Higher homotopy categories, higher derivators, and K-theory
aa r X i v : . [ m a t h . K T ] O c t HIGHER HOMOTOPY CATEGORIES,HIGHER DERIVATORS, AND K -THEORY GEORGE RAPTIS
Abstract.
For every ∞ -category C , there is a homotopy n -category h n C anda canonical functor γ n : C → h n C . We study these higher homotopy categories,especially in connection with the existence and preservation of (co)limits, byintroducing a higher categorical notion of weak colimit. Based on the ideaof the homotopy n -category, we introduce the notion of an n -derivator andstudy the main examples arising from ∞ -categories. Following the work ofMaltsiniotis and Garkusha, we define K -theory for ∞ -derivators and provethat the canonical comparison map from the Waldhausen K -theory of C tothe K -theory of the associated n -derivator D ( n ) C is ( n + 1)-connected. We alsoprove that this comparison map identifies derivator K -theory of ∞ -derivatorsin terms of a universal property. Moreover, using the canonical structure ofhigher weak pushouts in the homotopy n-category, we define also a K -theoryspace K (h n C , can) associated to h n C . We prove that the canonical comparisonmap from the Waldhausen K -theory of C to K (h n C , can) is n -connected. Contents
1. Introduction 12. Higher homotopy categories 63. Higher weak colimits 104. Higher derivators 175. K -theory of higher derivators 246. K -theory of homotopy n -categories 29References 351. Introduction
It has been long understood in homotopy theory that the homotopy category isonly a crude invariant of a much richer homotopy–theoretic structure. The prob-lem of finding a suitable formalism for this additional structure, one that encodeshomotopy–theoretic extensions of ordinary categorical notions, led to several foun-dational approaches, each with its own distinctive advantages and characteristics.The theory of ∞ -categories (or quasi–categories) [19, 20, 21, 4] is one of several suc-cessful approaches to develop useful foundations for the study of homotopy theoriesand it has led to groundbreaking new perspectives and results in the field.Even though passing to the homotopy category neglects this additional homotopy–theoretic structure, the general problem of understanding how much informationthis process retains still poses interesting questions in specific contexts. This has inspired many important developments, for example, in the context of rigidity theo-rems for homotopy theories [11, 33, 31, 29], derived/homotopical Morita and tiltingtheory (see [34] for a nice survey), or in connection with K -theory regarded as aninvariant of homotopy theories [8, 27, 32, 23].The theory of derivators, first introduced and developed by Grothendieck [15],Heller [18] and Franke [11], is a different foundational approach, based on the ideaof considering the homotopy categories of all diagram categories as a remedy tothe defects of the homotopy category (see also [17]). By supplementing the homo-topy category with the network of all these (homotopy) categories, it is possible toencode the collection of homotopy (co)limit functors and general homotopy Kanextensions as an enhancement of the homotopy category. This approach provides adifferent, lower (=2–) categorical formalism for expressing homotopy-theoretic uni-versal properties (see [5] and [35, 7] for some interesting applications). Moreover,Maltsiniotis [23] defined K -theory in the context of derivators with a view towardspartially recovering Waldhausen K -theory from the derivator. The K -theory ofderivators and its comparison with Waldhausen K -theory has been studied exten-sively in [6, 13, 14, 23, 25, 26]. In the context of the theory of derivators, thequestion about the information retained by the homotopy category is then up-graded to the analogous question for the derivator. However, the classical theory ofderivators still does not provide in general a faithful representation of a homotopytheory, even though it is possible in certain cases to recover in a non–canonical waythe homotopy theory from the derivator (see [30]).The purpose of this paper is to extend these ideas on the comparison betweenhomotopy theories and homotopy categories or derivators to n -categories (=( n, n -category of an ∞ -category. More specifically:(a) Higher homotopy categories.
Using the definition of the higher homotopycategories by Lurie [21], we consider the tower of homotopy n -categories { h n C } n ≥ associated to an ∞ -category C , and analyse the properties of thecomparison maps γ n : C → h n C . (Sections 2–3)(b) Higher derivators.
We introduce a higher categorical notion of a derivatorwhich takes values in n -categories. Then we develop the basic theory ofhigher derivators with a special emphasis on the examples which arise from ∞ -categories. (Section 4)(c) K-theory of higher derivators.
We extend the definition of derivator K -theory by Maltsiniotis [23] and Garkusha [13, 14] to n -derivators and studythe comparison map from Waldhausen K -theory. Our main result showsthat the comparison map is ( n + 1)-connected (Theorem 5.5). Moreover,following [25], we prove that this comparison map has a universal property(Theorem 5.13).(d) K-theory of homotopy n-categories.
In analogy with the K -theory of tri-angulated categories [27], we introduce K -theory for n -categories equippedwith distinguished squares. In the case of a homotopy n -category, we studythe comparison map from Waldhausen K -theory and prove that it is n -connected (Theorem 6.5).(a) Higher homotopy categories.
Every ∞ -category C has an associated ho-motopy n -category h n C and a canonical functor γ n : C → h n C . The construction IGHER HOMOTOPY CATEGORIES AND K -THEORY 3 of the homotopy n -category and its properties are studied in [21]. We review thisconstruction and its properties in Section 2. Intuitively, for n ≥
1, h n C is an ∞ -category with the same objects as C and whose mapping spaces are the appropriatePostnikov truncations of the mapping spaces in C . For n = 1, the homotopy cat-egory h C is the ordinary homotopy category of C . The collection of homotopy n -categories defines a tower of ∞ -categories: C γ n (cid:15) (cid:15) γ n − ❍❍❍❍❍❍❍❍❍ γ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲ · · · / / h n C / / h n − C / / · · · / / h C which approximates C . One of the defects of the homotopy category h C , which isessentially what the theory of derivators tries to rectify, is that it does not in generalinherit (co)limits from C . As a general rule, if C admits (co)limits, then h C admitsonly weak (co)limits – which may or may not be induced from C . (We recall that aweak colimit of a diagram is a cocone on the diagram which admits a morphism toevery other such cocone, but this morphism may not be unique in general.) Do thehigher homotopy categories have better inheritance properties for (co)limits, and inwhat sense? This question is closely related to the problem of understanding howmuch information h n C retains from C . We introduce a higher categorical versionof weak (co)limit in order to address this question. In analogy with ordinary weak(co)limits, a higher weak colimit is a cocone for which the mapping spaces to othercocones are highly connected, but not necessarily contractible. The relative strengthof the weak colimit is measured by how highly connected these mapping spaces are;the connectivity of these mapping spaces is called the order of the weak colimit. Theproperties of higher weak (co)limits are discussed in Section 3. For any simplicialset K , there is a canonical functorΦ Kn : h n ( C K ) → h n ( C ) K which is usually not an equivalence. The properties of this functor are relevant forunderstanding the interaction of K -colimits in C and in h n C . One of our conclusions(Corollary 3.17) is the following:Φ Kn induces an equivalence: h n − dim( K ) ( C K ) ≃ h n − dim( K ) (cid:0) h n ( C ) K (cid:1) .Moreover, in connection with higher weak colimits in h n C , we also conclude (Corol-lary 3.23):Suppose that C admits finite colimits. Then h n C admits finite coproducts andweak pushouts of order n −
1. In addition, the functor γ n : C → h n C preservescoproducts and sends pushouts in C to weak pushouts of order n − Higher derivators.
The main example of a (pre)derivator is given by the2-functor which sends a small category I to the homotopy category h ( C N ( I ) ),for suitable choices of an ∞ -category C . Using homotopy n -categories instead,we may consider more generally the example of the enriched functor which sendsa simplicial set K to the homotopy n -category h n ( C K ). Following the axiomaticdefinition of ordinary derivators, we introduce the definition of an n -derivator which G. RAPTIS encapsulates the salient, abstract properties of this example. The basic definitionsand properties of (left, right, pointed, stable) n -(pre)derivators, 1 ≤ n ≤ ∞ , arediscussed in Section 4. For any ∞ -category C , there is an associated n -prederivator D ( n ) C : Dia op → Cat n , K h n ( C K )where Dia denotes a category of diagram shapes and
Cat n is the ∞ -category of n -categories. These assemble to define a tower of ∞ -prederivators: D ( ∞ ) C (cid:15) (cid:15) " " ❋❋❋❋❋❋❋❋ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ · · · / / D ( n ) C / / D ( n − C / / · · · / / D (1) C which approximates D ( ∞ ) C : K C K . The n -prederivator D ( n ) C is an n -derivatorif certain homotopy Kan extensions exist in C and the corresponding base changetransformations satisfy the Beck–Chevalley condition. We prove the following factwhich can be used to obtain many examples of n -derivators (Proposition 4.17 andTheorem 4.18): C admits limits and colimits indexed by diagrams in Dia if and only if D ( n ) C is an n -derivator.The motivation for higher derivators is to bridge the gap between ∞ -categories andderivators by introducing a hierarchy of intermediate notions, a different one foreach categorical level, starting with ordinary derivators. For any fixed 1 ≤ n < ∞ ,the theory of n -derivators is still not a faithful representation of homotopy theories;it dwells in an ( n + 1)-categorical context in the same way the classical theory ofderivators involves a 2-categorical context. In this respect, our approach using n -derivators remains close to the original idea of a derivator, and differs from otherrecent perspectives on (pre)derivators in which (pre)derivators are reconstructedinto a model for the theory of ∞ -categories [12, 3, 25]. We will address the problemof comparing suitable nice classes of ∞ -categories with n -derivators in future jointwork with D.–C. Cisinski.(c) K-theory of higher derivators. K -theory for (pointed, left) derivators wasintroduced by Maltsiniotis [23] and Garkusha [13, 14]. The basic feature of aderivator that allows this definition of K -theory is that there is a natural notion ofcocartesian square for a derivator. The motivation for introducing this K -theory isconnected with the problem of recovering Waldhausen K -theory from the derivator.Maltsiniotis [23] conjectured that derivator K -theory satisfies additivity , localiza-tion , and that it agrees with Quillen K -theory for exact categories. Cisinski andNeeman [6] proved additivity for the derivator K -theory of stable derivators. Injoint work with Muro [26], we proved that localization and agreement with Quillen K -theory cannot both hold. On the other hand, Muro [24] proved that agreementwith Waldhausen K -theory holds for K and K (see also [23, Section 6]), andGarkusha [14] obtained further positive results in the case of abelian categories. InSection 5, after a short review of the K -theory of ∞ -categories, we define derivator K -theory for general (pointed, left) ∞ -derivators. For any pointed ∞ -category C with finite colimits, there is a comparison map to derivator K -theory, µ n : K ( C ) → K ( D ( n ) C ) , IGHER HOMOTOPY CATEGORIES AND K -THEORY 5 and these comparison maps define a tower of derivator K -theories: K ( C ) µ n (cid:15) (cid:15) µ n − & & ▲▲▲▲▲▲▲▲▲▲ µ + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ · · · / / K ( D ( n ) C ) / / K ( D ( n − C ) / / · · · / / K ( D (1) C )which approximates K ( C ). Our main result on the comparison map µ n is thefollowing connectivity estimate (Theorem 5.5): µ n is ( n + 1)-connected.We believe that this connectivity estimate is best possible in general (Remarks5.7 and 5.8). Following the ideas of [25], we also consider the Waldhausen K -theory K W, Ob ( D ) of a general (pointed, left) ∞ -derivator D . This K -theory alwaysagrees with Waldhausen K -theory (Propositon 5.10), but it is not invariant underequivalences of derivators in general. Similarly to the case of ordinary derivatorstreated in [25], we prove that the comparison map to derivator K -theory, µ : K W, Ob ( D ) → K ( D ) , identifies derivator K -theory in terms of a universal property (Theorem 5.13): µ is the initial natural transformation to a functorwhich is invariant under equivalences of ∞ -derivators.(d) K-theory of homotopy n -categories. The motivation for introducing K -theory for homotopy n -categories is to identify the part of Waldhausen K -theorywhich may be reconstructed from the homotopy n -category. As a basic instanceof this phenomenon, we recall that K ( C ) can be recovered from the triangulatedcategory h C for any stable ∞ -category C . The main feature of the homotopy n -category that is needed for our definition of K -theory is the collection of higher weakpushouts which come from pushouts in C . We revisit the properties of homotopy n -categories in Section 6 and discuss possible axiomatizations of these properties. Weconsider a general notion of K -theory for pointed n -categories with distinguishedsquares – this is a higher categorical, but more elementary, version of Neeman’s K -theory of categories with squares [27]. For a pointed ∞ -category C with finitecolimits, we consider the K -theory K (h n C , can) associated to h n C with the canon-ical structure of higher weak pushouts as distinguished squares. For every n ≥ ρ n : K ( C ) → K (h n C , can)and these assemble to define a tower of K -theories: K ( C ) ρ n (cid:15) (cid:15) ρ n − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ρ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ · · · / / K (h n C , can) / / K (h n − C , can) / / · · · / / K (h C , can)which approximates K ( C ). Our main result on the comparison map ρ n is thefollowing connectivity estimate (Theorem 6.5): ρ n is n -connected. G. RAPTIS
This connectivity estimate is best possible in general (Remark 6.10). Let P n X denote the Postnikov n -truncation of a topological space X , that is, the homotopygroups of P n X vanish in degrees > n and the canonical map X → P n X is ( n + 1)-connected. Based on the connectivity estimate above, we conclude (Corollary 6.11): P n − K ( C ) depends only on (h n C , can).This confirms a recent conjecture of Antieau [1, Conjecture 8.36] in the case ofconnective K -theory. Acknowledgements.
I would like to thank Benjamin Antieau, Denis–CharlesCisinski, Fernando Muro, and Hoang Kim Nguyen for interesting discussions andtheir interest in this work. I also thank Martin Gallauer and Christoph Schradefor their interest and helpful comments. This work was partially supported by the
SFB 1085 – Higher Invariants (University of Regensburg) funded by the DFG.2.
Higher homotopy categories n -categories. We recall the definition and basic properties of n -categoriesfollowing [21, 2.3.4]. Let C be an ∞ -category and let n ≥ − C is an n - category if it satisfies the following conditions:(1) Given f, f ′ : ∆ n → C , if f and f ′ are homotopic relative to ∂ ∆ n , then f = f ′ .(We recall that the notion of homotopy employed here means that the twomaps are homotopic via equivalences in C .)(2) Given f, f ′ : ∆ m → C , for m > n , if f | ∂ ∆ m = f ′| ∂ ∆ m , then f = f ′ .These conditions say that C has no morphisms in degrees > n and any two mor-phisms in degree n agree if they are equivalent. The conditions can be equivalentlyexpressed as follows: C is an n -category, n ≥
1, if for every diagramΛ mi / / (cid:15) (cid:15) C ∆ m > > ⑥⑥⑥⑥ where m > n and 0 < i < m , there exists a unique dotted arrow which makesthe diagram commutative [21, Proposition 2.3.4.9]. Using an inductive argument(see [21, Proposition 2.3.4.7]), it can also be shown that conditions (1) and (2) areequivalent to:(3) Given a simplicial set K and maps f, f ′ : K → C such that f | sk n ( K ) and f ′| sk n ( K ) are homotopic relative to sk n − ( K ), then f = f ′ .An important immediate consequence of (3) is that for every n -category C , the ∞ -category Fun( K, C ) is again an n -category for any simplicial set K [21, Corollary2.3.4.8]. Example 2.1.
The only ( − ∅ and ∆ . An ∞ -category is a 0-category if and only if it is isomorphic to (the nerve of) a poset. 1-categories are up to isomorphism (nerves of) ordinary categories. See [21, Examples2.3.4.2-2.3.4.3, Proposition 2.3.4.5].The property of being an n -category is not invariant under equivalences of ∞ -categories. The following proposition gives a characterization of the invariantproperty that an ∞ -category is equivalent to an n -category. We recall that an IGHER HOMOTOPY CATEGORIES AND K -THEORY 7 ∞ -groupoid (= Kan complex) X is n-truncated , where n ≥ −
1, if it has vanish-ing homotopy groups in degrees > n . We say that X is (-2)-truncated if it iscontractible. Proposition 2.2.
Let C be an ∞ -category and let n ≥ − be an integer. Then C is equivalent to an n -category if and only if Map C ( x, y ) is ( n − -truncated for all x, y ∈ C .Proof. See [21, Proposition 2.3.4.18]. (cid:3)
Higher homotopy categories.
Let C be an ∞ -category and let n ≥ n -category h n C of C .Given a simplicial set K , we denote by [ K, C ] n the set of mapssk n ( K ) → C which extend to sk n +1 ( K ). Two elements f, g ∈ [ K, C ] n are called equivalent,denoted f ∼ g , if the maps f, g : sk n ( K ) → C are homotopic relative to sk n − ( K ).The equivalence classes of such maps for K = ∆ m define the m -simplices of asimplicial set h n C , i.e., (h n C ) m : = [∆ m , C ] n / ∼ . Clearly an m -simplex of C defines an m -simplex in h n C , so we have a canonicalmap γ n : C → h n C . Note that this map is a bijection in simplicial degrees < n andsurjective in degrees n and n + 1.The following proposition summarises some of the basic properties of this con-struction. Proposition 2.3.
Let C be an ∞ -category and n ≥ .(a) The set of maps K → h n C is in natural bijection with the set [ K, C ] n / ∼ .(b) h n C is an n -category. In particular, it is an ∞ -category.(c) C is an n -category if and only if the map γ n : C → h n C is an isomorphism.(d) Let D be an n -category. Then the restriction functor along γ n : C → h n C , γ ∗ n : Fun(h n C , D ) → Fun( C , D ) , is an isomorphism.Proof. See [21, Proposition 2.3.4.12]. (cid:3)
Example 2.4.
For n = 1, the 1-category h C is isomorphic to the (nerve of the)usual homotopy category of C . Remark 2.5.
For an ∞ -category C , the homotopy 0-category h C can be describedin the following way. For x, y ∈ C , we write x ≤ y if Map C ( x, y ) is non-empty. Thisdefines a reflexive and transitive relation. We say that two objects x and y areequivalent if x ≤ y and y ≤ x . Then the relation ≤ descends to a partial orderon the set of equivalence classes of objects in C . The homotopy 0-category h C isisomorphic to the nerve of this poset. We will usually ignore the case n = 0 andrestrict to homotopy n -categories for n ≥ G. RAPTIS
Proposition 2.6.
The functor γ n : C → h n C is a categorical fibration between ∞ -categories. In addition, for every lifting problem ∂ ∆ m u / / (cid:15) (cid:15) C γ n (cid:15) (cid:15) ∆ m σ / / ; ; ✇✇✇✇✇ h n C where m ≤ n + 1 (resp. m < n ), there is a (unique) filler ∆ m → C which makesthe diagram commutative.Proof. Clearly for any object c in C and any equivalence f : c → c ′ in h n C , we mayfind a lift ˜ f : c → c ′ of f in C (uniquely if n > γ n is an inner fibration. Consider a lifting problemΛ mi (cid:15) (cid:15) u / / C γ n (cid:15) (cid:15) ∆ m / / h n C where 0 < i < m . For m ≤ n , there is a diagonal filler ∆ m → C because γ n isa bijection in simplicial degrees < n and surjective on n -simplices. For m > n ,any map v : ∆ m → C which extends u defines a diagonal filler for the diagram. Tosee this, note that v is unique up to homotopy relative to Λ mi ( ⊇ sk n − ∆ m ), andthen recall that the n -category h n C satisfies condition (3). Therefore γ n is an innerfibration and this completes the proof of the first claim.The second claim for m ≤ n follows again from the fact that γ n is bijective insimplicial degrees < n and surjective in degree n . For m = n + 1, we may find amap σ ′ : ∆ n +1 → C such that γ n σ ′ = σ , since γ n is surjective on ( n + 1)-simplices.The maps u, σ ′| ∂ ∆ n +1 : ∂ ∆ n +1 → C become equal after postcomposition with γ n . By Proposition 2.3(a), this meansthat they are homotopic relative to sk n − ( ∂ ∆ n +1 ). Using standard arguments, wemay extend this homotopy to a homotopy (relative to sk n − (∆ n +1 )) between σ ′ and a map v : ∆ n +1 → C such that v | ∂ ∆ n +1 = u . Moreover, since σ ′ and v arehomotopic relative to sk n − (∆ n +1 ), it follows that γ n v = γ n σ ′ = σ , and therefore v defines a diagonal filler for the diagram. (cid:3) Proposition 2.7.
Let C be an ∞ -category and let n ≥ be an integer. There is a(non-canonical) map ǫ : sk n +1 h n C → sk n +1 C such that the following diagram commutes: sk n +1 C / / C γ n (cid:15) (cid:15) sk n +1 h n C ǫ O O / / h n C where the horizontal maps are the canonical inclusions. IGHER HOMOTOPY CATEGORIES AND K -THEORY 9 Proof.
We have a diagram as follows:sk n − C ∼ = (cid:15) (cid:15) / / sk n C (cid:15) (cid:15) (cid:15) (cid:15) / / sk n +1 C / / (cid:15) (cid:15) (cid:15) (cid:15) C γ n (cid:15) (cid:15) sk n − h n C / / sk n h n C / / sk n +1 h n C / / h n C . We may choose a section ǫ ′ : sk n h n C → sk n C – uniquely up to equivalence. Weclaim that this section can be extended further to a section ǫ as required. Let [ σ ′ ]be an ( n + 1)-simplex in h n C , represented by:sk n ∆ n +1 σ ′ / / (cid:15) (cid:15) C . ∆ n +1 σ ; ; ✈✈✈✈✈ Using Proposition 2.3(a) for K = sk n (∆ n +1 ), the compositionsk n ∆ n +1 σ ′ −→ sk n C sk n ( γ n ) −−−−−→ sk n h n C ǫ ′ −→ sk n C → C is homotopic to σ ′ relative to sk n − ∆ n +1 . Let H : sk n (∆ n +1 ) × ∆ → C be such ahomotopy and consider the map:∆ n +1 × { } ∪ sk n (∆ n +1 ) ×{ } sk n (∆ n +1 ) × ∆ σ ∪ H −−−→ C . This map extends to a homotopy H ′ : ∆ n +1 × ∆ → C , which restricts to an ( n + 1)-simplex τ ′ = H ′| ∆ n +1 ×{ } : ∆ n +1 → C . We set ǫ ([ σ ′ ]) : = τ ′ . Repeating this processfor each [ σ ′ ], we obtain the required extension ǫ : sk n +1 h n C → sk n +1 C . (cid:3) Example 2.8.
Let C be an ∞ -category. The functor γ : C → h C is bijectiveon objects, so there is a unique section sk h C → sk C . By making choices ofmorphisms, one from each homotopy class, this map extends to a section sk h C → sk C . The last map extends further to a section sk h C → sk C by making (non-canonical) choices of homotopies for compositions.The functor h n ( − ) preserves categorical equivalences of ∞ -categories. Using Propo-sition 2.3, it follows that there is a tower of ∞ -categories: C → · · · → h n C → h n − C → · · · → h C . By construction, the canonical map C −→ lim( · · · → h n C → h n − C → · · · → h C )is an isomorphism, and by Proposition 2.6, this inverse limit defines also a homotopylimit in the Joyal model structure. Example 2.9.
As a consequence of Proposition 2.2, an ∞ -groupoid X is categor-ically equivalent to an n -category if and only if it is n -truncated. For example, aKan complex is equivalent to a 0-category if and only if it is homotopically discreteand to an 1-category if and only if it is equivalent to the nerve of a groupoid. Givenan ∞ -groupoid X , the canonical tower of ∞ -groupoids: X → · · · h n X → h n − X → · · · → h X → π X models the Postnikov tower of X and the map X → h n X is ( n + 1)-connected (i.e.,for every x ∈ X , the map π k ( X, x ) → π k (h n X, x ) is a bijection for k ≤ n andsurjective for k = n + 1.) Remark 2.10. An n -category C is n -truncated in the ∞ -category of ∞ -categories,that is, the ∞ -groupoid Map( K, C ) is n -truncated for any ∞ -category K (see [21,5.5.6] for the definition and properties of truncated objects in an ∞ -category). Tosee this, recall that Fun( K, C ) is again an n -category and then apply Proposition2.2. But an n -truncated ∞ -category C is not equivalent to an n -category in general,so the analogue of Example 2.9 fails for general ∞ -categories. An ∞ -category C is n -truncated if and only if C is equivalent to an ( n + 1)-category and the maximalKan subgroupoid C ≃ ⊆ C is n -truncated. Indeed, given an n -truncated ∞ -category C , then C ≃ ≃ Map(∆ , C ) is n -truncated. Moreover, since n -truncated objects areclosed under limits, it follows that Map C ( x, y ) is n -truncated for every x, y ∈ C ,using the fact that there is a pullback in the ∞ -category of ∞ -categories:Map C ( x, y ) (cid:15) (cid:15) / / C ∆ (cid:15) (cid:15) ∆ x,y ) / / C × C . Conversely, if C is an ( n + 1)-category and C ≃ is n -truncated, then it is possibleto show that Map(∆ k , C ) is n -truncated by induction on k ≥
0, from which itfollows that C is n -truncated. (I am grateful to Hoang Kim Nguyen for interestingdiscussions related to this remark.)Let Cat ∞ denote the category of ∞ -categories, regarded as enriched in ∞ -categories, and let Cat n denote the full subcategory of Cat ∞ which is spanned by n -categories. Proposition 2.11.
Let C and D be ∞ -categories and let n ≥ be an integer.(a) The natural map h n ( C × D ) ∼ = −→ h n C × h n D is an isomorphism.(b) There is a functor h C , D n : Fun( C , D ) → Fun(h n C , h n D ) which is natural in C and D . In particular, h n : Cat ∞ → Cat n is an enrichedfunctor.Proof. (a) follows directly from the definition of h n . For (b), we define the functorh C , D n as follows: a k -simplex F : C × ∆ k → D is sent to the compositeh n C × ∆ k ∼ = h n C × h n ∆ k ∼ = h n ( C × ∆ k ) h n F −−−→ h n D . The functor h C , D n is natural in C and D and turns h n into an enriched functor. (cid:3) Higher weak colimits
Basic definitions and properties.
It is well known that homotopy cat-egories do not admit small (co)limits in general, even when the underlying ∞ -category has small (co)limits. On the other hand, if the ∞ -category C admits, forexample, pushouts (resp. coproducts), then the homotopy category h C admitsweak pushouts (resp. coproducts), which are induced from pushouts (resp. co-products) in C . Moreover, if C admits small colimits, then h C admits small weak IGHER HOMOTOPY CATEGORIES AND K -THEORY 11 colimits – which may or may not be induced from C . These observations suggest thefollowing questions: does h n C , n >
1, have in some sense more or better (co)limitsthan the homotopy category, and how do these compare with (co)limits in C ?We introduce a notion of higher weak (co)limit in the context of ∞ -categorieswhich is both a higher categorical version of the classical notion of weak (co)limitand a weak version of the higher categorical notion of (co)limit. We will restrictto higher weak colimits as the corresponding definitions and results about higherweak limits are obtained dually.We begin with the definition of a weakly initial object. We fix t ∈ Z ≥ ∪ {∞} . Definition 3.1.
An object x of an ∞ -category C is weakly initial of order t if themapping space Map C ( x, y ) is ( t − y ∈ C . Example 3.2. If C is an ordinary category, a weakly initial object x ∈ C of order 0is a weakly initial object in the classical sense. For a general n -category C , a weaklyinitial object of order n is an initial object. Proposition 3.3.
Let C be an ∞ -category and x ∈ C . The following are equivalent:(1) x is weakly initial in C of order t .(2) x is weakly initial in h n C of order t for n > t .(3) x is initial in h t C .These imply:(4) x is initial in h n C for n < t .Proof. This follows from the fact that the functor γ n : C → h n C restricts to an n -connected map Map C ( x, y ) → Map h n C ( x, y ) for every x, y ∈ C . (cid:3) Proposition 3.4.
Let C be an ∞ -category and let t > . The full subcategory C ′ of C which is spanned by the weakly initial objects of order t is either empty or a t -connected ∞ -groupoid.Proof. Suppose that the full subcategory C ′ is non-empty. Then the mapping spacesof C ′ are ( t − t − ≥
0. It follows that every morphism in C ′ is an equivalence, therefore C ′ is an ∞ -groupoid. (cid:3) Remark 3.5.
The full subcategory C ′ of weakly initial objects in C of order 0 is notan ∞ -groupoid in general. In this case, we only have that Map C ( x, y ) is non-emptyfor every x, y ∈ C ′ . Definition 3.6.
Let C be an ∞ -category, K a simplicial set, and let F : K → C be a K -diagram in C . A weakly initial object G ∈ C F/ of order t is called a weakcolimit of F of order t . Example 3.7. If C is an n -category and G : K ⊲ → C is a weak colimit of F = G | K : K → C order t ≥ n , then G is a colimit diagram. This follows from Example3.2 using the fact that C F/ is again an n -category (see [21, Corollary 2.3.4.10]).In particular, a weak colimit of order ∞ is a colimit diagram. If C is an ordinarycategory, then a weak colimit of order 0 is a weak colimit diagram in the classicalsense. Remark 3.8.
There is an important difference between weak colimits of order 0and weak colimits of order >
0: as a consequence of Proposition 3.4, any two weak colimits of F of order > G ∈ C F/ is a weak colimitof order t > G ′ ∈ C F/ is a weak colimit of order >
0, then G ′ is also a weakcolimit of order t .The following proposition gives an alternative characterization of higher weakcolimits following the analogous characterization for colimits in [21, Lemma 4.2.4.3]. Proposition 3.9.
Let C be an ∞ -category, K a simplicial set, and let G : K ⊲ → C be a diagram with cone object x ∈ C . Then G is a weak colimit of F = G | K oforder t if and only if the canonical restriction map: Map C ( x, y ) ≃ Map C K⊲ ( G, c y ) → Map C K ( F, c y ) is t -connected for every y ∈ C , where c y denotes the constant K -diagram at y ∈ C .Proof. The fiber of the restriction map over F ′ : K ⊲ → C with cone object y isidentified with Map C F/ ( G, F ′ ) (see the proof of [21, Lemma 4.2.4.3]). (cid:3) The basic rules for the manipulation of higher weak colimits can be establishedsimilarly as for colimits. The following procedure shows that higher weak colimitscan be computed iteratively, exactly like colimits, but with the difference that theorder of the weak colimit may decrease with each iteration.
Proposition 3.10.
Let C be an ∞ -category and let K = K ∪ K K be a simplicialset where K ⊆ K is a simplicial subset. Let F : K → C be a diagram and denoteits restrictions by F i : = F | K i , i = 0 , , . Suppose that G i : K ⊲i → C is a weakcolimit of F i of order t i , i = 0 , , .(a) There are morphisms G → G | K ⊲ and G → G | K ⊲ in C F / . These to-gether with G and G determine a diagram in C as follows, H : K ⊲ ∪ K ∗ ∆ { } ( K ∗ ∆ ) ∪ K ∗ ∆ { } ( K ∗ ∆ ) ∪ K ∗ ∆ { } K ⊲ → C . (b) Let H p : ∆ ∪ ∆ ∆ → C be the restriction of H to the cone objects. Supposethat H ′ : ∆ × ∆ → C is a weak pushout of H p of order k . Then H ′ determines a cocone G : K ⊲ → C over F , with the same cone object as H ′ ,which is a weak colimit of F of order ℓ : = min( k, t , t − , t ) .Proof. (a) is clear by the properties of higher weak colimits. For (b), we first explainthe construction of the cocone G : K ⊲ → C . The functor H ′ is represented by adiagram x v (cid:15) (cid:15) u / / x f (cid:15) (cid:15) x g / / y where x i is the cone object of G i , and the morphisms u and v are given respectivelyby the morphisms G → G | K ⊲ and G → G | K ⊲ in C F / . The morphisms f and g produce two new cocones essentially uniquely: G ′ : K ⊲ → C over F , and G ′ : K ⊲ → C over F , with common cone object y . The restrictions G ′ | K ⊲ and G ′ | K ⊲ areequivalent as cocones over F . We may then extend G ′ | K ⊲ in an essentially uniqueway to a new cocone G ′′ : K ⊲ → C over F , which is equivalent to G ′ . The resultingcocones G ′ | K ⊲ : K ⊲ → C , G ′′ : K ⊲ → C , and G ′ : K ⊲ → C IGHER HOMOTOPY CATEGORIES AND K -THEORY 13 assemble to define the required cocone G : K ⊲ → C . Then the claim in (b) is shownby applying Proposition 3.9, first for weak pushouts and then for K i -diagrams, andusing the equivalence of ∞ -categories: C K ≃ C K × C K C K . (cid:3) Example 3.11.
Let C be an ordinary category that admits small coproducts andweak pushouts. By Proposition 3.10, every diagram F : K → C where K is 1-dimensional admits a weak colimit (of order 0). Now suppose that F : I → C is adiagram where I is an arbitrary ordinary small category. Since C is an ordinarycategory, a cocone G : I ⊲ → C over F is determined uniquely by its restriction to acocone G ′ : (sk I ) ⊲ → C over F | sk I , and similarly for morphisms between cocones.As a consequence, we may deduce the well known fact that C has weak I -colimits. Example 3.12.
Let T ⊂ ∆ × ∆ be the full subcategory spanned by the objects(0 , i ), for i = 0 , ,
2, and (1 , C be an ∞ -category with weak pushouts of order t and let F : T → C be a T -diagram in C . Write T = T ∪ T T where T is spannedby (0 , i ), i = 0 ,
1, and (1 , T is spanned by (0 , i ), i = 1 ,
2, and T = { (0 , } .Using Proposition 3.10, we may compute a weak colimit of F of order t in terms ofiterated weak pushouts of order t .3.2. Homotopy categories and (co)limits.
Let C be an ∞ -category and let K be a simplicial set. By the universal property of h n ( − ), the functorFun( K, C ) → Fun( K, h n C ) , which is given by composition with γ n : C → h n C , factors canonically through thehomotopy n -category:(3.13) Φ Kn : h n Fun( K, C ) → Fun( K, h n C ) . The comparison between K -colimits in C and in h n C is essentially a question aboutthe properties of the functor Φ Kn . Note that for n = 1, Φ K is simply the canonicalfunctor of ordinary categories: h ( C K ) → h ( C ) K . Lemma 3.14.
Let C be an ∞ -category, K a finite dimensional simplicial set ofdimension d > , and let n ≥ be an integer. The functor Φ Kn : h n Fun( K, C ) → Fun( K, h n C ) satisfies the following:(a) Φ Kn is a bijection in simplicial degrees < n − d .(b) Φ Kn is surjective in simplicial degree n − d ; it identifies ( n − d ) -simpliceswhich are homotopic relative to the ( n − -skeleton of ∆ n − d × K .(c) Φ Kn is surjective in simplicial degree n − d + 1 .Proof. The m -simplices of Fun( K, h n C ) are equivalence classes of mapssk n (∆ m × K ) → C that extend to sk n +1 (∆ m × K ). On the other hand, the m -simplices of h n Fun( K, C )are equivalence classes of maps sk n (∆ m ) × K → C that extend to sk n +1 (∆ m ) × K . The functor Φ Kn is induced by the canonical mapsk n (∆ m × K ) → sk n (∆ m ) × K which is an isomorphism if d ≤ max( n − m, Kn issurjective in simplicial degrees ≤ n − d + 1. Similarly, the mapsk n − (∆ m × K ) → sk n − (∆ m ) × K is an isomorphism if d ≤ max( n − m − , m < n − d . (cid:3) Remark 3.15.
The case d = 0 is both special and essentially trivial, since thefunctor Φ Kn is an isomorphism in this case. Proposition 3.16.
Let C be an ∞ -category, K a finite dimensional simplicial setof dimension d > , and let n ≥ be an integer. Then for every lifting problem ∂ ∆ m u / / (cid:15) (cid:15) h n ( C K ) Φ Kn (cid:15) (cid:15) ∆ m σ / / : : ✈✈✈✈✈ (h n C ) K where m ≤ n − d + 1 (resp. m < n − d ), there is a (unique) filler ∆ m → h n ( C K ) which makes the diagram commutative.Proof. The case m < n − d + 1 is a direct consequence of Lemma 3.14. For m = n − d + 1 ≤ n , Lemma 3.14 shows that there is a lift τ : ∆ m → h n ( C K ) of σ ,represented by a map τ ′ : ∆ m × K → C . Since the maps u, γ n ◦ τ ′| ∂ ∆ m : ∂ ∆ m → h n ( C K )become equal after composition with Φ Kn , it follows that they correspond to theequivalence classes of maps e u, τ ′| ∂ ∆ m × K : ∂ ∆ m × K → C which are homotopic relative to sk n − ( ∂ ∆ m × K ). Let H : ( ∂ ∆ m × K ) × J → C bea homotopy from τ ′| ∂ ∆ m × K to e u , where J = N (0 ⇄
1) denotes the Joyal intervalobject. Using standard arguments, this homotopy can be extended to a homotopy H ′ : (∆ m × K ) × J → C from τ ′ = H ′| ∆ m × K ×{ } which is constant on sk n − (∆ m × K ). Then the map e τ : = H ′| ∆ m × K ×{ } : ∆ m × K → C extends e u and its equivalence class in h n ( C ) Km is equal to σ , so it defines a filler asrequired. (cid:3) As a consequence of Proposition 3.16, we obtain the following result about thecomparison between the n -categories h n ( C K ) and h n ( C ) K . Corollary 3.17.
Let C be an ∞ -category, K a finite dimensional simplicial set ofdimension d > , and let n ≥ d be an integer. The functor Φ Kn : h n ( C K ) → (h n C ) K is essentially surjective and for every pair of objects F, G in h n ( C K ) , the inducedmap between mapping spaces Map h n ( C K ) ( F, G ) −→ Map (h n C ) K (Φ Kn ( F ) , Φ Kn ( G )) is ( n − d ) -connected. As a consequence, Φ Kn induces an equivalence: (3.18) h n − d (cid:0) Fun( K, C ) (cid:1) ≃ h n − d (cid:0) Fun( K, h n C ) (cid:1) . IGHER HOMOTOPY CATEGORIES AND K -THEORY 15 Now assume that C is an ∞ -category which admits K -colimits, where K is asimplicial set. We have colimit-functors:Fun( K, C ) colim K / / γ n (cid:15) (cid:15) C γ n (cid:15) (cid:15) h n Fun( K, C ) h n (colim K ) / / h n C . Assuming also that C and K are as in Corollary 3.17, and passing to the homotopy( n − d )-categories as in (3.18), we obtain the following corollary. Corollary 3.19.
Let C be an ∞ -category, K a finite dimensional simplicial set ofdimension d > , and let n ≥ d be an integer. Suppose that C admits K -colimits.Then there is an adjoint pair h n − d (colim K ) : h n − d (cid:0) Fun( K, h n C ) (cid:1) ⇄ h n − d ( C ) : h n − d ( c ) where c denotes the constant K -diagram functor. Remark 3.20.
Corollary 3.19 produces a truncated K -colimit functor for h n C :h n − d (colim K ) : h n − d (cid:0) Fun( K, h n C ) (cid:1) → h n − d ( C ) . According to Proposition 2.7, there is a (non-canonical) section ǫ : sk n − d +1 h n − d ( C K ) → sk n − d +1 ( C K ) . Using this section ǫ , we may consider the partial K -colimit functor for h n C (whichdepends on C and ǫ ): sk n − d +1 (cid:0) Fun( K, h n C ) (cid:1) → h n ( C )that is defined by the following diagramsk n − d +1 (cid:0) Fun( K, h n C ) (cid:1) / / (cid:15) (cid:15) sk n − d +1 ( C ) / / C γ n / / h n ( C )sk n − d +1 (cid:0) h n − d Fun( K, h n C ) (cid:1) (3.18) (cid:15) (cid:15) sk n − d +1 (cid:0) h n − d ( C K ) (cid:1) ǫ / / sk n − d +1 (cid:0) C K (cid:1) O O / / C K / / colim K O O h n ( C K ) . h n (colim K ) O O Proposition 3.21.
Let C be an ∞ -category with weak K -colimits of order k , where K is a simplicial set of dimension d > , and let n ≥ be an integer. Then h n C hasweak K -colimits of order ℓ = min( n − d, k ) . Moreover, the functor γ n : C → h n C sends weak K -colimits of order k to weak K -colimits of order ℓ .Proof. We may assume that n ≥ d and therefore the functor C K → (h n C ) K issurjective on objects. Then it suffices to prove the second claim. Let G : K ⊲ → C be a weak colimit of F = G | K of order k with cone object x ∈ C . We claim thatthe canonical map Map (h n C ) K⊲ ( G, c y ) → Map (h n C ) K ( F, c y )is ℓ -connected for every y ∈ C . Note that there is a k -connected map:Map (h n C ) K⊲ ( G, c y ) ≃ Map h n C ( x, y ) ≃ Map h n ( C K⊲ ) ( G, c y ) → Map h n ( C K ) ( F, c y ) . Hence it suffices to show that the canonical mapMap h n ( C K ) ( F, c y ) → Map (h n C ) K ( F, c y )is ( n − d )-connected. This follows from Corollary 3.17. (Alternatively, note thatthe last map is identified with the canonical map from the ( n − K op -limit of ∞ -groupoids to the K op -limit of the ( n − ∞ -groupoids:h n − (cid:0) lim K op Map C ( F ( − ) , y ) (cid:1) → lim K op h n − (cid:0) Map C ( F ( − ) , y ) (cid:1) . An inductive argument on d shows that the map is ( n − d )-connected.) (cid:3) Remark 3.22.
A different proof of Proposition 3.21 is also possible using elemen-tary lifting arguments based on Proposition 3.16.
Corollary 3.23.
Let C be an ∞ -category which admits finite colimits.(a) The homotopy n -category h n C admits finite coproducts and weak pushoutsof order n − . Moreover, the functor γ n : C → h n C preserves coproductsand sends pushouts in C to weak pushouts of order n − .(b) Suppose that γ n : C → h n C preserves finite colimits. Then C is equivalentto an n -category.Proof. (a) h n C admits finite coproducts by Remark 3.15. The existence and preser-vation of higher weak pushouts is a consequence of Proposition 3.21. (b) is a con-sequence of [28, Corollary 3.3.5]. (cid:3) Example 3.24.
Let C be an ∞ -category which has pushouts and let K denote the(nerve of the) “corner” category p : = ∆ ∪ ∆ ∆ . The functorΦ K : h ( C K ) → (h C ) K is surjective on objects and full. By Corollary 3.17, the pushout-functor on C induces a truncated pushout-functor:h (colim K ) : h (cid:0) Fun( K, h C ) (cid:1) → h ( C )which is a left adjoint to the constant diagram functor. Furthermore we have amap as follows,(3.25) sk (cid:0) Fun( K, h C ) (cid:1) → h ( C )which sends F : K → h C to the pushout of a choice of a lift e F : K → C . This issimply regarded as a map from the set of 0-simplices. Moreover, (3.25) extendsfurther to the 1-skeleton, but this involves non-canonical choices which are notunique even up to homotopy. As explained in Remark 3.20, an extension of this typecan be obtained from a section ǫ : sk h ( C K ) → sk ( C K ) . The fact that this processcannot be continued to higher dimensional skeleta relates to the non-functorialityof the weak pushouts in the homotopy category.More generally, for n ≥
1, there is a left adjoint truncated pushout-functor:h n − (cid:0) Fun( K, h n C ) (cid:1) → h n − ( C )and partial pushout-functors:sk n (cid:0) Fun( K, h n C ) (cid:1) → h n ( C )which define weak pushouts of order n − IGHER HOMOTOPY CATEGORIES AND K -THEORY 17 Higher derivators
Basic definitions and properties.
We recall that
Cat ∞ denotes the cate-gory of ∞ -categories, regarded as enriched in ∞ -categories. Let Dia denote a fullsubcategory of
Cat ∞ which has the following properties:(Dia 0) Dia contains the (nerves of) finite posets.(Dia 1)
Dia is closed under finite coproducts and under pullbacks along an innerfibration.(Dia 2) For every X ∈ Dia and x ∈ X , the ∞ -category X /x is in Dia .(Dia 3)
Dia is closed under passing to the opposite ∞ -category.The main examples of such subcategories of Cat ∞ are the following:(i) The full subcategory of (nerves of) finite posets.(ii) The full subcategory of (ordinary) finite direct categores D ir f . (We recallthat an ordinary category C is called finite direct if its nerve is a finitesimplicial set.)(iii) The full subcategory Cat n ⊂ Cat ∞ of n -categories for any n ≥ Cat ∞ .We denote by Dia op the opposite category taken 1-categorically, that is, the enrich-ment of Dia op is given by:Hom Dia op ( X , Y ) = Hom Dia ( Y , X ) = Fun( Y , X ) . Definition 4.1. An ∞ - prederivator with domain Dia is an enriched functor D : Dia op → Cat ∞ . An ∞ -prederivator D with domain Dia is an n - prederivator if it factors through theinclusion Cat n ⊂ Cat ∞ , that is, D is an enriched functor D : Dia op → Cat n . A strict morphism of ∞ -prederivators is a natural transformation F : D → D ′ between enriched functors. Thus, we obtain a category of ∞ -prederivators, denotedby PreDer ∞ , which is enriched in ∞ -categories. For any n ≥
1, there is a full sub-category
PreDer n ⊂ PreDer ∞ spanned by the n -prederivators. In the same waythat the classical theory of (pre)derivators is founded on a ((2,2)=)2-categoricalcontext, the general theory of ∞ -prederivators involves an ( ∞ , ∞ -prederivators, because these em-ploy non-trivial 2-morphisms in Cat ∞ , but these will not be needed in this paper. Notation.
Let D be an ∞ -prederivator with domain Dia and let u : X → Y be afunctor in Dia . We will often denote the induced functor D ( u ) : D ( Y ) → D ( X ) by u ∗ . Moreover, if i Y ,y : ∆ → Y is the inclusion of the object y ∈ Y and F ∈ D ( Y ),we will often denote the object D ( i Y ,y )( F ) in D (∆ ) by F y . Example 4.2.
An 1-prederivator with domain
Cat is a prederivator in the usualsense [23, 17]. Example 4.3.
Let C be an ∞ -category. There is an associated ∞ -prederivator(with domain Dia ) defined by D ( ∞ ) C : Dia op → Cat ∞ , X Fun( X , C ). Moreover, forany n ≥ D ( n ) C : Dia op → Cat n , X h n (cid:0) Fun( X , C ) (cid:1) defines an n -prederivator. Definition 4.4. An ∞ -prederivator D : Dia op → Cat ∞ is a left ∞ - derivator if itsatisfies the following properties:(Der 1) For every pair of ∞ -categories X and Y in Dia , the functor induced by theinclusions of the factors to the coproduct X ⊔ Y , D ( X ⊔ Y ) → D ( X ) × D ( Y ) , is an equivalence. Moreover, D ( ∅ ) is the final ∞ -category ∆ .(Der 2) For every ∞ -category X in Dia , the functor( i ∗ X ,x = D ( i X ,x )) x ∈ X : D ( X ) → Y x ∈ X D (∆ )is conservative, i.e., it detects equivalences. We recall that i X ,x : ∆ → X isthe functor that corresponds to the object x ∈ X .(Der 3) For every morphism u : X → Y in Dia , the functor u ∗ = D ( u ) : D ( Y ) → D ( X )admits a left adjoint: u ! : D ( X ) → D ( Y ) . (Der 4) Given u : X → Y in Dia and y ∈ Y , consider the following pullback diagramin Dia , u /yp u/y (cid:15) (cid:15) j u/y / / X u (cid:15) (cid:15) Y /y q Y /y / / Y . Then the canonical base change natural transformation: c u,y : ( p u/y ) ! j ∗ u,y −→ q ∗ Y /y u ! is a natural equivalence of functors.We define right ∞ -derivators dually. Definition 4.5. An ∞ -prederivator D : Dia op → Cat ∞ is a right ∞ - derivator if itsatisfies (Der1)–(Der2) as stated above together with the following dual versions of(Der3)–(Der4):(Der3)* For every morphism u : X → Y in Dia , the functor u ∗ = D ( u ) : D ( Y ) → D ( X )admits a right adjoint: u ∗ : D ( X ) → D ( Y ) . (Der4)* Given u : X → Y in Dia and y ∈ Y , consider the following pullback diagramin Dia , u y/p y/u (cid:15) (cid:15) j y/u / / X u (cid:15) (cid:15) Y y/ q y/ Y / / Y . IGHER HOMOTOPY CATEGORIES AND K -THEORY 19 Then the canonical base change natural transformation: c ′ u,y : q ∗ y/ Y u ∗ −→ ( p y/u ) ∗ j ∗ y/u is a natural equivalence of functors. Example 4.6.
Let D : Dia op → Cat ∞ be a left ∞ -derivator. Then the ∞ -prederivator D ( − op ) op : Dia op → Cat ∞ , X D ( X op ) op is a right ∞ -derivator. Definition 4.7. An ∞ -prederivator D : Dia op → Cat ∞ is an ∞ - derivator if it isboth a left and a right ∞ -derivator.We also specialize these definitions to n -prederivators as follows. Definition 4.8. An ∞ -prederivator D : Dia op → Cat ∞ is a (left, right) n - derivator if it is an n -prederivator and a (left, right) ∞ -derivator. Example 4.9.
A (left, right) 1-derivator with domain
Cat is a (left, right) deriva-tor in the usual sense [23, 17]. Example 4.10.
Let D : Dia op → Cat n be an n -prederivator, where n ∈ Z ≥ ∪ {∞} .For any k < n , there is an associated k -prederivator:h k D : Dia op → Cat n h k −→ Cat k . If D is a left (right) n -derivator, then h k D is a left (right) k -derivator.The axioms of Definition 4.4 have the following consequence which is a usefulstrong version of (Der4) and (Der4)* and identifies a larger class of squares forwhich the base change transformations are equivalences. Similar results are knownfor ∞ -categories and for ordinary (1-)derivators (see, for example, [4] and [22, 17]). Proposition 4.11.
Let D be an ∞ -derivator with domain Dia . Consider a pullbacksquare in
Dia : Z p (cid:15) (cid:15) j / / X u (cid:15) (cid:15) W q / / Y . (1) The canonical base change transformation: p ! j ∗ −→ q ∗ u ! is a natural equivalence if u is a cocartesian fibration or if q is a cartesianfibration.(2) The canonical base change transformation: q ∗ u ∗ −→ p ∗ j ∗ is a natural equivalence if u is a cartesian fibration or if q is a cocartesianfibration.Proof. Using the duality D D ( − op ) op , it suffices to prove only (1). Supposethat u is a cocartesian fibration. Applying (Der 2) and the naturality properties of base change transformations, it suffices to prove the claim only in the case where W = ∆ : X yp (cid:15) (cid:15) j / / X u (cid:15) (cid:15) ∆ i y / / Y . Using the following factorization of this square and applying (Der4), it suffices toprove the claim for the left square in the following diagram X y j / / p (cid:15) (cid:15) X /yp u/y (cid:15) (cid:15) j u,y / / X u (cid:15) (cid:15) ∆ i ( y = y ) / / Y /y q Y ,y / / Y . Note that the horizontal functors in the left square admit left adjoints ℓ : X /y → X y ℓ : Y /y → ∆ because u is a cocartesian fibration and Y /y has a terminal object ( y = y ). Thus,after applying D , we obtain pairs of adjoint functors ( j ∗ , ℓ ∗ ) and ( i ∗ ( y = y ) , ℓ ∗ ). There-fore the base change transformation between compositions of left adjoints:(4.12) p ! j ∗ −→ i ∗ ( y = y ) ( p u/y ) ! is conjugate to the natural equivalenceid : p ∗ u/y ( ℓ ) ∗ ≃ ( ℓ ) ∗ p ∗ which then implies that (4.12) is an equivalence, as claimed.Suppose now that q is a cartesian fibration. We show that the conjugate basechange transformation(4.13) u ∗ q ∗ → j ∗ p ∗ is a natural equivalence. Again, by (Der2), it suffices to restrict to the case where X = ∆ : Z j (cid:15) (cid:15) p / / W q (cid:15) (cid:15) ∆ u / / Y . The proof of (4.13) in this case is obtained similarly by dualizing the arguments inthe previous proof. (cid:3)
As in the theory of ordinary derivators, there is an additional axiom which is veryuseful in practice. Before we state this axiom, we recall that as in the classical case,for every ∞ -prederivator D and X , Y ∈ Dia , there is an underlying ( X − ) diagramfunctor : dia X , Y : D ( X × Y ) → Fun( X , D ( Y )) IGHER HOMOTOPY CATEGORIES AND K -THEORY 21 which is the adjoint of the following composition: X ∼ = Fun(∆ , X ) ( −× Y ) −−−−→ Fun( Y , X × Y ) D −→ Fun( D ( X × Y ) , D ( Y )) . We say that a functor F : C → D between ∞ -categories is n – full if it restricts to( n − Definition 4.14.
Let D be an ∞ -prederivator and let n ∈ Z ≥ ∪ {∞} . We saythat D is n-strong if the following axiom is satisfied:(Der 5 n ) For every X ∈ Dia , the underlying diagram functordia ∆ , X : D (∆ × X ) → Fun(∆ , D ( X ))is n -full and essentially surjective. Remark 4.15.
Similarly to the case of ordinary (pre)derivators, we may also con-sider a stronger form of the last axiom (cf. [18]) which states that the underlyingdiagram functor dia I , X : D ( I × X ) → Fun( I , D ( X ))is n -full and essentially surjective for every I in Dia which is equivalent in the Joyalmodel structure to a finite 1-dimensional simplicial set.Following the definitions of pointed and stable (1-)derivators [23, 17], we alsodefine pointed and stable ∞ -(pre)derivators as follows. Definition 4.16. An ∞ -prederivator D : Dia op → Cat ∞ is called pointed if it liftsto the ∞ -category Cat ∞ , ∗ of pointed ∞ -categories and functors which preserve zeroobjects. An ∞ -derivator D : Dia op → Cat ∞ is called stable if it is pointed and theassociated 1-derivator h D is stable.4.2. Examples of n -derivators. The examples of n -derivators we are mainly in-terested in are those where the underlying prederivator arises from an ∞ -categoryas in Example 4.3. Let Dia ⊂ Cat ∞ be a fixed full subcategory satifying the condi-tions (Dia 0)–(Dia 3). Proposition 4.17.
Let C be an ∞ -category and let n ∈ Z ≥ ∪ {∞} .(a) The n -prederivator D ( n ) C satisfies (Der 1), (Der 2) and (Der 5 n ). Moreover,the underlying diagram functor dia I , X : D ( n ) C ( I × X ) → Fun( I , D ( n ) C ( X )) is n -full and essentially surjective for every I in Dia which is equivalent (inthe Joyal model structure) to an 1-dimensional simplicial set.(b) Suppose that C admits X -colimits (resp. X -limits) for any X ∈ Dia . Then D ( n ) C satisfies (Der 3) and (Der 4) (resp. (Der 3)* and (Der 4)*). As aconsequence, D ( n ) C is a left (resp. right) n -derivator which is n -strong.Proof. (a) (Der 1) is obvious. (Der 2) says that the equivalences in h n ( C X ), X ∈ Dia ,are given pointwise, which holds by a theorem of Joyal [19, Chapter 5]. (Der 5 n )follows from Corollary 3.17. The second claim also follows from Corollary 3.17because we may replace I by an 1-dimensional simplicial set. (b) It suffices to showthat (Der 3) and (Der 4) hold for the ∞ -prederivator D ( ∞ ) C , that is, it suffices toshow that C admits Kan extensions along functors in Dia and that Kan extensions are given pointwise by the usual formulas as axiomatized in (Der 4). These claimsare established in [4, 6.4.7] (see also [21, 4.3.2-4.3.3]). (cid:3)
It is often possible to reduce statements about ∞ -derivators to correspondingknown statements about 1-derivators. This happens, for example, in the case ofstatements which involve the detection of equivalences. The following propositionshows another instance of this phenomenon and it produces many examples of n -derivators from known examples of 1-derivators. Theorem 4.18.
Let C be an ∞ -category. The following are equivalent:(1) C admits X -colimits and X -limits for every X ∈ Dia .(2) D ( ∞ ) C is an ∞ -derivator.(3) D C = D (1) C is an -derivator.Proof. (1) ⇒ (2) follows from Proposition 4.17 and (2) ⇒ (3) is obvious. Weprove (3) ⇒ (1). We restrict to showing that C admits X -colimits as the case of X -limits can be treated similarly by duality. Let F : X → C be an X -diagram andlet G : X ⊲ → C be the diagram which is the image of F ∈ D C ( X ) under u ! : D C ( X ) −→ D C ( X ⊲ )where u : X → X ⊲ is the canonical inclusion. As a consequence of (Der 4), the functor u ! is fully faithful because u is so. In particular, there is a canonical isomorphism F ∼ = u ∗ u ! ( F ) and therefore we may assume that G is an extension of the diagram F . We claim that G is a colimit diagram in C for the functor F . For this, it sufficesto prove that for each sieve between finite posets v : Y ′ → Y in Dia , the canonicalmap(4.19) Map( Y , Map C X ⊲ ( G, c y )) (cid:15) (cid:15) Map( Y ′ , Map C X ⊲ ( G, c y )) × Map( Y ′ , Map C X ( F,c y )) Map( Y , Map C X ( F, c y ))is a π -isomorphism, where c y denotes the constant functor at y ∈ C in the respectivefunctor ∞ -category. We will do this by expressing π of the domain and the targetof this map (4.19) in terms of morphism sets in the (1-)category D C ( X ⊲ × Y ) andthen using the derivator properties of D C .First, note that we have a canonical isomorphism of morphism sets: D C ( X ⊲ × Y )( π ∗ X ⊲ , Y ( G ) , c y ) = π (cid:0) Map C X ⊲ × Y ( π ∗ X ⊲ , Y ( G ) , c y ) (cid:1) ∼ = π (cid:0) Map( Y , Map C X ⊲ ( G, c y )) (cid:1) (4.20)where π X ⊲ , Y : X ⊲ × Y → X denotes the projection functor. Similarly, we have canon-ical isomorphisms D C ( X × Y )( π ∗ X , Y ( F ) , c y ) ∼ = π (cid:0) Map( Y , Map C X ( F, c y )) (cid:1) D C ( X × Y ′ )( π ∗ X , Y ′ ( F ) , c y ) ∼ = π (cid:0) Map( Y ′ , Map C X ( F, c y )) (cid:1) (4.21)where π X , Y and π X , Y ′ denote again the projection functors. IGHER HOMOTOPY CATEGORIES AND K -THEORY 23 Then consider the following pullback diagram in
Dia : X × Y π X , Y / / u × (cid:15) (cid:15) X u (cid:15) (cid:15) X ⊲ × Y π X ⊲, Y / / X ⊲ . Since the horizontal functors are cartesian fibrations, it follows from (Der 4) andProposition 4.11(1) that the canonical base change morphism in D C ( X ⊲ × Y ):(4.22) π ∗ X ⊲ , Y ( G ) = π ∗ X ⊲ , Y u ! ( F ) ∼ = ← ( u × ! π ∗ X , Y ( F )is an isomorphism.Let i : A = X ⊲ × Y ′ ∪ X × Y ′ X × Y ⊂ X ⊲ × Y denote the full subcategory. This isagain in Dia by (Dia 1) and (Dia 3) because it can be described as the pullback of afunctor X ⊲ × Y → ∆ × ∆ along the ‘upper corner’ inclusion ∆ ∪ ∆ ∆ → ∆ × ∆ .Note that using (4.21), we can identify the set of components of the target of (4.19)canonically with the morphism set: D C ( A )( i ∗ π ∗ X ⊲ , Y ( G ) , c y ) = D C ( A )( i ∗ π ∗ X ⊲ , Y u ! ( F ) , c y ) ∼ = π ( target of (4.19))(4.23)and that the map (4.19) on π agrees using the identifications (4.20) and (4.23)with the map defined by the restriction functor i ∗ : D C ( X ⊲ × Y ) → D C ( A )Since i is full, it follows from (Der 4) that the unit transformation 1 → i ∗ i ! of theadjunction ( i ! , i ∗ ) is a natural isomorphism. Therefore it suffices to show that thecounit morphism(4.24) i ! i ∗ (cid:0) π ∗ X ⊲ , Y ( G ) (cid:1) → π ∗ X ⊲ , Y ( G )is an isomorphism.Consider the following pullback diagram in Dia : X × Y j (cid:15) (cid:15) π X , Y / / X u (cid:15) (cid:15) A q = π X ⊲, Y i / / X ⊲ . The bottom functor q is a cartesian fibration because it is the composition of carte-sian fibrations. Therefore it follows from (Der 4) and Proposition 4.11(1) that thecanonical base change morphism in D C ( A ):(4.25) q ∗ ( G ) = q ∗ u ! ( F ) ∼ = ← j ! π ∗ X , Y ( F )is an isomorphism. As a consequence of (4.22) and (4.25), we obtain canonicalisomorphisms as follows, π ∗ X ⊲ , Y ( G ) = π ∗ X ⊲ , Y u ! ( F ) ∼ = ( u × ! π ∗ X , Y ( F ) ∼ = i ! j ! π ∗ X , Y ( F ) ∼ = i ! q ∗ u ! ( F )= i ! i ∗ π ∗ X ⊲ , Y ( G ) . This implies that (4.24) is an isomorphism and therefore, using the adjunction( i ! , i ∗ ) as explained above, it follows that the map (4.19) is a π -isomorphism. (cid:3) Remark 4.26.
We point out a significant simplification of the proof of Theorem4.18 in the case where
Dia is large enough so that it can detect equivalences of ∞ -groupoids, i.e., in the case where the following holds: a map of ∞ -groupoids X → Z is an equivalence if and only if π (Map( Y , X )) −→ π (Map( Y , Z ))is an isomorphism for every Y ∈ Dia . This happens, for example, when
Dia containsall posets – and not just the finite ones. Assuming that
Dia is large enough in thissense, then we may restrict to the case Y ′ = ∅ in the proof above, in which casethe proof becomes more immediate.5. K -theory of higher derivators Recollections.
We first recall the ∞ -categorical version of Waldhausen’s S • -construction [37, 2]. Let C be a pointed ∞ -category which admits finite colimits.For every n ≥
0, let Ar[ n ] denote the (nerve of the) category of morphisms of theposet [ n ]. The ∞ -category S n C is the full subcategory of Fun(Ar[ n ] , C ) spanned bythe objects F : Ar[ n ] → C such that:(i) F ( i → i ) is a zero object for all i ∈ [ n ].(ii) For every i ≤ j ≤ k , the following diagram in C , F ( i → j ) / / (cid:15) (cid:15) F ( i → k ) (cid:15) (cid:15) F ( j → j ) / / F ( j → k )is a pushout.The construction is clearly functorial in [ n ], n ≥
0, and S • C defines a simplicialobject of pointed ∞ -categories, which is functorial in C with respect to functorswhich preserve zero objects and finite colimits. We denote by S ≃• C the associatedsimplicial object of pointed ∞ -groupoids, which is obtained by passing pointwise tothe maximal ∞ -subgroupoids of S • C . For n ≥
1, the ∞ -groupoid S ≃ n C is equivalentto Map(∆ n − , C ). Moreover, we have S ≃ C ≃ ∆ and we may regard the geometricrealization | S ≃• C | as canonically pointed by a zero object in C . The Waldhausen K -theory of C is defined to be the space: K ( C ) : = Ω | S ≃• C | . If C arises from a good Waldhausen category, this definition of K -theory agreesup to homotopy equivalence with the Waldhausen K -theory of the correspondingWaldhausen category (see [2]). The definition of K -theory is functorial with respectto functors F : C → C ′ which preserve zero objects and finite colimits.Following [37, Lemma 1.4.1] and [25, Proposition 4.2.1], we consider also thefollowing simpler model for Waldhausen K -theory. Restricting to the objects ofS • C pointwise, we obtain a simplicial set s • C : ∆ op → Set , [ n ] s n C : = (S n C ) . There is a canonical comparison map, given by the inclusion of objects, ι : Ω | s • C | −→ Ω | S ≃• C | = K ( C ) . IGHER HOMOTOPY CATEGORIES AND K -THEORY 25 Proposition 5.1.
The comparison map ι is a weak equivalence.Proof. The proof is essentially the same as the proof of [37, Lemma 1.4.1 andCorollary] (see also [25, Proposition 4.2.1]). (cid:3)
Derivator K -theory. We extend the definition of derivator K -theory ofMaltsiniotis [23] and Garkusha [13, 14] to general pointed left ∞ -derivators. Asin the case of ordinary derivators, this definition is based on the following intrinsicnotion of cocartesian square.Let i : p = ∆ ∪ ∆ ∆ → (cid:3) = ∆ × ∆ denote the ‘upper corner’ inclusion. Forany left ∞ -derivator D , we have an adjunction: i ! : D ( p ) ⇄ D ( (cid:3) ) : i ∗ . Definition 5.2.
Let D : Dia op → Cat ∞ be a left ∞ -derivator with domain Dia . Anobject F ∈ D ( (cid:3) ) is called cocartesian if the canonical morphism i ! i ∗ ( F ) → F is an equivalence in D ( (cid:3) ).Let D be a pointed left ∞ -derivator (with domain Dia ). Before we define the K -theory of D , we first need to introduce some more notation: for every 0 ≤ i ≤ j ≤ k ≤ n , we denote by i i,j,k : (cid:3) → Ar[ n ] the inclusion of the following square inAr[ n ]: ( i → j ) / / (cid:15) (cid:15) ( i → k ) (cid:15) (cid:15) ( j → j ) / / ( j → k ) . We define S n D to be the full subcategory of D (Ar[ n ]) which is spanned by theobjects F ∈ D (Ar[ n ]) such that:(i) F ( i → i ) is a zero object for all i ∈ [ n ].(ii) For every i ≤ j ≤ k , the object i ∗ i,j,k ( F ) ∈ D ( (cid:3) ), which may be depicted asfollows: F ( i → j ) / / (cid:15) (cid:15) F ( i → k ) (cid:15) (cid:15) F ( j → j ) / / F ( j → k ) , is cocartesian in D ( (cid:3) ).The assignment [ n ] S n D defines a simplicial object of pointed ∞ -categories.Moreover, it is natural with respect to strict morphisms between pointed left ∞ -derivators which preserve the zero objects and cocartesian squares.Let S ≃• D denote the simplicial object of pointed ∞ -groupoids, which is obtainedby passing pointwise to the maximal ∞ -subgroupoids of S • D . We have S ≃ D ≃ ∆ and we regard the geometric realization | S ≃• D | as based at a zero object of D (∆ ). Definition 5.3.
Let D : Dia op → Cat ∞ be a pointed left ∞ -derivator with domain Dia . The derivator K -theory of D is defined to be the space: K ( D ) : = Ω | S ≃• D | . We note that the definition of derivator K -theory is functorial with respectto strict morphisms F : D → D ′ which preserve the zero objects and cocartesiansquares. Moreover, derivator K -theory is invariant under those strict morphismswhich are pointwise equivalences of ∞ -categories. Remark 5.4.
Waldhausen’s Additivity Theorem [37] establishes one of the funda-mental properties of Waldhausen K -theory. The analogue of this theorem has beenestablished for the derivator K -theory of stable 1-derivators by Cisinski–Neeman[6], confirming one of Maltsiniotis’ conjectures in [23]. It would be interesting toknow if the additivity theorem holds more generally for the derivator K -theory ofpointed left ∞ -derivators.5.3. Comparison with Waldhausen K -theory. Let C be a pointed ∞ -categorywith finite colimits. Applying the homotopy n -category functor pointwise to thesimplicial object [ k ] S ≃ k C , we obtain a new simplicial object of (pointed) ∞ -groupoids, h n S ≃• C : ∆ op → Grpd ∞ , [ k ] h n (cid:0) S ≃ k C (cid:1) , and there is a canonical comparison map:S ≃• C −→ h n (cid:0) S ≃• C (cid:1) . Let D ( n ) C be the pointed left n -derivator associated to C with domain D ir f (seeProposition 4.17). As a consequence of the natural identificationh n (cid:0) S ≃• C (cid:1) ≃ S ≃• D ( n ) C , we obtain a canonical comparison map from Waldhausen to derivator K -theory: µ n : K ( C ) → K ( D ( n ) C ) . In addition, the natural morphisms of simplicial objects h n (cid:0) S ≃• C (cid:1) → h n − (cid:0) S ≃• C (cid:1) ,for n >
1, define a tower of derivator K -theories for C which is compatible with thecomparison maps µ n : K ( C ) µ n (cid:15) (cid:15) µ n − & & ▲▲▲▲▲▲▲▲▲▲ µ , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ · · · / / K ( D ( n ) C ) / / K ( D ( n − C ) / / · · · / / K ( D (1) C ) . In the case of ordinary derivators, Maltsiniotis conjectured in [23] that the com-parison map µ is a weak equivalence for exact categories. The comparison map µ was subsequently studied in [14, 24, 25, 26]. It is known (see [26]) that µ isnot a weak equivalence for general C , and moreover, that it will fail to be a weakequivalence even for exact categories if derivator K -theory satisfies localization –a property which was also conjectured by Maltsiniotis [23].We prove a general result about the connectivity of the comparison map µ n forgeneral C and n ≥
1. The connectivity estimate is also a small improvement of theknown estimate for n = 1 that was shown by Muro [24]. Theorem 5.5.
Let C be a pointed ∞ -category which admits finite colimits. Thenthe comparison map µ n : K ( C ) → K ( D ( n ) C ) is ( n + 1) -connected. We will need the following useful elementary fact about simplicial spaces.
IGHER HOMOTOPY CATEGORIES AND K -THEORY 27 Lemma 5.6.
Let f • : X • → Y • be a map of simplicial spaces. Suppose that f k is ( m − k ) -connected for every k ≥ . Then the map || f • || : || X • || → || Y • || is m -connected. (Here || − || denotes the fat geometric realization of a simplicial space.)Proof. See [10, Lemma 2.4]. (cid:3)
Proof. (of Theorem 5.5) By Lemma 5.6, it suffices to show that the map of ∞ -groupoids S ≃ k C → h n (cid:0) S ≃ k C (cid:1) is ( n + 2 − k )-connected for all k ≥
0. This holds since the map is an equivalencefor k = 0 and ( n + 1)-connected for k > (cid:3) Remark 5.7.
The main result of [26, Theorem 1.2] shows that the comparisonmap µ is not a π -isomorphism in general. (In addition, a closer inspection ofthe proofs in [26] also shows that the map µ will not be 3-connected if derivator K -theory satisfies localization.) It seems likely that the connectivity estimate ofTheorem 5.5 is best possible in general. Remark 5.8.
By [14, Theorem 7.1], the comparison map µ is π ∗ -split injectivein the case where C is the bounded derived category of an abelian category. Infact, it is shown [14] that there is a retraction map to µ . As a consequence, thecomparison map µ n also admits a retraction in these cases for all n ≥
1. Relatedto this, an interesting problem suggested by B. Antieau is whether µ n is a weakequivalence when C is a stable ∞ -category which admits a bounded t-structure.5.4. Waldhausen K -theory of derivators. Waldhausen K -theory for pointedleft 1-derivators was defined in [25] and it was shown that it agrees with Waldhausen K -theory for all well–behaved Waldhausen categories [25, Theorem 4.3.1]. Weconsider an analogous definition of K -theory for general pointed left ∞ -derivators.Let D be a pointed left ∞ -derivator with domain Dia . Let S •• D be the bisimpli-cial set whose set of ( n, m )-simplices S n,m D is the set of objects F ∈ Ob (cid:0) D (∆ m × Ar[ n ]) (cid:1) such that:(1) for every j : [0] → [ m ] the object ( j × id) ∗ ( F ) ∈ Ob (cid:0) D (Ar[ n ]) (cid:1) is in S n D .(2) the underlying diagram functor associated to F dia ∆ m , Ar[ n ] ( F ) : ∆ m → D (Ar[ n ])takes values in equivalences.The bisimplicial operators of S •• D are again defined using the structure of theunderlying ∞ -prederivator. Moreover, it is easy to see that the construction isfunctorial in D with respect to strict morphisms which preserve the zero objects andcocartesian squares. Since S ,m D ≃ ∆ , we may regard the geometric realization | S •• D | as based at a zero object of D (∆ ). Definition 5.9.
Let D : Dia op → Cat ∞ be a pointed left ∞ -derivator with domain Dia . The
Waldhausen K -theory of D is defined to be the space: K W ( D ) := Ω | S •• D | . Following [25], we consider also the analogue of the s • -construction in this con-text. Restricting to the objects of S • D pointwise, we obtain a simplicial set s • D : ∆ op → Set , [ n ] s n D : = S n, D = (S n D ) . There is a canonical comparison map, given by the inclusion of objects, ι : Ω | s • D | −→ Ω | S •• D | = K W ( D ) . This is the analogue of the comparison map in Proposition 5.1 for pointed left ∞ -derivators. Proposition 5.10. (a) The comparison map ι is a weak equivalence. (b) Let C be a pointed ∞ -category with finite colimits and let n ∈ Z ≥ ∪ {∞} . There is acommutative diagram of weak equivalences: Ω | s • C | ≃ (cid:15) (cid:15) Ω | Ob S • D ( n ) C | ≃ (cid:15) (cid:15) K ( C ) = Ω | S ≃• C | ≃ / / Ω | S •• D ( n ) C | = K W ( D ( n ) C ) . Proof. (a) The proof is essentially the same as the proof of [25, Proposition 4.2.1](see also [37, Lemma 1.4.1 and Corollary]). (b) The bottom map is a weak equiva-lence (independently of n !) because we have (S ≃ k C ) m = S k,m D ( n ) C . The left verticalmap is a weak equivalence by Proposition 5.1. The result follows. (cid:3) Universal property of derivator K -theory. The comparison maps { µ n } from Waldhausen K -theory to derivator K -theory can be defined more generallyfor pointed left ∞ -derivators. Given a pointed left ∞ -derivator D (with domain Dia ), the underlying diagram functors define a bisimplicial map as follows,dia ∆ m , Ar[ n ] : S n,m D → (S ≃ n D ) m which after passing to the geometric realization and taking loop spaces defines acomparison map µ : Ω | s • D | . ≃ K W ( D ) → K ( D ) . A universal property of this comparison map in the case of 1-derivators was shown in[25, Theorem 5.2.2]. More specifically, it was shown that it is homotopically initialamong all natural transformations from Waldhausen K -theory to a functor whichis invariant under (pointwise) equivalences of pointed left derivators. The proof ofthis universal property extends similarly to our present ∞ -categorical context.Let Der denote the (ordinary) category of pointed left ∞ -derivators and strictmorphisms which preserve zero objects and cocartesian squares. It will be conve-nient to work with the simpler model for the Waldhausen K -theory of derivatorsusing the s • -construction. This will be denoted by K W, Ob : Der → Top , D Ω | s • D | , where Top denotes the ordinary category of topological spaces. Then we may regardthe comparison map µ as a natural transformation between functors K W, Ob → K defined on Der . IGHER HOMOTOPY CATEGORIES AND K -THEORY 29 Definition 5.11.
The category E of invariant approximations to Waldhausen K -theory is the full subcategory of the comma category K W, Ob ↓ Top
Der spanned bythe objects η : K W, Ob → F such that F : Der → Top sends pointwise equivalencesin
Der to weak equivalences. A morphism in E K W, Ob η { { ①①①①①①①①① η ′ ●●●●●●●●● F u / / F ′ is a weak equivalence if the components of u are weak equivalences. Remark 5.12.
The category E (denoted by App in [25]) is not even locally smallin general as defined in Definition 5.11. This set–theoretical issue can be addressedby restricting to suitable small subcategories of
Der as was done in [25].We recall from [9] that an object x ∈ C in an (ordinary) category with weakequivalences ( C , W ) (satisfying in addition the “2-out-of-6” property) is homotopi-cally initial if there are functors F , F : C → C which preserve the weak equivalencesand a natural transformation φ : F ⇒ F such that:(i) F is naturally weakly equivalent to the constant functor at x ∈ C .(ii) F is naturally weakly equivalent to the identity functor on C .(iii) φ x : F ( x ) → F ( x ) is a weak equivalence.A homotopically initial object in ( C , W ) defines an initial object in the associated ∞ -category. Theorem 5.13.
The object ( µ : K W, Ob → K ) ∈ E is homotopically initial.Proof. (Sketch) The proof is similar to [25, Theorem 5.2.2] so we only sketch thedetails. Given D ∈ Der and m ≥
0, let D ≃ m denote the ∞ -prederivator whose valueat X ∈ Dia is the full subcategory Fun ≃ (∆ m , D ( X )) ⊂ Fun(∆ m , D ( X )) spanned bythe functors ∆ m → D ( X ) which take values in equivalences. This ∞ -prederivatoris pointwise equivalent to D and therefore also a pointed left ∞ -derivator. Varying m ≥
0, we obtain a simplicial object ( D ≃ m ) m ≥ in Der with D ≃ = D .For the proof of the theorem, it suffices to note that every object in E ,( η D : K W, Ob ( D ) → F ( D )) D ∈ Der is naturally weakly equivalent (as object in E ) to the composite (cid:0) K W, Ob ( D ) → || K W, Ob ( D ≃• ) || φ −→ || F ( D ≃• ) || (cid:1) D ∈ Der . Moreover, the first map above defines a natural transformation which is canonicallyidentified with µ . As a result, we have constructed a zigzag of natural transforma-tions from the constant endofunctor at µ to id E satisfying (i)–(iii). (cid:3) K -theory of homotopy n -categories Revisiting the properties of homotopy n -categories. Let C be a pointed ∞ -category with finite colimits. Then the associated homotopy n -category h n C satisfies the following:(a) h n C is a pointed n -category.(b) The suspension functor Σ C : C → C induces a functor Σ : h n C → h n C . Thisis an equivalence if and only if C is stable (see [20, Corollary 1.4.2.27]). (c) h n C admits finite coproducts and weak pushouts of order n −
1. Moreover,these are preserved by the functor γ n : C → h n C (Proposition 3.21).(d) For every x ∈ h n C , there is a natural weak pushout of order n − x (cid:15) (cid:15) / / (cid:15) (cid:15) / / Σ x. Assuming that C is a stable ∞ -category, then the adjoint equivalence (Σ C , Ω C )induces an adjoint equivalence Σ : h n C ⇄ h n C : Ω. Moreover, by duality, h n C alsosatisfies in this case the following dual versions of (c)–(d):(c) ′ h n C admits finite products and weak pullbacks of order n −
1. Moreover,these are preserved by the functor γ n : C → h n C .(d) ′ For every x ∈ h n C , there is a natural weak pullback of order n − x (cid:15) (cid:15) / / (cid:15) (cid:15) / / x. In addition, if C is a stable ∞ -category, h n C satisfies the following property:(e) A square in h n C is a weak pushout of order n − n − n >
1, weak pushouts (resp. weak pullbacks) of order n − n >
1. The validity of (e) for n = 1 can be verified by a direct argument.An attempt towards an axiomatization of the properties (a)–(d) would naturallylead to considering triples ( D , Σ : D → D , σ : D → D (cid:3) )where:(1) D is a pointed n -category.(2) Σ : D → D is an endofunctor.(3) D admits finite coproducts and weak pushouts of order n − σ sends an object x ∈ D to a weak pushout of order n − x (cid:15) (cid:15) / / (cid:15) (cid:15) / / Σ x. Specializing to the stable context, it would be natural to require in addition:(2) ′ Σ : D → D is an equivalence.(3) ′ D admits finite products and weak pullbacks of order n − D is a weak pushout of order n − n − stable n -categories , 1 ≤ n ≤ ∞ , satisfying the following properties: IGHER HOMOTOPY CATEGORIES AND K -THEORY 31 (i) Stable n -categories, exact functors and natural transformations form an( n, Cat n .(ii) For each n ≥ k , the homotopy k -category functor defines a functor fromstable n -categories to stable k -categories.(iii) For n = ∞ , the theory recovers the theory of stable ∞ -categories, exactfunctors, and natural transformations.(iv) For n = 1, the theory recovers the theory of triangulated categories, exactfunctors, and natural transformations.It seems reasonable to take properties (1)–(5) (incl. (2) ′ –(3) ′ ) as a minimal basisfor such a notion of stable n -category. Firstly, these properties are preserved afterpassing to lower homotopy categories (Proposition 3.21). Moreover, for any pointed n -category C which satisfies these properties for n >
2, the associated homotopycategory h C can be equipped with a canonical triangulated structure (the proof isessentially the same as for stable ∞ -categories in [20, 1.1.2], using the propertiesof weak pushouts of order n − > n -angulated categoryin the sense of [16]. Finally, for n = ∞ , these properties characterize stable ∞ -categories.On the other hand, concerning the case n = 1, the notion of a triangulatedstructure includes more structure than what is required in (1)–(5). This could beregarded either as a singularity that arises at the lowest level of coherence – sinceweak pushouts (of order 0) are not unique up to equivalence, they do not yieldcanonical connecting “boundary” maps, not even up to homotopy, and thereforethey do not suffice for defining distinguished triangles –, or it may in fact be de-sirable to consider additional structure in the form of fixing choices of higher weakcolimits and stipulate their properties. We will not attempt to give an axiomaticdefinition of a stable n -category in this paper, but we will suggest to consider a triplesatisfying (1)–(4) as a basic invariant of any good notion of a stable n -category.6.2. K -theory of pointed n -categories with distinguished squares. We de-fine K -theory for certain n -categories equipped with distinguished squares thatare meant to play the role of pushout squares in the definition of Waldhausen K -theory. In the case of ordinary categories, this notion of a category with distin-guished squares and its K -theory correspond to a more basic version of Neeman’s K -theory of a category with squares as defined in [27, Sections 5–7].The main example we are interested in is the homotopy n -category h n C of apointed ∞ -category C which admits finite colimits, equipped with the squares whichcome from pushout squares in C as the distinguished squares. The purpose ofintroducing K -theory for h n C is in order to identify a part of Waldhausen K -theory K ( C ) which may be recovered from h n C , regarded as an n -category withdistinguished squares. In particular, our main result (Theorem 6.5) generalizes thewell–known fact that K ( C ) can be recovered from h C , regarded as a categoryequipped with those squares which arise from pushouts in C . Definition 6.1. A pointed n - category with distinguished squares , n ≥
1, is a pair( C , T ) where C is a pointed n -category and T is a collection of weak pushout squaresin C of order n − An exact functor F : ( C , T ) → ( C ′ , T ′ ) between pointed n -categories with dis-tinguished squares is a functor F : C → C ′ which preserves zero objects and distin-guished squares. Definition 6.2.
Let C be a pointed ∞ -category with finite colimits and let n ≥ canonical structure of distinguished squares in h n C to be the collectionof squares in h n C which are equivalent in h n C to the image of a pushout square in C . By Proposition 3.21, these are weak pushout squares of order n −
1. For n > n −
1. We will denote this pointed n -category with distinguished squares by (h n C , can). Remark 6.3.
The canonical structure on h n C for n > n C and therefore depends only on h n C . For n = 1, thecanonical structure is an additional structure on h C that is canonically inducedfrom C .Let ( C , T ) be a pointed n -category with distinguished squares. Let S q ( C , T )denote the full subcategory of Fun(Ar[ q ] , C ) which is spanned by F ∈ Fun(Ar[ q ] , C )such that:(i) F ( i → i ) is a zero object for all i ∈ [ q ].(ii) For every 0 ≤ i ≤ j ≤ k ≤ m ≤ q , the following diagram in C , F ( i → k ) / / (cid:15) (cid:15) F ( i → m ) (cid:15) (cid:15) F ( j → k ) / / F ( j → m )is a distinguished square in ( C , T ).Note that the construction is functorial in q ≥ • ( C , T ) defines a simpli-cial object of pointed n -categories. Moreover, this is functorial with respect toexact functors between pointed n -categories with distinguished squares. We denoteby S ≃• ( C , T ) the associated simplicial object of ∞ -groupoids which is obtained bypassing to the maximal ∞ -subgroupoids pointwise. We have S ≃ ( C , T ) ≃ ∆ andtherefore we may regard the geometric realization | S ≃• ( C , T ) | as based at a zeroobject of C . Definition 6.4.
Let ( C , T ) be a pointed n -category with distinguished squares.The K - theory of ( C , T ) is defined to be the space: K ( C , T ) : = Ω | S ≃• ( C , T ) | . Comparison with Waldhausen K -theory. Let C be a pointed ∞ -categorywhich admits finite colimits. By Proposition 3.21, it follows that passing from S • C to the homotopy n -category pointwise defines map of simplicial objects,S • C → S • (h n C , can) , and therefore also a comparison map between K -theory spaces ρ n : K ( C ) → K (h n C , can) . Note that this comparison map factors canonically through the comparison map µ n : K ( C ) → K ( D ( n ) C ) (Subsection 5.3). By Proposition 3.21, the canonical functors γ n − : h n C → h n − C , n >
1, define exact functors (h n C , can) → (h n − C , can). IGHER HOMOTOPY CATEGORIES AND K -THEORY 33 Therefore, we obtain a tower of K -theories for C which is compatible with thecomparison maps ρ n : K ( C ) ρ n (cid:15) (cid:15) ρ n − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ρ , , ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ · · · / / K (h n C , can) / / K (h n − C , can) / / · · · / / K (h C , can) . The next result gives a general connectivity estimate for the comparison map ρ n . Theorem 6.5.
Let C be a pointed ∞ -category which admits finite colimits. For n ≥ , the comparison map ρ n : K ( C ) → K (h n C , can) is n -connected.Proof. We write h n C for (h n C , can) when the canonical structure is understoodfrom the context in order to simplify the notation. By Lemma 5.6, it suffices toshow that the map of ∞ -groupoids(6.6) S ≃ q C → S ≃ q h n C is ( n + 1 − q )-connected for every q ≥
0. The claim is obvious for q = 0. For q = 1,the map is ( n + 1)-connected. For q = 2 and n = 1, the map is 0-connected bydefinition. This completes the proof for n = 1 and we may now restrict to the case n >
1. We show that for every q >
1, the map (6.6) is ( n − ≃ q C / / ≃ (cid:15) (cid:15) S ≃ q h n C (cid:15) (cid:15) ( C ∆ q − ) ≃ / / h n ( C ∆ q − ) ≃ / / (cid:0) (h n C ) ∆ q − (cid:1) ≃ where the vertical maps are given by restriction along the inclusion map of posets[ q − ⊆ Ar[ q ] , j (0 → j + 1) . The left vertical map in (6.7) is an equivalence.The lower left map is ( n + 1)-connected (Example 2.9). The lower right map in(6.7) is n -connected by Corollary 3.17 since we may replace ∆ q − by its spine whichis 1-dimensional. Thus, it suffices to show that the right vertical map in (6.7) is n -connected for any q >
0. This claim is obvious for q = 1. For q >
1, we proceedby induction and consider the following diagramS ≃ q h n C d q / / (cid:15) (cid:15) S ≃ q − h n C (cid:15) (cid:15) (cid:0) (h n C ) ∆ q − (cid:1) ≃ d q / / (cid:0) (h n C ) ∆ q − (cid:1) ≃ . For 0 ≤ k ≤ q , let T qk ⊆ Ar[ q ] be the full subcategory which contains the subposetAr[ q − ⊆ Ar[ q ] and the elements { ( j → q ) | ≤ j ≤ k } . Moreover, let T qk denotethe full subcategory of Map( T qk , h n C ) which satisfies properties (i)–(ii) (Subsection6.2) restricted to T qk . In other words, this is the full subcategory which is spanned bythe image of S ≃ q h n C under the restriction functor Map(Ar[ q ] , h n C ) → Map( T qk , h n C ). Then we may factorize the square above as the composition of the following com-posite squareS ≃ q h n C = T qq ≃ / / (cid:15) (cid:15) T qq − / / (cid:15) (cid:15) · · · (cid:15) (cid:15) / / T q (cid:15) (cid:15) / / T q (cid:15) (cid:15) (cid:0) (h n C ) ∆ q − (cid:1) ≃ (cid:0) (h n C ) ∆ q − (cid:1) ≃ · · · · · · (cid:0) (h n C ) ∆ q − (cid:1) ≃ followed by the pullback square T q / / (cid:15) (cid:15) S ≃ q − h n C (cid:15) (cid:15) (cid:0) (h n C ) ∆ q − (cid:1) ≃ d q / / (cid:0) (h n C ) ∆ q − (cid:1) ≃ . We note that all the horizontal and vertical maps in these diagrams are given by therespective restriction functors. We claim that each map T qk → T qk − is n -connected,for any 1 ≤ k < q , from which the required result follows. To see this, we considerthe pullback of ∞ -groupoids(6.8) T qk (cid:15) (cid:15) / / (cid:0) h n ( C ) (cid:3) , can (cid:1) ≃ (cid:15) (cid:15) T qk − / / (cid:0) h n ( C ) p (cid:1) ≃ where the bottom map is given by the restriction to the subposet of T qk − ( k − → q − (cid:15) (cid:15) / / ( k − → q )( k → q − (cid:0) h n ( C ) (cid:3) , can (cid:1) ≃ ⊂ (cid:0) h n ( C ) (cid:3) (cid:1) ≃ is the full ∞ -subgroupoid that is spanned by theweak pushouts of order n −
1. The right vertical map in (6.8) is given by restrictionalong the upper corner inclusion in ∆ × ∆ . The fiber of this map at F ∈ (cid:0) h n ( C ) p (cid:1) ≃ is exactly the ∞ -groupoid of weak colimits of F of order n −
1. Since h n C admitsweak pushouts of order n − >
0, it follows that the fibers of the right verticalmap in (6.8) are ( n − n -connected and the result follows. (cid:3) Example 6.9.
Theorem 6.5 for n = 1 shows that the map ρ : K ( C ) → K (h C , can)is 1-connected. In particular, this recovers the well–known fact that K ( C ) can beobtained from h C equipped with the canonical structure of distinguished squares. Remark 6.10.
The connectivity estimate in Theorem 6.5 is best possible in gen-eral. Indeed, for n = 1 and E an exact category, the comparison map ρ for the ∞ -category associated to the Waldhausen category of bounded chain complexes in E factors through the comparison map to Neeman’s K -theory of the triangulatedcategory D b ( E ), that is, we have maps ρ : K ( E ) βα −−→ K ( d D b ( E )) → K ( D b ( E ) , can) IGHER HOMOTOPY CATEGORIES AND K -THEORY 35 where the last map is induced by the forgetful map of simplicial objects d S • D b ( E ) → S • ( D b ( E ) , can). (We refer to [27] for a nice overview of the K -theory of triangulatedcategories and details about the comparison maps α and β .) The map induced on K by βα is not injective in general by [27, Section 11, Proposition 1], see [36,Sections 2 and 5].We write P n X for the Postnikov n -section of a topological space X , that is, thecanonical map X → P n X is ( n + 1)-connected – this agrees with the homotopy n -category of an ∞ -groupoid. Theorem 6.5 implies that the functor C P n − K ( C )descends to a functor defined for (h n C , can). The following immediate corollaryconfirms a conjecture of Antieau in the case of connective K -theory [1, Conjecture8.36]. Corollary 6.11.
Let C and C ′ be stable ∞ -categories such that there is an equiv-alence (h n C , can) ≃ (h n C ′ , can) , as pointed n -categories with distinguished squares.Then there is a weak equivalence P n − K ( C ) ≃ P n − K ( C ′ ) . Remark 6.12.
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G. RaptisFakult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany
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