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Higher K-theory via universal invariants

Using the formalism of Grothendieck's derivators, we construct `the universal localizing invariant of dg categories'. By this, we mean a morphism U_l from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator M^loc such that U_l commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle and is universal for these properties. Similary, we construct the `the universal additive invariant of dg categories', i.e. the universal morphism of derivators U_a to a strong triangulated derivator M^add which satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen K-theory appears as a mapping space in the target of the universal additive invariant. This is the first conceptual characterization of Quillen-Waldhausen's K-theory since its definition in the early 70's. As an application we obtain for free the higher Chern characters from K-theory to cyclic homology.


Introduction
Differential graded categories (=dg categories) enhance our understanding of triangulated categories appearing in algebra and geometry, see [19].
They are considered as non-commutative schemes by Drinfeld [9] [10] and Kontsevich [23] [24] in their program of non-commutative algebraic geometry, i.e. the study of dg categories and their homological invariants.
In this article, using the formalism of Grothendieck's derivators, we construct 'the universal localizing invariant' of dg categories cf. [21]. By this, we mean a morphism U l from the pointed derivator HO(dgcat) associated with the Morita homotopy theory of dg categories, see [37] [38] [39], to a triangulated strong derivator M loc dg such that U l commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle and is universal for these properties. Because of its universality property reminiscent of motives, see section 4.1 of Kontsevich's preprint [25], we call M loc dg the (stable) localizing motivator of dg categories.
Similary, we construct 'the universal additive invariant' of dg categories, i.e. the universal morphism of derivators U a from HO(dgcat) to a strong triangulated derivator M add dg which satisfies the first two properties but the third one only for split exact sequences. We call M add dg the additive motivator of dg categories. We prove that Waldhausen's K-theory spectrum appears as a spectrum of morphisms in the base category M add dg (e) of the additive motivator. This shows us that Waldhausen's K-theory is completely characterized by its additive property and 'intuitively' it is the universal construction with values in a stable context which satisfies additivity.
To the best of the author's knowledge, this is the first conceptual characterization of Quillen-Waldhausen's K-theory [34] [44] since its definition in the early 70's. This result gives us a completely new way to think about algebraic K-theory and furnishes us for free the higher Chern characters from K-theory to cyclic homology [26].
The co-representation of K-theory as a spectrum of morphisms extends our results in [37] [38], where we co-represented K 0 using functors with values in additive categories rather than morphisms of derivators with values in strong triangulated derivators.
For example, the mixed complex construction [22], from which all variants of cyclic homology can be deduced, and the non-connective algebraic K-theory [35] are localizing invariants and factor through U l and through U a . The connective algebraic K-theory [44] is an example of an additive invariant which is not a localizing one. We prove that it becomes co-representable in M add dg , see theorem 16.10. Our construction is similar in spirit to those of Meyer-Nest [31], Cortiñas-Thom [8] and Garkusha [12]. It splits into several general steps and also offers some insight into the relationship between the theory of derivators [14] [13] [20] [29] [3] and the classical theory of Quillen model categories [33]. Derivators allow us to state and prove precise universal properties and to dispense with many of the technical problems one faces in using model categories.
In chapter 3 we recall the notion of Grothendieck derivator and point out its connexion with that of small homotopy theory in the sense of Heller [14]. In chapter 4, we recall Cisinski's theory of derived Kan extensions [4] and in chapter 5, we develop his ideas on the Bousfield localization of derivators [7]. In particular, we characterize the derivator associated with a left Bousfield localization of a Quillen model category by a universal property, see theorem 5.4. This is based on a constructive description of the local weak equivalences.
In chapter 6, starting from a Quillen model category M satisfying some compactness conditions, we construct a morphism of prederivators

HO(M)
Rh −→ L Σ Hot M f which commutes with filtered homotopy colimits, has a derivator as target and is universal for these properties. In chapter 7 we study morphisms of pointed derivators and in chapter 8 we prove a general result which garantees that small weak generators are preserved under left Bousfield localizations. In chapter 9, we recall Heller's stabilization construction [14] and we prove that this construction takes 'finitely generated' unstable theories to compactly generated stable ones. We establish the connection between Heller's stabilization and Hovey/Schwede's stabilization [17] [36] by proving that if we start with a pointed Quillen model category which satisfies some mild 'generation' hypotheses, then the two stabilization procedures yield equivalent results. This allows us to characterize Hovey/Schwede's construction by a universal property and in particular to give a very simple characterisation of the classical category of spectra in the sense of Bousfield-Friedlander [2]. In chapter 10, by applying the general arguments of the previous chapters to the Morita homotopy theory of dg categories [37] [38] [39], we construct the universal morphism of derivators where B/A denotes Drinfeld's dg quotient. By applying the localization techniques of section 5 and using the fact that the derivator St(L Σ,P Hot dgcat f ) admits a stable Quillen model, we invert the morphisms S K and obtain finally the universal localizing invariant of dg categories U l : HO(dgcat) −→ M loc dg . We establish a connection between the triangulated category M loc dg (e) and Waldhausen's K-theory by showing that Waldhausen's S • -construction corresponds to the suspension functor in M loc dg (e). In section 12, we prove that the derivator M loc dg admits a stable Quillen model given by a left Bousfield localization of a category of presheaves of spectra. In section 13, we introduce the concept of upper triangular dg category and construct a Quillen model structure on this class of dg categories, which satisfies strong compactness conditions. In section 14, we establish the connection between upper triangular dg categories and split short exact sequences and use the Quillen model structure of section 13 to prove an 'approximation result', see proposition 14.2. In section 15, by applying the techniques of section 5, we construct the universal morphism of derivators U u : HO(dgcat) −→ M unst dg which commutes with filtered homotopy colimits, preserves the point and sends each split short exact sequence to a homotopy cofiber sequence. We prove that Waldhausen's K-theory space construction appears as a fibrant object in M unst dg . This allow us to obtain the weak equivalence of simplicial sets In section 16 we stabilize the derivator M unst dg , using the fact that it admits a Quillen model and obtain finally the universal additive invariant of dg categories U a : HO(dgcat) −→ M add dg .
Connective algebraic K-theory is additive and so factors through U a . We prove that for a small dg category A its connective algebraic K-theory corresponds to a fibrant resolution of U a (A) [1], see theorem 16.9. Using the fact that the Quillen model for M add dg is enriched over spectra, we prove our main co-representability theorem.
Let A and B be small dg categories with A ∈ dgcat f .

For any morphism of functors
one has a morphism of functors Of course, all these data have to verify some coherence conditions, namely: (a) For any composable maps in Cat, X (vu) * = u * v * and 1 * X = 1 D(X) .
respects weak equivalences, so that it induces a well defined functor The 2-functoriality of localization implies that we have a prederivator D M .
Let X be a small category and let x be an object of X. Given an object F ∈ D(X) we will write F x = x * (F ). The object F x will be called the fiber of F at the point x.
For a prederivator D, define its oposite to be the prederivator D op given by the formula D op (X) = D(X op ) op for all small categories X.
called the cohomological direct image functor (resp. homological direct image functor) associated to u. Notation 3.6. Let X be a small category and p = p X : X → e. If p has a cohomological direct image functor in D, one defines for any object F of D(X) the object of global sections of F as Γ * (X, F ) = p * (F ) . Dually, if p has a homological direct image in D then, for any object F of D(X), one sets Notation 3.7. Let D be a prederivator, and let be a 2-diagram in Cat. By 2-functoriality one obtains the following 2-diagram If we assume that the functors u and u ′ both have cohomological direct images in D one can define the base change morphism induced by α as follows. The counit u * u * → 1 D(X) induces a morphism v * u * u * → v * , and by composition with α * u * , a morphism u ′ * w * u * → v * . This gives β by adjunction. This construction will be used in the following situation: let u : X → Y be a map in Cat and let y be a point of Y . According to notation 3.1 we have a functor j : X/y → X, defined by the formula j(x, f ) = x, where f : u(x) → y is a morphism in Y . If p : X/y → e is the canonical map one obtains the 2-diagram below, where α denotes the 2-cell defined by the formula α (x,f ) = f Using notation 3.6, the associated base change morphism gives rise to a canonical morphism u * (F ) y → Γ * (X/y, F/y) for any object F ∈ D(X), where F/y = j * (F ). Dually one has canonical morphisms where y\F = k * (F ) and k denotes the canonical functor from y\X to X.
Notation 3.8. Let X and Y be two small categories. Using the 2-functoriality of D, one defines a functor as follows. Setting X ′ = X × Y , we have a canonical functor which defines a functor and then a functor Using the canonical functor this gives the desired functor. In particular, for any small category X, one gets a functor .
Definition 3.9. A derivator is a prederivator D with the following properties.
Der1 (Non-triviality axiom). For any finite set I and any family {X i , i ∈ I} of small categories, the canonical functor is an equivalence of categories.
Der2 (Conservativity axiom). For any small category X, the family of functors , then ϕ is an isomorphism in D(X). Der3 (Direct image axiom). Any functor in Cat has a cohomological direct image functor and a homological direct image functor in D (see definition 3.5). Der4 (Base change axiom). For any functor u : X → Y in Cat, any point y of Y and any object F in D(X), the canonical base change morphisms (see notation 3.7)

see notation 3.8, is full and essentially surjective.
Notice that a strong derivator is the same thing as a small homotopy theory in the sense of Heller [14]. Notice also that by proposition 2.15 in [6], HO(M) is a strong derivator. Definition 3.12. A derivator D is regular if in D, sequential homotopy colimits commute with finite products and homotopy pullbacks. Notation 3.13. Let X be a category. Remember that a sieve (or a crible) in X is a full subcategory U of X such that, for any object x of X, if there exists a morphism x → u with u in U then x is in U . Dually a cosieve (or a cocrible) in X is a full subcategory Z of X such that, for any morphism z → x in X, if z is in Z then so is x.
A functor j : U → X is an open immersion if it is injective on objects, fully faithful, and if j(U ) is a sieve in X. Dually a functor i : Z → X is a closed immersion if it is injective on objects, fully faithful, and if i(Z) is a cosieve in X. One can easily show that open immersions and closed immersions are stable by composition and pullback. Definition 3.14. A derivator D is pointed if it satisfies the following property. Der6 (Exceptional axiom). For any closed immersion i : Z → X in Cat the cohomological direct image functor has a right adjoint called the exceptional inverse image functor associated to i. Dually, for any open immersion j : U → X the homological direct image functor has a left adjoint called the coexceptional inverse image functor associated to j.
Let be the category given by the commutative square We are interested in two of its subcategories. The subcategory is We thus have two inclusion functors σ : → and τ : → (σ is an open immersion and τ a closed immersion). A global commutative square in D is an object of D( ). A global commutative square C in D is thus locally of shape A global commutative square C in D is cartesian (or a homotopy pullback square) if, for any global commutative square B in D, the canonical map is bijective. Dually, a global commutative square B in D is cocartesian (or a homotopy pushout square) if, for any global commutative square C in D, the canonical map is bijective. As is isomorphic to its opposite op , one can see that a global commutative square in D is cartesian (resp. cocartesian) if and only if is cocartesian (resp. cartesian) as a global commutative square in D op . Let D and D ′ be derivators. We denote by Hom(D, D ′ ) the category of all morphisms of derivators, by Hom ! (D, D ′ ) the category of morphisms of derivators which commute with homotopy colimits [4, 3.25] and by Hom f lt (D, D ′ ) the category of morphisms of derivators which commute with filtered homotopy colimits, see [3,5].

Derived Kan extensions
Let A be a small category and Fun(A op , Sset) the Quillen model category of simplicial pre-sheaves on A, endowed with the projective model structure, see [40]. We have at our disposal the functor where Hom A (?, X) is considered as a constant simplicial set.
The functor h gives rise to a morphism of pre-derivators A h −→ HO(Fun(A op , Sset)) .
Using the notation of [4], we denote by Hot A the derivator HO(Fun(A op , Sset)). The following results are proven in [4].
Let D be a derivator.
Proof. We construct a universal 2-morphism of functors Let F be a morphism of derivators belonging to Hom(Hot A , D). Let L be a small category and X an object of Hot A (L). Recall from [4] that we have the diagram Now, let p be the functor π op and q the functor ̟ op . By the dual of proposition 1.15 in [4], we have the following functorial isomorphism Finally let ǫ L (X) be the composed morphism and notice, using theorem 4.1, that ǫ induces an adjunction. √

Localization: model categories versus derivators
Let M be a left proper, cellular Quillen model category, see [15]. We start by fixing a frame on M, see definition 16.6.21 in [15]. Let D be a small category and F a functor from D to M. We denote by hocolim F the object of M, as in definition 19.1.2 of [15]. Let S be a set of morphisms in M and denote by L S M the left Bousfield localization of M by S.
Notice that the Quillen adjunction which commutes with homotopy colimits.
Then the class W S equals the class of S-local equivalences in M, see [15]. Let us now show that conversely, each S-local equivalence is in W S . Let be an S-local equivalence in M. Without loss of generality, we can suppose that X is cofibrant. Indeed, let Q(X) be a cofibrant resolution of X and consider the diagram Notice that since π is a weak equivalence, g is an S-local equivalence if and only if g • π is one. By theorem 4.3.6 in [15], g is an S-local equivalence if and only if the morphism L S (g) appearing in the diagram is a weak equivalence in M. This shows that it is enough to prove that j(X) and j(Y ) belong to W S . Apply the small object argument to the morphism X −→ * using the set Λ(S), see proposition 4.2.5 in [15]. We have the factorization where j(X) is a relative Λ(S)-cell complex.
We will now prove two stability conditions concerning the class W S : S1) Consider the following push-out where W 0 , W 1 and W 2 are cofibrant objects in M and f is a cofibration which belongs to W S . Observe that f * corresponds to the colimit of the morphism of diagrams Now, proposition 19.9.4 in [15] and property d) imply that f * belongs to W S . S2) Consider the following diagram in M, where the objects are cofibrant and the morphisms are cofibrations which belong to the class W S . Observe that the transfinite composition of X corresponds to the colimit of the morphism of diagrams . . .

Now
, since X is a Reedy cofibrant diagram on category with fibrant constants, see definition 15.10.1 in [15], theorem 19.9.1 from [15] and property d) imply that the transfinite composition of X belongs to W S . Notice that the above argument can be immediatly generalized to a transfinite composition of a λ-sequence, where λ denotes an ordinal, see section 10.2 in [15]. Now, the construction of the morphism j(X) and the stability conditions S1) and S2) shows us that it is enough to prove that the elements of Λ(S) belong to W S . By proposition 4.2.5 in [15], it is sufficient to show that the set of horns in S is contained in W S . Recall from definition 4.2.1 in [15] thatg :Ã →B denotes a cosimplicial resolution of g : A → B andg is a Reedy cofibration. We have the diagramÃ Observe that the morphismÃ identifies withÃ ng n −→B n , and so belongs to W S . Now, the morphism corresponds to the induced map of latching objects which is a cofibration in M by proposition 15 Notice that the morphism I belongs to W S by the stability condition S1) applied to the morphismÃ which is a cofibration and belongs to W S . Since the morphismg ⊗ 1 belongs to W S so does Λ(g). This proves the proposition. √ Let D be derivator and S a class of morphisms in D(e).
which commutes with homotopy colimits, sends the elements of S to isomorphisms in L S D(e) and satisfies the universal property: for every derivator D ′ the morphism γ induces an equivalence of categories where Hom !,S (D, D ′ ) denotes the category of morphisms of derivators which commute with homotopy colimits and send the elements of S to isomorphisms in D ′ (e). Therefore, the induced functor admits a left adjoint σ * and σ * γ * = (γσ) * is isomorphic to the identity. Therefore γ * is fully faithful. We now show that γ * is essentially surjective. Let sends the elements of W S to isomorphisms. This proves the theorem since by proposition 5.1 the class W S equals the class of S-local equivalences in M. √

Filtered homotopy colimits
Let M be a cellular Quillen model category, with I the set of generating cofibrations. Suppose that the domains and codomains of the elements of I are cofibrant, ℵ 0 -compact, ℵ 0 -small and homotopically finitely presented, see definition 2.1.1 in [41].
Recall that a dg functor F : C → E is a quasi-equivalence, resp. quasi-equiconic, resp. a Morita dg functor, if it satisfies one of the following conditions C1) or C2): C1) The dg category C is empty and all the objects of E are contractible. C2) For every object c 1 , c 2 ∈ C, the morphism of complexes from Hom C (c 1 , c 2 ) to Hom E (F (c 1 )), F (c 2 )) is a quasi-isomorphism and the functor H 0 (F ), resp. H 0 (pre-tr(F )), resp. H 0 (pre-tr(F )) ∋ , is essentially surjective. Observe that the domains and codomains of the set I of generating cofibrations in dgcat satisfy the conditions above for all the Quillen model structures.
The following proposition is a simplification of proposition 2.2 in [41].

Proposition 6.2. Let M be a Quillen model category which satisfies the conditions above. Then 1) A filtered colimit of trivial fibrations is a trivial fibration.
2) For any filtered diagram X i in M, the natural morphism is an isomorphism in Ho(M).

3) Any object X in M is equivalent to a filtered colimit of strict finite I-cell
objects.

4) An object X in M is homotopically finitely presented if and only if it is equivalent to a rectract of a strict finite I-cell object.
Proof. The proof of 1), 2) and 3) is exactly the same as that of proposition 2.2 in [41]. The proof of 4) is also the same once we observe that the domains and codomains of the elements of the set I are already homotopically finitely presented by hypothesis. √ In everything that follows, we fix: -A co-simplicial resolution functor in the model category M, see definition 16.1.8 in [15]. This means that for every object X in M, Γ(X) is cofibrant in the Reedy model structure on in the model category M, see [15].
is a cylinder object for Γ(X) 0 , see proposition 16.1.6. from [15]. Since M f is also stable under the functors Γ(−) n , n ≥ 0, this cylinder object also belongs to M f . This implies that if in the construction of the homotopy category Ho(M), as in  [15]. We denote it by L Σ Fun(M op f , Sset). We have a composed functor that we still denote by h We also have a composed functor that we still denote by h which is functorial in X.
Lemma 6.7. The functor h preserves weak equivalences between fibrant objects.
Proof. Let X be a fibrant object in M. We have an equivalence see [15]. This implies the lemma. √ Remark 6.8. The previous lemma implies that the functor h admits a right derived functor Rh : , sends, by definition, the elements of S to weak equivalences, we have an induced morphism Remark 6.9. Notice that lemma 4.2.2 from [40] implies that for every X in M f , the morphism Ψ(X) is an isomorphism in Ho(L Σ Fun(M op f , Sset)).

This shows that the functors
are canonically isomorphic and so we have the following diagram which is commutative up to isomorphism.
Lemma 6.10. The functor Rh commutes with filtered homotopy colimits.
Proof. Let {Y i } i∈I be a filtered diagram in M. We can suppose, without loss of generality, that Y i is fibrant in M. By proposition 6.2, the natural morphism is an isomorphism in Ho(M) and colim i∈I Y i is also fibrant. Since the functor , commutes with homotopy colimits and in Ho(Fun(M op f , Sset)) they are calculated objectwise, it is sufficient to show that the morphism is an isomorphism in Ho(Sset), for every object X in M f . Now, since every object X in M f is homotopically finitely presented, see proposition 6.2, we have the following equivalences: This proves the lemma. Observe that the morphism of functors Lemma 6.11. The 2-morphism Ψ is an isomorphism.
Proof. For the terminal category e, the 2-morphism Ψ coincides with the morphism of functors of remark 6.9. Since this one is an isomorphism, so is Ψ by conservativity. This proves the lemma. √ As before, we have the following diagram Rh v v l l l l l l l l l l l l l l L Σ Hot M f which is commutative up to isomorphism in the 2-category of pre-derivators. Notice that by lemma 6.10, Rh commutes with filtered homotopy colimits.
Let D be a derivator.
Lemma 6.12. The morphism of pre-derivators induces an equivalence of categories . This last category identifies, under the equivalence given Recall from section 9.5 in [11] that the co-simplicial resolution functor Γ(−) that we have fixed in the beginning of this section allows us to construct a Quillen adjunction: Since the functor Re sends the elements of Σ to weak equivalences in M, we have the following Quillen adjunction and a natural weak equivalence see [11].
This implies that we have the following diagram We now claim that LRe • Rh is naturally isomorphic to the identity. Indeed, by proposition 6.2, each object of M is isomorphic in Ho(M), to a filtered colimit of strict finite I-cell objects. Since Rh and LRe commute with filtered homotopy colimits and LRe • Ho(h) ≃ Id, we conclude that LRe • Rh is naturally isomorphic to the identity. This implies that the morphism Rh is fully faithful. Now, observe that the natural weak equivalence η induces a 2-isomorphism and so we obtain the following diagram which is commutative up to isomorphism in the 2-category of pre-derivators. Notice that LRe • Rh is naturally isomorphic to the identity (by conservativity) and so the morphism of derivators Rh is fully faithful.
Let D be a derivator.
Theorem 6.13. The morphism of derivators induces an equivalence of categories where Hom f lt (HO(M), D) denotes the category of morphisms of derivators which commute with filtered homotopy colimits.
Proof. We have the following adjunction with Rh * a fully faithful functor. Now notice that the adjunction of lemma 4.2 induces naturally an adjunction This implies that the composed functor is fully faithful. We now show that this functor is essentially surjective. Let F be an object of Hom f lt (HO(M), D). Consider the morphism Notice that this morphism does not necessarily commute with homotopy colimits. Now, by the above adjunction, we have a universal 2-morphism

Consider the 2-morphism
Now, we will show that this 2-morphism is a 2-isomorphism. By conservativity, it is sufficient to show this for the case of the terminal category e. For this, observe that Rh * (ϕ) induces an isomorphism Now each object of M is isomorphic, in Ho(M), to a filtered colimit of strict finite I-cell objects. Since F and Ψ(LRe * (F )) commute with filtered homotopy colimits, Rh * (ϕ) induces an isomorphism. This shows that the functor Rh * is essentially surjective.
This proves the theorem. √

Pointed derivators
Recall from the previous section that we have constructed a derivator L Σ Hot M f associated with a Quillen model category M satisfying suitable compactness assumptions. Now suppose that Ho(M) is pointed, i.e. that the morphism in M, where ∅ denotes the initial object and * the terminal one, is a weak equivalence. Consider the morphism where ∅ denotes the initial object in L Σ Fun(M op f , Sset). Observe that, since Rh admits a left adjoint, h(∅) identifies with the terminal object in We denote by L Σ,P Fun(M op f , Sset) , the left Bousfield localization of L Σ Fun(M op f , Sset) at the morphism P . Notice that the category is now a pointed one. We have the following morphisms of derivators By construction, we have a pointed morphism of derivators which commutes with filtered homotopy colimits and preserves the point.
Let D be a pointed derivator.

Proposition 7.1. The morphism of derivators Φ • Rh induces an equivalence of categories
where Hom f lt,p (HO(M), D) denotes the category of morphisms of derivators which commute with filtered homotopy colimits and preserve the point.
Proof. By theorem 5.4, we have an equivalence of categories By theorem 6.13, we have an equivalence of categories We now show that under this last equivalence, the category Hom !,P (L Σ Hot M f , D) identifies with Hom f lt,p (HO(M), D). Let F be an object of Hom !,P (L Σ Hot M f , D).
Since F commutes with homotopy colimits, it preserves the initial object. This implies that F • Rh belongs to Hom f lt,p (HO(M, D) . Let now G be an object of Hom f lt,p (HO(M), D). Consider, as in the proof of theorem 6.13, the morphism Since Ψ(LRe * (G)) commutes with homotopy colimits, by construction, it sends ∅ to the point of D. Observe also that h(∅) is also sent to the point of D because This proves the proposition. √

Small weak generators
Let N be a pointed, left proper, compactly generated Quillen model category as in definition 2.1 of [41]. Observe that in particular this implies that N is finitely generated, as in section 7.4 in [16]. We denote by G the set of cofibers of the generating cofibrations I in N . By corollary 7.4.4 in [16], the set G is a set of small weak generators for Ho(N ), see definitions 7.2.1 and 7.2.2 in [16]. Let S be a set of morphisms in N between objects which are homotopically finitely presented, see [41], and L S N the left Bousfield localization of N by S. We have an adjunction

Proof. The previous adjunction is equivalent to
where Ho(N ) S denotes the full subcategory of Ho(M) formed by the S-local objects of N and (−) f denotes a fibrant resolution functor in L S N , see [15]. Clearly, this implies that the image of the set G under the functor (−) f is a set of weak generators in Ho(L S N ). We now show that the S-local objects in N are stable under filtered homotopy colimits. Let {X i } i∈I be a filtered diagram of S-local objects. By proposition 6.2, we have an isomorphism hocolim in Ho(N ). We now show that colim i∈I X i is an S-local object. Let g : A → B be an element of S. We have at our disposal the following commutative diagram Now observe that since A and B are homotopically finitely presented objects, the vertical arrows in the diagram are isomorphisms in Ho(Sset). Since each object X i is S-local, the morphism g * i is an isomorphism in Ho(Sset) and so is colim i∈I g * i . This implies that colim i∈I X i is an S-local object. This shows that the inclusion This proves the lemma. √ Recall from the previous chapter that we have constructed a pointed derivator L Σ,P Hot M f . We will now construct a strictly pointed Quillen model category whose associated derivator is equivalent to L Σ,P Hot M f . Consider the pointed Quillen model category . We have the following Quillen adjunction

Stabilization
Let D be a regular pointed strong derivator.
In [14], Heller constructs a universal morphism to a triangulated strong derivator which commutes with homotopy colimits. This construction is done in two steps. First consider the following ordered set naturally as a small category. We denote by the full subcategory of 'points on the boundary'. Now, let Spec(D) be the full subderivator of D V , see definition 3.4 in [3], formed by the objects X in D V (L), whose image under the functor see section 8 in [14]. We have an evaluation functor ev (0,0) : Spec(D) → D, which admits a left adjoint L[0, 0]. Finally, let St(D) be the full reflexive subderivator of Spec(D) consisting of the Ω-spectra, as defined in [14].
We have the following adjunctions in the 2-category of derivators. Let T be a triangulated strong derivator. The following theorem is proved in [14].  Proof. Let X be an object of St(D)(e). Suppose that for each g in G and n in Z, we have Then by the following isomorphisms we conclude that, for all n in Z, we have ev (n,n) X = * .
By the conservativity axiom, X is isomorphic to * in St(D)(e). This shows condition A1). Now observe that condition A2) follows from the following isomorphisms

Sf inite
Hom D(e) (g, Proof. Suppose that for every g ∈ G and every point x in L. Then by adjunction x * M is isomorphic to * in T(e) and so by the conservativity axiom, M is isomorphic to * in T(L). This shows condition A1). Condition A2) follows from the following isomorphisms Relation with Hovey/Schwede's stabilization. We will now relate Heller's construction with the construction of spectra as it is done by Hovey in [17] and Schwede in [36]. Let M be a pointed, simplicial, left proper, cellular, almost finitely generated Quillen model category, see definition 4.1 in [17], where sequential colimits commute with finite products and homotopy pullbacks. This implies in particular that the associated derivator HO(M) will be regular. where Σ(X) denotes the suspension of an object X , i.e. the pushout of the diagram Recall also that in [17] and [36] the authors construct a stable Quillen model category Sp N (M) of spectra associated with M and with the left Quillen functor Σ(−).
We have the following Quillen adjunction, see [17],  [17]. Suppose also that each object of G considered in M is cofibrant, finitely presented, homotopy finitely presented and has a finitely presented cylinder object. is an equivalence.
The proof of theorem 9.7 will consist in verifying the conditions of the following general proposition.
Proposition 9.8. Let F : T 1 → T 2 be a morphism of strong triangulated derivators. Suppose that the triangulated categories T 1 (e) and T 2 (e) are compactly generated and that there is a set G ⊂ T 1 (e) of compact generators, which is stable under suspensions and satisfies the following conditions : a) F (e) induces bijections Hom T1(e) (g 1 , g 2 ) → Hom T2(e) (F g 1 , F g 2 ), ∀g 1 , g 2 ∈ G and b) the set of objects {F g | g ∈ G} is a set of compact generators in T 2 (e). Then the morphism F is an equivalence of derivators.
Proof. Conditions a) and b) imply that F (e) is an equivalence of triangulated categories, see [32]. Now, let L be a small category. We show that conditions a) and b) are also verified by F (L), T 1 (L) and T 2 (L). By lemma 9.3 the sets {x ! (g)| g ∈ G, x : e → L} and {x ! (F g)| g ∈ G, x : e → L} consist of compact generators for T 1 (L), resp. T 2 (L), which are stable under suspensions. Since F commutes with homotopy colimits F (x ! (g)) = x ! (F g) and so the following isomorphisms imply the proposition. √ Let us now prove theorem 9.7: Proof. Let us first prove condition b) of proposition 9.8. Since the set G of small generators in Ho(M) satisfies the conditions of lemma 9.2, we have a set {Σ n stab(g) | g ∈ G, n ∈ Z} of compact generators in St(HO(M))(e), which is stable under suspensions. We now show that the set {Σ n LΣ ∞ (g) | g ∈ G, n ∈ Z} is a set of compact generators in Ho(Sp N (M)). These objects are compact because the functor Rev 0 in the adjunction commutes with filtered homotopy colimits. We now show that they form a set of generators. Let Y be an object in Ho(Sp N (M)), that we can suppose, without loss of generality, to be an Ω-spectrum, see [17]. Suppose that Since Y is an Ω-spectrum, we have This implies that Y is isomorphic to * in Ho(Sp N (M)). We now show condition a). Let g 1 and g 2 be objects in G. Observe that we have the following isomorphisms, see [14] Hom St(HO(M))(e) (stab(g 1 ), stab(g 2 )) ≃ Hom Ho(M) (g 1 ,
This proves condition a) and so the theorem is proven. Since the category Sset • satisfies all the conditions of theorem 9.7, we have the following characterization of the classical category of spectra, after Bousfield-Friedlander [2], by a universal property. Proposition 9.11. We have an equivalence of categories Proof. By theorems 9.7 and 4.1, we have the following equivalences This proves the proposition. √ Remark 9.12. An analoguos caracterization of the category of spectra, but in the context of stable ∞-categories is proved in [27, 17.6].

DG quotients
Recall from [37] [38] that we have at our disposal a Morita Quillen model structure on the category of small dg categories dgcat, see example 6.1. As shown in [37] [38] the homotopy category Ho(dgcat) is pointed. In the following, we will be considering this Quillen model structure. We denote by I its set of generating cofibrations. Let us now make some general arguments. Let D be a pointed derivator. We denote by M the category associated to the graph 0 ← 1 .
Proof. Apply the small object argument to the morphism ∅ −→ B using the set of generating cofibrations I and obtain the factorization where i is an I-cell. Now consider the following fiber product

Notice that p −1 (A) is a full dg subcategory of Q(B).
Now, by proposition 6.2, we have an isomorphism where J is the filtered category of inclusions of strict finite sub-I-cells X j into Q(B). For each j ∈ J, consider the fiber product In this way, we obtain a morphism of diagrams such that for each j in J, the inclusion Y j ֒→ X j belongs to the set E and J is filtered.
in the category Fun( × J, dgcat). Now, notice that we have the isomorphism in Fun( , dgcat) and the weak equivalence in Fun( , dgcat), when endowed with the projective model structure, see [15]. Since Fun( , dgcat) is clearly also compactly generated, we have an isomorphism Finally, notice that D R is an object of HO(dgcat)( × J) and that p ! (D R ), where p : × J → J denotes the projection functor, identifies with This proves the proposition.
Let D K be the object of HO(dgcat)( × J) constructed in proposition 10.2. In particular p ′ ! (D K ) ∼ → K , where p ′ : × J → denotes the projection functor. The inclusion i : ֒→ , induces a commutative square and a morphism Ψ : We will now show that where p : × J → , denotes the projection functor.
The fact that we have the following commutative square and that the morphism of derivators U T commutes with filtered homotopy colimits implies the following equivalences We now show that Ψ is an isomorphism. For this, by conservativity, it is enough to show that for every object j : e → J in J, the morphism is an isomorphism in St(L Σ,P Hot dgcat f )( ). Recall from proposition 10 where L j belongs to E. We now show that (1 × j) * (Ψ) identifies with ϕ Lj , which by hypotheses, is an isomorphism. Now, the following commutative diagram and the dual of proposition 2.8 in [4] imply that we have the following equivalences Since by hypotheses ϕ Lj is an isomorphism and the morphism G commutes with homotopy colimits the theorem is proven. √

The universal localizing invariant
Recall from theorem 9.7 and remark 9.9 that if we consider for the category M the category L Σ,P Fun(dgcat op f , Sset • ), see example 9.6, we have an equivalence of triangulated derivators ) . Now, stabilize the set E st defined in the previous section under the functor loop space and choose for each element of this stabilized set a representative in the category Sp N (L Σ,P Fun(dgcat op f , Sset • )). We denote the set of these representatives by E st . Since Sp N (L Σ,P Fun(dgcat op f , Sset • )) is a left proper, cellular Quillen model category, see [17], its left Bousfield localization by E st exists. We denote it by L f Est Sp N (L Σ,P Fun(dgcat op f , Sset • )). By lemma 5.3 it is a stable Quillen model category. Observe that the morphism of derivators U l is pointed, commutes with filtered homotopy colimits and satisfies the following condition: Dr) For every inclusion A K ֒→ B of a full dg subcategory the canonical morphism . We now give a conceptual characterization of condition Dr). Let us now denote by I be the category associated with the graph 0 ← 1.
where A f denotes the full dg subcategory of B f whose objects are those in the image by the dg functor F f . Since F f is a quasi-fully faithful dg functor, the dg functor π is a quasi-equivalence. This proves the lemma. Let D be a strong triangulated derivator.
complex. If A is an arbitrary dg k-category, its Hochschild chain complex is defined as the one of a k-flat (e.g. a cofibrant) resolution of A. The following theorem is proved in [22].
Theorem 11.7. The map A → C(A) yields a morphism of derivators which commutes with filtered homotopy colimits, preserves the point and satisfies condition Dr).
Remark 11.8. By theorem 11.5 the morphism of derivators C factors through U l and so gives rise to a morphism of derivators C : M loc dg → HO(Λ − M od) . Non-connective K-theory. Let A be a small dg category. Its non-connective K-theory spectrum K(A) is defined by applying Schlichting's construction [35] to the Frobenius pair associated with the category of cofibrant perfect A-modules (to the empty dg category we associate 0). Recall that the conflations in the Frobenius category of cofibrant perfect A-modules are the short exact sequences which split in the category of graded A-modules.

Theorem 11.9. The map A → K(A) yields a morphism of derivators
to the derivator associated with the category of spectra, which commutes with filtered homotopy colimits, preserves the point and satisfies condition Dr).
Proof. Proposition 11.15 in [35], which is an adaption of theorem 1.9.8 in [43], implies that we have a well defined morphism of derivators HO(dgcat) → HO(Spt) . Lemma 6.3 in [35] implies that this morphism commutes with filtered homotopy colimits and theorem 11.10 in [35] implies that condition Dr) is satisfied. √ Remark 11.10. By theorem 11.5, the morphism of derivators K factors through U l and so gives rise to a morphism of derivators K : M loc dg → HO(Spt) . We now establish a connection between Waldhausen's S • -construction, see [44] and the suspension functor in the triangulated category M loc dg (e). Let A be a Morita fibrant dg category, see [37] [38]. Notice that Z 0 (A) carries a natural exact category structure obtained by pulling back the garded-split structure on C dg (A) along the Yoneda functor h : Notation 11.11. Remark that the simplicial category S • A, obtained by applying Waldhausen's S • -construction to Z 0 (A), admits a natural enrichissement over the complexes. We denote by S • A this simplicial Morita fibrant dg category obtained.
Recall that ∆ denotes the simplicial category and p : ∆ → e the projection functor.

Proposition 11.12. There is a canonical isomorphism in
Proof. As in [28, 3.3], we consider the sequence in HO(dgcat)(∆) where A • denotes the constant simplicial dg category with value A and P S • A the path object of S • A. For each point n : e → ∆, the nth component of the above sequence is the following short exact sequence in Ho(dgcat) Since the morphism of derivators U l satisfies condition Dr), the conservativity axiom implies that we obtain a triangle [1] in M loc dg (∆). By applying the functor p ! , we obtain the following triangle in M loc dg (e). We now show that we have natural isomorphisms and where 0 denotes the zero object in the triangulated category M loc dg (e). This clearly implies the proposition. Since the morphisms of derivators Φ, stab and γ commute with homotopy colimits it is enough to show that we have isomorphisms and where * denotes the terminal object in Hot dgcat f (e). Notice that since A and P S n A, n ≥ 0 are Morita fibrant dg categories, we have natural isomorphisms Now, since homotopy colimits in Fun(dgcat op f , Sset) are calculated objectwise and since h(A • ) is a constant simplicial object in Fun(dgcat op f , Sset), corollary 18.7.7 in [15] implies that we have an isomorphism in M loc dg (e). Notice also that since P S • A is a contractible simplicial object, see [28], so is h(P S • A). Since homotopy colimits in Fun(dgcat op f , Sset) are calculated objectwise, we have an isomorphism This proves the proposition. √

A Quillen model in terms of presheaves of spectra
In this section, we construct another Quillen model category whose associated derivator is the localizing motivator of dg categories M loc dg . Consider the Quillen adjunction Proof. Observe that theorems 5.4 and 9.7 imply that both derivators have the same universal property. This proves the proposition. √ Remark 12.3. Notice that the stable Quillen model category identifies with Fun(M op f , Sp N (Sset • )) endowed with the projective model structure.
The above considerations imply the following proposition.

Upper triangular DG categories
In this section we study upper triangular dg categories using the formalism of Quillen's homotopical algebra. In the next section, we will relate this important class of dg categories with split short exact sequences in Ho(dgcat).
Definition 13.1. An upper triangular dg category B is given by an upper triangular matrix where A and C are small dg categories and X is a A-C-bimodule. A morphism F : B → B ′ of upper triangular dg categories is given by a triple F := (F A , F C , F X ), where F A , resp. F C , is a dg functor from A to A ′ , resp. from C to C ′ , and F X is a morphism of A-C-bimodules from X to X ′ (we consider X ′ endowed with the action induced by F A and F C ). The composition is the natural one.
Notation 13.2. We denote by dgcat tr the category of upper triangular dg categories.
Let B ∈ dgcat tr . Definition 13.3. Let |B| be the totalization of B, i.e. the small dg category whose set of objects is the disjoint union of the set of objects of A and C and whose morphisms are given by We have the following adjunction We have at our disposal the following adjunction [38][39] that dgcat admits a structure of cofibrantly generated Quillen model category whose weak equivalences are the Morita dg functors. This structure clearly induces a componentwise model structure on dgcat × dgcat which is also cofibrantly generated.
(1) Since the functor E is also a right adjoint to p 1 × p 2 , the functor p 1 × p 2 commutes with colimits and so condition (1) is verified. (2) Let J, resp. J × J, be the set of generating trivial cofibrations in dgcat, resp. in dgcat × dgcat. Since the functor p 1 × p 2 commutes with filtered colimits it is enough to prove the following: let G : B ′ → B ′′ be an element of the set E(J × J), B an object in dgcat tr and B ′ → B a morphism in dgcat tr . Consider the following push-out in dgcat tr : We now prove that (p 1 × p 2 )(G * ) is a weak-equivalence in dgcat × dgcat.
Observe that the image of the previous push-out under the functors p 1 and p 2 correspond to the following two push-outs in dgcat : Since G A ′ * and G C ′ * belong to J the morphism is a weak-equivalence in dgcat × dgcat. This proves condition (2). The proposition is then proven. √ Let B, B ′ ∈ dgcat tr .
Definition 13.7. A morphism F : B → B ′ is a total Morita dg functor if F A and F C are Morita dg functors, see [37][38] [39], and F X is a quasi-isomorphism of A-C-bimodules.
Remark 13.8. Notice that if F is a total Morita dg functor then |F | is a Morita dg functor in dgcat but the converse is not true.
Theorem 13.9. The category dgcat tr admits a structure of cofibrantly generated Quillen model category whose weak equivalences W are the total Morita dg functors and whose fibrations are the morphisms F : B → B ′ such that F A and F C are Morita fibrations, see [37] [38], and F X is a componentwise surjective morphism of bimodules.
Proof. The proof is based on enlarging the set E(I×I), resp. E(J ×J), of generating cofibrations, resp. generating trivial cofibrations, of the Quillen model structure of proposition 13.6. LetĨ be the set of morphisms in dgcat tr where S n−1 is the complex k[n − 1] and D n the mapping cone on the identity of S n−1 . The k-k-bimodule S n−1 is sent to D n by the identity on k in degree n − 1.
Consider also the setJ of morphisms in dgcat tr Observe that a morphism F : B → B ′ in dgcat tr has the right lifting property (=R.L.P.) with respect to the setJ, resp.Ĩ, if and only if F X is a componentwise surjective morphism, resp. surjective quasi-isomorphism, of A-C-bimodules. Define I := E(I × I) ∪Ĩ as the set of generating cofibrations in dgcat tr and J := E(J × J) ∪J as the set of generating trivial cofibrations. We now prove that conditions (1)-(6) of theorem 2.1.19 from [16] are satisfied. This is clearly the case for conditions (1)-(3).
(4) We now prove that J-cell ⊂ W, see [15]. Since by proposition 13.6 we have E(J × J)-cell ⊂ W ′ , where W ′ denotes the weak equivalences of proposition 13.6, it is enough to prove that pushouts with respect to any morphism inJ belong to W. Let n be an integer and B an object in dgcat tr . Consider the following push-out in dgcat tr : Notice that the morphism T corresponds to specifying an object A in A and an object C in C. The upper triangular dg category B ′ is then obtained from B by gluing a new morphism of degree n from A to C. Observe that R A and R C are the identity dg functors and that R X is a quasi-isomorphism of bimodules. This shows that R belongs to W and so condition (4) is proved.  (5) and (6). This proves the theorem. √ Remark 13.10. Notice that the Quillen model structure of theorem 13.9 is cellular, see [15] and that the domains and codomains of I (the set of generating cofibrations) are cofibrant, ℵ 0 -compact, ℵ 0 -small and homotopically finitely presented, see definition 2.1.1. from [41]. This implies that we are in the conditions of proposition 6.2 and so any object B in dgcat tr is weakly equivalent to a filtered colimit of strict finite I-cell objects.
Proposition 13.11. If B be a strict finite I-cell object in dgcat tr , then p 1 (B), p 2 (B) and |B| are strict finite I-cell objects in dgcat.
Proof. We consider the following inductive argument: -notice that the initial object in dgcat tr is ∅ 0 0 ∅ and it is sent to ∅ (the initial object in dgcat) by the functors p 1 , p 2 and | − |. -suppose that B is an upper triangular dg category such that p 1 (B), p 2 (B) and |B| are strict finite I-cell objects in dgcat. Let G : B ′ → B ′′ be an element of the set I in dgcat tr , see the proof of theorem 13.9, and B ′ → B a morphism. Consider the following push-out in dgcat tr : We now prove that p 1 (PO), p 2 (PO) and |PO| are strict finite I-cell objects in dgcat. We consider the following two cases : 1) G belongs to E(I × I): observe that p 1 (PO), p 2 (PO) and |PO| correspond exactly to the following push-outs in dgcat: Since G A ′ and G C ′ belong to I this case is proved. 2) G belongs toĨ: observe that p 1 (PO) identifies with A, p 2 (PO) identifies with C and |PO| corresponds to the following push-out in dgcat : where S(n) is a generating cofibration in dgcat, see section 2 in [39]. This proves this case. The proposition is proven. √

Split short exact sequences
In this section, we establish the connexion between split short exact sequences of dg categories and upper triangular dg categories.
Definition 14.1. A split short exact sequence of dg categories is a short exact sequence of dg categories, see [19], which is Morita equivalent to one of the form To a split short exact sequence, we can naturally associate the upper triangular dg category Conversely to an upper triangular dg category B such that C admits a zero object (for instance if C is Morita fibrant), we can associate a split short exact sequence where P and R are the projection dg functors. Moreover, this construction is functorial in B and sends total Morita equivalences to Morita equivalent split short exact sequences. Notice also that by lemma 13.4 this functor preserves colimits.
Proposition 14.2. Every split short exact sequence of dg categories is weakly equivalent to a filtered homotopy colimit of split short exact sequences whose components are strict finite I-cell objects in dgcat.
be a split short exact sequence of dg categories. We can supose that A, B and C are Morita fibrant dg categories, see [37] [38]. Consider the upper triangular dg category Now by remark 13.10, B is equivalent to a filtered colimit of strict finite I-cell objects in dgcat tr . Consider the image of this diagram by the functor, described above, which sends an upper triangular dg category to a split short exact sequence. By proposition 13.11 the components of each split short exact sequence of this diagram are strict finite I-cell objects in dgcat. Since the category dgcat satisfies the conditions of proposition 6.2, filtered homotopy colimits are equivalent to filtered colimits and so the proposition is proven. √

Quasi-Additivity
Recall from section 11 that we have at our disposal the Quillen model category L Σ,P Fun(dgcat op f , Sset) which is homotopically pointed, i.e. the morphism ∅ → * , from the initial object ∅ to the terminal one * , is a weak equivalence. We now consider a strictly pointed Quillen model. This follows from the fact that the category L Σ,P Fun(dgcat op f , Sset) is homotopically pointed and from the following general argument.
where f ♯ is the morphism which corresponds to f , considered as a morphism in M, under the adjunction and i : ∅ ∼ → * . Since the morphism i ∐ 1 corresponds to the homotopy colimit of i and 1, which are both weak equivalences, proposition 5.1 implies that i ∐ 1 is a weak equivalence. Now, by the '2 out of 3' property, the morphism f is a weak equivalence if and only if f ♯ is one. This proves the proposition.
-Notice that by construction the domains and codomains of the set E s un are homotopically finitely presented objects. Therefore by lemma 8.1 the set in Sset. This proves the proposition. √ Let A be a Morita fibrant dg category. Recall from notation 11.11 that S • A denotes the simplicial Morita fibrant dg category obtained by applying Waldhausen's S • -construction to the exact category Z 0 (A) and remembering the enrichment in complexes. Proof. As in [28, 3.3], we consider the sequence in HO(dgcat)(∆) where A • denotes the constant simplicial dg category with value A and P S • A the path object of S • A. By applying the morphism of derivators U u to this sequence, we obtain the canonical morphism in M unst dg (∆). We now prove that for each point n : e → ∆, the nth component of For each point n : e → ∆, we have a split short exact sequence in Ho(dgcat): 0 → 0 → A 0 → · · · → A n−1 and R n maps a sequence 0 → A 0 → A 1 → · · · → A n−1 to A 0 . Now, by construction of L g E s un L Σ,P Fun(dgcat op f , Sset • ) the canonical morphisms S In : cone(U u (A In → P S n A)) → U u (S n A), n ∈ N are isomorphisms in M unst dg (e). Since homotopy colimits in L g E s un L Σ,P Fun(dgcat op f , Sset • ) are calculated objectwise the nth component of S I identifies with S In and so by the conservativity axiom S I is an isomorphism in M unst dg (∆). This implies that we obtain the homotopy cocartesian square As in the proof of proposition 11.12, we show that p ! U u (A • ) identifies with N.wrep mor (?, A) = U u (A) and that p ! (U u (P S • A)) is contractible. Since we have the equivalence In particular, we have the following isomorphisms Proof. This follows from propositions 15.10, 15.11 and from the following weak equivalences   We now will prove that this morphism of derivators is co-representable in M add dg . Let A be a small dg category. Proof. We prove first that K(A) c is a fibrant object in Sp N (L g E s un L Σ,P Fun(dgcat op f , Sset • )). By [16] [36] we need to show that K(A) c is an Ω-spectrum, i.e. that K(A) c n is a fibrant object in L g in Fun(dgcat op f , Sset • ). Now observe that for every integer n, K c (A) n satisfies conditions (1)-(3) of proposition 15.10. Condition (4) follows from Waldhausen's fibration theorem, as in the proof of proposition 15.11, applied to the S.-construction. This shows that K(A) c is an Ω-spectrum.
We now prove that η is a (componentwise) weak equivalence in Sp N (L g E s un L Σ,P Fun(dgcat op f , Sset • )). For this, we prove first that the structural morphisms β n : S 1 ∧ N.wS . We now show that the structure morphism β n , which corresponds to β ♯ n by adjunction, see [44], is also a weak equivalence. The derived adjunction (S 1 ∧ −, RΩ(−)) induces the following commutative diagram in Ho(L g E s un L Σ,P Fun(dgcat op f , Sset • )) where the vertical arrow is an isomorphism since the previous square is homotopy bicartesian. This shows that the induced morphism is an isomorphism in Ho(L g E s un L Σ,P Fun(dgcat op f , Sset • )) and so β n is a weak equivalence. Now to prove that η is a componentwise weak equivalence, we proceed by induction : observe that the zero component of the morphism η is the identity. Now suppose that the n-component of η is a weak equivalence. The n + 1-component of η is the composition of β n+1 , which is a weak equivalence, with the suspension of the n-component of η, which by proposition 5.1 is also a weak equivalence.
Proof. By theorem 11.9 the morphism of derivators   Finally notice that the canonical element 1 ∈ k furnishes us the higher Chern characters and so the theorem is proven. √

Concluding remarks
By the universal properties of U u , U a and U l , we obtain the following diagram: Notice that Waldhausen's connective K-theory is an example of an additive invariant which is NOT a derived one, see [19]. Waldhausen's connective K-theory becomes co-representable in M add dg by theorem 16.10. An analogous result should be true for non-connective K-theory and the morphism Φ : M add dg / / M loc dg . should be thought of as co-representing 'the passage from additivity to localization'.