A Simple Criterion for Extending

Natural Transformations to Higher K-Theory

Gonçalo Tabuada^{1}

Received: January 26, 2011

Communicated by Max Karoubi

Abstract. In this article we introduce a very simple an widely applicable criterion for extending natural transformations to higher K-theory. More precisely, we prove that every natural transformation defined on the Grothendieck group and with values in an additive theory admits a unique extension to higher K-theory. As an application, the higher trace maps and the higher Chern characters originally constructed by Dennis and Karoubi, respectively, can be obtained in an elegant, unified, and conceptual way from our general results.

2010 Mathematics Subject Classification: 19D55, 18G55

Keywords and Phrases: Higher K-theory, higher trace maps, higher Chern characters, non-commutative motives

Introduction

In his foundational work, Grothendieck [10] introduced a very simple and elegant construction K_{0}, the Grothendieck group, in order to formulate a farreaching generalization of the Riemann-Roch theorem. Since then, this versatile construction spawned well-beyond the realm of algebraic geometry to become one of the most important (working) tools in mathematics. Latter, through revolutionary topological techniques, Quillen [23] extended the Grothendieck group to a whole family of higher K-theory groups K_{n}, n ≥ 0. However, in contrast with K_{0}, these higher K-theory groups are rather misterious and their computation is often out of reach. In order to capture some of its flavour, Connes, Dennis, Karoubi, and others, constructed natural transformations towards simpler theories E making use of a variety of highly involved techniques; see [6, 7, 15]. Typically, the construction of a natural transformation K_{0} ⇒ E_{0} is very simple, while its extension K_{n} ⇒ E_{n} to higher K-theory is a real “tour-de-force”. For example, the trace map K_{0} ⇒ HH_{0} consists simply in taking the trace of an idempotent, while its extension K_{n} ⇒ HH_{n} makes use of an array of tools (Hurewicz maps, group homology, assembly maps, etc) coming from topology, algebra, representation theory, etc. These phenomena motivate the following general questions:

1The author was partially supported by the grant FCT-Portugal PTDC/MAT/098317/2008.

Questions: Given a natural transformation K_{0} ⇒ E_{0}, is it possible to extend it to higher K-theory K_{n} ⇒ E_{n} ? If so, is such an extension unique ?

In this article we prove that if E verifies three very simple conditions, not only such an extension exists, but it is moreover unique. The precisely formulation of our results makes use of the language of Grothendieck derivators, a formalism which allows us to state and prove precise universal properties; see Appendix A.

1. Statement of results

A differential graded (=dg) category, over a fixed commutative base ring k, is a category enriched over cochain complexes of k-modules (morphisms sets are such complexes) in such a way that composition fulfills the Leibniz rule : d(f ∘ g) = (df) ∘ g + (−1)^{deg(f)}f ∘ (dg). Dg categories extend the classical notion of (dg) k-algebra and solve many of the technical problems inherent to triangulated categories; see Keller’s ICM address [16]. In non-commutative algebraic geometry in the sense of Bondal, Drinfeld, Kaledin, Kontsevich, Van den Bergh, and others, they are considered as differential graded enhancements of (bounded) derived categories of quasi-coherent sheaves on a hypothetic noncommutative space; see [1, 8, 9, 14, 17, 18]. Let E : dgcat → Spt be a functor, defined on the category of dg categories, and with values in the category of spectra [2]. We say that E is an additive functor if it verifies the following three conditions:

(i) filtered colimits of dg categories are mapped to filtered colimits of spectra;

(ii) derived Morita equivalences (i.e. dg functors which induce an equivalence on the associated derived categories; see [16, §4.6]) are mapped to weak equivalences of spectra;

(iii) split exact sequences (i.e. sequences of dg categories which become split exact after passage to the associated derived categories; see [24, §13]) are mapped to direct sums

\xymatrix{0 \ar[r] & A \ar[r] & B \ar[r] \ar[l] & C \ar[r] \ar[l] & 0} ↦ E(A) ⊕ E(C) ≃ E(B)

in the homotopy category of spectra. Examples of additive functors include Hochschild homology (HH), cyclic homology (HC), and algebraic K-theory (K); see [16, §5]. Recall from [25] that the category dgcat carries a Quillen model structure whose weak equivalences are the derived Morita equivalences. Given an additive functor E, we obtain then an induced morphism of derivators 𝔼 : HO(dgcat) → HO(Spt). Associated to E, we have also the composed functors

Extending natural transformations to K-theory 659

E_{n} : dgcat ⟶^{E} Spt ⟶^{π_{n}^{s}} Ab n ≥ 0 , where π_{n}^{s} denotes the n^{th} stable homotopy group functor and Ab the category of abelian groups. Our answer to the questions stated in the Introduction is:

Theorem 1.1. For any additive functor E, the natural map

(1.2)  Nat(𝕂, 𝔼) →^{∼} Nat(K_{0}, E_{0})

is bijective. In particular, every natural transformation ϕ : K_{0} ⇒ E_{0} admits a canonical extension ϕ_{n} : K_{n} ⇒ E_{n} to all higher K-theory groups.

Intuitively speaking, Theorem 1.1 show us that all the information concerning a natural transformation is encoded on the Grothendieck group. Its proof relies in an essential way on the theory of non-commutative motives, a subject envisioned by Kontsevich [17, 19] and whose development was initiated in [3, 4, 24, 25, 27, 28]. In the next section we illustrate the potential of this general result by explaining how the highly involved constructions of Dennis and Karoubi can be obtained as simple instantiations of the above theorem. Due to its generality and simplicity, we believe that Theorem 1.1 will soon be part of the toolkit of any mathematician whose research comes across the above conditions (i)-(iii).

2. Applications

2.1. Higher trace maps. Recall from [16, §5.3] the construction of the Hochschild homology complex HH(A) of a dg category A. This construction is functorial in A and so by promoting it to spectra we obtain a well-defined functor

(2.1)  HH : dgcat ⟶ Spt .

As explained in loc. cit., this functor verifies conditions (i)-(iii) and hence it is additive. Now, given a k-algebra A, recall from [20, Example 8.3.6] the construction of the classical trace map

K_{0}(A) → HH_{0}(A) = A/[A, A] .

Roughly, it is the map induced by sending an idempotent matrix to the image of its trace (i.e. the sum of the diagonal entries) in the quocient A/[A, A]. This construction extends naturally from k-algebras to dg categories (see [26]) giving rise to a natural transformation

(2.2)  K_{0} ⇒ HH_{0} .

Proposition 2.3. In Theorem 1.1 let E be the additive functor (2.1) and let ϕ be the natural transformation (2.2). Then, for every k-algebra A, the canonical extension ϕ_{n} : K_{n}(A) → HH_{n}(A) of ϕ agrees with the n^{th} trace map constructed originally by Dennis (see [20, §8.4 and §11.4]).

2.2. Higher Chern characters. Recall also from [16, §5.3] the construction of the cyclic homology complex HC(A) of a dg category A. By promoting this construction to spectra we obtain a functor

(2.4)  HC : dgcat ⟶ Spt

which verifies conditions (i)-(iii). Given a k-algebra A, recall from [20, Theorem 8.3.4] the construction of the Chern characters

ch_{0,i} : K_{0}(A) ⟶ HC_{2i}(A) i ≥ 0 .

Morally, these are the non-commutative analogues of the classical Chern character with values in even dimensional de Rham cohomology. As shown in [26] this construction extends naturally from k-algebras to dg categories giving rise to natural transformations

K_{0} ⇒ HC_{2i} i ≥ 0 .

Proposition 2.5. In Theorem 1.1 let E be the additive functor Ω^{2i}HC (obtained by composing (2.4) with the (2i)^{th}-fold looping functor on Spt) and let ϕ be the natural transformation K_{0} ⇒ (Ω^{2i}HC)_{0} = HC_{2i}. Then, for every kalgebra A, the canonical extension ϕ_{n} : K_{n}(A) → (Ω^{2i}HC)_{n}(A) = HC_{n+2i}(A) of ϕ agrees with the higher Chern character ch_{n,i} constructed originally by Karoubi (see [15, §2.27-2.36]).

3. Proof of Theorem 1.1

We start by describing the natural map (1.2). As mentioned in §1, the category dgcat carries a Quillen model structure whose weak equivalences are the derived Morita equivalences; see [25, Theorem 5.3]. Let us write Hmo for the associated homotopy category and l : dgcat → Hmo for the localization functor. According to our notation the map (1.2) sends a natural transformation Φ ∈ Nat(𝕂, 𝔼) to the natural transformation π_{0}^{s} ∘ Φ(e) ∘ l ∈ Nat(K_{0}, E_{0}). Pictorially, we have:

(3.1)  [figure: dgcat; l; Hmo; 𝔼(e); 𝕂(e); ⇓ Φ(e); Ho(Spt); π_{0}^{s}; Ab .]

The functors K, E : dgcat → Spt are additive and so the following diagrams

\xymatrix{dgcat \ar[d]_{l} \ar[r]^{K} & Spt \ar[d] & dgcat \ar[d]_{l} \ar[r]^{E} & Spt \ar[d] \\ Hmo \ar[r]_{𝕂(e)} & Ho(Spt) & Hmo \ar[r]_{𝔼(e)} & Ho(Spt)}

are commutative. Moreover, the 0^{th} stable homotopy group functor π_{0}^{s} descends to the homotopy category Ho(Spt). These facts show us that the composed horizontal functors in the above diagram (3.1) are in fact K_{0} and E_{0}. We now study the set Nat(K_{0}, E_{0}). Recall from [16, §5.1] the notion of additive invariant. Intutively, it consists of a functor defined on dgcat and with values in an additive category which verifies conditions similar to (ii)-(iii). Since by hypothesis E is additive, the composed functor

E_{0} : dgcat ⟶^{l} Hmo ⟶^{𝔼(e)} Ho(Spt) ⟶^{π_{0}^{s}} Ab

is an additive invariant. Hence, as proved in [26, Proposition 4.1], we have the following natural bijection

(3.2)  Nat(K_{0}, E_{0}) ⟶^{∼} E_{0}(\underline{k})  η ↦ η(\underline{k})([k]) .

Some explanations are in order: \underline{k} denotes the dg category naturally associated to the base ring k, i.e. the dg category with only one object and with k as the dg algebra of endomorphisms (concentrated in degree zero); the symbol [k] stands for the class of k (as a module over itself) in the Grothendieck group K_{0}(\underline{k}) = K_{0}(k). Let us now turn our attention to Nat(𝕂, 𝔼). Recall from [24, §15] the notion of additive invariant of dg categories. Roughly speaking, it consists of a morphism of derivators defined on HO(dgcat) and with values in a triangulated derivator which verifies conditions analogous to (i)-(iii). Since the functor E is additive, the induced morphism of derivators

𝔼 : HO(dgcat) ⟶ HO(Spt)

is an additive invariant of dg categories. Following [3, Theorem 8.1] we have then a natural bijection^{2}

Nat(𝕂, 𝔼) ⟶^{∼} π_{0}^{s} 𝔼(\underline{k}) = E_{0}(\underline{k}) . (3.3)

A careful inspection of the proof of [3, Theorem 8.1] show us that (3.3) sends a natural transformation Φ ∈ Nat(𝕂, 𝔼) to the element π_{0}^{s}(Φ(e)(\underline{k}))([k]) of the abelian group E_{0}(\underline{k}). Note that this element is simply the image of [k] by the abelian group homomorphism

K_{0}(k) = π_{0}^{s}(𝕂(e)(\underline{k})) ⟶^{π_{0}^{s}(Φ(e)(\underline{k}))} π_{0}^{s}(𝔼(e)(\underline{k})) = E_{0}(\underline{k}) .

We now prove that the following diagram

(3.4)  \xymatrix{Nat(𝕂, 𝔼) \ar[r]^{(1.2)} \ar[dr]_{(3.3)} & Nat(K_{0}, E_{0}) \ar[d]^{(3.2)} \\  & E_{0}(\underline{k})}

commutes. Let Φ ∈ Nat(𝕂, 𝔼). On the one hand, we observe that the composed map (3.2) ∘ (1.2) sends Φ to the element (π_{0}^{s} ∘ Φ(e) ∘ l)(\underline{k})([k]) of the abelian group E_{0}(\underline{k}). On the other hand, the following equalities hold:

(Φ(e) ∘ l)(\underline{k}) = Φ(e)(\underline{k})  (π_{0}^{s} ∘ Φ(e))(\underline{k}) = π_{0}^{s}(Φ(e)(\underline{k})) .

2In [3] this bijection was established for a localizing invariant 𝔼. However, the arguments in the additive case are completely similar.

Therefore, we have

(π_{0}^{s} ∘ Φ(e) ∘ l)(\underline{k})([k]) = π_{0}^{s}(Φ(e)(\underline{k}))([k]) .

Finally, since the right-hand side in this latter equality coincides with the image of Φ by the map (3.3), we conclude that (3.3) = (3.2) ∘ (1.2). Theorem 1.1 now follows from diagram (3.4) and the fact that both maps (3.2) and (3.3) are bijective. The canonical extension ϕ_{n} : K_{n} ⇒ E_{n} of ϕ : K_{0} ⇒ E_{0} is then the composition π_{n}^{s} ∘ Φ(e) ∘ l, where Φ is the unique natural transformation associated to ϕ under the bijection (1.2).

4. Proof of Proposition 2.3

The essence of the proof consists in describing the unique natural transformation Φ ∈ Nat(𝕂, ℍℍ) which corresponds to (2.2) under the bijection (1.2). Recall from [24, §15] the construction of the universal additive invariant of dg categories

U_{A} : HO(dgcat) ⟶ Mot_{A} .

Given any Quillen model category M we have an induced equivalence of categories

(4.1)  (U_{A})^{∗} : \underline{Hom}_{!}(Mot_{A}, HO(M)) ⟶^{∼} \underline{Hom}_{A}(HO(dgcat), HO(M)) ,

where the left-hand side denotes the category of homotopy colimit preserving morphisms of derivators and the right-hand side the category of additive invariants of dg categories. The algebraic K-theory functor K is additive and so the induced morphism 𝕂 is an additive invariant of dg categories. Thanks to equivalence (4.1), it factors then uniquely through U_{A}. Recall from [24, Theorem 15.10] that for every dg category A we have a weak equivalence of spectra

ℝHom(U_{A}(\underline{k}), U_{A}(A)) ≃ K(A) ,

where ℝHom(−, −) denotes the spectral enrichment of Mot_{A} (see [3, §A.3]). Therefore, we conclude that 𝕂 can be expressed as the following composition

(4.2)  HO(dgcat) ⟶^{U_{A}} Mot_{A} ⟶^{ℝHom(U_{A}(\underline{k}),−)} HO(Spt) .

The Hochschild homology functor, with values in the projective Quillen model category C(k) of cochain complexes of k-modules (see [12, Theorem 2.3.11]), verifies conditions (i)-(iii). Hence, it gives rise to an additive additive invariant of dg categories which we denote by

ℍH : HO(dgcat) ⟶ HO(C(k)) .

Note that, according to our notation, ℍℍ can be expressed as the following composition

(4.3)  HO(dgcat) ⟶^{ℍH} HO(C(k)) ⟶^{ℝHom(k,−)} HO(Spt) .

Equivalence (4.1) provide us then the following commutative diagram

\xymatrix{HO(dgcat) \ar[d]_{U_{A}} \ar[r]^{ℍH} & HO(C(k)) \\ _{Mot_{A}} \ar[ur]_{\overline{ℍH}} & .}

By construction, the morphism \overline{ℍH} maps U_{A}(\underline{k}) to ℍH(\underline{k}) = k. Hence, by making use of the above factorizations (4.2) and (4.3), we conclude that it induces a natural transformation Φ ∈ Nat(𝕂, ℍℍ). We now show that the image of this natural transformation Φ by the map (1.2) is the natural transformation (2.2). By taking E = HH in bijection (3.2) we obtain:

(4.4)  Nat(K_{0}, HH_{0}) ≃ HH_{0}(\underline{k}) ≃ k  η ↦ η(\underline{k})([k]) .

Under this bijection, the natural transformation (2.2) corresponds to the unit of the base ring k; see [26, Theorem 1.3]. Hence, it suffices to show that the same holds for the natural transformation π_{0}^{s}∘Φ(e)∘l associated to Φ. The class [k] of k (as a module over itself) in the Grothendieck group K_{0}(k) corresponds to the identity morphism in

Hom_{Mot_{A}(e)}(U_{A}(\underline{k}), U_{A}(\underline{k})) ≃ K_{0}(\underline{k}) ≃ K_{0}(k) .

By functoriality, \overline{ℍH}(e) maps this identity morphism to the identity morphism in Hom_{D(k)}(ℍH(\underline{k}), ℍH(\underline{k})). Under the natural isomorphisms

Hom_{D(k)}(ℍH(\underline{k}), ℍH(\underline{k})) ≃ Hom_{D(k)}(k, k) ≃ HH_{0}(k) ≃ k

the identity morphism corresponds to the unit of the base ring k and so we conclude that π_{0}^{s} ∘ Φ(e) ∘ l agrees with (2.2). This implies that Φ is in fact the unique natural transformation which corresponds to (2.2) under the bijection (1.2). Finally, let A be a k-algebra. As proved in [27, Theorem 2.8], the canonical extension ϕ_{n} : K_{n}(A) → HH_{n}(A) of ϕ (i.e. the abelian group homomorphism (π_{n}^{s} ∘ Φ(e) ∘ l)(A)) agrees with the n^{th} trace map constructed by Dennis and so the proof is finished.

5. Proof of Proposition 2.5

We prove first the particular case (i = 0). Let us start by describing the unique natural transformation Φ ∈ Nat(𝕂, ℍℂ) which corresponds to ϕ : K_{0} ⇒ HC_{0} under the bijection (1.2). Observe that ℍℂ can be expressed as the following composition

HO(dgcat) ⟶^{𝕄} HO(C(Λ)) ⟶^{ℙ} HO(k[u]-Comod) ⟶^{𝕌} HO(C(k)) ⟶^{ℝHom(k,−)} HO(Spt) .

Some explanations are in order: C(Λ) is the projective Quillen model category of mixed complexes and 𝕄 the morphism induced by the mixed complex construction^{3} (see [4, Example 7.10]); k[u]-Comod is the Quillen model category of k[u]-comodules (where k[u] is the Hopf algebra of polynomials in one variable of degree 2) and ℙ the morphism induced by the perioditization construction (see [4, Example 7.11]); 𝕌 is the morphism induced by the natural forgetful construction. Moreover, as explained in [4, Examples 8.10 and 8.11], negative cyclic homology and periodic cyclic homology admit the following factorizations:

3Denoted by C in [4, Example 7.10].

(5.1)  ℍℂ^{−} : HO(dgcat) ⟶^{𝕄} HO(C(Λ)) ⟶^{ℝHom(k,−)} HO(Spt)
ℍℙ : HO(dgcat) ⟶^{(ℙ∘𝕄)} HO(k[u]-Comod) ⟶^{ℝHom(k[u],−)} HO(Spt) .

Therefore, since ℙ maps k to k[u] and 𝕌 maps k[u] to k, we obtain the classical natural transformations

(5.2)  ℍℂ^{−} ⇒ ℍℙ ⇒ ℍℂ

between the cyclic homology variants; see [20, §5.1]. The mixed complex morphism 𝕄 is an additive invariant of dg categories and so by equivalence (4.1) it factors uniquely through U_{A}. We obtain then a commutative diagram

\xymatrix{HO(dgcat) \ar[r]^{𝕄} \ar[d]_{U_{A}} & HO(C(Λ)) \\ _{Mot_{A}} \ar[ur]_{\overline{𝕄}} & .}

By construction, the morphism \overline{𝕄} maps U_{A}(\underline{k}) to 𝕄(\underline{k}) = k. Therefore, making use of the factorizations (4.2) and (5.1), we conclude that \overline{𝕄} induces a natural transformation Φ_{1} : 𝕂 ⇒ ℍℂ^{−}. Its composition with (5.2) gives rise to a natural transformation which we denote by Φ ∈ Nat(𝕂, ℍℂ). We now show that the image of Φ by the map (1.2) is the natural transformation ϕ : K_{0} ⇒ HC_{0}. Recall from [26, Theorem 1.7(ii)] that ϕ admits the following factorization

K_{0} ⇒^{ch_{0}^{−}} HC_{0}^{−} ⇒ HP_{0} ⇒ HC_{0} ,

where ch_{0}^{−} is the negative Chern character and the other natural transformations are the ones associated to (5.2). Hence, it suffices to show that the natural transformation π_{0}^{s} ∘ Φ_{1}(e) ∘ l, associated to Φ_{1} : 𝕂 ⇒ ℍℂ^{−}, agrees with ch_{0}^{−}. This fact is proved in [28, Proposition 4.2] and so we conclude that Φ is the unique natural transformation which corresponds to ϕ under the bijection (1.2). Now, let A be a k-algebra. As explained in [20, §11.4.3], Karoubi’s Chern character ch_{n,0}(A) can be expressed as the following composition

K_{n}(A) ⟶^{ch_{n}^{−}(A)} HC_{n}^{−}(A) ⟶ HP_{n}(A) ⟶ HC_{n}(A) .

Note that the right-hand side maps coincide the ones associated to (5.2). Therefore, it suffices to show that the abelian group homomorphism

(π_{n}^{s} ∘ Φ_{1}(e) ∘ l)(A) : K_{n}(A) ⟶ HC_{n}^{−}(A)

agrees with ch_{n}^{−}(A). This latter fact is proved in [27, Theorem 2.8] and so the proof of the particular case (i = 0) is finished.

We now prove the case (i > 0). Recall from [13, §1] that for any dg category A we have a natural periodicity map S : Ω^{2}𝕄(A) → 𝕄(A) in the category C(Λ) of mixed complexes. This construction is natural in A and so by iterating it we obtain an infinite sequence of maps

(5.3)  ⋯ ⟶ Ω^{2i}𝕄(A) ⟶ ⋯ ⟶ Ω^{2}𝕄(A) ⟶ 𝕄(A) .

Under the natural equivalences

ℝHom(k, Ω^{2i}𝕄(−)) ≃ Ω^{2i}ℍℂ^{−}
ℝHom(k[u], ℙ(Ω^{2i}𝕄(−))) ≃ Ω^{2i}ℍℙ
ℝHom(k, 𝕌(ℙ(Ω^{2i}𝕄(−)))) ≃ Ω^{2i}ℍℂ ,

the above sequence of maps (5.3) gives rise to the following commutative diagram of natural transformations

(5.4)  \xymatrix{𝕂 \ar@{=>}[r]^{Φ_{1}} & ℍℂ^{−} \ar@{=>}[r] & ℍℙ \ar@{=>}[r] & ℍℂ \\  & Ω^{2}ℍℂ^{−} \ar@{=>}[r] \ar@{=>}[u] & Ω^{2}ℍℙ \ar@{=>}[u]^{≃} \ar@{=>}[r] & Ω^{2}ℍℂ \ar@{=>}[u] \\  & ⋮ \ar@{=>}[u] & ⋮ \ar@{=>}[u]^{≃} & ⋮ \ar@{=>}[u] \\  & Ω^{2i}ℍℂ^{−} \ar@{=>}[r] \ar@{=>}[u] & Ω^{2i}ℍℙ \ar@{=>}[u]^{≃} \ar@{=>}[r] & Ω^{2i}ℍℂ \ar@{=>}[u] \\  & ⋮ \ar@{=>}[u] & ⋮ \ar@{=>}[u]^{≃} & ⋮ \ar@{=>}[u]}

The periodicity map S becomes invertible in periodic cyclic homology and so the middle column in (5.4) consists of natural isomorphisms. Hence, we obtain the classical sequence of natural transformations

ℍℙ ⇒ ⋯ ⇒ Ω^{2i}ℍℂ ⇒ ⋯ ⇒ Ω^{2}ℍℂ ⇒ ℍℂ

which relates periodic cyclic homology with the even dimensional loopings of cyclic homology; see [20, §5.1.8]. Let us then take for Φ the composed natural transformation

𝕂 ⇒^{Φ_{1}} ℍℂ^{−} ⇒ ℍℙ ⇒ Ω^{2i}ℍℂ .

The fact that its image by the map (1.2) is the natural transformation ϕ : K_{0} ⇒ HC_{2i} is now an immediate consequence of the following factorization

ϕ : K_{0} ⇒^{ch_{0}^{−}} HC_{0}^{−} ⇒ HP_{0} ⇒ HC_{2i} ,

see [26, Theorem 1.7(ii)], and the particular case (i = 0). Similarly, the fact that the canonical extension ϕ_{n} : K_{n}(A) → HC_{n+2i}(A) agrees with Karoubi’s higher Chern character ch_{n,i}(A) follows from the following factorization

ch_{n,i}(A) : K_{n}(A) ⟶^{ch_{n}^{−}(A)} HC_{n}^{−}(A) ⟶ HP_{n}(A) ⟶ HC_{n+2i}(A) ,

see [20, §11.4.3], and the particular case (i = 0). This achieves the proof.

Appendix A. Grothendieck Derivators

In order to make this article more self-contained we give a brief introduction to Grothendieck’s theory of derivators [11]; this language can easily be acquired by skimming through [21], [5, §1] or [3, 4, Appendix A]. Derivators originate in the problem of higher homotopies in derived categories. Given a triangulated category T and a small category I, it essentially never happens that the diagram category Fun(I^{op}, T ) remains triangulated; this already fails for the category of arrows in T . However, our triangulated category T often appears as the homotopy category T = Ho(M) of some Quillen model category M (see [22]). In this case we can consider the category Fun(I^{op}, M) of diagrams in M whose homotopy category Ho(Fun(I^{op}, M)) is triangulated and provides a reasonable approximation to Fun(I^{op}, T ). More importantly, one can let I vary. This “nebula” of categories Ho(Fun(I^{op}, M)), indexed by small categories I, and the various (adjoint) functors between them is what Grothendieck formalized into the concept of a derivator. A derivator consists of a strict contravariant 2-functor, from the 2-category of small categories to the 2-category of all categories, subject to five natural conditions. We shall not list these conditions here for it would be too long; see [5, §1]. The essential example to keep in mind is the (triangulated) derivator HO(M) associated to a (stable) Quillen model category M and defined for every small category I by

HO(M)(I) := Ho(Fun(I^{op}, M)) .

We will write e for the 1-point category with one object and one identity morphism. Note that HO(M)(e) is the homotopy category Ho(M). Given Quillen model categories M_{1} and M_{2} and weak equivalence preserving functors E, F : M_{1} → M_{2}, we will denote by 𝔼, 𝔽 : HO(M_{1}) ⟶ HO(M_{2}) the induced morphisms of derivators and by Nat(𝔼, 𝔽) the set of natural transformations from 𝔼 to 𝔽; see [5, §5]. Note that given Φ ∈ Nat(𝔼, 𝔽), Φ(e) is a natural transformation between the induced functors 𝔼(e), 𝔽(e) : Ho(M_{1}) → Ho(M_{2}).

Acknowledgments: The author is very grateful to Bruno Kahn for motivating questions, and to Paul Balmer, Denis-Charles Cisinski and Christian Haesemeyer for useful discussions. He would also like to thank the Departments of Mathematics of UCLA, MIT and UIC for its hospitality and the Clay Mathematics Institute, the Midwest Topology Network, and the Calouste Gulbenkian Foundation for financial support.

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Gonçalo Tabuada

Department of Mathematics MIT Cambridge, MA 02139 USA tabuada@math.mit.edu Dep. de Matemática FCT-UNL Quinta da Torre, 2829-516 Caparica Portugal tabuada@fct.unl.pt