesa form of additivity in [CN08], but their proof involves Neeman’s theory of regionsand does not admit an obvious analogy in the non-triangulated situation. We provethe following broader theorem.
Main Result. (Theorems 5.5 and 5.20)Let D and E be left pointed derivators. Then the following are equivalent and true:(1) The map D cof (0 , ∗ × (1 , ∗ / / D × D induces a homotopy equivalence on derivator K-theory, where D cof is the leftpointed derivator of cofiber sequences in D .(2) If Ξ : D → E cof is a cofibration morphism of derivators, then there exists ahomotopy between the target of the cofibration morphism and the source plusquotient. Specifically, K ( T ) ≃ K ( S ) ⊔ K ( Q ) ( ∼ = K ( S ⊔ Q ))where ⊔ denotes the coproduct.The first statement is the form of additivity proven by Cisinski-Neeman and con-jectured by Maltsiniotis. The second statement is the one proven in Theorem 5.20and uses techniques of Grayson in [Gra11] that have a more diagrammatic flavorappropriate for general derivator theory. We obtain as a corollary the delooping of HE K-THEORY OF LEFT POINTED DERIVATORS 5 the K-theory space K ( D ), and so conclude that K ( D ) is an infinite loop space fora general left pointed derivator D , which was not known in any cases before. Thisanswers two questions of Muro-Raptis posed in [MR17].We first recall the necessary results from [Col20] that establish the domain ofderivator K-theory. We then give the construction of derivator K-theory and presentpreviously-known results. We conclude with the main new additivity theorem ofderivator K-theory and the important consequences thereof.2. Preliminaries
Recall that a prederivator is just a strict 2-functor D : Cat op → CAT , wherethe domain is the 2-category of small categories and the codomain the ‘2-category’of not-necessarily-small categories. For a morphism u : J → K in Cat we denoteby u ∗ the functor D ( u ) : D ( K ) → D ( J ) in CAT , and for α : u ⇒ v in Cat we de-note by α ∗ the natural transformation D ( α ) : u ∗ ⇒ v ∗ . Composition is respectedstrictly, so that ( vu ) ∗ = u ∗ v ∗ and ( α ⊙ β ) ∗ = α ∗ ⊙ β ∗ (here ⊙ is the pastingof natural transformations). Identities are also preserved, so that (id J ) ∗ = id D ( J ) and (id u ) ∗ = id u ∗ .A derivator is a prederivator that models a system of diagram categories which ishomotopically bicomplete. We give the definition in two parts.Let D be a prederivator, K a small category, and k ∈ K be any object. Recallthat we have a functor that classifies the object k which we denote k : e → K ,where e is the category with one object and one (identity) morphism. Then forany X ∈ D ( K ), we have an object k ∗ X ∈ D ( e ). Suppose that f : k → k is a mapin K . Then we have a corresponding natural transformation f ∗ : k ∗ ⇒ k ∗ and thusa map f ∗ X : k ∗ X → k ∗ X in D ( e ). Repeating this process for all objects and mapsin K , we obtain a functor dia K : D ( K ) → Fun( K, D ( e ))which sends X ∈ D ( K ) to the functor which assembles all the above data. We callthis an underlying diagram functor , and its existence implies that the prederivator D should be modelling K -shaped diagrams in D ( e ), which we call the underlyingcategory or the base of the prederivator. We will refer to the categories D ( K ) as coherent diagrams, as opposed to the incoherent diagrams Fun( K, D ( e )). Definition 2.1. A semiderivator is a prederivator D satisfying the following twoaxioms:(Der1) Coproducts are sent to products. Explicitly, consider any set { K a } a ∈ A ofsmall categories, and let i b : K b → a a ∈ A K a be the inclusion for any b ∈ A . IAN COLEY
Pulling back along this inclusion gives a functor i ∗ b : D a a ∈ A K a ! → D ( K b )which induces a map to the product Y b ∈ A i ∗ b : D a a ∈ A K a ! → Y b ∈ A D ( K b )We require this map to be an equivalence of categories for anycollection { K a } a ∈ A .(Der2) Isomorphisms are detected pointwise. That is, for any K ∈ Cat , the underly-ing diagram functor dia K is conservative. More specifically, a map f : X → Y is an isomorphism in D ( K ) if and only if the map k ∗ f : k ∗ X → k ∗ Y is anisomorphism for all k ∈ K .These two axioms comprise the ‘system of diagram categories’ part of the defini-tion. For the next two axioms, we need the following notation. Definition 2.2.
Let u : J → K be any functor, and let k ∈ K be any object.We define the comma category ( u/k ) as follows: its objects are pairs j ∈ J witha map f : u ( j ) → k , and a map ( j, f ) → ( j ′ , f ′ ) in the comma category is amap g : j → j ′ in J making the obvious diagram commute:(2.3) u ( j ) u ( g ) / / f (cid:28) (cid:28) ✽✽✽✽✽✽ u ( j ′ ) f ′ (cid:2) (cid:2) ✆✆✆✆✆✆✆ k For any category K ∈ Cat , we write π K for the unique functor K → e . Definition 2.4.
A semiderivator D is a left derivator if it satisfies the following twoaxioms:(Der3L) The base of the semiderivator D ( e ) is (homotopically) cocomplete. Specif-ically, for every functor u : J → K , the pullback u ∗ admits a left adjoint,which we denote u ! : D ( J ) → D ( K ) and call the (homotopy) left Kan exten-sion along u . As a special case, this includes π K : K → e and thus D ( e )admits all (coherent) colimits. HE K-THEORY OF LEFT POINTED DERIVATORS 7 (Der4L) Left Kan extensions can be computed pointwise. Let u : J → K and k ∈ K .Then we have the following lax pullback square in Cat :( u/k ) pr / / π (cid:15) (cid:15) J u (cid:15) (cid:15) ⇒ α e k / / K where we let π = π ( u/k ) for brevity. Applying the semiderivator D to thissquare, we obtain the following square in CAT , remembering that functorsare reversed and natural transformations are not: D (( u/k )) D ( J ) ⇒ α ∗ pr ∗ o o D ( e ) π ∗ O O D ( K ) k ∗ o o u ∗ O O By Der3L, both vertical functors admit left adjoints, so we may constructthe left mate of α ∗ as the pasting of the below diagram, which we denoteby α ! (rather than the ‘official’ notation ( α ∗ ) ! ): D ( e ) D (( u/k )) π ! o o ⇒ D ( J ) ⇒ α ∗ pr ∗ o o ⇒ D ( e ) = R R π ∗ O O D ( K ) k ∗ o o u ∗ O O D ( J ) u ! o o = n n In total we have the natural transformation α ! : π ! pr ∗ ⇒ k ∗ u ! . We requirethis map to be a natural isomorphism.A semiderivator D is a right derivator if it satisfies the analogous axioms Der3R andDer4R, which together say that every functor u ∗ admits a right adjoint u ∗ satisfyinga pointwise computation formula. A derivator is just a left and right derivator.There is a relative construction that we need to introduce at this point. Supposethat D is a prederivator and I ∈ Cat is a category. Then we can define anotherprederivator D I by D I ( K ) = D ( I × K ); for u : J → K , D I ( u ) = D (id I × u ); andsimilar for natural transformations. If D is a (left/right/full) derivator, so is D I .This is often called a shifted derivator .There is a ‘fifth axiom’ for derivators that is not needed in all contexts, but isneeded for ours. IAN COLEY
Definition 2.5.
A prederivator is strong if for any finite free category I and for anycategory K ∈ Cat , the partial underlying diagram functordia
K,I : D ( K × I ) → Fun( K, D ( I ))is full and essentially surjective. In some derivator literature this axiom is calledDer5.The functor dia K,I is related to the underlying diagram functor defined above,except in this case we leave the I -dimension of all coherent diagrams intact. Thecontent of this axiom is that any incoherent diagram of a simple shape is liftable to acoherent one; further, any map of these incoherent diagrams lifts to a map betweenthe coherent ones. This axiom is asking for the same sort of thing as lifting a mapbetween objects in some homotopy category to a map of bifibrant replacements. Remark 2.6.
The strongness axiom usually only asks for the case I = [1] (andsometimes all finite ordinals [ n ], see Notation 3.1), but all known examples eithersatisfy this ‘strong strongness’ version of the axiom or fail for I = [1]. Derivatorsfailing the case I = [1] are constructed in [LN17], so any version of strongness isa non-extraneous axiom, but all derivators arising from some sort of model satisfyDer5 as above (see Lemma 3.8 below).We have one more adjective to attach to our derivators. Definition 2.7.
A derivator D is pointed if its underlying category D ( e ) is pointed,i.e. the unique morphism from the initial to the final object is an isomorphism. Wewill write 0 ∈ D ( e ) for its zero object.This definition is easy to check but does not tell the whole story. One immediateconsequence is that each category D ( J ) has a zero object, given by 0 J := π ∗ J (0)where π J : J → e is the projection. The more interesting corollary requires somedefinitions first: Definition 2.8.
Let u : J → K be a fully faithful functor that is injective on objects.(1) The functor u is a sieve if for any morphism k → u ( j ) in K , k lies in theimage of u .(2) The functor u is a cosieve if for any morphism u ( j ) → k in K , k lies in theimage of u . Proposition 2.9 (Proposition 1.23, [Gro13]) . Let D be a pointed derivator, andlet u : J → K be a sieve (resp. cosieve). Then u ∗ : D ( J ) → D ( K ) (resp. u ! ) is fullyfaithful, with essential image X ∈ D ( K ) such that k ∗ X ∼ = 0 for all k ∈ K \ u ( J ).These adjoints are called extension by zero morphisms and are essential in theproofs of derivator K-theory below. HE K-THEORY OF LEFT POINTED DERIVATORS 9
Proposition 2.10 (Corollaries 3.5 and 3.8, [Gro13]) . A derivator is pointed if andonly if extension by zero morphisms admit exceptional adjoints. Specifically, everyright extension by zero u ∗ along a sieve admits a right adjoint u ! and every leftextension by zero u ! along a cosieve admits a left adjoint u ? .The existence of exceptional adjoints is crucial for functoriality properties onderivator K-theory, which we now address. Definition 2.11.
Let D , E : Cat op → CAT be prederivators. A morphism of pred-erivators
Φ : D → E is a pseudonatural transformation of the associated 2-functors.This consists of the following data: for each K ∈ Cat we have afunctor Φ K : D ( K ) → E ( K ) and for every u : J → K we have a natural isomor-phism γ Φ u : u ∗ Φ K ⇒ Φ J u ∗ D ( K ) Φ K / / u ∗ (cid:15) (cid:15) E ( K ) u ∗ (cid:15) (cid:15) ⇒ γ Φ u D ( J ) Φ J / / E ( J )where we have slightly abused notation by writing u ∗ for both D ( u ) and E ( u ). Theseare subject to certain coherence conditions which we leave to [Col20], [Gro13], or[Bor94, § K ( X ) for X ∈ D ( K )for all K ∈ Cat , we will usually write Φ X for X ∈ D . Our constructions will be notheavily dependent on specific K .A morphism of derivators is just a morphism of the underlying prederivators. Wesay that a morphism of (pre)derivators is an equivalence if each functor Φ K is anequivalence of categories. There is a subclass of morphisms that need singling out. Definition 2.12.
A morphism of (pre)derivators Φ : D → E is called strict if forevery u : J → K , the structure isomorphism γ Φ u : u ∗ Φ K ⇒ Φ J u ∗ is the identity.In 2-categorical language, a strict morphism Φ is a strict natural transformationof 2-functors, not just pseudo natural. A morphism being strict seems fairly unlikely,as it implies a great deal of rigidity in what is a fairly flexible homotopical con-text. Nonetheless, the model of derivator K-theory we use in this paper will requirestrict morphisms, and we will be able to obtain strict morphisms (up to equivalence)whenever we need.The main class of morphisms of derivators that we study involve shifted derivators.Suppose that u : J → K is a functor and D is a prederivator. Then we obtain amorphism of prederivators u ∗ : D K → D J which is actually strict, as the coherencedata γ u ∗ arise from the strict 2-functoriality of D : Cat op → CAT . Moreover, if D is a left or right derivator, we obtain morphisms u ! , u ∗ : D J → D K , but these are not strict. This is related to the fact that (co)limits are essentially unique, which allowsfor the construction of the structure isomorphisms, but not actually unique. Thesemorphisms enjoy other properties which we will describe now. Definition 2.13.
Let D , E be left derivators and u : J → K in Cat . We say that amorphism Φ : D → E preserves left Kan extensions along u if the left mate of ( γ Φ u ) − is a natural isomorphism. Specifically, we have the pasting D ( J ) = . . u ! / / D ( K ) ⇒ Φ K / / u ∗ (cid:15) (cid:15) E ( K ) u ∗ (cid:15) (cid:15) ⇒ ( γ Φ u ) − = (cid:17) (cid:17) ⇒ D ( J ) Φ J / / E ( J ) u ! / / E ( K )giving us a natural transformation ( γ Φ u ) − : u ! Φ J ⇒ Φ K u ! which we demand is anisomorphism, where again we slightly abuse notation by writing u ! for the left adjointto both D ( u ) and E ( u ). If the morphism Φ preserves left Kan extensions alongall u : J → K in Cat , we say that Φ is cocontinuous .There is an analogous notion of continuous morphism that we will not spell out(as we will not need it below).Cocontinuous morphisms of derivators can appear in the same way as colimit-preserving functors in category theory: via adjunctions.
Definition 2.14.
Given two morphisms of (pre)derivators Φ , Ψ : D → E , a naturaltransformation ρ : Φ ⇒ Ψ is given by a modification of pseudonatural transforma-tions. This is the data of a natural transformation ρ K : Φ K ⇒ Ψ K for every K ∈ Cat satisfying coherence conditions that we do not record here.
Definition 2.15.
Let Φ : D → E and Ψ : E → D be two morphisms of (pre)derivators.We say that Φ is left adjoint to Ψ (equivalently, Ψ is right adjoint to
Φ) if there ex-ist two modifications η : id D ⇒ ΨΦ and ε : ΦΨ ⇒ id E satisfying the usual triangleidentities.In particular, an adjunction of morphisms of derivators (Φ , Ψ) gives rise to anadjunction of functors (Φ K , Ψ K ) for each K ∈ Cat . However, this condition isnot sufficient. A morphism of derivators Φ : D → E may admit a right adjointto Φ K : D ( K ) → E ( K ) for all K ∈ Cat , but part of the data of a right adjointmorphism of derivators is the structure isomorphisms, which we have no way ofrecovering in this general situation.
Lemma 2.16 (Proposition 2.9, [Gro13]) . Let Φ : D → E be a morphism of leftderivators such that each Φ K admits a right adjoint Ψ K . Then the collection of HE K-THEORY OF LEFT POINTED DERIVATORS 11 functors { Ψ K } assembles to a morphism of derivators Ψ : E → D which is rightadjoint to Φ if and only if Φ is cocontinuous.The morphism Φ being cocontinuous allows us to construct the coherence isomor-phisms γ Ψ u and ‘glue together’ the various Ψ K . We are not claiming anything likean adjoint functor theorem for general derivators, so this lemma does not admit aconverse.There are two classes of examples that give us everything we need for this paper.Let u : J → K be a functor in Cat . If u admits a categorical rightadjoint v : K → J , then u ∗ : D ( K ) → D ( J ) is right adjoint to v ∗ : D ( J ) → D ( K )because (strict) 2-functors send adjunctions to adjunctions, though in our case whichis left and which is right swaps. We can upgrade this to, for any prederivator D , a(cocontinuous) left adjoint morphism v ∗ : D K → D J which preserves any left Kanextensions that D K happens to have.If D is a left derivator, then the left adjoint functor u ! : D ( J ) → D ( K ) lifts to a leftadjoint morphism of derivators u ! : D J → D K with right adjoint u ∗ . Similarly, if D isa right derivator, u ∗ : D J → D K is a right adjoint morphism of derivators with leftadjoint u ∗ . In fact, for any prederivator D , the morphism u ∗ : D K → D J preservesall left and right Kan extensions that D happens to have by [Gro13, Proposition 2.5]for categorical reasons.Finally, suppose D is a pointed derivator and u : J → K is a sieve. Then theright extension by zero u ∗ : D ( J ) → D ( K ) admits an exceptional right adjoint byProposition 2.10, so u ∗ is a left adjoint and hence cocontinuous.3. Left pointed derivators
Having set up the basic vocabulary of the theory of derivators, we can begin toexamine what we actually need for K-theory.To motivate the following definition, we recall the definition of K of an abeliancategory A . It is constructed as the free abelian group on (isomorphism classes of)objects A ∈ A , written [ A ] ∈ K ( A ), under the relation that if 0 → A → B → C → B ] = [ A ] + [ C ]. A short exact sequence isequivalently a cocartesian square A / / (cid:15) (cid:15) B (cid:15) (cid:15) / / C under the assumption that A → B is a monomorphism. Thus if we are to constructeven K for a derivator, it needs to admit a notion of (coherent) cocartesian squaresand a zero object. Notation 3.1.
For n ∈ N , let [ n ] denote the totally ordered set with n + 1 elements:0 → → · · · → n − → n Each of these are finite free categories.
Notation 3.2.
Let (cid:3) be the category [1] × [1], with labelling(0 , / / (cid:15) (cid:15) (1 , (cid:15) (cid:15) (0 , / / (1 , i p : p → (cid:3) be the full subcategory of (cid:3) lacking the element (1 , Definition 3.3.
Let D be a left derivator and X ∈ D ( (cid:3) ). We say that X is co-cartesian (i.e. a pushout square) if X is in the essential image of i p , ! : D ( p ) → D ( (cid:3) ).Otherwise put, X is cocartesian if the counit i p , ! i ∗ p X → X of the ( i p , ! , i ∗ p ) adjunctionis an isomorphism.This is where Muro and Raptis obtained their domain for derivator K-theory:they considered left derivators which admit a zero object. However, would like tobe able to construct pushouts appropriate for computing K as above. This meanscoherently making cocartesian squares starting from an element in D ( p ) of the form(3.4) a / / (cid:15) (cid:15) b p -shaped diagram starting from a coherentarrow ( a → b ) ∈ D ([1]), we need more than the structure of a left derivator. Definition 3.5.
A prederivator D : Dia op → CAT is a left pointed derivator if it isa strong left derivator, D ( e ) is pointed, and for every sieve u : J → K , u ∗ admits aright adjoint u ∗ satisfying Der4R.Indeed, the inclusion i [1] : [1] → p is a sieve, so by Proposition 2.9 we can computethat(3.6) ( a f → b ) i [1] , ∗ −→ a f / / (cid:15) (cid:15) b i p , ! −→ a f / / (cid:15) (cid:15) b (cid:15) (cid:15) / / C ( f )where we have named the object at the (1 ,
1) position the cone of the (coherent)morphism f ∈ D ([1]). This means that there is a morphism of derivators D [1] → D (cid:3) realising the above diagram. HE K-THEORY OF LEFT POINTED DERIVATORS 13
Remark 3.7.
Given that we can lift incoherent diagrams in the shape of finite freecategories, it should be pointed out that p is such a shape. Therefore since wecan build diagrams of shape Diagram 3.4 incoherently, we can lift them to coherentobjects of D ( p ). From that point we can take the coherent pushout via i p , ! . Wecannot lift ‘incoherent pushout squares’ because (cid:3) is not finite free.However, this process spoils any hope of functoriality in the construction of thecoherent pushout of a morphism starting from D ([1]), and this functoriality is essen-tial. The requirement that our left pointed derivators be strong is used only to checkthe the computation at Diagram 5.17; it requires using ‘incoherent reasoning’ thatmust be lifted up to the derivator and does not interfere with any functoriality.It may be possible that the computation can be made without strongness, butthis author does not have a proof. It may also be that this computation requires strongness, and a proof is also lacking for this possibility. This small point does nottake away from the main result of the paper, so we leave it for future consideration.The key example of a left pointed derivator, and indeed the motivation of theabstract defintion, is the following, drawn from Corollary 2.24, Proposition 3.4, andLemma 4.3 in [Cis10]. Lemma 3.8.
Let W be a saturated Waldhausen category satisfying the cylinderaxiom. Then the associated prederivator D W : K Ho(Fun( K, W )) defined on Dir f is a (strong) left pointed derivator. Moreover, an exact functor of Waldhausencategories induces a cocontinuous morphism of the corresponding derivators. Inparticular, these morphisms preserve cocartesian squares and the zero object.Recall that derivators need not be defined on all of Cat , but on sub-2-categories
Dia ⊂ Cat satisfying some closure properties. One key example is
Dir f , whichconsists of all finite direct categories , i.e. categories whose nerve has only finitely manynondegenerate simplices. These are also called homotopy finite categories by [Arl20]and [GPS14]. General Waldhausen categories will not admit arbitrary colimits andwill admit no (non-empty) limits whatsoever.For the purposes of K-theory, Dir f is an ideal domain for our left pointed deriva-tors. Homotopical cocompleteness for all of Cat means the existence of infinitecoproducts. This allows for a derivator version of the usual Eilenberg swindle onK-theory, e.g. [Wei13, V.1.9.1]. Since this trick requires additivity, we will prove itbelow as a corollary of the main theorem at Proposition 6.1.Hereafter we let
Der K be the 1-category with objects strong left pointed deriva-tors on Dir f and morphisms cocontinuous morphisms of derivators up to invertiblemodification. That is, we consider Φ , Ψ : D → E to be the same if there exists azig-zag of invertible modifications from Φ to Ψ. We do this because such morphismswill induce homotopic maps on K-theory, as we will show in Corollary 4.2 shortly.We will leave the adjective ‘strong’ implicit throughout. Derivator K-theory
It is helpful at this point to recall Waldhausen’s K-theory for a category withcofibrations and weak equivalences from [Wal85]. To such a category W we assign asimplicial object in Waldhausen categories S • W , where S n W is the category of exactfunctors from the arrow category Ar[ n ] to W . Taking the wide subcategory withonly maps the weak equivalences w S • W , we obtain again a simplicial Waldhausencategory. Then we define K-theory as follows: K ( W ) := Ω | N • w S • W| , the loop space of the (diagonal) geometric realisation of the bisimplicial set given bythe nerve.In [MR17], Muro and Raptis improved upon a construction of Garkusha in [Gar05]which generalises Waldhausen’s S • construction. First, we can restate the S • con-struction in the language of derivators. To help with notation, for a category [ n ] ∈ ∆,let the elements of its arrow category Ar[ n ] be written ( i, j ) for i → j .Let D be a left pointed derivator. We let S n D be the full subcategory of D (Ar[ n ])of objects X such that:(1) For every 0 ≤ i ≤ n , ( i, i ) ∗ X ∈ D ( e ) is a zero object.(2) For every fully faithful inclusion ι : (cid:3) → Ar[ n ], the object ι ∗ X ∈ D ( (cid:3) ) iscocartesian.In (2), it suffices to check only the inclusions such that ι (0 ,
1) = ( i, i ) by [Gro13,Proposition 3.13]. We then define derivator K-theory by K ( D ) = Ω | N • i S • D | where i S n D ⊂ S n D is the wide subcategory consisting only of isomorphisms, inanalogy with w S n C , and the geometric realization is taking diagonally.To give a few examples, first we have that S D ⊂ D (Ar[0]) = D ( e ) is trivial: ithas only one object and property (1) above requires it to be a zero object of D ( e ).The category S D ⊂ D (Ar[1]) is slightly more interesting: it is a staircase with onenontrivial object: 0 / / a (cid:15) (cid:15) (cid:3) → Ar[1], so there is nothing else to require.We can see that S D = D ( e ) as a category, an observation we will use later. The HE K-THEORY OF LEFT POINTED DERIVATORS 15 first interesting category is S D ⊂ D (Ar[2]), whose objects have the form0 / / a f / / (cid:15) (cid:15) b g (cid:15) (cid:15) / / c (cid:15) (cid:15) K ( D )being encoded: the zero simplices of K ( D ) = Ω | N • i S • D | come from S D (becauseΩ gives a dimension shift) and these zero simplices are identified in π K ( D ) due tothe existence of a path, i.e. an element of S D , relating them. Thus three objectsappearing a cocartesian square in D ( (cid:3) ) leads to a relation on the homotopy classesof zero simplices in π K ( D ). The same thing happens at each π n K ( D ), but therelationship is more difficult to describe for large values of n .We said above that in order for a morphism Φ : D → E to induce a map on K-theory, it needs to preserve cocartesian squares and the zero object, which we couldthen conclude was equivalent to asking for Φ to be cocontinuous. However, there isanother problem. If we have a cocontinuous morphism Φ : D → E that is not strict ,then Φ may not induce a map of simplicial sets S • D → S • E . If we take (for example)the face map d i : [ n ] → [ n + 1], then we obtain a diagram S n +1 D d ∗ i (cid:15) (cid:15) Φ Ar[ n +1] / / S n +1 E d ∗ i (cid:15) (cid:15) ⇒ γ Φ di S n D Φ Ar[ n ] / / S n E But this diagram commutes only up to natural isomorphism, so Φ does not give us anhonest natural transformation of the bisimplicial sets N • i S • D → N • i S • E . However,we have the following proposition to aid us. Proposition 4.1 (Proposition 10.14, [CN08]) . Let Φ : D → E be a morphism ofprederivators. Then there exists a prederivator e D , a strict equivalence of derivatorsΠ Φ : e D → D , and a strict morphism e Φ : e D → E such that the following diagramcommutes: e D Π Φ ∼ { { ✈✈✈✈✈✈ e Φ ❍❍❍❍❍❍ D Φ / / E We name the equivalence Π Φ because it is some sort of projection, though we willnot need the precise formula. If Φ is cocontinuous, e Φ is also cocontinuous because it is the composition of cocontinuous morphisms. The following corollary is foundat [CN08, Corollary 10.19] or [MR17, Remark 5.1.4].
Corollary 4.2.
Any cocontinuous morphism of left pointed derivators Φ : D → E gives rise to a map on derivator K-theory K (Φ) : K ( D ) → K ( E ) in S , the homotopycategory of spaces. Moreover, this association is functorial, in the sense that wehave a 1-functor Der K → S after inverting isomodifications in Der K to obtain a1-category.An immediate consequence is that equivalent left pointed derivators have equiva-lent K-theories.There are some first results that are worth collecting. Maltsiniotis in [Mal07]proved that, if D is a triangulated derivator, then K ( D ( e )) of the underlying tri-angulated category is equivalent to K ( D ). He also established a comparison mapfrom Quillen’s K-theory of an exact category to derivator K-theory, which was sub-sequently extended to Waldhausen categories by Garkusha in [Gar06].Specifically, let W be a saturated Waldhausen category satisfying the cylinderaxiom as in Lemma 3.8 so that it gives rise to a left pointed derivator D W (hereafterwe will leave these assumptions on W implicit). Then we have an obvious mapFun(Ar[ n ] , W ) → D W (Ar[ n ]) = Ho(Fun(Ar[ n ] , W ))sending a diagram to its homotopy class. This map restricts to S n W → S n D W , sendsweak equivalences to isomorphisms, and behaves well with respect to the simplicialstructures, so we obtain a map of spaces µ : K ( W ) → K ( D W )Maltsiniotis’ proof is easily rewritten to imply that µ := π µ is an isomorphism,and Muro in [Mur08] proved that µ is an isomorphism as well. Muro’s techniquesare a bit ad hoc, but recent work of Raptis [Rap19, Theorem 5.5] proves that thecomparison map µ is 2-connected, recovering Muro’s result on µ and proving that µ is surjective. Maltsiniotis conjectured that µ should be a weak homotopy equivalencein the case that D W is triangulated, and the same question can be asked in general.Unfortunately, the conjecture fails almost totally. Muro and Raptis togetherin [MR11] show that µ will generally not be an equivalence for triangulated deriva-tors arising from stable module categories. In that same work, the authors use theexample of Schlichting in [Sch02] to prove that any K-theory of derivators invari-ant under equivalences of derivators cannot satisfy both agreement and send Verdierlocalizations of triangulated derivators to homotopy fibrations in K-theory. Such alocalization theorem was also conjectured by Maltsiniotis. Raptis conjectures that µ should not be more than 2-connected in great generality. HE K-THEORY OF LEFT POINTED DERIVATORS 17
One positive result is a theorem of Cisinski and Neeman [CN08] that derivatorK-theory of triangulated derivators satisfies an additivity theorem, to be made moreexplicit shortly. Muro and Raptis in their second paper on derivator K-theory [MR17]asked whether this additivity proof could be adapted to the more general contextof left pointed derivators. The positive answer to this question occupies the nextsection. 5.
Additivity
Rather than approach the problem as Cisinski and Neeman did in [CN08] usingNeeman’s method of regions, we will prove additivity in a novel way. Throughout, let D be a left pointed derivator defined on Dia = Dir f . We adapt this first definitionfrom [CN08, Definition 11.7]. Definition 5.1.
Let D be a left pointed derivator. We define the corresponding cofiber sequence category for each K ∈ Dir f by D cof ( K ) ⊂ D K ( (cid:3) ) the full subcategoryof cocartesian squares X such that (0 , ∗ X = 0 ∈ D ( K ). Lemma 5.2.
The cofiber sequence categories assemble to a prederivator D cof . More-over, there is an equivalence D [1] → D cof which is pseudonatural with respect tococontinuous morphisms of derivators, which makes D cof a left pointed derivator aswell. Proof.
For any u : J → K in Dir f , u ∗ : D (cid:3) ( K ) → D (cid:3) ( J ) is cocontinuous by [Gro13,Proposition 2.5]. Therefore u ∗ preserves cocartesian squares and the zero object,so restricts to u ∗ : D cof ( K ) → D cof ( J ). There are no modifications needed for thenatural transformations, so this makes D cof a prederivator.The equivalence between D [1] and D cof is given by the composition of Diagram 3.6: D [1] i [1] , ∗ / / D p i p , ! / / D cof ⊂ D (cid:3) By definition the image of this composite consists of cocartesian squares with thezero object in the (0 ,
1) position. Since i [1] and i p are fully faithful, their left andright Kan extensions are fully faithful by [Gro13, Proposition 1.20] (which still holdsfor left pointed derivators), hence the above composite induces an equivalence ontoits image, which is precisely D cof ⊂ D (cid:3) .For the pseudonaturality, consider a morphism of derivators Φ : D → E . Thenfor Φ (cid:3) : D (cid:3) → E (cid:3) to restrict to Φ : D cof → E cof , it would have to send cocartesiansquares to cocartesian squares, and it would have to send the zero object of D to thezero object of E . As we have assumed Φ is cocontinuous, this property holds and weare done. (cid:3) Remark 5.3.
The pseudonaturality with respect to cocontinuous morphisms is themost important takeaway of the preceding lemma. In the below constructions, wewill construct morphisms Φ : D cof → ( D cof ) K for various categories K ∈ Dir f , butoften we will have to define these morphisms first as Φ : D → D K . We may thenextend Φ to a morphism D cof → ( D cof ) K if Φ is a cocontinuous morphism. Definition 5.4.
Let D , E be left pointed derivators. We define a cofibration mor-phism of derivators to be a strict cocontinuous morphism Ξ : D → E cof . To Ξ weassociate three strict cocontinuous morphisms D → E S := (0 , ∗ Ξ T := (1 , ∗ Ξ Q := (1 , ∗ Ξand two strict cocontinuous morphisms α, β : D → E [1] given by restricting to thetop and right arrows of the coherent square, respectively. Incoherently, we have a ∈ D S ( a ) α a / / (cid:15) (cid:15) T ( a ) β a (cid:15) (cid:15) / / Q ( a ) ∈ E cof This is a coherent version of a cofibration sequence of exact morphisms of Wald-hausen categories in [Wal85, p. 331]. We prove a similar theorem to [Wal85, Propo-sition 1.3.2].
Theorem 5.5.
Let D and E be left pointed derivators. The following are equivalent:(1) The map D cof (0 , ∗ × (1 , ∗ / / D × D induces a homotopy equivalence on derivator K-theory.(2) If Ξ : D → E cof is a cofibration morphism of derivators, then there exists ahomotopy K ( T ) ≃ K ( S ) ⊔ K ( Q ) ( ∼ = K ( S ⊔ Q ))The first statement is the statement of additivity `a la Garkusha, Maltsiniotis,and Cisinski-Neeman, first found in [Mal07, Conjecture 3] and similar to [Wal85,Proposition 1.3.2(2)]. The latter is a reinterpretation of [Wal85, Proposition 1.3.2(4)].Our proof follows the strategy set out by Waldhausen.To expand a little on (1), it will be helpful to use that S • D × S • D ∼ = S • D e ⊔ e . First,we know that D (Ar[ n ]) × D (Ar[ n ]) ∼ = D (Ar[ n ] ⊔ Ar[ n ]) ∼ = D e ⊔ e (Ar[ n ])Second, for a diagram X ∈ D e ⊔ e (Ar[ n ]), the condition of being in S n D e ⊔ e coincideswith each projection to D (Ar[ n ]) being in S n D . This makes it easier for us to definemaps into S • D × S • D ; they can arise from (strict) cocontinuous morphisms into D e ⊔ e . HE K-THEORY OF LEFT POINTED DERIVATORS 19
In particular, in the spirit of Remark 5.3, morphisms arising from left adjointfunctors are cocontinuous, so for any functor u : J → K , u ∗ , u ! induce maps on K-theory. The extension by zero morphisms u ∗ for any sieve u are also cocontinuous,as they admit an exceptional right adjoint. Put another way, Der K contains all leftand right Kan extension morphisms available to left pointed derivators. Proof. (2) = ⇒ (1)The map ρ := (0 , ∗ × (1 , ∗ admits a section σ : D e ⊔ e → D cof on K-theory.Incoherently for ( a, c ) ∈ D e ⊔ e , the functor σ is roughly (but not precisely)( a, c ) a / / (cid:15) (cid:15) a ⊔ c (cid:15) (cid:15) / / c We will write σ as a composite of morphisms of derivators coming from diagramfunctors, but we will need some diagram notation first.Recall from Notation 3.2 the functor i p : p → (cid:3) . Further, let i : e ⊔ e → p be theinclusion into (1 ,
0) and (0 , × [2], with labelling(0 , / / (cid:15) (cid:15) (1 , (cid:15) (cid:15) (0 , / / (cid:15) (cid:15) (1 , (cid:15) (cid:15) (0 , / / (1 , J be the full subcategory of [1] × [2] without the element (1 , i (cid:3) : (cid:3) → J and j : J → [1] × [2] be the obvious inclusions, and let r : (cid:3) → [1] × [2]be the inclusion of the bottom square. Note that i is a cosieve, and i p , i (cid:3) , and j aresieves.At the level of the derivators, the section σ is given by(5.6) D e ⊔ e i ! / / D p i p , ! / / D (cid:3) i (cid:3) , ∗ / / D J j ! / / D [1] × [2] r ∗ / / D (cid:3) ⊃ D cof All of these maps are cocontinuous, though not all are strict. As we mentioned inCorollary 4.2, this means that σ will give rise to a well-defined map in S .For ( a, c ) ∈ D e ⊔ e , σ ( a, c ) is explicitly( a, c ) / / (cid:15) (cid:15) ca / / (cid:15) (cid:15) c (cid:15) (cid:15) a / / a ⊔ c / / (cid:15) (cid:15) c (cid:15) (cid:15) a / / (cid:15) (cid:15) a ⊔ c / / (cid:15) (cid:15) c (cid:15) (cid:15) a / / (cid:15) (cid:15) a ⊔ c (cid:15) (cid:15) / / c ′ a / / (cid:15) (cid:15) a ⊔ c (cid:15) (cid:15) / / c ′ By construction, the image of this composite lands in D cof ⊂ D (cid:3) , as the square r ∗ j ! X is easily shown to be cocartesian using Proposition 6.3 for any X ∈ D J , and(1 , ∗ r ∗ j ! X = 0 as long as X is in the image of i (cid:3) , ∗ (as is our case). Note furtherthat the composite map c → a ⊔ c → c ′ is an isomorphism, as it is the pushout ofthe isomorphism 0 → σ : D e ⊔ e → D cof . Hence we obtainan isomodification id D e ⊔ e ⇒ ρσ , as the canonical isomorphism c → c ′ gives rise toan isomorphism ( a, c ) → ρσ ( a, c ) = ( a, c ′ ) natural in ( a, c ) ∈ D e ⊔ e . On K-theory(after strictifying the non-strict morphisms and passing to S ), this gives a homotopy K ( ρσ ) ≃ K (id D e ⊔ e ). Therefore it suffices to construct a homotopy in the reversedirection, i.e. K ( σρ ) ≃ K (id D cof ).To that end, we use our assumption. We will construct a cofibration morphism ofderivators such that S ⊔ Q ∼ = σρ and T ∼ = id D cof . Our morphism Ξ : D cof → ( D cof ) cof will have the form(5.7) a f / / (cid:15) (cid:15) b g (cid:15) (cid:15) / / c a = / / (cid:15) (cid:15) a (cid:15) (cid:15) a (cid:15) (cid:15) f / / b g (cid:15) (cid:15) / / / / c / / (cid:15) (cid:15) (cid:15) (cid:15) / / (cid:15) (cid:15) c = (cid:15) (cid:15) / / / / c id a ,f / / (cid:15) (cid:15) g, id c (cid:15) (cid:15) / / where all squares commute.We can accomplish this as the pullback along a single functor in Dir f . In thediagram above of shape (cid:3) × (cid:3) , let the first (cid:3) denote the outer square coordinatesand the second the inner coordinates. For example, the entry c in the top right is at(1 , , ,
1) We now define ξ : (cid:3) × (cid:3) → (cid:3) by ξ ( a , b , a , b ) = (0 ,
0) ( a , b , a , b ) = (0 , , , , (0 , , , , (1 , , , ,
0) ( a , b , a , b ) = (1 , , , ,
1) ( a , b , a , b ) = (1 , , , , (1 , , , , (1 , , , ,
1) otherwiseIn plain language, we make this definition so that ξ behaves on objects as sketchedin Diagram 5.7, and it is also the appropriate functor for the maps. For example,consider the map (1 , , , → (1 , , ,
0) in (cid:3) × (cid:3) . Diagram 5.7 says that we want(1 , , , ∗ ξ ∗ X → (1 , , , ∗ ξ ∗ X = a f −→ b HE K-THEORY OF LEFT POINTED DERIVATORS 21
We see that ξ (1 , , ,
0) = (0 ,
0) and ξ (1 , , ,
0) = (1 , ξ (1 , , , ∗ ⇒ ( ξ (1 , , , ∗ induces the map f : a → b when applied to X ∈ D cof . Checking that we also have g and identities where required can be donesimilarly.We define a strict cocontinuous morphism of derivators Ξ := ξ ∗ : D (cid:3) → D (cid:3) × (cid:3) . Byconstruction, assuming we restrict our domain to D cof (as illustrated), the image willbe a global cocartesian square in D cof ( (cid:3) ) with zero in the bottom-left corner, so weobtain the required (strict cocontinuous) morphism Ξ : D cof → ( D cof ) cof .Our assumption gives us that K ( T ) ≃ K ( S ⊔ Q ), and clearly T = id D cof . It is alsoevident that we have an isomodification σρ ⇒ S ⊔ Q using again the naturality ofthe comparison isomorphism c ′ → c . This shows that K (id D cof ) ≃ K ( σρ ) as required,which proves additivity in the historical sense for derivator K-theory.(1) = ⇒ (2)First, consider the two maps (1 , ∗ : E cof → E and ρ : E cof → E e ⊔ e → E , where ρ is defined to be the composite cocontinuous morphism (1 , ∗ i p , ! i ! ρ which computesthe coproduct of (0 , ∗ X and (1 , ∗ X for any X ∈ E cof (see Equation 5.6).We claim that these maps are homotopic. If we precompose with themap σ : E e ⊔ e → E cof , it is immediate that (1 , ∗ σ and ρσ are (canonically) isomor-phic, as they both compute the coproduct of ( a, c ) ∈ E e ⊔ e . Thus if σ is a homotopyequivalence on K-theory, (1 , ∗ and ρ are still homotopic. But by our assumption(1), σ is a section of the homotopy equivalence ρ , so it too is a homotopy equivalence.Statement (2) then follows immediately by precomposing these two homotopicmaps by any cofibration morphism Ξ : D → E cof , which yields K ( ρ Ξ) ∼ = K ( S ⊔ Q ) ≃ K ( T ) = K ((1 , ∗ Ξ) (cid:3) This new reformulation of the additivity theorem produces a proof that differsgreatly from [CN08] and [Gar05], the latter of which includes a gap which seemsirreparable, see [MR17, § Y be a simplicial set. Without any loss of generality, we mayextend Y : ∆ op → Sets to Y : Ord op → Sets , where
Ord is the category of(nonempty) finite totally ordered sets with order-preserving maps. For A ∈ Ord , welet Y ( A ) := Y ([ n ]), where [ n ] is the unique element of ∆ ⊂ Ord isomorphic to A .We do this in order to introduce a binary operation ∗ on Ord , which otherwise would cause us problems. We let A ∗ B be concatenation, that is, A ∗ B := ( { } × A ) ∪ ( { } × B ) ⊂ [1] × ( A ∪ B )with lexicographical ordering. This results in each element of A being smaller thaneach element of B , but within A and B the ordering does not change. Definition 5.8.
Let Y be a simplicial set (on Ord ). The two-fold edge-wise subdi-vision sub Y of Y is the simplicial set defined by sub Y ( A ) := Y ( A ∗ A ).There is a natural homeomorphism | sub Y | → | Y | defined in [Gra89, §
4] whoseconstruction we do not recall here. The important thing to note is that we do notchange the homotopy type (or even homeomorphism type) of our simplicial set bysubdividing.Now we can begin to bring derivators back into the conversation. Let Φ , Ψ : D → E be two strict cocontinuous morphisms of derivators. These induce morphisms ofsimplicial categories S • Φ , S • Ψ : S • D → S • E . We define a new map of simplicialcategories ∇ Φ , Ψ : sub S • D → S • E in the following way.The totally ordered set A ∗ A has two full subcategories i , i : A → A ∗ A , givenby a (0 , a ) , (1 , a ) respectively. These extend to functors on the arrow categoriesAr( A ) → Ar( A ∗ A ), and give i ∗ , i ∗ : D (Ar( A ∗ A )) → D (Ar( A )). Since restrictionmorphisms are strict and cocontinuous, i ∗ and i ∗ define morphisms of simplicialcategories S • D ( A ∗ A ) → S • D ( A ), where we adopt the notation S • D ( A ) = S n D forthe unique [ n ] such that A ∼ = [ n ].Consider an object X ∈ sub S • D ( A ) = S • D ( A ∗ A ). Then let ∇ Φ , Ψ ( X ) := Φ( i ∗ ( X )) ⊔ Ψ( i ∗ ( X ))This indeed lands in S • E ( A ) as the coproduct of any two cocartesian squares is againcocartesian. We have, in essence, doubled S • D and applied Φ to the first half and Ψto the second, then taken the coproduct of the results.Unfortunately, the described map does not exist on the level of simplicial cate-gories. While the functors Φ i ∗ and Ψ i ∗ are still strict, taking the coproduct is nota strict operation. Therefore we will need to strictify this last map, as in Proposi-tion 4.1, so we do not honestly get a map of simplicial sets with codomain S • E . Buton K-theory the map we want does exist, given by the zig-zag(5.9)Ω | sub N • i S • D | K (Φ i ∗ ) × K (Ψ i ∗ ) / / K ( E ) × K ( E ) ∼ = K ( E e ⊔ e ) K (cid:16) g E e ⊔ e (cid:17) ∼ o o e ⊔ / / K ( E )where g E e ⊔ e is the prederivator constructed in Proposition 4.1 to strictify the coprod-uct map ⊔ = (1 , ∗ i p , ! i ! : E e ⊔ e → E .Now, we need to construct a cylinder object for S • D that will allow us to use ∇ Φ , Ψ as a replacement for Φ ⊔ Ψ. We need three definitions to get us there.
HE K-THEORY OF LEFT POINTED DERIVATORS 23
Definition 5.10 (Definition 1.2, [Gra11]) . For
A, B ∈ Ord , let A ⋉ B be the set A × B with lexicographic ordering. That is, ( a, b ) ≤ ( a ′ , b ′ ) if and only if a < a ′ or a = a ′ and b ≤ b ′ . Note that the map A ⋉ B → A is order-preserving, hence amorphism in Ord , but the other ‘projection’ A ⋉ B → B is generally not. Definition 5.11 (Definition 1.3, [Gra11]) . Given two maps ϕ : A → C and s : B → C in Ord , define ϕ − ( s ) ∈ Ord to be the subset of A ⋉ B given by { ( a, b ) : ϕ ( a ) = s ( b ) } . Definition 5.12 (Definition 1.4, [Gra11]) . Let s : [2] → [1] be the morphism definedby s (0) = 0 and s (1) = s (2) = 1. For any simplicial set Y , define a new simplicialset IY by IY ( A ) := { ( ϕ, y ) : ϕ : A → [1] , y ∈ Y ( ϕ − ( s )) } . The definition of IY onmorphisms in Ord extends by naturality.
Remark 5.13.
To see how IY is a useful object, notice that ϕ − ( s ) = ϕ − (0) ∗ ϕ − (1) ∗ ϕ − (1)Therefore the choice ϕ = 0 gives ϕ − ( s ) = A , and the choice ϕ = 1gives ϕ − ( s ) = A ∗ A . The simplicial subset of IY at ϕ = 0 is isomorphic to Y ,and the simplicial subset of IY at ϕ = 1 is isomorphic to sub Y . Any other mor-phism ϕ : A → [1] gives a totally ordered set ϕ − ( s ) interpolating between these twoendpoints. Lemma 5.14 (Lemma 1.6, [Gra11]) . There is a homeomorphism | IY | → | ∆ | × | Y | .We do not include the proof because nothing is changed in the context of derivators.This shows that IY is indeed a cylinder object for Y , so we may prove the mainproposition. Proposition 5.15.
Let Ξ : D → E cof be a cofibration morphism of derivators, andlet S, T, Q : D → E be the corresponding morphisms of derivators. Then there is amap of simplicial categories Θ : I S • D → S • E such that Θ agrees with T on the simpli-cial subcategory where ϕ = 0 and Θ agrees with ∇ Q,S on the simplicial subcategorywhere ϕ = 1. Remark 5.16.
Just as ∇ Q,S is not an honest morphism of simplicial categories, Θwill not be well-defined per se but will induce a map on K-theory via a zigzag comingfrom strictification. We will continue to abuse notation in this fashion.
Proof.
We will construct Θ in two steps.First, we define a morphism P : D [1] → E . For a coherent morphism( f : a → b ) ∈ D [1] , we may apply Ξ [1] to obtain an object in E [1]cof ⊂ E [1] × (cid:3) . Specifically, its underlying diagram takes the form (where we do not label every arrow) S ( a ) S ( f ) (cid:15) (cid:15) ❆❆❆❆❆ α a / / T ( a ) β a $ $ ❍❍❍❍❍ (cid:15) (cid:15) / / (cid:15) (cid:15) Q ( a ) Q ( f ) (cid:15) (cid:15) S ( b ) ❆❆❆❆❆ / / T ( b ) $ $ ❍❍❍❍❍ / / Q ( b )We may consider the functor p → [1] × (cid:3) by the inclusion into the upper-leftcorner of the back face of the cube given above. Restriction along this functor givesthe coherent diagram in E p (5.17) S ( a ) α a / / S ( f ) (cid:15) (cid:15) T ( a ) S ( b )We may then apply (1 , ∗ i p , ! to first take the pushout of Diagram 5.17 and thenrestrict to the new object. The composition is a cocontinuous morphism which wedenote P : D [1] → E .We point out two special cases. If f ∈ D [1] is a coherent isomorphism,then P ( f ) ∼ = T ( a ), as pushing out along an isomorphism is still an isomorphism.Second, if f is a zero map, then there is a natural isomorphism P ( f ) ∼ = Q ( a ) ⊔ S ( b ).This arises from a factorization of f as follows S ( a ) / / (cid:15) (cid:15) S ( f ) % % T ( a )0 (cid:15) (cid:15) S ( b )If we take pushouts one square at a time, we first obtain an object isomorphicto Q ( a ) and second compute the pushout of Q ( a ) with S ( b ): S ( a ) / / (cid:15) (cid:15) T ( a )0 (cid:15) (cid:15) S ( b ) S ( a ) / / (cid:15) (cid:15) T ( a ) (cid:15) (cid:15) (cid:15) (cid:15) / / Q ( a ) S ( b ) S ( a ) / / (cid:15) (cid:15) T ( a ) (cid:15) (cid:15) (cid:15) (cid:15) / / Q ( a ) (cid:15) (cid:15) S ( b ) / / S ( b ) ⊔ Q ( a ) HE K-THEORY OF LEFT POINTED DERIVATORS 25
But the composition of two pushouts is again a pushout, which impliesthat S ( b ) ⊔ Q ( a ) is the pushout of the total upper-right corner, which is exactlyDiagram 5.17. Thus we conclude P ( f ) ∼ = S ( b ) ⊔ Q ( a ). Remark 5.18.
The phrase ‘coherent isomorphism’ is justified in any (not strong)left derivator D : an object f ∈ D ([1]) has underlying diagram an isomorphism ifand only if the cone C ( f ) ∼ = 0 ∈ D ( e ). As another option, f ∈ D ([1]) is a coher-ent isomorphism if and only if the counit of the (0 ! , ∗ ) adjunction is an isomor-phism by [Gro13, Proposition 3.12]. This is actually true in any prederivator, as thefunctor 0 ! : D ( e ) → D ([1]) is canonically isomorphic to π ∗ [1] , where π [1] : [1] → e is theprojection.Unfortunately, we do not have a way to make sense of a ‘coherent zero map’,i.e. an object in D ([1]) whose underlying diagram factors through the 0 in D ( e ).Just as there is no morphism D ([1]) ⊔ D ( e ) D ([1]) → D ([2]) that would ‘coherentlycompose’ two coherent maps with the appropriate domain and codomain, there is nomorphism D ([1]) → D ([2]) which ‘inserts’ the zero object.Instead, what we do is take the underlying diagram of Diagram 5.17, and noticethat the vertical map factors through zero. We may then lift the ‘tall p ’ to acoherent object as this category is still finite free. The pushouts maybe taken onestep at a time, similar to the construction at Equation 5.6, to give us a coherentobject of D ([1] × [2]), whose restriction to the outer square gives us the originalpushout of Diagram 5.17.The step of passing to the incoherent diagram, inserting zero, and lifting back toa slightly-larger coherent diagram is not functorial, but it does allow us to identify(up to isomorphism) the pushout of our original diagram. It would be helpful toknow if there is an appropriate notion of ‘coherent zero map’, in which case we couldeliminate the assumption of strongness from our left pointed derivators. It would beequally interesting to know if the notion of ‘coherent zero map’ does not exist fornon-strong derivators, thereby validating all the axioms on the domain of derivatorK-theory. We leave this for future work.Now, we construct the second step of Θ. Recall the definition of s : [2] → [1].Now we define two sections of s : first, d : [1] → [2] with d (0) = 0, d (1) = 1; second, e : [1] → [2] with e (0) = 0, e (1) = 2. For any a ∈ A , ( dϕ ( a ) , a ) and ( eϕ ( a ) , a ) are bothelements of ϕ − ( s ) as se = sd = id [1] . This gives two inclusions from A into ϕ − ( s ),to which we also name d and e (as the other functors will not be returning). Thereis also a unique natural transformation ζ : d ⇒ e as dϕ ( a ) ≤ eϕ ( a ) for any a ∈ A .We may first extend d, e : Ar( A ) → Ar( ϕ − ( s )) by naturality, and we still have thenatural transformation ζ : d ⇒ e . We can better view ζ as afunctor ζ : Ar( A ) × [1] → Ar( ϕ − ( s )), where (in particular) the original data of the natural transformation is retained by the maps in the [1]-dimension ζ (cid:16) (( a → b ) , → (( a → b ) , (cid:17) = d ( a → b ) ζ ( a → b ) −−−→ e ( a → b ) . Therefore we obtain a strict cocontinuous morphism ζ ∗ : D (Ar( ϕ − ( s ))) → D (Ar( A ) × [1])As we vary ϕ : A → [1], we obtain a map out of I S • D ( A ), and by cocontinuity wecan restrict the codomain to S • D [1] ( A ). This tells us how to extend the various ζ ∗ to a map Z : I S • D → S • D [1] (read: capital ζ ).We now need to check that Z is a map of simplicial categories. Suppose we have amap ψ : B → A in Ord . Then we need to check that the following square commutes:(5.19) I S • D ( A ) ψ ∗ / / Z A (cid:15) (cid:15) I S • D ( B ) Z B (cid:15) (cid:15) S • D [1] ( A ) ψ ∗ / / S • D [1] ( B )where we use ψ ∗ , ψ ∗ to denote the maps induced by ψ on the two different simplicialcategories I S • D and S • D [1] .Let ( ϕ : A → [1] , X ∈ S • D ( ϕ − ( s ))) ∈ I S • D ( A ). Then taking the upper composi-tion of Diagram 5.19 we first get the object ψ ∗ ( ϕ, X ) = ( ϕψ, ( ψ ′ ) ∗ X ∈ S • D (( ϕψ ) − ( s ))where ψ ′ is the natural map ( ϕψ ) − ( s ) → ϕ − ( s ) induced by ψ , ( n, b ) ( n, ψ ( b )).If we apply Z B to ( ϕψ, ( ψ ′ ) ∗ X ), we obtain the coherent object in S • D [1] ( B ) d ∗ B ( ψ ′ ) ∗ X ζ ∗ B / / e ∗ B ( ψ ′ ) ∗ X where d B and e B are the specific instances of d, e in this case.If we traverse Diagram 5.19 using the lower composition, we first take Z A ( ϕ, X )to obtain ζ ∗ A : d ∗ A X → e ∗ A X . Then applying ψ ∗ we obtain ψ ∗ d ∗ A X ψ ∗ ζ ∗ A / / ψ ∗ e ∗ A X. But by strict 2-functorality, these two coherent maps are equal on the nose, as theyare just pullbacks induced by maps in
Dir f . Therefore Z is indeed a morphism ofsimplicial categories.This finishes the definition of Θ = ( S • P ) Z : I S • D → S • D [1] → S • E . We now needto check that Θ in fact interpolates between T and ∇ Q,S . Suppose we take some A ∈ Ord , and let ( ϕ, X ) ∈ I S • D ( A ). HE K-THEORY OF LEFT POINTED DERIVATORS 27 If ϕ = 0 is the zero map, then ϕ − ( s ) = A by Remark 5.13. In this case, ζ : d ⇒ e is the identity natural transformation, as dϕ ( a ) = eϕ ( a ) always. So for an element(0 , X ∈ S • D ( A )) ∈ I S • D ( A ), we see that Z A (0 , X ) is nothing else but the constantmap in the [1]-direction. Thus Z A (0 , X ) = id X . This is an isomorphism, one of thespecial cases we noted following Diagram 5.17, and we conclude that S • P ( Z A (0 , X ))is naturally isomorphic to S • T ( X ).If ϕ = 1, then ϕ − ( s ) = A ∗ A by the same remark. We now consider an objectof the form (1 , X ∈ S • D ( A ∗ A )) ∈ I S • D ( A ). The resulting Z A (1 , X ) is an elementof S • D [1] ( A ), which is a subcategory of D (Ar( A ) × [1]). Incoherently, if we write X = ( x → y ), as X ∈ D (Ar( A ∗ A )), then Z A (1 , X ) looks like d ∗ x ζ ∗ x / / (cid:15) (cid:15) e ∗ x (cid:15) (cid:15) d ∗ y ζ ∗ y / / e ∗ y where the vertical maps are the elements of Ar( A ∗ A ). But since ϕ = 1, we knowthat e ∗ x ≥ d ∗ y . Therefore the map above factors as d ∗ x / / (cid:15) (cid:15) d ∗ y / / (cid:15) (cid:15) e ∗ x (cid:15) (cid:15) d ∗ y / / d ∗ y / / e ∗ y Therefore our overall (coherent) map Z A (1 , X ) factors through the object( d ∗ y → d ∗ y ) ∈ S • D ( A ). But this is a zero object by assumption, as it corresponds toan object of the form ( i, i ) ∈ Ar([ n ]) (where [ n ] ∼ = A ∗ A ), so Z A (1 , X ) is a zero map.This was our second special case, and we conclude that S • P ( Z A (1 , X )) is naturallyisomorphic to S • ∇ Q,S ( X ). This completes the proof. (cid:3) Proving the additivity of derivator K-theory is now immediate.
Theorem 5.20.
Let Ξ : D → E cof be a cofibration morphism of derivators, and let S, T, Q be the corresponding morphisms of derivators. Then S ⊔ Q and T inducehomotopic maps K ( D ) → K ( E ). Proof.
From Lemma 5.14 and Proposition 5.15, the morphism of simplicial cate-gories Θ above gives a homotopy between S • T and ∇ S • Q, S • S on | S • D | → | S • E | .We now need to prove that there exists a cofibration morphism of derivators whoseassociated functors are S, S ⊔ Q, Q .From the proof of Proposition 5.5, recall that we constructed afunctor σ : E e ⊔ e → E cof whose image is precisely what we desire. Let us precom-pose σ by the morphism ( S ⊔ Q ) : D → E e ⊔ e . The target map of σ ( S ⊔ Q ) : D → E cof is isomorphic to S ⊔ Q , and so ∇ S • Q, S • S and S • S ⊔ S • Q induce homotopic maps onK-theory as well. Combining these homotopies, we see that T and S ⊔ Q inducehomotopic maps K ( D ) → K ( E ). (cid:3) In particular, by Proposition 5.5, this proves additivity as it was conjectured byMaltsiniotis in [Mal07]. 6.
Further properties
As additivity sits as the central property of any flavor of algebraic K-theory, wecan now see what else we have obtained. First, as promised, we can perform theEilenberg swindle for left pointed derivators with ‘large’ domains.
Proposition 6.1.
Suppose that D is a left pointed derivator defined on Dia ⊂ Cat such that the countable discrete set ω is in Dia . Then K ( D ) ∼ = 0. Proof.
If we look at the functor π ω : ω → e , we see that the left Kan extension π ω, ! isthe colimit of this discrete set, i.e. the coproduct. From an object X ∈ D ( e ), we canobtain the countable coproduct of copies of X via the functor π ω, ! π ∗ ω : D ( e ) → D ( e ),and we can promote this to a cocontinuous morphism of derivators ` : D → D .We now want to construct a cofibration morphism of derivators with this con-struction in mind. Let s : ω → ω denote the successor function, i.e. s ( n ) = n + 1We build another category Γ s as the diagram underlying the function s . That is,the objects of Γ s are two disjoint copies of ω , which we label 0 and 1, with anarrow n → s ( n ) = ( n + 1) . To sketch the initial segment of Γ s :0 (cid:31) (cid:31) ❅❅❅❅❅❅❅ (cid:31) (cid:31) ❅❅❅❅❅❅❅ (cid:31) (cid:31) ❅❅❅❅❅❅❅ ❆❆❆❆❆❆❆❆ · · · · · · This category admits a functor p : Γ s → [1] which sends n i to i ∈ [1]. We claim thatthe cofibration morphism of derivators Ξ : D → D cof induced by D π ∗ Γ s / / D Γ s p ! / / D [1] i [1] , ∗ / / D p i p , ! / / D cof provides us with a null homotopy of the identity on D , giving us our swindle.We need to identify the source, target, and quotient morphisms associated to thiscofibration morphism. Let X ∈ D . To begin, (0 , ∗ Ξ is isomorphic to 0 ∗ p ! π ∗ Γ s , asthe last two functors i [1] , ∗ and i p , ! are fully faithful and do not change the under-lying diagram of [1] ⊂ (cid:3) . Using Der4L, we can compute 0 ∗ p ! π ∗ Γ s X via the commacategory ( p/ ∗ p ! π ∗ Γ s X ∼ = π ( p/ , ! pr ∗ π ∗ Γ s X HE K-THEORY OF LEFT POINTED DERIVATORS 29 where pr : ( p/ → Γ s is the usual projection to the comma category as in Defini-tion 2.4. The composition pr ∗ π ∗ Γ s is the pullback along ( p/ → Γ s → e , which is thesame thing as the pullback π ∗ ( p/ directly. To complete this computation, we needto identify the comma category.The objects of the comma category are n i ∈ Γ s along with a map p ( n i ) = i → p ( n i ) = 0 as there are no other maps in [1]. Therefore ( p/
0) consistssolely of the subcategory ω ⊂ Γ s . Therefore we conclude(0 , ∗ Ξ X ∼ = 0 ∗ p ! π ∗ Γ s X ∼ = π ( p/ , ! pr ∗ π ∗ Γ s X ∼ = π ω , ! π ∗ ω X = ` X The computation for (1 , ∗ Ξ X is similar: it is isomorphic to 1 ∗ p ! π ∗ Γ s X . We havean isomorphism by Der4L1 ∗ p ! π ∗ Γ s X ∼ = π ( p/ , ! pr ∗ π ∗ Γ s X ∼ = π ( p/ , ! π ∗ ( p/ X We now need to identify ( p/
1) as a category. As 1 ∈ [1] is the terminal object, thiscomma category is equal to Γ s itself. In order to compute the colimit of shape Γ s ,we note that it admits a reflective subcategory. Consider ω ⊂ Γ s as a subcategory.Then this inclusion admits a left adjoint ℓ : Γ s → ω such that ℓ ( n ) = n and ℓ ( m ) = S ( m ) = ( m + 1) . To double check that this is actually an adjunction, wecheck for m , n ∈ Γ s Hom Γ s ( m , n ) = Hom ω (( m + 1) , n )The lefthand side is nonempty if and only if m + 1 = n , which is the same for therighthand side. In the case that we are looking at the maps between m and n or m and n , the hom-set equality is straightforward as both sides are empty.By [Gro13, Proposition 1.18], right adjoint functors preserve colimits; that is,if r : ω → Γ s is the inclusion, we have π ω , ! r ∗ Y ∼ = π Γ s , ! Y for any Y ∈ D (Γ s ). Not-ing finally that the composition r ∗ π ∗ ω = π ∗ ( p/ , we can now complete the chain ofisomorphisms to finish the computation:(1 , ∗ Ξ X ∼ = 1 ∗ p ! π ∗ Γ s X ∼ = π ( p/ , ! pr ∗ π ∗ Γ s X ∼ = π ( p/ , ! π ∗ ( p/ X ∼ = π ω , ! π ∗ ω X ∼ = ` X The last step is to compute (1 , ∗ Ξ X , which takes a little more work. In or-der to identify it, we modify the construction of the cofibration morphism Ξ up toisomorphism. Instead of immediately applying p ! : D Γ s → D [1] , we note that Γ s isnearly ω × [1], if we think of this latter category as − (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) · · · · · · Let i : Γ s → ω × [1] be the inclusion as indicated by the labelling in the diagram. Thecategory ω × [1] has the canonical projection to [1] which we denote p [1] . Then wehave a factorisation p = p [1] i , so we have an isomorphism p ! ∼ = p [1] , ! i ! . The functor i is a cosieve, so the left Kan extension i ! is extension by zero, and the morphism p [1] , is similar to p ! in that it computes the infinite coproduct of the top row and thebottom row separately. We do not need to make this computation, but in essencewe have0 ∗ p [1] , ! i ! π ∗ Γ s ∼ = 0 ⊔ X ⊔ X ⊔ · · · ∼ = ` X, ∗ p [1] , ! i ! π ∗ Γ s ∼ = X ⊔ X ⊔ · · · ∼ = ` X This reorganising allows us to make explicit the [1]-dimension of Γ s and reversethe order of the infinite coproduct and the pushout. More specifically, the followingdiagram commutes up to isomorphism as all the morphisms involved are cocontinuousand involve distinct dimensions of the overall diagram: D ω × [1] id ∗ ω × i p , ! i [1] , ∗ / / p [1] , ! (cid:15) (cid:15) D ω cof ` (cid:15) (cid:15) D [1] i p , ! i [1] , ∗ / / D cof Therefore to compute (1 , ∗ Ξ X , it suffices to compute the following:(1 , ∗ i p , ! i [1] , ∗ p ! π ∗ Γ s X ∼ = (1 , ∗ i p , ! i [1] , ∗ p [1] , ! i ! π ∗ Γ s X ∼ = (1 , ∗ ` (id ∗ ω × i p , ! i [1] , ∗ ) i ! π ∗ Γ s X We can commute the coproduct morphism with (1 , ∗ , so the final challenge isto understand the (1 ,
1) entry of each of the cocartesian squares in D ω cof that weobtain before taking the infinite coproduct. We have already made the computationof i ! π ∗ Γ s X ∈ D ω × [1] and found that it is0 (cid:15) (cid:15) X (cid:15) (cid:15) X (cid:15) (cid:15) X (cid:15) (cid:15) · · · X X X X · · · where all but the first map are (coherent) identities. The pushouts are accomplishedindependently of each other, which means (id ∗ ω × i p , ! i [1] , ∗ ) i ! π ∗ Γ s X has the form (wherewe rotate all our arrows from vertical to horizontal before pushing out)0 / / (cid:15) (cid:15) X (cid:15) (cid:15) X / / (cid:15) (cid:15) X (cid:15) (cid:15) X / / (cid:15) (cid:15) X (cid:15) (cid:15) · · · / / X / / / / · · · So we see that (1 , ∗ ` applied to this diagram yields a coproduct of X with count-ably many zero objects, so we conclude that (1 , ∗ Ξ X ∼ = X . HE K-THEORY OF LEFT POINTED DERIVATORS 31
By additivity, on K-theory we have K ( T ) ≃ K ( S ) ⊔ K ( Q ). This yields for us K ( ` ) ≃ K ( ` ) ⊔ K (id D ), so that 0 ≃ K (id D ). The only way for the identity to beequal to zero is if K ( D ) itself is zero, completing the proof. (cid:3) We now proceed in the spirit of Waldhausen in [Wal85], beginning first with thedelooping of the K-theory space K ( D ). In order to do so, we need a construction ofrelative K-theory.To begin, we note that the S • -construction on derivators not only gives us asimplicial category, but in fact a simplicial left pointed derivator. Proposition 6.2.
Let j : [ n − → Ar[ n ] denote the inclusion i (0 , i + 1).Then j ∗ : S n D → D ([ n − S n D ⊂ D Ar[ n ] the structure of a left pointed derivator via the equivalence with D [ n − . Proof.
We will construct the quasiinverse directly.First, consider the functor i : [ n − → [ n ] defined by i i + 1. This map is acosieve, so the morphism i , ! is extension by zero.Second, consider the subcategory D ⊂ Ar[ n ] which contains the top row (0 , i )as well as all diagonal entries ( i, i ). The inclusion i : [ n ] → J is a sieve, so i , ∗ isextension by zero.The last step is to take the inclusion i : D → Ar[ n ] and compute i , ! . We claimthat the image of this composition lies in S n D ( e ) ⊂ D (Ar[ n ]). To see this, we use[Gro13, Proposition 3.10] to detect cocartesian squares, whose statement we providenow (using notation more convenient for our purposes). Proposition 6.3.
Let ι : (cid:3) → K be a fully faithful functor and let u : J → K beany functor. We may consider the full category K \ ι (1 ,
1) obtained by removing theimage of the bottom-right corner of the square, and then form the comma category( K \ ι (1 , /ι (1 , / / (cid:15) (cid:15) K \ ι (1 , (cid:15) (cid:15) ⇒ e ι (1 , / / K where the righthand vertical map is the inclusion of the subcategory. This commacategory receives a functor from p induced by ι , which we denote ι .Assume that ι : p → ( K \ ι (1 , /ι (1 , ι (1 ,
1) doesnot lie in the image of u : J → K . Then for all X ∈ D ( J ), the square ι ∗ u ! X ∈ D ( (cid:3) )is cocartesian.We will apply this ‘detection lemma’ with J = [ n −
1] and K = Ar[ n ]. Let ι i,j : (cid:3) → Ar[ n ] be the square given by ( a, b ) ( i + b, j + a ) for 0 ≤ i < j ≤ n − a and b comes in because of an unfortunate inconsistency in the notation between (cid:3) and Ar[ n ]. It suffices to prove each of these squares iscocartesian, as any other square will be a composite of such squares. If a largersquare can be subdivided into cocartesian squares, then it too is cocartesian by[Gro13, Proposition 3.13(1)].Because Ar[ n ] is a poset, we can identify the comma category(Ar[ n ] \ ( i + 1 , j + 1) / ( i + 1 , j + 1))as the full subcategory of Ar[ n ] on ( p, q ) admitting a map to ( i + 1 , j + 1), i.e. p ≤ i + 1 and q ≤ j + 1, excluding ( p, q ) = ( i + 1 , j + 1) as it has been removed.Call the resulting category B i,j . We now construct a left adjoint for ι i,j directly. Wedefine ℓ : B i,j → p by ℓ ( p, q ) = (0 , p ≤ i and q ≤ j (0 , q = j + 1(1 , p = i + 1Direct computation shows that Hom( ℓ ( p, q ) , ( a, b )) = Hom(( p, q ) , ι i,j ( a, b )) for anyelements ( p, q ) ∈ B i,j and ( a, b ) ∈ p . We can also construct the unit and the counitdirectly; the counit is just the identity on p , and the unit can only be the uniquemap ( p, q ) → ι ( ℓ ( p, q )) at each ( p, q ) ∈ B i,j .This proves that ι ∗ i,j i , ! X is a cocartesian square for any X ∈ D ( D ). Pastingthese squares together shows that any square in i , ! X is cocartesian. Moreover, ifwe have X = i , ∗ i , ! Y for some Y ∈ D ([ n − i, i ) ∗ X = 0 by construction.Therefore X ∈ S n D ( e ). Call this total morphism Φ : D ([ n − → S n D .Because Φ is constructed as left and right Kan extensions of fully faithful functors,it too is fully faithful. Moreover, it is left adjoint to j ∗ , with the counit id D ([ n − = j ∗ Φthe identity modification. Because the left adjoint morphism is fully faithful, the unitis an isomorphism. Because both the unit and counit are invertible modifications,this gives an equivalence of categories. (cid:3)
Remark 6.4.
We can give an alternative proof that S n D has the structure of a leftpointed derivator. First, for K ∈ Dir f , we define S n D ( K ) ⊂ D Ar[ n ] ( K ) to be the fullsubcategory on objects X such that k ∗ X ∈ S n D for any k ∈ K . This makes S n D aprederivator on Dir f . Der1, Der2, and Der5 follow immediately from its definitionas a certain levelwise subcategories of a derivator, and it is also (weakly) pointedbecause the 0 object of D (Ar[ n ]) is in S n D ( e ).For the remainder of the axioms, it suffices to show that the left and right Kanextensions in D Ar[ n ] land in the appropriate subcategory. Let X ∈ S n D ( J ) andlet u : J → K be a functor. We only know for sure that u ! X ∈ D Ar[ n ] ( K ), sowe need to check that for all k ∈ K , ( i, i ) ∗ k ∗ u ! X = 0 for all i ∈ [ n ] and for allsquares ι : (cid:3) → Ar[ n ], ι ∗ k ∗ u ! X ∈ D (cid:3) is cocartesian. HE K-THEORY OF LEFT POINTED DERIVATORS 33
For the first point, let us just examine ( i, i ) ∗ u ! X ∈ D ( K ). Because ( i, i ) ∗ is co-continuous, we have ( i, i ) ∗ u ! X ∼ = u ! ( i, i ) ∗ X . We know that ( i, i ) ∗ X = 0 ∈ D ( J )because X ∈ S n D ( J ), so we have u ! ( i, i ) ∗ X = 0 ∈ D ( K ) because u ! is a pointed mor-phism. Therefore k ∗ ( i, i ) ∗ u ! X = 0 ∈ D ( e ), and these first two morphisms commutebecause they are pullback morphisms in unrelated diagrammatic directions, givingus ( i, i ) ∗ k ∗ u ! X = 0 for any i ∈ [ n ] and k ∈ K .For the second point, we have ι ∗ k ∗ u ! X ∼ = k ∗ ι ∗ u ! X ∼ = k ∗ u ! ι ∗ X for reasons identical to the above. We know that ι ∗ X is a cocartesian squarein D ( (cid:3) × J ), and u ! preserves cocartesian squares. This implies that each k ∗ u ! ι ∗ X is cocartesian in D ( (cid:3) ), and following the chain of isomorphisms backwards finishesthe proof.There is no difference if we consider the right Kan extension sieve u : J → K ,as the extension by zero morphism u ∗ : D Ar[ n ] ( J ) → D Ar[ n ] ( K ) is still cocontinuous.That was the only fact we used in the case of left Kan extensions u ! , and so wecomplete the proof.Because S n D has the structure of a left pointed derivator, it means that S • D isactually a simplicial object in left pointed derivators. This means we can iteratethe S • construction, and will do so shortly. But before that, we will define ourrelative K-theory construction. To do so, we need the following general simplicialconstructions.For any simplicial set Y , we may define a new simplicial set P Y (of paths in Y ) byprecomposing Y by the functor ∆ op → ∆ op which sends [ n ] to [ n + 1] via i i + 1. Lemma 6.5 (Lemma 1.5.1, [Wal85]) . P Y is simplicially homotopy equivalent to theconstant simplicial set on Y .There is a projection map P Y → Y induced by the 0-face map. Moreover, thereis a functor Y → P Y which is the inclusion of 0-simplices, as (
P Y ) = Y . Thisgives a sequence Y → P Y → Y for any simplicial set Y .Now suppose that Φ : D → E is a strict cocontinuous morphism of left pointedderivators. We then define the simplicial category S • Φ by the following 2-pullbackin
CAT , sometimes called the iso-comma, which lies between the lax pullback andthe strict pullback: S • Φ / / (cid:15) (cid:15) P S • E d (cid:15) (cid:15) ⇒ ∼ = S • D Φ / / S • E Specifically, at each [ n ] ∈ ∆ op , we have a square S n Φ / / (cid:15) (cid:15) ( P S • E ) n = S n +1 E d (cid:15) (cid:15) ⇒ ∼ = S n D Φ / / S n E in which the top-left corner is explicitly the following: an object in S n Φ is a pair( A ∈ S n D , B ∈ S n +1 E ) along with an isomorphism f A,B : Φ( A ) → d ( B ) ∈ S n E .Note that if Φ is not a strict morphism, then there are naturality problems with theface and degeneracy maps of S • Φ.However, if Φ is not a strict morphism, then by Proposition 4.1 there are twostrict (cocontinuous) morphisms Π Φ : e D → D and e Φ : e D → E such that Π Φ is a weakequivalence and ΦΠ Φ = e Φ. While S • Φ will not be defined directly, it will havethe same homotopy type as S • e Φ. Therefore it is not an issue for us to assume themorphism Φ is strict for the rest of this argument.In Waldhausen K-theory, S • F for F : C → D an exact morphism of Waldhausencategories is again a simplicial Waldhausen category. For us, it is not immediatelyclear that S • Φ should be a simplicial left pointed derivator, so we prove that now.
Proposition 6.6.
The simplicial category S • Φ underlies a simplicial object in leftpointed derivators (which we give the same name).
Proof.
For K ∈ Dir f , the category S n Φ( K ) will have objects a triple A ∈ S n D ( K ) , B ∈ S n +1 E ( K ) , f A,B : Φ( A ) ∼ = −→ d ( B ) , which we will shorten to ( A, B, f
A,B ). For u : J → K , we define u ∗ ( A, B, f
A,B ) to bethe triple ( u ∗ A, u ∗ B, g
A,B ), where g A,B is the map filling in the commutative diagramof isomorphisms below: u ∗ Φ( A ) γ Φ u (cid:15) (cid:15) u ∗ f A,B / / u ∗ d ( B ) γ d u (cid:15) (cid:15) Φ( u ∗ A ) g A,B / / ❴❴❴❴ d ( u ∗ B )The vertical isomorphisms are actually equalities because Φ is assumed to be strict(and d is in any case), so g A,B = u ∗ f A,B . We include the full picture for analogywith what follows. This proves that S n Φ has the structure of a prederivator.A fair question at this point is why we need the flexibility of an isomorphism f A,B if Φ is assumed to be strict. If we required f A,B to be the identity, then we knowthat u ∗ ( f A,B ) would also be the identity by strict 2-functoriality. The issue arisesfor the left and right Kan extensions, which we address now. Suppose now that
HE K-THEORY OF LEFT POINTED DERIVATORS 35 ( A, B, f
A,B ) ∈ S n Φ( J ). Then we define u ! ( A, B, f
A,B ) to be ( u ! A, u ! B, h
A,B ), where h A,B is the map filling in a similar commutative diagram of isomorphisms: u ! Φ( A ) ( γ Φ u ) ! (cid:15) (cid:15) u ! f A,B / / u ! d ( B ) ( γ d u ) ! (cid:15) (cid:15) Φ( u ! A ) h A,B / / ❴❴❴❴ d ( u ! B )Because Φ and d are both cocontinuous, the left mates of the structure isomor-phisms γ u are isomorphisms (by definition). However, even though γ d u may be theidentity, there is no reason to believe that its mates are also identities; they are onlyguaranteed to be isomorphisms. Therefore we have h A,B = ( γ d u ) ! ◦ u ! f A,B ◦ ( γ Φ u ) − This explains the definition of S • Φ as an iso-comma (simplicial) category instead ofa strict pullback as in Waldhausen’s original construction.These left Kan extensions for S n Φ are natural and (moreover) are the only onesthat makes sense. The construction of left Kan extensions also generalises to rightKan extensions along sieves u : J → K , as the right mates ( γ Φ u ) ∗ and ( γ d u ) ∗ will alsobe isomorphisms by Der4R.For strongness, because the (partial) underlying diagram functors for S n Φ areconstructed as a pullback of those for S n D and S n +1 E , they are still full and essentiallysurjective on finite free categories.Finally, S n Φ( e ) is pointed by (0 , , ∼ =), where the isomorphism is unique, whichgives S • Φ the structure of a simplicial left pointed derivator. (cid:3)
We can now formulate the statement of relative derivator K-theory. Let Φ : D → E be a strict cocontinuous morphism of left pointed derivators. First, there is aninclusion S E → P S • E as zero simplices, where we view S E as a constant simplicialleft pointed derivator. Composing this with the map d : P S • E → S • E we obtain asequence S E → P S • E → S • E the composition of which is constant. We can lift the map S E → P S • E to S E → S • Φ using the pullback defining S • Φ and the constant (at zero) map S E → S • D → S • E S E (cid:21) (cid:21) ) ) " " ❋❋❋❋ S • Φ / / (cid:15) (cid:15) P S • E d (cid:15) (cid:15) ⇒ ∼ = S • D Φ / / S • E Composing with the projection from S • Φ to S • D we obtain a sequence S E → S • Φ → S • D the composition of which is again constant. Iterating the S • construction, we havethe following theorem. Theorem 6.7.
The sequence(6.8) i S • S E → i S • S • Φ → i S • S • D is a homotopy fibration after (diagonally) geometrically realizing nerves and passingto the homotopy category of spaces S . Proof.
We proceed as in [Wal85, Proposition 1.5.5]. We use the following ‘realizationlemma’ from [Wal78]:
Lemma 6.9.
Let X •• → Y •• → Z •• be a sequence of bisimplicial sets so that X •• → Z •• is constant. Suppose that X • n → Y • n → Z • n is a homotopy fibrationfor every n . Suppose further that Z • n is connected for every n . Then the originalsequence is also a homotopy fibration.We are indeed in this situation, up to a little unpacking. We have a sequenceof bisimplicial categories, which we will turn into a sequence of trisimplicial sets bytaking the nerve: N • i S • S E → N • i S • S • Φ → N • i S • S • D . However, let us treat this as a bisimplicial set by considering the first two simplicialdirections as one diagonal direction, i.e. if we let • and ⋆ denote the two differentdirections, we have N • i S • S E → N • i S • S ⋆ Φ → N • i S • S ⋆ D which is now a sequence of bisimplicial sets. The first term appears the same be-cause S E was constant in the ⋆ direction. As geometric realization may be takenvariable-by-variable or diagonally (in fact, these give homeomorphic spaces), it suf-fices to show that this second sequence of bisimplicial sets is a homotopy fibration. HE K-THEORY OF LEFT POINTED DERIVATORS 37
The last thing to check is that N • i S • S n D is connected for all n . But N i S S n D consists of the zero objects in S n D ( e ), all of which are isomorphic, hence there is a 1-simplex in N i S S n D which is this isomorphism. Applying a degeneracy map in the S -direction will give us a 1-simplex in the diagonal simplicial set N i S S n D whichconnects these 0-simplices, showing that this simplicial set is indeed connected.Therefore let us fix an n and consider the sequence i S • S E → i S • S n Φ → i S • S n D of simplicial left pointed derivators. We will make our argument here and pass tothe nerve and the corresponding diagonal simplicial sets as we outlined above.We will show, as Waldhausen does, that this relative K-theory sequence is homo-topic to the trivial homotopy fibration. We will do so using the additivity theorem.We define a cofibration morphism of derivatorsΞ n : S n Φ → ( S n Φ) cof such that (0 , ∗ Ξ n takes values in a copy of S E inside S n Φ, (1 , ∗ Ξ n = id S n Φ ,and (1 , ∗ Ξ n takes values in a copy of S n D inside S n Φ.The sketch for the construction of Ξ n is the following: for ( A, B, f
A,B ) ∈ S n Φ,Ξ n ( A, B, f
A,B ) = (0 , s n · · · s (0 → (0 , ∗ B → , ∼ =) / / (cid:15) (cid:15) ( A, B, f
A,B ) (cid:15) (cid:15) (0 , , ∼ =) / / ( A, s d ( B ) , f A,B )where s i are the degeneracy maps of the simplicial set P S n E . The entry in the top leftis a degenerate n -simplex in P S n E which comes from (0 → (0 , ∗ B → ∈ P S E .The isomorphisms between zero objects on the lefthand side are more subtle thanthey appear, and we will address this below.To begin with the S n D component of S n Φ, we define a cofibration morphism S n D → ( S n D ) cof . Consider the map p n : Ar[ n ] × (cid:3) → Ar[ n ] defined as follows: for( a, b ) = (1 ,
0) or (1 , p n ( i, j, a, b ) = ( i, j ). Further, we let p n ( i, j, ,
0) = (0 , p n ( i, j, ,
1) = (0 ,
0) be constant. We illustrate the functor p : Ar[2] × (cid:3) → Ar[2],using bold arrows for the (cid:3) dimension of the diagram. We label the objects of thedomain according to where they map in the codomain, which shows better how the pullback p ∗ behaves:0 / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / a / / (cid:15) (cid:15) b (cid:15) (cid:15) / / (cid:15) (cid:15) + / / c (cid:15) (cid:15) (cid:11) (cid:19) (cid:11) (cid:19) / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / a / / (cid:15) (cid:15) b (cid:15) (cid:15) / / (cid:15) (cid:15) + / / c (cid:15) (cid:15) → / / a / / (cid:15) (cid:15) b (cid:15) (cid:15) / / c (cid:15) (cid:15) The horizontal arrows in the (cid:3) dimension are necessarily zero maps, and the ver-tical arrows are identity maps, as p n ( − , − , , − ) : Ar[ n ] × [1] → Ar[ n ] defining therighthand vertical map is just the projection id Ar[ n ] × π [1] and p n ( − , − , , − ) definingthe lefthand vertical map is the constant functor Ar[ n ] × [1] → e . This square iscocartesian, and establishes the construction for S n D . Note that the definition of p n implicitly uses that n ≥
1, but for the case n = 0, Ar[0] = e so p : (cid:3) → e is theconstant functor π (cid:3) by necessity.For the P S n E = S n +1 E component of the derivator S n Φ, we define amap q n : Ar[ n + 1] × (cid:3) → Ar[ n + 1] computing what we want. First, we willhave q n ( i, j, ,
0) = ( i, j ), just as p n was defined. To deal with the other ( a, b ) ∈ (cid:3) ,we start with ( a, b ) = (0 , q n ( i, j, ,
0) = (0 ,
0) ( i, j ) = (0 , , i = 0 , j ≥ ,
1) otherwiseWe want q n ( − , − , ,
1) to be a constant functor (as p n ( − , − , ,
1) was) but in thiscase we let q n ( i, j, ,
1) = (1 , a, b ) = (1 , q n ( i, j, ,
1) = (1 ,
1) ( i, j ) = (0 , , j ) i = 0 and j ≥ i, j ) otherwise HE K-THEORY OF LEFT POINTED DERIVATORS 39
To illustrate q , we have the following picture, where we again label the zeroes:0 ′ / / α = / / (cid:15) (cid:15) α (cid:15) (cid:15) = / / α (cid:15) (cid:15) ′ / / α / / (cid:15) (cid:15) β (cid:15) (cid:15) / / γ (cid:15) (cid:15) ′ / / ′ (cid:15) (cid:15) / / ′ (cid:15) (cid:15) ′ / / δ (cid:15) (cid:15) / / ε (cid:15) (cid:15) ′ / / ′ (cid:15) (cid:15) ′ / / ζ (cid:15) (cid:15) ′ ′ ′ / / ′ / / (cid:15) (cid:15) ′ (cid:15) (cid:15) / / ′ (cid:15) (cid:15) ′ / / ′ / / (cid:15) (cid:15) δ = (cid:15) (cid:15) / / ε = (cid:15) (cid:15) ′ / / ′ (cid:15) (cid:15) / / ′ (cid:15) (cid:15) ′ / / δ (cid:15) (cid:15) / / ε (cid:15) (cid:15) ′ / / ′ (cid:15) (cid:15) ′ / / ζ (cid:15) (cid:15) ′ ′ + + (cid:11) (cid:19) (cid:11) (cid:19) → ′ / / α / / (cid:15) (cid:15) β / / (cid:15) (cid:15) γ (cid:15) (cid:15) ′ / / δ / / (cid:15) (cid:15) ε (cid:15) (cid:15) ′ / / ζ (cid:15) (cid:15) ′ To check that this defines the functor Ξ n : S n Φ → ( S n Φ) cof , we need to constructthe isomorphisms Φ( p ∗ n A ) → d ( q ∗ n B ) corresponding to Ξ n ( A, B, f
A,B ), and we willdemonstrate how do so at each corner of (cid:3) . The isomorphisms at (1 ,
0) and (1 ,
1) arejust f A,B , but the morphisms at (0 ,
0) and (0 ,
1) are between specific zero objects.In the above diagrams, f A,B defines an isomorphism Φ(0 i ) → ′ i for each i . By con-struction, (0 , ∗ p ∗ n A and (0 , ∗ p ∗ n A are the constant diagram on the zero object 0 ,and d ((0 , ∗ q ∗ n B ) and d ((0 , ∗ q ∗ n B ) are the constant diagram on the zero object 0 ′ .Therefore (0 , ∗ f A,B is still the appropriate isomorphism of zero objects and no ad-ditional data is required. Indeed, if we were forced to compose isomorphisms of theform Φ(0 ) → ′ → ′ (or worse, Φ(0 ) → ′ ← ′ ), we could not complete thisconstruction coherently (as we discussed in Remark 5.18).Thus we define Ξ n ( A, B, f
A,B ) = ( p ∗ n A, q ∗ n B, p ∗ n f A,B ) . The structure isomorphisms γ Ξ n u come directly from the structure isomorphisms γ p n u and γ q n u .The source morphism (0 , ∗ Ξ n has image a subcategory of S n Φ equivalent to S E ,and the quotient morphism (1 , ∗ Ξ n has image a subcategory equivalent to S n D . Inparticular, every object ( A, s d ( B ) , f A,B ) is isomorphic to the object (
A, s Φ( A ) , id).The target morphism (1 , ∗ Ξ has image precisely S n Φ. These morphisms are more-over essentially surjective.
Now, there are two projections π : S n Φ → S E and π : S n Φ → S n D . Thefirst is defined by π ( A, B, f
A,B ) = (0 → (0 , ∗ B →
0) and the second definedby π ( A, B, f
A,B ) = A . This gives a total projection ρ : S n Φ → S E × S n D . This map has a section σ : S E × S n D → S n Φ given by σ ((0 → b → , A ) = ( A, s n · · · s (0 → b → ⊔ s Φ( A ) , id) . For example, for A = ( a → a → a ) ∈ S D , the component of σ ( b, A ) in P S E is0 / / b / / (cid:15) (cid:15) b ⊔ Φ( a ) / / (cid:15) (cid:15) b ⊔ Φ( a ) (cid:15) (cid:15) / / Φ( a ) / / (cid:15) (cid:15) Φ( a ) (cid:15) (cid:15) / / Φ( a ) (cid:15) (cid:15) σ relies on the additive structure of S n Φ. We firstdefine S E × S n D → S n Φ × S n Φ by the two morphisms S E × S n D / / S E S ′ / / S n Φ and S E × S n D / / S n D Q ′ / / S n Φwhere the first map in each case is the projection. The morphism S ′ is defined by S ′ (0 → b →
0) = (0 , s n · · · s (0 → b → , ∼ =)and the morphism Q ′ is defined by Q ′ ( A ) = ( A, s Φ( A ) , id) . We then compose ( S ′ , Q ′ ) : S E × S n D → S n Φ × S n Φ with the coproduct map S n Φ × S n Φ ∼ = S n Φ e ⊔ e → S n Φ. The definition of this map and how to strictifyit is contained in Equation 5.9 above.By construction, ρσ ∼ = id S E × S n D , and so obtain a homotopy after passing to K-theory. We now to show that the reverse composition σρ is homotopic to the identity.By applying the additivity theorem to Ξ n , the identity on S n Φ is homotopic tothe sum of the inclusion of S E and S n D , and this a morphism isomorphic to σρ .Therefore map of sequences i S • S E / / = (cid:15) (cid:15) i S • S n Φ / / ∼ (cid:15) (cid:15) i S • S n D = (cid:15) (cid:15) i S • S E / / i S • S E × i S • S n D / / i S • S n D HE K-THEORY OF LEFT POINTED DERIVATORS 41 has all vertical maps equivalences. The bottom sequence is a trivial homotopy fibra-tion, so the top sequence is also a homotopy fibration, completing the proof. (cid:3)
Definition 6.10.
For Φ : D → E a strict cocontinuous morphism of left pointedderivators, define K (Φ) := Ω | i S • S • Φ | . There is one more corollary of relative K-theory that bears mentioning before usingthis definition.
Corollary 6.11.
The topological space K ( D ) is an infinite loop space. Proof.
If we take the case Φ = id D : D → D , we can identify S • Φ with P S • D . Thereis certainly an equivalence of simplicial categories S • id D ( K ) → P S • D ( K ) for each K ∈ Dir f as the pullback of the equivalence id D : S • D ( K ) → S • D ( K ) is still anequivalence. But because the morphism S • id D → P S • D is defined globally and islevelwise an equivalence of categories, we get an equivalence of simplicial left pointedderivators.Using this replacement, we have a fibration i S • S D → P ( i S • S • D ) → i S • S • D , where P modifies the first simplicial direction. But now the middle term is con-tractible, giving a homotopy equivalence | i S • D | ∼ = | i S • S D | ∼ / / Ω | i S • S • D | We use here that the bisimplicial category i S • S D is homotopy equivalent to thesimplicial category i S • D . The equivalence is given (using morphisms of derivatorsand passing to maps up to homotopy in the geometric realization) by the forgetfulfunctor S D → D on the one hand and an iterated extension by zero morphism forthe inverse.Specifically, consider the cosieve t : e → [1] given by the inclusion of the targetand the sieve s : [1] → Ar[1] given by the inclusion into (0 , → (0 , s ∗ t ! : D → D (Ar[1]) is, for a ∈ D , a / / a / / a (cid:15) (cid:15) D by the simplicial left pointed derivator S • D and repeat theprocess, obtaining | i S • S • D | ∼ = | i S • S S • D | ∼ / / Ω | i S • S • S • D | which implies that | i S • D | is equivalent to Ω | i S • S • S • D | .By induction, we then conclude K ( D ) = Ω | i S • D | ∼ / / Ω ( n ) | i S ( n ) • D | and so K ( D ) is an infinite loop space. In particular, we can view K ( D ) as a connectiveΩ-spectrum. (cid:3) Corollary 6.12.
Let Φ : D → E be a strict cocontinuous morphism of left pointedderivators. Then there is a homotopy fibration K (Φ) → K ( D ) → K ( E ) Proof.
If we rotate the fibration sequence of Equation 6.8 to the left twice (andreplace S E by E ), we have a fibration sequenceΩ | i S • S • Φ | → Ω | i S • S • D | → | i S • E | . By the above corollary, | i S • D | → Ω | i S • S • D | is a homotopy equivalence. Replacingthe middle term and applying Ω everywhere, the corollary follows. (cid:3) The next logical step is to use Theorem 6.7 to prove a localization theorem inK-theory and answer (positive or negatively) Maltsiniotis’ conjecture that Verdierquotients of triangulated derivators get sent to long exact sequences in K-theory.Unfortunately, the necessity of coherent diagrams in derivator K-theory obstructs[Wal85, Theorem 1.6.4] from proceeding verbatim. In particular, that technique be-ing directly translatable would mean that Waldhausen K-theory agrees with derivatorK-theory in general, which has been proven false in [TV04] and [MR11]. Nonetheless,future work will address the issue of localization in a novel way which should avoidthis obstruction.As a coda, we can relate derivator K-theory to the K-theory of stable ∞ -categoriesas described in [BGT13]. Proposition 6.13.
Derivator K-theory is an additive invariant of stable ∞ -categories.Specifically, there is a functor K D : Cat ex ∞ → S ∞ from the category of small stable ∞ -categories and exact functors to the category of spectra that inverts Morita equiv-alences, preserves filtered colimits, and sends split-exact sequences to fibrations. Proof.
First, how does one obtain a derivator from an ∞ -category? Arlin (n´e Carl-son) in [Arl20] gives the following natural definition [Definition 9] using the quasicat-egory model for ∞ -categories. For Q a quasicategory, define the prederivator HO( Q )byHO( Q ) : J Ho (cid:0) Q N • J (cid:1) , u : J → K Ho( N • u ∗ ) : Ho (cid:0) Q N • K (cid:1) → Ho (cid:0) Q N • J (cid:1) where Q N • J is the quasicategory of simpicial maps N • J → Q . The action of HO( Q )on natural transformations is the only sensible one given the above. Arlin proves HE K-THEORY OF LEFT POINTED DERIVATORS 43 that all such prederivators satisfy Der1, Der2, and Der5. Moreover, if Q admits(homotopy) limits and colimits, so does HO( Q ). In particular, if Q is a stablequasicategory, it admits all homotopy finite limits and colimits, so HO( Q ) is a strongderivator on Dir f . Moreover, an exact functor of stable ∞ -categories preserves allfinite limits and colimits, so induces a cocontinuous morphism of the correspondingderivators. Therefore HO( Cat ex ∞ ) ⊂ Der K .Arlin further proves in [Arl20, Corollary 20] that quasicategories embed simpliciallyfully-faithfully into the simplicial enrichment of (pre)derivators developed by Muro-Raptis in [MR17], wherein the authors also prove that the derivator K-theory functor Der K → S admits a simplicial enrichment in [MR17, Proposition 5.1.3]. We willnot reiterate the details of these simplicial enrichment on derivators here, but it canbe noted that it requires working with strict morphisms of derivators only. Sincethese are the only morphisms that pass honestly to K-theory, this is no problem.Part of Arlin’s proof is that all functors of quasicategories give strict morphisms ofderivators, so we do not have an invisible ‘strictification’ step in the middle.As derivator K-theory takes values in infinite loop spaces, i.e. connective spec-tra, we can postcompose the K-theory functor with the ( ∞ -categorical) suspensionspectrum functor Σ ∞ + : S → S ∞ without changing anything. The total definitionbecomes K D : Cat ex ∞ ♮ −→ Cat perf ∞ ⊂ QCat HO −→ Der
K K −→ S → S ∞ We consider all categories above to be simplicial categories (as our model for ∞ -categories). Thus what we have above is not literally a functor between ∞ -categories,but induces one once enough (co)fibrant replacement is incorporated.The functor ♮ is the idempotent completion functor, which also appears as the firststep of the universal additive invariant in [BGT13]; this ensures that Morita equiv-alences are inverted. The category Cat perf ∞ is just the full subcategory idempotentcomplete ∞ -categories. That split exact sequences are sent to fibrations is exactlythe additivity theorem.Finally, we need to address filtered colimits. First, the idempotent completionfunctor is left adjoint to the inclusion Cat perf ∞ ⊂ Cat ex ∞ by (for example) the com-ments near [BGT13, Definition 2.14], so preserves all colimits. Therefore assumethat Q : I → Cat perf ∞ is a filtered diagram. For any K ∈ Dir f , we can consider thecomparison map colim I HO( Q i )( K ) → HO(colim I Q i )( K )which, if we unwind the definition, iscolim I Ho (cid:0) Q N • Ki (cid:1) → Ho (cid:16) (colim I Q i ) N • K (cid:17) The functor Ho :
Cat ∞ → Cat which gives the underlying category of a quasicat-egory is left adjoint the nerve functor, so preserves all colimits. Therefore we needonly consider the comparison map between quasicategoriescolim I Q N • Ki → (colim I Q i ) N • K Here we may use the fact that K ∈ Dir f , so that N • K has only finitely manynondegenerate simplices. In particular, this is a compact object in simplicial sets, sothe functor ( − ) N • K commutes with filtered colimits.We therefore obtain an equivalencecolim I HO( Q i )( K ) → HO(colim I Q i )( K )for all K ∈ Dir f , which assemble to an equivalence of the corresponding derivators.This equivalence passes through to K-theory, completing the argument. (cid:3) In light of this proposition, we should expect that the comparison map betweenWaldhausen and derivator K-theory that Muro-Raptis studied in [MR17] agrees withthe ‘universal trace map’ K ∞ → K D guaranteed by the results of [BGT13, Theo-rem 10.3]. In particular, the spectrum of natural transformations Nat( K ∞ , K D ) isisomorphic to K D ( S ω ∞ ), the K-theory associated to the derivator of (compact) spec-tra, i.e. the derivator K-theory of modules over the sphere spectrum S .We know that π K ∞ ( S ) ∼ = Z , π K ∞ ( S ) ∼ = Z / Z , and π K ∞ ( S ) ∼ = Z / Z , whichcan be found in [BM19]. Using the results of [Rap19] (building on [Mur08]), thecomparison map K ∞ → K D is 2-connected. This means that π K D ( S ) ∼ = Z , sothat the derivator trace map should correspond to an integer x ∈ Z . In [BGT13,Theorem 10.6], the authors prove that the Dennis trace K → THH corresponds to1 ∈ THH ( S ) ∼ = Z , but this relies on earlier (classical) computations by Waldhausen.Future work will explore this perspective more closely and identify the integer cor-responding to the ‘derivator trace’. In particular, Raptis’ conjecture of whether thederivator trace is generally not an isomorphism on π might start by showing that π K D ( S ) = 0, though we have no evidence for this at present. References [Arl20] Kevin Arlin. A higher Whitehead theorem and the embedding of quasicategories in pred-erivators.
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