$t$-Structures on stable derivators and Grothendieck hearts
Manuel Saorín Jan Šťovíček Simone Virili












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	lemma[theorem]Lemmaproposition[theorem]Propositioncorollary[theorem]Corollaryremarkremark[theorem]Remarkdefinitionexample[theorem]Exampleexercise[theorem]Exercisequestion[theorem]Questiondefinition[theorem]Definitionnotation[theorem]Notationt-Structures on stable derivators and Grothendieck hearts
    Manuel Saorín[The first named author was supported by the research projects from
the Ministerio de Economía y Competitividad of Spain (MTM2016-77445-P) and the Fundación `Séneca' of
Murcia (19880/GERM/15), both with a part of FEDER funds.] Jan Šťovíček[The second named author was supported by Neuron Fund for Support of Science.] Simone Virili[The third named author was supported by the Ministerio de Economía y Competitividad of
Spain via a grant `Juan de la Cierva-formación'. He was also supported by the
Fundación `Séneca' of Murcia (19880/GERM/15) with a part of FEDER funds.]
    
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We prove that, given any strong and stable derivator and a t-structure in its base triangulated category D, the t-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one.
We use this to show that the t-structure being compactly generated implies that the coaisle is closed under directed homotopy colimits which, in turn, implies that the heart is an (Ab.5) Abelian category. If, moreover, D is a well-generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) t-structure has a generator.
As a consequence, it follows that the heart of any compactly generated t-structure of a well-generated algebraic or topological triangulated category is a Grothendieck Abelian category.
t-structures, heart, derivator, Grothendieck Abelian category.
  18E30, 18E40, 18E15.

tocsectionIntroduction

§ INTRODUCTIONt-Structures in triangulated categories were introduced by Beilinson, Bernstein and Deligne <cit.> in their study of perverse
sheaves on an algebraic or analytic variety. At-structure in a triangulated category𝒟is a pair of full
subcategories satisfying a suitable set of axioms (see the precise definition
in Subsection <ref>) which guarantees that their intersection is an
Abelian categoryℋ, called the heart of thet-structure. One then naturally defines a cohomological functor

    H𝒟⟶ℋ,
 
which allows to develop an intrinsic (co)homology theory,
where the homology “spaces” are objects of𝒟itself.t-Structures have been used in many
branches of mathematics, with special impact in algebraic geometry, algebraic topology
and representation theory of algebras. 


Given at-structure in a triangulated category D, and considering the induced Abelian category$̋, a natural problem consists in finding necessary and sufficient conditions on
the t-structure and on the ambient category for the heart to be a “nice” Abelian
category. When our triangulated category has (co)products, the category $̋ is known to be (co)complete (see <cit.>) and, using the classical hierarchy of Abelian categories due to Grothendieck <cit.>, the natural question  
 is the following: 


When is the heart $̋ a Grothendieck Abelian category?


As one might expect, the real issue is to prove that$̋ has exact directed colimits. In this respect, we encounter a phenomenon which seems invisible to the triangulated category D alone, namely directed homotopy colimits and the question on whether $̋ is closed under these. To work with homotopy colimits, we need a certain enhancement of Dand we choose Grothendieck derivators, as this is in some sense the minimal homotopy theoretic framework where a well-behaved calculus of homotopy (co)limits is available. This said, we are immediately led to the second main problem of the paper:


How do t-structures interact with strong and stable Grothendieck derivators?


The study of Question <ref> has a long tradition in algebra. In its initial steps, the focus was almost exclusively put on the case of the so-called Happel-Reiten-Smalø introduced in <cit.> and treated in much greater generality in <cit.>.
These aret-structures on a derived category( G)of an Abelian category Ginduced by a torsion pair in G.
The study of conditions for the heart of the Happel-Reiten-Smaløt-structure in(𝒢), for a Grothendieck or module category𝒢, to be again a Grothendieck or a module category, has received a lot of attention in recent years (see <cit.> and <cit.>).
Let us remark that the first named author with C. Parra <cit.> gave a complete answer to Question <ref> in this particular case: the heart of the Happel-Reiten-Smaløt-structure_τ, associated to a torsion pairτ=(𝒯,ℱ)in a Grothendieck Abelian category𝒢, is again a Grothendieck Abelian category if, and only if, the torsion-free classℱis closed under taking directed colimits in𝒢.  


When more generalt-structures are considered, the answers to Question <ref> are more scarce and they are often inspired in one or another way by tilting theory. In a sense, the classical derived Morita Theorems of Rickard <cit.> (for the bounded setting) and Keller <cit.> (for the unbounded case) can be seen as the first examples where an answer to the problem is given. Namely, ifAandBare ordinary algebras
 and_BT_Ais a two-sided tilting complex (see <cit.>), then the triangulated equivalence-⊗_B^𝕃T(B)⟶̃(A)takes the canonicalt-structure(^≤ 0(B),^≥ 0(B))to the pair(T^⊥_>0,T^⊥_<0), which is then at-structure in(A), whose heart is equivalent to B. This includes the case of a classical (n-)tilting module in the sense of <cit.>. The dual of a (not necessarily classical) (n-)tiltingA-module is that of a (big) (n-)cotiltingA-moduleQ, in which case the second named author proved that(_^⊥_<0Q,_^⊥_>0Q)is at-structure in(A)whose heart is a Grothendieck Abelian category (see <cit.>). These two  results have recently been extended to include all silting sets of compact objects and all pure-injective cosilting sets in a compactly generated triangulated category (see <cit.>, also for the used terminology). Results saying that cosiltingt-structures have Grothendieck hearts under appropriate assumptions also include <cit.> and <cit.>, whereas conditions under which thet-structure(T^⊥_>0,T^⊥_<0)obtained from a non-classical tilting moduleThas a Grothendieck heart were given in <cit.>.
A common feature of the results in <cit.> is that the heart is proved to be a Grothendieck Abelian category rather indirectly, using the pure-injectivity of certain cotilting or cosilting objects and ideas from model theory of modules.


Last but not least, there is a family of results which provides evidence thatt-structures generated by a set of compact objects should, under all reasonable circumstances, have Grothendieck hearts.
Briefly summarizing, <cit.> establishes this result for any compactly generatedt-structure in the derived category(R)of a commutative ringRwhich is given by a left bounded filtration by supports, <cit.> gives the same result for any non-degenerate compactly generatedt-structure in an algebraic compactly generated triangulated category,
and finally Bondarko establishes such a result in <cit.> for practically all triangulated categories which one encounters in practice.


Our approach here is rather different from the ones above and, in some sense, much more direct. It is more in the spirit of Lurie's <cit.>, where a criterion for at-structure to have a Grothendieck heart is given in the language of∞-categories. Our aim is to reach the corresponding criterion for exactness of directed colimits in the heart faster and in a way hopefully more accessible to the readers concerned with the representation theory of associative algebras and related fields.


To outline our strategy, consider at-structure = (,Σ)on a triangulated category Dwith coproducts and let=̋∩Σbe the heart of. Then we have the following well-known chain of implications:


 is compactly        is closed       $̋ has exact 

generated (see p.compactly_gen_t_st)        under coproducts         coproducts.

However,being closed under coproducts is not enough to have exact directed colimits in$̋. This follows from the main results of <cit.> (see also Example <ref> below).
What we need instead is a stronger condition: we assume that  is closed under directed homotopy colimits. For instance, in the case of the derived category D = () of a Grothendieck Abelian category , the homotopy colimit is the total left derived functor


    _I = 𝐋_I(^I) ⟶()

of the usual colimit functor _I^I →. The problem is, of course, that albeit there always exists a canonical comparison functor (^I) →()^I, it is typically far from being an equivalence.
At this point, the language of derivators naturally enters the scene, as the categories of the form (^I) can be naturally assembled to form a derivator (for all the undefined terminology we refer to Section <ref>). To see this, one should recall that, when ℳ is a cofibrantly generated model category with 𝒲 its class of weak equivalences (e.g., the usual injective model structure of (), with  the class of quasi-isomorphisms), the functor category ℳ^I admits a cofibrantly generated model structure, with weak equivalences calculated level-wise, for each small category I, and the assignment I↦𝔻(I):=Ho(ℳ^I) gives a well-defined derivator (see<cit.>), i.e. a 2-functor

    ^⟶

which satisfies certain axioms, where  is the 2-category of small categories and  is the 2-“category" of all categories. Furthermore,  is strong and stable provided ℳ is a stable model category. The axioms in particular imply that the natural range of this 2-functor is naturally the 2-“category” of all triangulated categories, that is, 𝔻(I) is a triangulated category for each I∈. A prototypical example is precisely the assignment I ↦(^I) for a Grothendieck Abelian category .


The main results of the paper will be proven in general for such a strong and stable derivator . Note also that, denoting by  the one-point category with the identity morphism only, and letting 𝒟:=() the base category of , one usually views  as an enhancement of the triangulated category D, which is in some sense the minimalistic enhancement which allows for a well-behaved calculus of homotopy (co)limits or, more generally, homotopy Kan extensions (see <cit.>). More precisely, a homotopy colimit functor _I(I)→() is simply a left adjoint to (_I)()→(I), where _I is the unique functor I→ in . The existence of the adjoint is ensured by the axioms of a derivator and this notion of homotopy colimit is consistent with our previous definition of _I as the total left derived functor of _I.


The advantage of derivators is that we now can give  a precise meaning to what it means that  is closed under directed homotopy colimits but, on the other hand, we now fully hit Question <ref>. We started with a t-structure =(,Σ) in the base category (), and it is a natural question whether this t-structure lifts to the other triangulated categories (I), with I∈. Even when it does, what we are really concerned with is the relation between the heart in (I) and the I-shaped diagrams in the heart of (). Luckily, both these problems have a very natural solution that is condensed in the following theorem (for the proof see Section <ref>):


Let ^→ be a strong and stable derivator, and =(,Σ) a t-structure in (). If we let 

    _I:={X∈(I):X_i∈, ∀ i∈ I},
       _I:={Y∈(I):Y_i∈, ∀ i∈ I},
 
then _I:=(_I,Σ_I) is a t-structure in (I). Furthermore, the diagram functor 

    _I(I)⟶()^I
 
induces an equivalence _̋I≅^̋I between the heart _̋I of _I and the category ^̋I of diagrams of shape I in the heart of .


Now we can state our main answer to the part of Question <ref> which is concerned with the exactness of directed colimits in the heart of a t-structure. This result will be proved in Section <ref>.


Let ^→ be a strong stable derivator and let =(,Σ) be a t-structure in (). Then we have the implications:


 is compactly        is closed under       $̋ has exact 

generated        directed homotopy colimits        directed colimits.


What remains is to give a criterion for the heart to have a generator. As it turns out, this is, unlike the (Ab.5) condition, a problem of mostly a technical nature. For most triangulated categories (or derivators) andt-structures arising in practice, the answer is affirmative. In the next theorem we give a general criterion in the setting of what one may call “accessible stable derivators”, that is, the ones associated with a stable combinatorial model category. This will be treated in Section <ref>.


Let (,, B, F) be a stable combinatorial model category, D=() the triangulated homotopy category and =(,Σ) a λ-accessibly embedded t-structure in D for some infinite regular cardinal λ (e.g., D a well-generated algebraic or topological triangulated category, with a t-structure generated by a small set of objects).


Then the heart =̋∩Σ of  has a generator. If, in particular,  is homotopically smashing (equivalently, λ=ℵ_0), then $̋ is a Grothendieck Abelian category.


As an immediate consequence, we obtain the following corollary which provides an alternative to <cit.> in showing that the heart of a compactly generatedt-structure is, in practice, always a Grothendieck Abelian category (see also Remark <ref>).


Let D=(), where  is a combinatorial stable model category. If =(𝒰,Σ𝒱) is a compactly generated t-structure in D, then the heart =̋𝒰∩Σ𝒱 is a Grothendieck Abelian category.
 Acknowledgement. 
It is a pleasure for us to thank Fritz Hörmann and Moritz Groth for helpful discussions and suggestions. We are also grateful to Rosie Laking and Gustavo Jasso for  pointing out some analogies with results of Jacob Lurie in the∞-categorical setting, and to Mikhail Bondarko for telling us about his results in <cit.>. Finally, we are indebted to an anonymous referee whose numerous remarks and suggestions have helped to greatly improve the exposition and clarity of our results.





§ PRELIMINARIES AND NOTATION


In this first section we fix most of the notations and conventions about general category theory that will be needed in the rest of the paper. This includes basic facts about additive and Abelian categories, torsion pairs, triangulated categories,t-structures, categories with weak equivalences and model categories. All the results included in this section are known, so the proofs are omitted in favor of suitable references to the existing literature.



 §.§ Conventions and basic results in category theory

Given a categoryand two objectsxandyin, we denote by(x,y):=_(x,y)the-set of all morphismx→ yin. Throughout the paper, all the subcategories will be assumed to be full, so we just say “subcategory” to mean “full subcategory”. Similarly, we  generally drop the distinction between a subcategory⊆and a subclass⊆(), as the two things univocally determine each other. 

We denote bythe category of Abelian groups.


Ordinals.
Any ordinalλcan be viewed as a category in the following way: the objects ofλare the ordinalsα<λand, givenα,β<λ, the-setλ(α,β)is a point ifα≤β, while it is empty otherwise. Following this convention,

  * ={0} is the category with one object and no non-identity morphisms;

  * ={0→ 1} is the category with one non-identity morphism;

  *  in general, 𝐧={0→ 1→…→ (n-1)}, for any n∈_>0.
Functor categories, limits and colimits.
A categoryIis said to be ( skeletally)  small when (the isomorphism classes of) its objects form a set. IfandIare an arbitrary and a small category, respectively, a functorI→is said to be a  diagram inof shapeI. The category of diagrams inof shapeI, and natural transformations between them, is denoted by^I.
A diagramXof shapeI, is also denoted as(X_i)_i∈ I, whereX_i
:= X(i)for eachiinI. When every diagram of shapeIhas a limit (resp., colimit), we say
thathas allI-limits (resp.,I-colimits). In this case,lim_I ^I→(resp.,_I ^I→) will denote
the (I-)limit (resp., (I-)colimit) functor, which is the right (resp., left) adjoint to the constant diagram functorκ_I→^I. A particular case, very important for us,  comes whenIis a directed set, viewed as
a small category in the usual way. The corresponding colimit
functor is the (I-)directed colimit functor_I^I→. TheI-diagrams inare usually called directed
systems  of shapeI.



The categoryis said to be
 complete (resp.,  cocomplete, bicomplete) when allI-limits (resp.,I-colimits, both) exist in, for any small categoryI.


2-Categories of categories.
We denote bythe2-category of small categories and by^the2-category obtained by reversing the direction of the functors in(but letting the direction of natural transformations unchanged). Similarly, we  denote bythe  of all categories. This, when taken literally, may originate some set-theoretical problems that, for our constructions, can be safely ignored: see the discussion after <cit.>.

Given two natural transformationsα F⇒ G→andβ G⇒ H→, we denote byβ∘α F⇒ Htheir vertical composition, that is,(β∘α)_C:=β_C∘α_Cfor eachC∈. Furthermore, given two natural transformationsα F_1⇒ F_2→andβ G_1⇒ G_2→ℰ, we denote byβ⊛α G_1∘ F_1⇒ G_2∘ F_2their horizontal composition, that is,(β⊛α)_C:=β_F_2(C)∘ G_1(α_C)=G_2(α_C)∘β_F_1(C), for eachC∈. With a slight abuse of notation, we also letβ⊛ F_1:=β⊛𝕀_F_1andG_1⊛α:=𝕀_G_1⊛ α.



Categories of adjoint functors and mates.
Letandbe two categories. Given a pair of adjoint functorsL:⇄:R, we use the following compact notation 

    LηϵR
 
to mean thatLis left adjoint toR, with unitη𝕀_⇒ R∘ Land counitϵ L∘ R⇒𝕀_. In particular, the following relations hold

    𝕀_L=(ϵ⊛ L)∘( L⊛η)   and  𝕀_R=(R⊛ϵ) ∘ (η⊛ R)

Furthermore, we let(,)(resp.,(,)) be the (full) subcategory of(,)spanned by the left (resp., right) adjoint functors.

Given two categories  and , there is an equivalence of categories 

    Φ(,)⟶(,)^,
 
which is constructed as follows:

  *  for each left adjoint L fix a right adjoint Φ(L) of L (that is, L⊣Φ(L));

  *  given two left adjoints L and L', where LηϵΦ(L) and L'η'ϵ'Φ(L'), and a natural transformation α L⇒ L' one defines Φ(α)Φ(L')⇒Φ(L) as follows:

    Φ(α):=(Φ L⊛ϵ')∘ (Φ L⊛α⊛Φ L')∘ (η⊛Φ L').

Furthermore, the bijection Φ_L,L'(,)(L,L')→(,)(Φ(L'),Φ(L)) respects compositions and identities. In particular, a given α L⇒ L' is a natural isomorphism if and only if Φ(α) is a natural isomorphism.

Given a natural transformation between left adjointsα L⇒ L', its imageΦ(α)in(,)via the equivalence described above is said to be the  total mate of, or the natural transformation  conjugated to,α.
Consider now a square in, that commutes up to the natural transformationα:

    @C=10pt@R=6pt[dd]_L[rr]^F      '[dd]^L'@[ddll]|-α
    
    [rr]_G      '

If there are adjunctionsLηϵR→, andL'η'ϵ'R''→', then we can consider the following pasting diagram:

    @C=10pt@R=6pt@<-[dd]_R[rr]^F      '@<-[dd]^R'@[ddll]|-α_*
    
    [rr]_G      '
        @C=11pt@R=2pt   @/_10pt/@=[ddrr][rr]^R      [dd]_L[rr]^F      '[dd]^L'@/_-10pt/@=[ddrr]      
    
    :=              ϵ      α      η'           
             [rr]_G      '[rr]_R'      '

that is,α_*:=(R'G⊛ϵ)∘ (R'⊛α⊛ R)∘ (η'⊛ FR). Dually, given  adjunctionsF_!η^Fϵ^FF'→, andG_!η^Gϵ^GG'→, we can consider the following pasting:

    @C=10pt@R=6pt@<-[dd]_F_![rr]^L      @<-[dd]^G_!@[ddll]|-α_!
                :=
    '[rr]_L'      '
        @C=11pt@R=2pt'@/_10pt/@=[ddrr][rr]^F_!      [dd]_F[rr]^L      [dd]^G@/_-10pt/@=[ddrr]      
               η^F      α      ϵ^G           
          '[rr]_L'      '[rr]_G_!

that is,α_!:=(ϵ^G⊛ LF_!)∘ (G_!⊛α⊛ F_!)∘ (G_!L'⊛η^F). In fact, there is a close relation betweenα_!andα_*:


Let , ',  and ' be categories, consider the adjunctions:

    LηϵR→,       L'η'ϵ'R''→',      F_!η^Fϵ^FF'→,      G_!η^Gϵ^GG'→.

and a natural transformation α L'∘ F⇒ G∘ L. Then α_*  F∘ R⇒ R'∘ G is the natural transformation conjugated to α_! G_!∘ L'⇒ L∘ F_!. In particular, α_* is a natural isomorphism if, and only if, α_! is a natural isomorphism.
Furthermore, the operations (-)_* and (-)_! are each other's inverse, in the sense that 

    (α_*)_!=α=(α_!)_*.


 §.§ Additive categories, Abelian categories and torsion pairs
Additive categories.
Recall that a category is said to be additive if it is an-enriched category with a0-object and finite products and coproducts. An important property of additive categories is that, in this context, finite products and coproducts coincide. More precisely, given an additive categoryand two objectsA_1andA_2, any coproduct(A, (ϵ_i A_i→ A)_i=1,2)(resp., any product(A, (π_i A→ A_i)_i=1,2)) can be completed to a biproduct, that is, a triple(A, (ϵ_i A_i→ A)_i=1,2, (π_i A→ A_i)_i=1,2)such that the following identities hold:

    π_1ϵ_1=𝕀_A_1,  π_2ϵ_2=𝕀_A_2,  ϵ_1π_1+ϵ_2π_2=𝕀_A,  π_2ϵ_1=0=π_1ϵ_2.

This allows one to interpret (column-finite) matrices as morphisms between (infinite) coproducts.


Let  be an additive category and consider the following sequence in :

    A[r]^α    B[r]^β    C.

We say that the above sequence is  split-exact if there exist α' B→ A and β' C→ B such that the triple (B,(α,β'),(α',β)) is a biproduct of A and C.
 
Note that a sequence Aα⟶Bβ⟶C in an additive category 𝒞 is split-exact if, and only if, either one of the following equivalent conditions holds:

  *  it is a  kernel-cokernel sequence (i.e., α is the kernel of β, and β is the cokernel of α) such that α is a section (resp., β is a retraction);

  *  there are morphisms α' B→ A and β' C→ B such that α'α =𝕀_A, ββ'=𝕀_C and 𝕀_B=αα'+β'β (i.e., the conditions βα=0=α'β' are redundant). Indeed, notice for example that βα=βαα'α=β(𝕀_B-β'β)α=(𝕀_C-ββ')βα=0.


Whenα A→ A'andβ B'→ Bare isomorphisms, the following are trivial examples of split-exact sequences:

    @C=33pt
    A[r]^α    A'[r]^0    0   0[r]^0    B'[r]^β    B.

Let  be an additive category. 

  *  Consider the following sequence in :

    @C=55pt
    A[r]^-α:=[α_1
    α_2]    A'⊔ B'[r]^-β:=[β_1   β_2]   B.

This sequence is split-exact if β∘α=0 and both α_1 A→ A' and β_2 B'→ B are isomorphisms as, in this case, it is enough to choose α':=[ccα_1^-1   0] and β':=[c0 
β_2^-1]. Similarly, the sequence is split-exact if β∘α=0 and both α_2 and β_1 are isomorphisms.

  *  If X∈ if an object such that the countable coproduct X^() of copies of X exists in , then the following sequence is split-exact:

    @C=130pt
    X^()[r]^α=
    [cccc
    1         
    
    -1   1      0
       -1   1   
    0      ⋱   ⋱]
       X^()[r]^β=[cccc
    1   1   1   ⋯]   X,
 
where α':=[ccccc
0   -1   -1   -1   ⋯
   0   -1   -1   ⋯
      0   -1   ⋯
0         ⋱   ⋱] and β':=[c1 
0 
0 
⋮].
Subcategories of additive categories.
Given an additive categoryand a (always full) subcategory⊆, we shall denote by_()(resp.,_()), or simply()(resp.,()) if no confusion is possible, the subcategory ofspanned by the direct summands of finite (resp., small) coproducts of objects in.

We use the notation=to mean thatis closed under taking directed colimits, that is, givenF I→forIa directed set, such thatF(i)∈for alli∈ I, whenever the colimit_IFexists in the ambient category, it is an object of.

A family of objectsis said to be a  set of generators when the functor

    ∐_S∈(S,-) ⟶
 
is faithful. An objectGis a  generator ofwhen{G}is a set of
generators. 




(Ab.5) and Grothendieck (Abelian) categories.
Letbe an Abelian category. 
Recall from  <cit.> thatis called (Ab.5)  when it is (Ab.3) (=cocomplete)  and the directed colimit functor_I^I→is exact, for any directed setI.
An (Ab.5) Abelian categoryhaving a set of generators (equivalently, a generator), is said to be a  Grothendieck Abelian category. Such a category always has enough injectives, and any of its objects has an injective envelope
(see <cit.>). Moreover, it is always a complete (and cocomplete) category (see <cit.>). 


Finitely presented objects.
Whenis a cocomplete additive category, an objectXofis called  finitely presented when(X, -) →preserves directed colimits, that is, for any directed setIand any diagram(Y_i)_i∈ Iin^I, the following canonical map is an isomorphism

    _I (X,Y_i)⟶(X,_I Y_i)

where the first colimit is taken inand the second in. Whenis a Grothendieck Abelian category with a set of finitely presented generators which, in this setting, is equivalent to say that each object ofis a directed colimit of finitely
presented objects, we say thatis  locally finitely presented.


Torsion pairs.
A  torsion pair in an Abelian categoryis a pairτ = (𝒯 , ℱ)of subcategories satisfying the following
two conditions:

  * (T, F) = 0, for all T∈𝒯 and F∈ℱ;

  *  for any object X of  there is a short exact sequence ε_X 0 → T_X→ X→ F_X → 0, where T_X∈𝒯 and
F_X ∈ℱ.

In such case,

the short exact sequenceε_Xfrom (TP.2) is
uniquely determined, up to a unique isomorphism, and the assignmentX↦ T_X(resp.X↦ F_X) underlies a functor→𝒯(resp.,→ℱ) which is right
(resp., left) adjoint to the inclusion functor𝒯→(resp.,ℱ→). We say thatτis of  finite type providedℱ=ℱ.





 §.§ Triangulated categories and t-structures
Triangulated categories.
We refer  to <cit.> for the precise definition of triangulated category. In particular, given a triangulated category𝒟, we  always denote byΣ𝒟→𝒟the  suspension functor, and we denote ( distinguished)  triangles either byX → Y→  Z+→orX→ Y → Z→Σ X.
Unlike the terminology used in the abstract setting of additive categories, in the
context of triangulated categories a weaker version of the notion of “set of generators” is commonly used. Namely,
a set⊆𝒟is called a  set of generators of𝒟if an objectXof𝒟is zero whenever𝒟(Σ^kS, X) = 0,
for allS ∈andk ∈. In case𝒟has coproducts, we  say that an objectXis  compact when the functor𝒟(X,-) 𝒟→preserves coproducts. We  say that𝒟is  compactly generated when it has a set of compact generators. 

Given a set𝒳of objects in𝒟and a subsetI⊆, we let

    𝒳^⊥_I   :={Y∈𝒟:𝒟(X,Σ^iY)=0, for all X∈𝒳 and i∈ I}
    ^⊥_I𝒳   :={Z∈𝒟:𝒟(Z,Σ^iX)=0, for all X∈𝒳 and i∈ I}.

IfI={i}for somei∈, then we let𝒳^⊥_i:=𝒳^⊥_Iand^⊥_i𝒳:=^⊥_I𝒳. Ifi=0, we even let𝒳^⊥_:=𝒳^⊥_0and^⊥_𝒳:=^⊥_0𝒳. In particular,is a set of generators if, and only if^⊥_=0.

Cohomological functors.
Given a triangulated category𝒟and an Abelian category, an additive functorH^0 𝒟→is said to be a  cohomological functor if, for any given triangleX→ Y→ Z→Σ X, the sequenceH^0(X)→ H^0(Y)→ H^0(Z)is exact in. In particular, one obtains a long exact sequence as follows:

    …→ H^n-1(Z)→ H^n(X)→ H^n(Y)→ H^n(Z)→ H^n+1(X)→…

whereH^n := H^0 ∘Σ^n, for anyn ∈.

t-Structures.
A t-structure in𝒟is a pair=(, )of subcategories satisfying the following axioms:

  * 𝒟(U, Σ^-1W) = 0, for all U ∈ and W ∈;

  * Σ⊆;

  *  for each X ∈𝒟, there are U_X ∈, V_X ∈Σ^-1 and a triangle 

    U_X → X → V_X→Σ U_X    in .

One can see that, in such case, both classes are closed under taking direct summands in𝒟, that = Σ(^⊥)and = ^⊥(Σ^-1) = ^⊥(^⊥). For this reason, we  write at-structure as=(, Σ(^⊥))or=(, Σ), meaning that:=^⊥. We will calland^⊥the  aisle and the  co-aisle of thet-structure, respectively.

The triangle of the above axiom (t-S.3) is
uniquely determined byX, up to a unique isomorphism, and defines
functorsτ_𝒟→andτ^^⊥𝒟→^⊥which are right and left adjoints to the respective inclusions. We call them the  left and  right truncation functors with respect to the givent-structure. Furthermore, the above triangle will be referred to as the  truncation triangle ofXwith respect to. 


We say that at-structure(,Σ)is generated by a set, whenΣ=^⊥_<0(equivalently, =^⊥_≤ 0). 
 When𝒟has coproducts, we say that thet-structure is  compactly generated when  it is generated by a setconsisting of compact objects in𝒟; in this case, we say thatis a  set of compact generators of the aisleor of thet-structure.


Hearts.
Given at-structure=(, Σ)in a triangulated category, the subcategoryℋ := ∩Σ⊆is called the  heart of thet-structure and it is an Abelian category, where the short exact sequences “are” the triangles of𝒟with the three terms in$̋. Moreover, with an obvious abuse of notation, the assignments X↦τ_U ∘τ^Σ(^⊥)(X) and X↦τ^Σ(^⊥)∘τ_U (X) define two naturally isomorphic cohomological functors H^0_𝒟→$̋ (see <cit.>). 
 



Let D be a triangulated category together with a t-structure =(,Σ) and heart :̋=∩Σ. Given a torsion pair τ=(𝒯,ℱ) in $̋ we can define a newt-structure_τ=(_τ,Σ_τ)on D, called the  Happel-Reiten-Smalø tilt of  with respect to τ (see <cit.>), where 

    _τ:=Σ*𝒯,  and    _τ:=ℱ*,

with the convention that, given two classes𝒳, 𝒴⊆ D,Z∈𝒳*𝒴if and only if there exists a triangleX→ Z→ Y→Σ Xin𝒟, withX∈𝒳andY∈𝒴.


 §.§ Categories with weak equivalences
Categories with weak equivalences. Letbe a category and letbe a collection of morphisms containing all the isomorphisms in. The pair(,)is said to be a  category with weak equivalences (or a  relative category) if, given two composable morphismsϕandψ, whenever two elements of{ϕ,ψ,ψϕ}belong inso does the third. The elements ofare called  weak equivalences.

Given a category with weak equivalences(,), its  universal localization is a pair([^-1],F)consisting of a category[^-1]and a  functorF→[^-1]such thatF(ϕ)is an isomorphism for allϕ∈, and it is universal with respect to these properties, that is, ifG→is a functor such thatG(ϕ)is an isomorphism for allϕ∈, then there exists a unique functorG'[^-1]→such thatG'F=G(see <cit.>).


Derived functors.
Let(,),(',')be categories with weak equivalences and suppose that their universal localizations exist. A functor𝐋G[^-1]→'['^-1]together with a natural transformationα𝐋G ∘ F⇒ F'∘ Gis called the  total left derived functor of  if the pair(𝐋G,α)is terminal between all pairs(H,β)such that  andβ H ∘ F⇒ F'∘ G. That is, given any(H,β), there is a unique natural transformationγ H ⇒𝐋Gsuch thatβ = α∘γ F.
The notion of  total right derived functor𝐑GofGis defined dually.



Model categories. A model structure on a bicomplete category𝒞is a triple(, B, F)of classes of morphisms, closed under retracts, called respectively the weak equivalences, cofibrations, and fibrations, such that(,)is a category with weak equivalence and satisfying a series of axioms, for which we refer to <cit.>. 
A model category is then a bicomplete categoryequipped with a model structure; i.e. a quadruple(; ,  B,  F).

The mere existence of a model structure for a category with weak equivalences allows one to give an explicit construction of the universal localization[^-1], which is traditionally called the homotopy category ofin this context, and denoted by().
Even better, model structures allow to construct and compute derived functors as well. To this end, an adjunction(F,G)⇄'between two model categories,'with model structures(, B, F)and(', B', F'), respectively, is called a Quillen adjunction if it satisfies one of the following equivalent conditions (see <cit.>):


  *  the left adjoint F→' preserves cofibrations and trivial cofibrations;

  *  the right adjoint G' → preserves fibrations and trivial fibrations.


Given a Quillen adjunction, the total derived functors(𝐋F,𝐑G)exist and form an adjunction between()and('). Moreover,𝐋F(X)for an objectXofcan be computed by applyingFto a cofibrant replacement ofXand dually for𝐑G.

In the context of algebraic examples, model structures on Abelian (even Grothendieck) categories play a prominent role. In particular, many of our examples will arise from the so-called Abelian model structures (see <cit.>).
The following example allows one to encode the machinery of classical homological algebra in the framework of model categories. 


Given a Grothendieck Abelian category , we will denote by (), 𝒦() and () the category of (unbounded) cochain complexes
of objects of , the homotopy category of  and the derived category of , respectively (see <cit.>). 
Recall that () is a bicomplete category. With the class  of quasi-isomorphisms in (), the pair ((),) is a category with weak equivalences. Furthermore, taking ℱ be the class of all the epimorphisms with dg-injective kernels and let ℬ be the class of monomorphisms, then () with (, B, F) is a model category (see for example <cit.>, <cit.> or <cit.> for details and a proof). The homotopy category in this case is (). 


§ PRELIMINARIES ON DERIVATORS


In this section we fix some notational conventions and we recall the basic definitions about prederivators, derivators, morphisms and2-morphisms between them. Furthermore, we collect some useful facts about pointed, additive, strong and stable derivators. All these results are probably known to experts but, in several cases, we have not been able to find suitable references in the existing literature. In those cases we have included a short proof.



 §.§ (Pre)Derivators
Prederivators. 
A  pre-derivator is a strict2-functor

    ^→.

All along this paper, we will use the following notational conventions:

  *  the letter  will denote a (pre)derivator;

  *   for any natural transformation α u⇒ v J→ I in , we will use the notation α^*:=(α) u^*⇒ v^*(I)→(J). Furthermore, we denote respectively by u_! and u_* the left and the right adjoint to u^* (whenever they exist), and call them respectively the  left and  right homotopy Kan extension of u;

  *  the letters U, V, W, X, Y, Z,  will be used either for objects in the base () or for (incoherent) diagrams in (), that is, functors I→(), for some small category I;

  *  the letters 𝒰, 𝒱, 𝒲, 𝒳, 𝒴, 𝒵,  will be used for objects in some image (I) of the derivator, for I a category (possibly) different from . Such objects will be usually referred to as  coherent diagrams of shape I;

  *  given I∈, consider the unique functor _I I→. We usually denote by _I:=(_I)_!(I)→() and _I:=(_I)_*(I)→() respectively the left and right homotopy Kan extensions of _I; these functors are called respectively  homotopy colimit and  homotopy limit. 


For a given objecti∈ I, we also denote byithe inclusioni→ Isuch that0↦ i. We obtain an evaluation functori^*(I)→(). For an object𝒳∈(I), we let𝒳_i:=i^*𝒳. Similarly, for a morphismα i→ jinI, one can interpretαas a natural transformation fromi→ Itoj→ I. In this way, to any morphismαinI, we can associateα^* i^*⇒ j^*. For an object𝒳∈(I), we let𝒳_α:=α^*_𝒳𝒳_i→𝒳_j.
For anyIin, we denote by 

    _I(I)⟶()^I

the  diagram functor, such that, given𝒳∈(I),_I(𝒳) I→()is defined by_I(𝒳)(iα→ j)=(𝒳_i𝒳_α→𝒳_j). We will refer to_I(𝒳)as the  underlying (incoherent)  diagram of the coherent diagram𝒳.


Let ^→ be a prederivator. Given I∈, consider the unique functor _I I→, let X∈() and consider 𝒳:=_I^*X∈(I). Then the underlying diagram _I(𝒳)∈()^I is constant, that is, 𝒳_i= X for all i∈ I and, for all α i→ j in I, the map 𝒳_α𝒳_i→𝒳_j is the identity of X. The objects isomorphic to one of the form _I^*X∈(I) for some X∈() are called   constant (coherent) diagrams. 








The 2-category of prederivators. 
Prederivators can be organized into a2-category, as is explained in <cit.>. Given prederivatorsand', a morphism of prederivatorsF→'consists of functorsF_I(I)→'(I), one for eachI∈, and natural equivalencesγ^F_u u^*∘ F_I≅ F_J∘ u^*, one for each functoru J→ I, subject to coherence relations (as in <cit.>, we will be sloppy and use the same symbol forγ^F_uand its inverse). Given morphisms of prederivatorsF,  G→', a2-morphism (also called a natural transformation)α F⇒ Gconsists of a family of natural transformationsα_I F_I⇒ G_I, one for eachI∈, which are in a standard way compatible with the natural equivalencesγ^F_uandγ^G_uas above for all functorsuin. More details can be found in our references.


Any ordinary category  gives rise to a prederivator 𝕐_^→ by the rule 𝕐_(I)=^I. Such prederivators are called represented, <cit.>, or Yoneda prederivators. The construction shows that  embeds into the 2-category of prederivators (up to usual set-theoretical issues, see <cit.> for details).
Derivators. 
A pre-derivatoris a  derivator if it satisfies the following four axioms:

  *  if ∐_i∈ I J_i is a disjoint union of small categories, then the canonical functor (∐_I J_i) →∏_I(J_i) is an equivalence of categories;

  *  for each I∈ and  f𝒳→𝒴 a morphism in (I), f is an isomorphism if, and only if, i^*(f) i^*(𝒳) → i^*(𝒴) is an isomorphism in () for each i∈ I;

  *  for each functor u I → J in , the functor u^* has both a left adjoint u_! and a right adjoint u_* (i.e.  homotopy Kan extensions are required to exist).


Before stating the last axiom (Der.4), let us introduce the following notation. Suppose we have a natural transformation:α v∘ u'⇒ u∘ win:

    @C=10pt@R=6pt
    I@<-[dd]_-u@<-[rr]^-v      I'@<-[dd]^-u'@[ddll]|-α
    
    
    J@<-[rr]_-w      J',

and let^→be a prederivator that satisfies (Der.3). Applyingto the above square, we get a natural transformationα^* (u')^*∘ v^*⇒ w^*∘ u^*@C=10pt@R=6pt(I)[dd]_-u^*[rr]^-v^*      (I')[dd]^-(u')^*@[ddll]|-α^*
    
    (J)[rr]_-w^*      (J').

In this setting, we defineα_!:=(α^*)_! w_!∘(u')^*⇒ u^*∘ v_!andα_*:=(α^*)_* v^*∘ u_*⇒ u'_*∘ w^*, where(-)_!and(-)_*are defined right before Lemma <ref>. Hence, we deduce by that lemma thatα_!is an isomorphism if and only ifα_*is, and, furthermore, 

    (α_!)_*=α^*=(α_*)_!.

In order to properly state (Der.4), we need to start with a special case of the square (<ref>). Indeed, letu J→ Ibe a functor inand leti∈ I. We define the  slice categoryu/i(resp.,i/u) whose objects are pairs(j,a u(j)→ i)(resp.,(j,a i→ u(j))) wherej∈ Jandais a morphism inI. Furthermore, a morphismf (j,a)→ (j',a')inu/i(resp., ini/u) is a morphismf j→ j'inJsuch thata=a'∘ u(f)(resp.,u(f)∘ a=a'). Then, “forgetting the second component of objects" gives well-defined functors

    _i u/i⟶ J     and    ^i i/u⟶ J.

We then get the following two squares inthat commute up to a natural transformation:

    @C=18pt@R=8pt
    u/i[dd]_-_u/i[rr]^-_i      J[dd]^-u@[ddll]|-γ
    
    [rr]_-i      I
            and        @C=18pt@R=8pt
    i/u[dd]_-_u/i[rr]^-^i      J[dd]^-u@[ddll]|-δ
    
    [rr]_-i      I,

whereγ_(j,a):=a u(j)→ ifor each(j,a)∈ u/iandδ_(j,b):=b i→ u(j)for each(j,b)∈ i/u. We finally have all the ingredients to properly state the axiom (Der.4): 

  *  the homotopy Kan extensions can be computed pointwise, that is, given u J→ I in  and i∈ I, the following natural transformations are invertible

    @C=18pt@R=8pt(u/i)[dd]_-(_u/i)_!@<-[rr]^-^*_i      (J)[dd]^-u_!@[ddll]|-γ_!
    
    ()@<-[rr]_-i^*      (I)
            and        @C=18pt@R=8pt(i/u)[dd]_-(_i/u)_*@<-[rr]^-(^i)^*      (J)[dd]^-u_*@[ddll]|-δ_*
    
    ()@<-[rr]_-i^*      (I).



Finally, let us note that1- and2-cells in the2-category of derivators are defined exactly as for prederivators. In other words, derivators form a full sub-2-category of the2-category of prederivators. 




Let  be a prederivator that satisfies (Der.3), u J→ I in  and i∈ I. Consider the natural transformation γ u∘_i⇒ i∘_u/i described in the above discussion. Now remember that, by (<ref>), we have that (γ_!)_*=γ^*. Expanding this equality we get:

    γ^*=(γ_!)_*   =(_u/i^*i^*⊛ϵ_u)∘(_u/i^*⊛γ_!⊛ u^*)∘(η_u/i⊛_i^*u^*)

where ϵ_u u_!u^*⇒𝕀_(I) and η_u/i𝕀_(u/i)⇒_u/i^*(_u/i)_! are the counit and unit, respectively, of the corresponding adjunctions. A similar observation can be done regarding the natural transformation δ.

The Yoneda prederivator (Example <ref>) is a derivator if and only if  is a complete and cocomplete category.

Given a derivator  and a small category A, we can define a prederivator ^A^→ by ^A(I):=(A× I) (and in the obvious way on functors and natural transformations). Then ^A is in fact a derivator by <cit.> and it is called the A-shift of . Moreover, given a functor u I→ J in , one obtains a morphism of derivators u^*^J→^I (see <cit.>).

Let ^→ be a derivator, I∈  and j∈ I. Then, by, the following triangle commutes up to a canonical isomorphism:

    @R=5pt(I)[rrrr]^_I            ()^I
       ≅   
    ()@/_1.0pc/[uurrrr]_-⊗ j[uu]^j_!

where -⊗ j is the left adjoint to the obvious “evaluation at j" functor (-)_j()^I→(). Furthermore, given X∈(), we have that (X⊗ j)(k)≅∐_I(j,k)X, for all k∈ I.



The following computations will be useful later on:


Let ^→ be a derivator, I a small category, J⊆ I a (full) subcategory and u J→ I the inclusion functor. Then, the following assertions hold true:

  *  letting u_!η_uϵ_uu^*(J)⇄(I) be the induced adjunction, the following natural transformation is invertible: u^*⊛ϵ_u=(η_u⊛ u^*)^-1 u^*u_!u^*⟹̃u^*;



  *  if k∈ J⊆ I, so that k=u∘ k→ I, the following natural transformation is invertible: k^*u^*⊛ϵ_u=k^*⊛ϵ_u k^*u_!u^*⟹̃k^*.

(1). By <cit.>, as u is fully faithful, also u_! is fully faithful and, equivalently, η_u is a natural isomorphism.  Then, by the triangle unit-counit identities (<ref>) we obtain that u^*⊛ϵ_u=(η_u⊛ u^*)^-1 u^*u_!u^*⟹̃u^* is a natural isomorphism.


(2) follows from (1) and the fact that  is a strict 2-functor.

An important special case of a full subcategory is a cosieve:
it is a full subcategory J⊆ I such that whenever j∈ J and j→ k is a morphism in I, then k∈ J. In that case, we also know what k^*u_!𝒳 looks like for k∈ I∖ J. Namely, k^*u_!𝒳 is, by <cit.>, the initial object of () for every 𝒳. Thus, in the case of pointed derivators (see Subsection <ref> below), u_! is called the left extension by zero functor.
















Preservation of homotopy Kan extensions. 
Given a morphism of derivators 

    F=(F_I,γ^F_u)⟶'

 andu J→ I, we have the following natural transformations:
 
    (γ_u^F)_! u_!F_J⟹ F_Iu_!    and      (γ_u^F)_* F_Iu_*⟹ u_*F_J.

 If either of these is a natural isomorphism, we say thatFpreserves homotopy left or right Kan extensions alongu, respectively. Recall from <cit.> thatFis left exact if it preserves homotopy pullbacks and final objects and right exact if it preserves homotopy pushouts and initial objects. 


The following statement is well-known in ordinary category theory. Here we extend it to arbitrary morphisms of derivators:


A morphism of derivators which preserves directed homotopy colimits and finite
coproducts also preserves arbitrary coproducts.

Let G→' be a morphism of derivators that preserves directed homotopy colimits and finite
coproducts, and let I be a set, viewed as a discrete category. We consider the directed set P:=𝒫^<ω(I) of the finite parts of I, ordered by inclusion, and consider it as a posetal category. There is an obvious inclusion functor u  I→ P, such that i↦{i}. Furthermore,

    (@C=40pt I[r]^-u   P[r]^-_P   ) = (I[r]^-_I   ).

By the compatibility of mates with pasting (see <cit.>) to prove that G commutes with coproducts indexed by I, that is, with homotopy left Kan extensions along _I (use (Der.1) for this equivalence), it is enough to verify that G commutes, separately, with homotopy left Kan extensions along u and along _P, the latter being trivial as P is directed. It remains to prove that (γ_u^G)_! u_!∘ G_I⟹ G_P∘ u_! is invertible. By (Der.2), (γ_u^G)_! is an isomorphism if, and only if, F^*((γ_u^G)_!) is an isomorphism for each F∈ P. Fix then an F∈ P and consider the following natural isomorphisms

    F^*∘  u_!∘ G_I≅_u/F∘_F^*∘ G_I≅_u/F∘ G_u/F∘_F^*,

where the first isomorphism comes from the axiom (Der.4), while the second is induced by the natural isomorphism γ^G__F. Similarly we have the following two isomorphisms, the first one induced by γ^G_F and the second one by (Der.4): 

    F^*∘ G_P∘ u_!=G_∘ F^*∘ u_!≅ G_∘_u/F∘_F^*.

Hence, to prove that F^*((γ_u^G)_!) is an isomorphism, it is enough to prove that 

    (γ__u/F^G)_!_u/F∘ G_u/F⟹ G_∘_u/F
 
is invertible. On the other hand, u/F≅ F is a finite discrete category, so that _u/F is a finite coproduct, and G commutes with finite coproducts by hypothesis.





Homotopical epimorphism. 
Following <cit.>, we call a functorv J→ Ibetween small categories a  homotopical epimorphism if, for any derivator, the induced functorv^*(I)→(J)is fully faithful. As this holds for any derivator, in particular it holds for all the shifts^K(see Example <ref>) of a given derivator, showing thatv J→ Iis a  homotopical epimorphism if, and only if, for any derivator, it induces a fully faithful morphism of derivatorsv^*^I→^J. 


If v J→ I is a homotopical epimorphism between small categories, then it is also a lax epimorphism in the sense of <cit.>, that is, for any category , the functor (-)∘ v^I→^J is fully faithful. 

By the equivalence “(1)⇔(2)” of <cit.>, it is enough to check that (-)∘ v^I→^J is fully faithful. This last condition follows by the definition of homotopical epimorphism applied to the derivator 𝕐_ (see Example <ref>).



The following is a useful criterion to identify homotopical epimorphisms:



Let u A→ I be an essentially surjective functor in , let  be a derivator, and let 𝔼⊆^A be a sub-prederivator of ^A
(in the sense that 𝔼(J) is a subcategory of ^A(J) for each J∈)
that satisfies the following two conditions:

  * Im(u^*^I→^A)⊆𝔼;

  *  for any J∈ and 𝒳∈𝔼(J), the component of the counit ϵ_𝒳 u^*u_*𝒳→𝒳 of the adjunction u^*⊣ u_* is an isomorphism.

Then, u^*^I→^A is fully faithful and  Im(u^*)= 𝔼. In particular, 𝔼 is a derivator. 


 §.§ Pointed and additive derivators

We refer to <cit.> for a detailed discussion, as well as for the precise definitions of  pointed derivators (i.e., derivatorsfor which()has a zero object). Similarly, we refer to <cit.> for a discussion of  additive derivators (i.e., derivatorsfor which()is additive). In fact, for a pointed (resp., additive) derivator, the condition imposed on()implies that the categories(I)(for eachIin) are automatically pointed (resp., additive). 

Ifand'are pointed (resp., additive) derivators, a morphismF→'is called  pointed (resp.,  additive) ifF_Iis a pointed (resp., additive) functor, for eachIin. Note that, for eachu J→ Iin, the three morphismsu^*^I→^Jandu_!,  u_*^J→^Iare always pointed (resp., additive). 


Let ^→ be an additive derivator and I∈. A sequence

    𝒳[r]^ϕ   𝒴[r]^ψ   𝒵    in (I),

is said to be  pointwise split-exact if, for each i∈ I, the sequence 𝒳_i→𝒴_i→𝒵_i, obtained by applying i^* to  (<ref>), is split-exact in the additive category ().

Let 𝔻^→ be an additive derivator. For each I∈, the functor _I𝔻(I)→𝔻(1)^I takes pointwise split-exact sequences to kernel-cokernel sequences.
 
Let 𝒳ϕ⟶𝒴ψ⟶𝒵 be a pointwise split-exact sequence in 𝔻(I). It follows that 
  𝒳_i=(dia_I𝒳)_iϕ_i⟶𝒴_i=(dia_I𝒴)_iψ_i⟶𝒵_i=(dia_I𝒵)_i is a split-exact sequence, whence a kernel-cokernel one, in 𝔻(1),  for all i∈ I. Now use the fact a morphism in 𝔻(1)^I has a co/kernel if, and only if, so do all of its components and, in this case, the co/kernel is computed componentwise.
 

 §.§ Strong and stable derivators


We refer to <cit.> for a detailed discussion, as well as for the precise definitions of  stable derivators (they are the pointed derivators in which a coherent commutative square is cartesian if, and only if, it is cocartesian), and of  strong derivators (the ones in which the partial diagram functors(2× I) →(I)^2are full and essentially surjective for eachI∈). 
The key fact, which can be found in <cit.>, is that, given a strong and stable derivator, each(I)is in fact a triangulated category.


Furthermore, ifand'are stable derivators, left and right exactness of a morphismF→'are equivalent and left (=right) exact morphisms are simply called exact in this context. IfF→'is an exact morphism of strong and stable derivators, allF_I(I)→'(I)are naturally triangulated functors by <cit.>. In particular, all the morphisms of the formu^*and the Kan extensionsu_!,u_*, for someu J→ Iin, are naturally triangulated functors. 


Let ^→ be a strong and stable derivator, I∈ and let

    𝒳[r]^ϕ   𝒴[r]^ψ   𝒵

be a pointwise split-exact sequence in (I). Then, this sequence can be completed to a distinguished triangle in (I). 

Consider the morphism ϕ𝒳→𝒴 and complete it to a triangle in (I):

    𝒳[r]^ϕ   𝒴[r]^ϕ'   𝒞[r]^ϕ”   Σ𝒳.

Since ψϕ=0 by hypothesis, there is a morphism f𝒞→𝒵 such that fϕ'=ψ; let us verify that f is an isomorphism. By (Der.2), it is enough to verify that f_i:=i^*(f) is an isomorphism, for each i in I. But this is clearly true since ϕ_i is a split monomorphism (by hypothesis) and both ϕ'_i and ψ_i are cokernels of ϕ_i (the first by <cit.> and the second by hypothesis). Hence, by the universality of cokernels, f_i is the unique morphism such that f_iϕ_i'=ψ_i and it is an isomorphism. 






In the next example, we mention some classes of strong and stable derivators that will appear frequently in the rest of the paper:


Let (,𝒲,,ℱ) be a model category. For any small category I, let _I be the class of morphisms in ^I which belong pointwise to . By <cit.>, the universal localization ^I[_I^-1] can always be constructed and, furthermore, the assignment I↦^I[_I^-1] underlies a derivator _(,)^→. Furthermore, _(,) is always strong and it is pointed (resp., stable) if  has the same property in the sense of model categories. For such derivators, homotopy co/limits and, more generally, homotopy Kan extensions, are just the total derived functors of the usual co/limit and Kan extension functors. 

Given a Grothendieck Abelian category , 
we refer to the strong and stable derivator arising as above from the injective model structure on ()
as the  canonical derivator enhancing the derived category(). 


§ CATEGORIES OF FINITE AND OF COUNTABLE LENGTH


In this section we introduce the concept of category of finite length and that of category of countable length. Then, given such a categoryI, we study canonical presentations for objects in(I)(first in the finite and then in the countable length case), whereis allowed to be an arbitrary additive derivator. Finally, we verify that every small category is the homotopically surjective image of a suitable category of countable length. In this way the results about standard presentations for coherent diagrams over categories of countable length can be applied to arbitrary shapes.





 §.§ Categories of finite length

A small category I is said to be of  finite length if there is n∈_>0 and

    d I→𝐧^,

 a functor such that, for any pair of objects i,  j∈ I, if there is a non-identity morphism in I(i,j), then d(i)>d(j). If I is a category of finite length, the smallest n∈_>0 for which there exists a functor d I→𝐧^ as above is called the  length of I, in symbols ℓ(I)=n. 


Note that, for a category of finite lengthI, we have thatI(i,i)={𝕀_i}for anyi∈ I. Furthermore, giveni≠ j∈ I, at most one ofI(i,j)andI(j,i)is not empty. Moreover, the length ofIis exactly the length of a maximal path inI.



Given a category I of finite length, we  say that an object i in I is  minimal if there is no non-identity morphism starting at i. 


Throughout this subsection, we fix the following notation:


We let ^→ be an additive derivator, I a category of finite length and u J→ I the inclusion of the (full) subcategory J of I of all the non-minimal objects. We also denote by I∖ J⊆ I the subcategory of minimal objects. We denote by u_!η_uϵ_uu^* the induced adjunction. Furthermore, for each i∈ I∖ J we have the following adjunctions:

    i_!η_iϵ_ii^*    and    _u/iη_u/iϵ_u/i_u/i^*.
 
Finally, we denote by γ_! (_u/i)_!_i^*⇒ i^*u_! the canonical natural transformation, which is invertible by (Der.4).


The following lemma is a trivial fact about categories of finite length that will be extremely important when trying to prove things by induction:


For each i∈ I∖ J, we have that ℓ(I)>ℓ(J)≥ℓ(u/i).

The fact that ℓ(I)>ℓ(J) follows since any maximal path in I ends in a minimal object that, by definition, does not belong in J. Furthermore, a maximal path in u/i is something of the form 

    (u(j_1)ϕ_1⟶i)ψ_1⟶(u(j_2)ϕ_2⟶i)ψ_2⟶…ψ_k-1⟶(u(j_k)ϕ_k⟶i)

where ψ_s j_s→ j_s+1 and ϕ_s+1ψ_s=ϕ_s, for s=1,…,k-1. Then, 
j_1ψ_1⟶j_2ψ_2⟶…ψ_k-1⟶j_k
is a path in J, so that ℓ(u/i)≤ℓ(J).


We can now introduce the standard presentation for coherent diagrams of finite length:


Given 𝒳 in (I), there is a pointwise split-exact sequence:

    @C=50pt@R=9pt∐_I∖ Ji_!i^*u_!u^*𝒳[rr]^-[ccc⋱      0
       i_!i^*ϵ_u   
    0      ⋱
    ⋯   -ϵ_i   ⋯]
    
          ∐_I∖ Ji_!𝒳_i       ⊔    u_!u^*𝒳[rr]^-[[   ⋯ ϵ_i   ⋯ ϵ_u ]]
          𝒳.

We have to verify that, for each k∈ I, when we apply the evaluation functor k^* to the sequence in the statement, we get a split-exact sequence in (). We distinguish two cases based on whether k is minimal or not.
 Indeed, if k∈ J then k^*i_!=0 for all i∈ I∖ J since then {i}⊆ I is a cosieve (cf. Remark <ref>), and k^*ϵ_u is an isomorphism by Lemma <ref>, so we get the following split-exact sequence:

    @C=30pt@R=9pt
    0[rrr]^-0          k^*u_!u^*𝒳[rrr]^-k^*ϵ_u          k^*𝒳.

Suppose now that k∈ I∖ J; using the natural isomorphism k^*ϵ_k k^*k_!k^* ⟹̃ k^* (see Lemma <ref> recalling that the functor k→ I is fully faithful), we get the following split-exact sequence:

    @C=29pt@R=9pt
    k^*k_!k^*u_!u^*𝒳[rrr]^-[[ k^*k_!k^*ϵ_u;             ;      -k^*ϵ_k ]]
             k^*k_!k^*𝒳⊔  k^*u_!u^*𝒳[rrr]^-[[ k^*ϵ_k k^*ϵ_u ]]
              k^*𝒳,

as this is an occurrence of the second part of Example <ref>(1).


The following two technical propositions will be essential in the following section. Note that, even if we state them in the setting of this subsection, they hold for any small categoryI, any full subcategoryJand anyi∈ I∖ J.


Given 𝒳,𝒴∈(I) and i∈ I∖ J, the following diagram commutes:

    @C=150pt@R=21pt(I)(i_!i^*𝒳,𝒴)[r]^-(-)∘ i_!i^*(ϵ_u)[ddd]^-[origin=c]90∼_-i^*(-)∘η_i   (I)(i_!i^*u_!u^*𝒳,𝒴)[d]_-[origin=c]90∼^-i^*(-)∘η_i
       ()(i^*u_!u^*𝒳,i^*𝒴)[d]_-[origin=c]90∼^-(-)∘γ_!
       ()(_u/i_i^*u^*𝒳,i^*𝒴)[d]_-[origin=c]90∼^-_u/i^*(-)∘η_u/i
    ()(𝒳_i,𝒴_i)[r]_^*_u/i(-)∘γ^*   (u/i)(_i^*u^*(𝒳),_u/i^*𝒴_i)

where the columns are natural isomorphisms. Furthermore, for each f𝒳_i→𝒴_i the corresponding map ^*_u/i(f)∘γ^*_i^*u^*(𝒳)→_u/i^*𝒴_i acts as follows: for each object (j,a j→ i) in u/i,

    (j,a)^*(^*_u/i(f)∘γ^*)=f∘𝒳_a𝒳_j→𝒴_i.

The left column and the first morphism in the right column are induced by the adjunction i_!⊣ i^* so they are clearly isomorphisms. The second and the third morphism in the right column are induced, respectively, by the natural transformation γ_!, which is invertible by (Der.4), and by the adjunction _u/i⊣_u/i^*, so it is invertible too. Finally, to see that the square commutes, consider the following equalities:

    ^*_u/i(i^*(-)∘η_i)∘γ^*   =^*_u/i(i^*(-)∘η_i∘ i^*(ϵ_u)∘γ_!)∘η_u/i
       =^*_u/i(i^*(-)∘ i^*i_!i^*(ϵ_u)∘η_i∘γ_!)∘η_u/i,

where the first equality holds since γ^*=(γ_!)_* (see Remark <ref>) and the second one by the naturality of η_i𝕀_()⇒ i^*i_!.

Given 𝒳,𝒴∈(I) and i∈ I∖ J, the following diagram commutes 

    @C=150pt@R=21pt(I)(u_!u^*(𝒳),𝒴)[r]^-(-)∘ϵ_i[dr]_-i^*(-)[ddd]^-[origin=c]90∼_-u^*(-)∘η_u   (I)(i_!i^*u_!u^*𝒳,𝒴)[d]_-[origin=c]90∼^-i^*(-)∘η_i
       ()(i^*u_!u^*𝒳,i^*𝒴)[d]_-[origin=c]90∼^-(-)∘γ_!
       ()(_u/i_i^*u^*𝒳,i^*𝒴)[d]_-[origin=c]90∼^-_u/i^*(-)∘η_u/i
    (J)(u^*(𝒳),u^*(𝒴))[r]_γ^*∘^*_i(-)   (u/i)(_i^*u^*(𝒳),_u/i^*𝒴_i)

where the columns are natural isomorphisms. Furthermore, for each f u^*𝒳→ u^*𝒴 the corresponding map γ^*∘^*_i(f)_i^*u^*(𝒳)→_u/i^*𝒴_i acts as follows: for each object (j,a j→ i) in u/i,

    (j,a)^*(γ^*∘^*_i(f))=𝒴_a∘ f_j𝒳_j→𝒴_i.

The three morphisms in the right column are the same as the ones in Proposition <ref>, so they are isomorphisms. Furthermore, the upper triangle commutes by the usual triangular equalities relative to the adjunction i_!⊣ i^*. On the other hand, the morphism in the left column is the natural isomorphism induced by the adjunction u_!⊣ u^*, whose inverse is the map ϵ_u∘ u_!(-)(J)(u^*(𝒳),u^*(𝒴))→(I)(u_!u^*(𝒳),𝒴). Hence, the diagram in the statement commutes if, and only if, the following diagram commutes:

    @C=150pt@R=22pt(I)(u_!u^*(𝒳),𝒴)[r]^-i^*(-)@<-[dd]^-[origin=c]90∼_-ϵ_u∘ u_!(-)   ()(i^*u_!u^*𝒳,i^*𝒴)[d]_-[origin=c]90∼^-(-)∘γ_!
       ()(_u/i_i^*u^*𝒳,i^*𝒴)[d]_-[origin=c]90∼^-_u/i^*(-)∘η_u/i
    (J)(u^*(𝒳),u^*(𝒴))[r]_γ^*∘^*_i(-)   (u/i)(_i^*u^*(𝒳),_u/i^*𝒴_i).

Finally, the above diagram commutes by the following series of equalities:

    ^*_u/i(i^*(ϵ_u∘ u_!(-))∘γ_!)∘η_u/i=
       =^*_u/i(i^*(ϵ_u)∘ i^*u_!(-)∘γ_!)∘η_u/i
       =^*_u/i(i^*(ϵ_u)∘γ_!∘(_u/i)_!_i(-))∘η_u/i   by naturality of γ_!;
       =^*_u/i(i^*(ϵ_u))∘^*_u/i(γ_!)∘^*_u/i(_u/i)_!_i(-)∘η_u/i
       =^*_u/i(i^*(ϵ_u))∘^*_u/i(γ_!)∘η_u/i∘_i(-)   by naturality of η_u/i;
       =γ^*∘^*_i(-)   since (γ_!)_*=γ^*.


 §.§ Categories of countable length

A small category I is said to be of  countable length if there is a functor

    d I→^

such that, for any pair of objects i,  j∈ I, if there is a non-identity morphism in I(i,j), then d(i)>d(j) in . 


As for the finite length case, ifIhas countable length, thenI(i,i)={𝕀_i}for anyi∈ I. Furthermore, giveni≠ j∈ I, at most one ofI(i,j)andI(j,i)is not empty. Throughout this subsection, we fix the following notation:


We let ^→ be an additive derivator and I a category of countable length with a fixed functor d I→^ like in Definition <ref>. For each n∈, we let 

    I_≤ n⊆ I  be the (full) subcategory of all { i∈ I: d(i)≤ n-1}.
 
Furthermore, we let ι_m,n I_≤ n→ I_≤ m and ι_n I_≤ n→ I be the canonical cosieve inclusions, for all m≥ n in . Finally, we consider the adjunctions:

    (ι_n)_!η_nϵ_nι_n^*    and    (ι_m,n)_!η_m,nϵ_m,nι_m,n^*

and we define
∂^n:=(ϵ_n⊛ (ι_n+1)_!ι_n+1^*)∘ ((ι_n)_!ι_n+1,n^*⊛η_n+1⊛ι_n+1^*) (ι_n)_!ι_n^*⇒(ι_n+1)_!ι_n+1^*.


Let us remark that, in the above notation, the restrictiond_n I_≤ n→𝐧^of the functord I→^shows that eachI_≤ nis a category of finite length. Furthermore, for eachm≥ n∈, we haveι_m∘ι_m,n=ι_n.
In particular, the equalityι_n+1,n^*∘ι_n+1^*=ι_n^*and Lemma <ref> provide us with a natural isomorphismψ_n (ι_n)_! ∼⟹(ι_n+1)_!(ι_n+1,n)_!given by


    ψ_n = (ϵ_n⊛(ι_n+1)_!(ι_n+1,n)_!) ∘
    ((ι_n)_!ι_n+1,n^*⊛η_n+1⊛(ι_n+1,n)_!) ∘
    ((ι_n)_!⊛η_n+1,n).

Let now𝒳∈(I)andn∈, the following commutative diagram helps visualize∂^n:

    @C=120pt@R=5pt
    (ι_n)_!ι_n+1,n^*ι_n+1^*𝒳[r]^-(ι_n)_!ι_n+1,n^*(η_n+1)@=[d]   (ι_n)_!ι_n+1,n^*  ι_n+1^*(  ι_n+1)_!  ι_n+1^*𝒳@=[d]
    
    (ι_n)_!ι_n^*𝒳[rddd]_-∂^n   (ι_n)_!ι_n^*(  ι_n+1)_!  ι_n+1^*𝒳[ddd]^-ϵ_n
    
    
       (ι_n+1)_!  ι_n+1^*𝒳.


It is also instructive to recall Lemma <ref> and Remark <ref> in the context
of the above definitions. For example, given𝒳∈(I), the component of((ι_n)_!ι_n^*𝒳)_kis isomorphic to𝒳_kifk∈ I_≤ nand((ι_n)_!ι_n^*𝒳)_k=0otherwise. The components of the counit(ι_n)_!ι_n^*𝒳→𝒳are isomorphisms or maps from the zero object of(), respectively. The next lemma shows that the map∂^nis of a similar nature (in fact, a direct but tedious computation also reveals that∂^ncoincides, up to the identification given byψ_nfrom (<ref>), with(ι_n+1)_!⊛ϵ_n+1,n⊛ι_n+1^* (ι_n+1)_!(ι_n+1,n)_!ι_n^*⟹(ι_n+1)_!ι_n+1^*).


In the above notation, ϵ_n+1∘∂^n=ϵ_n, for each n∈.

Let n∈ and note that, by the naturality of ϵ_n, we have:

    ϵ_n+1∘ (ϵ_n⊛ (ι_n+1)_!ι_n+1^*)=ϵ_n∘ ( (ι_n)_!ι_n^*⊛ϵ_n+1).

In other words, the following diagram commutes for each 𝒳∈(I):

    @C=70pt
    (ι_n)_!ι_n^*𝒳@<-[r]^-(ι_n)_!ι_n^*(ϵ_n+1)[d]_ϵ_n   (ι_n)_!ι_n^*(ι_n+1)_!ι_n+1^*𝒳[d]^ϵ_n
    𝒳@<-[r]_-ϵ_n+1   (ι_n+1)_!ι_n+1^*𝒳.

Therefore,  we have that 

    ϵ_n+1∘∂^n   =ϵ_n+1∘ (ϵ_n⊛ (ι_n+1)_!ι_n+1^*)∘ ((ι_n)_!ι_n+1,n^*⊛η_n+1⊛ι_n+1^*)   
       =ϵ_n∘ ( (ι_n)_!ι_n^*⊛ϵ_n+1)∘ ((ι_n)_!ι_n+1,n^*⊛η_n+1⊛ι_n+1^*)   
       =ϵ_n∘ ((ι_n)_!ι_n+1,n^*⊛ ((ι_n+1^*⊛ϵ_n+1)∘ (η_n+1⊛ι_n+1^*)))=ϵ_n,

where we have applied the triangular equality (ι_n+1^*⊛ϵ_n+1)∘ (η_n+1⊛ι_n+1^*)=𝕀_ι_n+1^* relative to the adjunction (ι_n+1)_!⊣ι_n+1^*.


We are now ready to introduce the standard presentation of a coherent diagram of countable length:



For any 𝒳∈(I), there is a pointwise split-exact sequence

    @C=130pt∐_(ι_n)_!ι_n^*𝒳[r]^α=
    [cccc
    1         
    
    -∂^0   1      0
       -∂^1   1   
    0      ⋱   ⋱]
       ∐_(ι_n)_!ι_n^*𝒳[r]^-β=
    [ccccϵ_0    ϵ_1    ϵ_2    ⋯]
       𝒳.










It is easily seen that, by Lemma <ref>, β∘α=0. Let now k∈ I_≤ n∖ I_≤ n-1⊆ I (for some integer n≥1) and let us show that, after applying k^* to the sequence in the statement, we get a split-exact sequence in (). By <cit.> we get that

    k^*(ι_m)_!ι_m^*=0,    whenever m<n,

while k^*(ϵ_m) is invertible for all m≥ n, by Lemma <ref>, and k^*(ϵ_m+1)∘ k^*( ∂^m)=k^*(ϵ_m), by Lemma <ref>. In particular, we obtain the following commutative diagram in (), where the three columns are isomorphisms

    @C=100pt∐_m≥ nk^*(ι_m)_!ι_m^*𝒳[r]^ k^*(α)[d]^-[origin=c]90∼_(k^*(ϵ_m))_m   ∐_m≥ nk^*(ι_m)_!ι_m^*𝒳[r]^ k^*(β)[d]_-[origin=c]90∼^(k^*(ϵ_m))_m   𝒳_k@=[d]
    
    𝒳_k^()[r]|-[cccc
    1         
    
    -1   1      0
       -1   1   
    0      ⋱   ⋱]
       𝒳_k^()[r]_[cccc
    1   1   1   ⋯]
       𝒳_k

showing that our sequence is (isomorphic to) the one in Example <ref>.


The following technical proposition will be essential in the following section:


Given 𝒳,𝒴∈(I) and n∈, the following diagram commutes 

    @C=100pt@R=30pt(I)((ι_n)_!ι_n^*(𝒳),𝒴)@<-[r]^-(-)∘∂^n[d]^-[origin=c]90∼_-ι_n^*(-)∘η_n   (I)((ι_n+1)_!ι_n+1^*(𝒳),𝒴)[d]_-[origin=c]90∼^-ι_n+1^*(-)∘η_n+1
    (I_≤ n)(ι_n^*(𝒳),ι_n^*(𝒴))@<-[r]^ι_n+1,n^*(-)   (I_≤ n+1)(ι_n+1^*(𝒳),ι_n+1^*(𝒴))

where the columns are natural isomorphisms.

The two columns are isomorphisms because they are the maps given by the adjunctions (ι_n)_!⊣ι_n^* and (ι_n+1)_!⊣ι_n+1^*, respectively. In particular, the inverse of the map in the leftmost column is ϵ_n∘(ι_n)_!(-) and, therefore, the commutativity of the original square is equivalent to the commutativity of the following diagram:

    @C=100pt@R=30pt(I)((ι_n)_!ι_n^*(𝒳),𝒴)@<-[r]^-(-)∘∂^n@<-[d]^-[origin=c]90∼_-ϵ_n∘(ι_n)_!(-)   (I)((ι_n+1)_!ι_n+1^*𝒳,𝒴)[d]_-[origin=c]90∼^-ι_n+1^*(-)∘η_n+1
    (I_≤ n)(ι_n^*(𝒳),ι_n^*(𝒴))@<-[r]^ι_n+1,n^*(-)   (I_≤ n+1)(ι_n+1^*(𝒳),ι_n+1^*(𝒴)).

That is, we have to prove that ϵ_n∘(ι_n)_!(ι_n+1,n^*(ι_n+1^*(-)∘η_n+1))=(-)∘∂^n. But, in fact,

    ϵ_n∘(ι_n)_!(ι_n+1,n^*(ι_n+1^*(-)∘η_n+1))   =ϵ_n∘(ι_n)_!ι_n^*(-)∘(ι_n)_!ι_n+1,n^*(η_n+1)
       =(-)∘ϵ_n∘(ι_n)_!ι_n+1,n^*(η_n+1)=(-)∘∂^n,

where the first equality holds since ι_n+1∘ι_n+1,n=ι_n, the second one is true by the naturality of ϵ_n (analogously to the commutativity of the diagram (<ref>) but with an arbitrary map instead of ϵ_n+1), and the last one follows just by definition of ∂^n.


 §.§ Homotopical epimorphic images of categories of countable length


LetIbe a small category. In this section we associate withIa new category of countable lengthΔ(I)and a homotopical epimorphismuΔ(I) → I. The idea of using the categoryΔ(I)in this way was suggested to us by Fritz Hörmann. A similar idea appears in his paper <cit.>, where he proves much more in this context—he uses the categoriesΔ(I)to extend the domain of definition of fibred (multi)derivators from directed categories of countable length to arbitrary shapes. Let us start with the following definition:


Given a small category I, the category Δ(I) is defined as follows:

  *  the objects of Δ(I) are the functors 𝐧→ I, for n∈_>0;

  *  given two objects a𝐧→ I and b𝐦→ I, we define

    Δ(I)(a,b):={φ𝐦→𝐧 strictly increasing and s.t. a∘φ=b}.

There is a canonical functor

    uΔ(I)⟶ I    such that     (a𝐧→ I)↦ a(0).

In what follows, our aim is to prove that the above functoruΔ(I)→ Iis a homotopical epimorphism. Giveni∈ I, define the  fiber ofuatito be the categoryu^-1(i), given by the following pullback diagram in:

    u^-1(i)@.>[r]^ι_i@[dr]|P.B.@.>[d]__u^-1(i)   Δ(I)[d]^u
    [r]_i   I

We can describeu^-1(i)as the subcategory of the slice categoryi/uof those objects(a,φ i→ u(a))such thatu(a)=iandφ=𝕀_i. The following lemma is taken from the discussion in <cit.>.


In the above notation, the following statements hold true:

  * Δ(I) has countable length;

  *  the inclusion u^-1(i)→ i/u has a right adjoint;

  * u^-1(i) has a terminal object.

(1). We define a functor dΔ(I)→^


as follows: d(𝐧→ I):=n-1 and, given a morphism (𝐧→ I)φ⟶ (𝐦→ I), we have that n≥ m, so we can map φ to the unique map (n-1)→ (m-1) in ^. It is now easy to verify that this functor satisfies the conditions of Definition <ref>.


(2). Let x=(a_0→ a_1→…→ a_n, φ i→ a_0)∈ i/u and suppose that φ𝕀_x (i.e., that x∉u^-1(i)). Let c(x) be the following object of u^-1(i):

    c(x)=(iφ⟶a_0→ a_1→…→ a_n, 𝕀_i i→ i).

Of course the morphism 𝐧+1→𝐧+2 such that k↦ k+1 gives a well-defined morphism c(x)→ x in i/u. It is easy to verify that this morphism is a coreflection of x onto u^-1(i).


(3). One checks directly that (i⟶I, 𝕀_i i→ i) is a terminal object in u^-1(i).


Before introducing the following definition, let us recall that an object𝒳∈(J)is said to be constant if, and only if,𝒳≅_J^*𝒳_jfor some (i.e., all)j∈ J. Note also that, ifJhas a terminal objectt, then there is a unique natural transformationα𝕀_J⇒ t∘_Jand𝒳is constant if, and only if,α^*_𝒳𝒳→_J^*𝒳_tis an isomorphism. By (Der.2) this is equivalent to ask thatj^*(α^*_𝒳)is an isomorphism for eachj∈ J. Furthermore,j^*(α^*_𝒳)=𝒳_a𝒳_j→𝒳_t, wherea j→ tis the unique morphism inJ(j,t). As a consequence, wheneverJhas a terminal object, a coherent diagram𝒳∈(J)is constant if, and only if,𝒳_a𝒳_j→𝒳_kis an isomorphism for each morphisma j→ kinJ. This observation is important because it reduces the property of being constant for the coherent diagram𝒳, to a condition that we can check on the incoherent diagram_J(𝒳)(that is, that all its transitions maps are invertible).


Let  be a derivator and 𝒳∈(Δ(I)). We say that 𝒳 is  constant on fibers if, for any i∈ I, ι_i^*𝒳 is a constant diagram in (u^-1(i)) (where ι_i u^-1(i)→Δ(I) is the functor introduced in (<ref>)).


We are now ready to state and prove the main result of this subsection:


In the above notation, uΔ(I)→ I is a homotopical epimorphism. Furthermore, given a derivator , the essential image of u^*^I→^Δ(I) is the sub-derivator 𝔼⊆^Δ(I), where 𝔼(J) is spanned by those 𝒳∈^Δ(I)(J) that are constant on fibers, for each J∈. 

Fix a derivator  and let us verify that the sub-prederivator 𝔼⊆^Δ (I) defined in the statement satisfies the conditions of Lemma <ref>. Indeed, the first condition of the lemma can be verified as follows: let J∈ and 𝒳∈^I(J), let us prove that u^*𝒳∈𝔼(J)(⊆^Δ (I)(J)). Indeed, choose i∈ I and a∈ u^-1(i). 
Applying our derivator to the pullback diagram in (<ref>), we get the following commutative square

    @C=40pt(Δ (I))[d]_ι^*_i   (I)[l]_u^*[d]^u(a)^*
    (u^-1(i))   ()[l]^(.35)^*_u^-1(i)

showing that ι_i^*(u^*𝒳)= ^*_u^-1(i)u(a)^*𝒳= ^*_u^-1(i)(u^*𝒳)_a, as desired.
As for the second condition, choose J∈ and 𝒴∈𝔼(J), and let us study the component of the counit 

    ϵ_𝒴 u^*u_*𝒴⟶𝒴.
 
By (Der.2), isomorphisms in (Δ (I)) can be detected pointwise, so it is enough to show that, for each a∈Δ (I),  the following map  is an isomorphism

    a^*ϵ_𝒴 a^*u^*u_*𝒴⟶𝒴_a.
 
Let a∈Δ (I) and i:=u(a); by (Der.4), there is an isomorphism 

    a^*u^*u_*𝒴=i^*u_*𝒴≅⟶_i/u_i^*𝒴,
 
where _i i/u→Δ (I) is the obvious projection. Hence, it suffices to prove that the canonical map _i/u_i^*𝒴→𝒴_a is an isomorphism (see also <cit.>).
Now let α_i u^-1(i)→ i/u be the obvious inclusion; by Lemma <ref>, α_i is a left adjoint and so, by the dual of <cit.>, the following canonical map is an isomorphism

    _i/u_i^*𝒴≅⟶_u^-1(i)α_i^*_i^*𝒴=_u^-1(i)ι_i^*𝒴≅_u^-1(i)_u^-1(i)^*𝒴_a

where the last isomorphism holds since we have started with 𝒴∈𝔼(J). To conclude recall that, by Lemma <ref>, there is a terminal object t_i→ u^-1(i), that is, t_i is right adjoint to _u^-1(i). Hence, _u^-1(i)^*≅ (t_i)_*, showing that 

    _u^-1(i)_u^-1(i)^*𝒴_a≅_u^-1(i)(t_i)_*𝒴_a≅ (_u^-1(i)∘ t_i)_*𝒴_a≅𝒴_a.


§ LIFTING INCOHERENT DIAGRAMS ALONG DIAGRAM FUNCTORS


Let^→be a strong and stable derivator. In this section we study the following problem: 

Given an (incoherent) diagram X I→(), under which conditions is it possible to lift it to a coherent diagram 𝒳 of shape I? That is, can we find an object 𝒳∈(I) such that _I(𝒳)≅ X?

The property ofbeing strong implies that we can always lift diagrams of shape. The main result of this section is the following theorem that gives sufficient conditions for a diagram of arbitrary shape to lift to a coherent diagram. These sufficient conditions consist in assuming that there are no “negative extensions" in()between the components of our diagramX I→(). Similar conditions are given to identify pairs of coherent diagrams𝒳, 𝒴∈(I)for which_I(-)(I)(𝒳, 𝒴)→()^I(_I𝒳, _I𝒴)is bijective.



Given a strong and stable derivator ^→, the following statements hold true for any small category I:

  *  given 𝒳, 𝒴∈(I), the canonical map (I)(𝒳,𝒴)→()^I(_I𝒳,_I𝒴) is an isomorphism provided ()(Σ^n 𝒳_i,𝒴_j)=0 for all i,  j∈ I and n>0;

  *  given X∈()^I, there is an object 𝒳∈(I) such that _I(𝒳)≅ X, provided ()(Σ^n X_i,X_j)=0 for all i,  j∈ I and n>0.


The proof of the above theorem is quite involved and it will occupy the rest of this section, which is divided in three subsections to reflect the main steps of the argument.







 §.§ Lifting diagrams of finite length


In this subsection we are going to verify the statement of Theorem <ref> for diagrams of shapeI, withIa category of finite length. These results are very close to some of the main results in <cit.>. We offer here a different and self-contained argument.



Let ^→ be a strong and stable derivator, I a small category of finite length, and  𝒳, 𝒴∈(I). The canonical map 

    _I(-)(I)(𝒳,𝒴)⟶()^I(_I𝒳,_I𝒴),
 
is an isomorphism provided ()(Σ^n 𝒳_i,𝒴_j)=0 for all i,  j∈ I and n>0.




We proceed by induction on ℓ(I). If ℓ(I)=1, then I is a disjoint union of copies of   and, by (Der.1), there is nothing to prove. For ℓ(I)>1, consider the setting of Notation <ref> and let 𝒳, 𝒴∈(I). By Lemmas <ref> and <ref>, there is a triangle in (I):

    ∐_I∖ Ji_!i^*u_!u^*𝒳[r]^-α   ∐_I∖ Ji_!i^*𝒳⊔ u_!u^*𝒳[r]^-β   𝒳[r]   Σ(∐_I∖ Ji_!i^*u_!u^*𝒳).

where the maps α and β are described explicitly in Lemma <ref>. We are going now to apply the functor (-,𝒴):=(I)(-,𝒴) to the above triangle and study the resulting long exact sequence in . We start noting that, by Propositions <ref> and <ref>, for each i∈ I, there is the following commutative diagram:

    @C=52pt@R=4pt
    (i_!i^*𝒳⊔ u_!u^*𝒳,𝒴)@=[d]
    
    (i_!i^*𝒳,𝒴)× (u_!u^*𝒳,𝒴)[rr]^-[ccc(-)∘ i_!i^*ϵ_u      
    -(-)∘ϵ_i
    ]
    [dddddd]^-[origin=c]90∼_[cc
    i^*(-)∘η_i   0
    
    
    0   u^*(-)∘η_u
    ]      (i_!i^*u_!u^*𝒳,𝒴)[dddddd]^(*)_-[origin=c]90∼
    
    
    
    
    
    ()(i^*𝒳,i^*𝒴)×(J)(u^*𝒳,u^*𝒴)[rr]_-[ccc^*_u/i(-)∘γ^*      
    -γ^*∘^*_i(-)
    ]      (u/i)(_i^*u^*(𝒳),_u/i^*𝒴_i)

where both columns are isomorphisms and the map (*) is the composition of the three vertical maps on the right-hand-side of the diagrams in both Propositions   <ref> and <ref>. We also have the following series of isomorphisms: 

    (Σ(∐_I∖ Ji_!i^*u_!u^*𝒳),𝒴)   ≅∏_I∖ J()(Σ i^*u_!u^*𝒳,𝒴_i)
       ≅∏_I∖ J()(Σ_u/i_i^*u^*𝒳,𝒴_i)
       ≅∏_I∖ J(u/i)(Σ_i^*u^*𝒳,_u/i^*𝒴_i),

showing that (Σ(∐_I∖ Ji_!i^*u_!u^*𝒳),𝒴)=0 by inductive hypothesis. Therefore, after all these observations, we have obtained the following commutative diagram in 
    @C=10pt@R=14pt
    0[d]      
    
    (𝒳,𝒴)[d]      
    ∏_I∖ J()   (i^*𝒳,i^*𝒴)
       ×(J)(u^*𝒳,u^*𝒴)[dddd]|
    -[ccc|c⋱      0   ⋮
       _u/i^*(-)∘γ^*       -γ^*∘_i^*(-)
    0      ⋱   ⋮]
    
    [rr]^-_-∼      ∏_I∖ J   ()(i^*𝒳,i^*𝒴)
       ×()^J(_J(u^*𝒳),_J(u^*𝒴))[dddd]^Φ
    
    
    
    ∏_I∖ J(u/i)(_i^*u^*(𝒳),_u/i^*𝒴_i)[rr]^-_-∼      ∏_I∖ J()^u/i(_u/i(_i^*u^*(𝒳)),_u/i(_u/i^*𝒴_i))

where the left column is exact, showing that (𝒳,𝒴) is the kernel of the following map in the diagram, while the rows consist in an application of the corresponding diagram functors and, therefore, they are isomorphisms by inductive hypothesis (as we are working with categories of shorter length than ℓ(I)). To conclude, it is enough to verify that (Φ)=()^I(_I𝒳,_I𝒴). Hence, take an element 

    f:=(f_k)_k∈ I∈∏_I∖ J()(i^*𝒳,i^*𝒴)×()^J(_J(u^*𝒳),_J(u^*𝒴))

where f_ J:=(f_j)_j∈ J satisfies the required compatibilities to be a morphism of diagrams _J(u^*𝒳)→_J(u^*𝒴). By Propositions <ref> and <ref>, this f is sent to

    Φ(f)=((f_i∘𝒳_a-𝒴_a∘ f_j)_(j,a)∈ u/i)_i∈ I∖ J.

That is, Φ((f_k)_k∈ I)=0 if, and only if, for each i∈ I∖ J and each (j,a j→ i)∈ u/i,  we have that f_i∘𝒳_a-𝒴_a∘ f_j=0, that is, each of the following diagrams commutes:

    @R=25pt@C=50pt𝒳_j[r]^f_j[d]_𝒳_a   𝒴_j[d]^𝒴_a
    𝒳_i[r]_f_i   𝒴_i.

Equivalently, (f_k)_k∈ I represents a morphism in ()^I from _I𝒳 to _I𝒴.

Let ^→ be a strong and stable derivator, I a small category of finite length, and  X∈()^I. Then, there is an object 𝒳∈(I) such that _I(𝒳)≅ X, provided ()(Σ^n X(i),X(j))=0 for all i,  j∈ I and n>0.

We proceed by induction on ℓ(I). If ℓ(I)=1, then I is a disjoint union of copies of   and, by (Der.1), there is nothing to prove. If ℓ(I)>1, consider the setting of Notation <ref>.
By inductive hypothesis we have  an object 𝒳_J∈(J) such that there exists an isomorphism

    ξ:=(ξ_j)_j∈ J_J(𝒳_J)[r]^-∼    X_ J      in (1)^J.
 
For each i∈ I, let 𝒳_u/i:=_i^*𝒳_J∈(u/i) and note  that 

    _u/i(𝒳_u/i)=_J(𝒳_J)∘_i[rrr]^-∼_-(ξ__i(a))_a∈ u/i          X_ J∘_i.

We also need to consider _u/i^*X(i), for which _u/i(_u/i^*X(i))=κ_u/i(X(i)) is constant, and C_i:=_u/i𝒳_u/i∈(), for which we have the usual isomorphism

    C_i=_u/i𝒳_u/i[rr]^-∼_-γ_!      i^*u_!𝒳_J.

Consider now the following morphism in ()^u/i:

    (X_a)_a∈ u/i X_ J∘_i[rr]      κ_u/i(X(i))

where, given a=(j,a j→ i)∈ u/i, the morphism X_a X(j)→ X(i) is the obvious connecting map in X∈()^I. By Proposition <ref>, the diagram functor _u/i induces the isomorphism (u/i)(𝒳_u/i, _u/i^*X(i))≅()^u/i(_u/i(𝒳_u/i), _u/i(_u/i^*X(i))), hence there is a unique isomorphism

    φ̅_i𝒳_u/i⟶_u/i^*X(i)

such that _u/i(φ̅_i)=(X_a∘ξ__i(a))_a∈ u/i. Furthermore,  the adjunction (_u/i)_!⊣_u/i^* and the axiom (Der.4) induce the following isomorphisms:

    @C=22pt@R=2pt()(i^*u_!𝒳_J,X(i))[rr]^-∼_-(-)∘γ_!      ()(C_i, X(i))[rr]^-∼_-_u/i^*(-)∘η_u/i      (u/i)(𝒳_u/i,_u/i^*X(i))
    φ_i@|->[rrrr]            φ̅_i

so we can define φ_i i^*u_!𝒳_J→ X(i) as the unique map such that _u/i^*(φ_i∘γ_!)∘η_u/i=φ̅_i. Now define 𝒳 as the cone in (I) of the following map α:

    @C=32pt@R=9pt∐_I∖ Ji_!i^*u_!𝒳_J[rr]^-α:=[ccc⋱      0
       i_!φ_i   
    0      ⋱
    ⋯   -ϵ_i   ⋯]
    
          ∐_I∖ Ji_!X(i)      ⊔    u_!𝒳_J   [r]^-β   𝒳[r]   Σ(∐_I∖ Ji_!i^*u_!𝒳_J).

Note also that the above triangle is pointwise split-exact: it is enough to verify that k^*(α) is a split monomorphism in () for all k∈ I, but this is trivially true for k∈ J, as in this case k^*(∐_I∖ Ji_!i^*u_!𝒳_J)=0, while for k=i∈ I∖ J, so the morphism i^*(α) becomes:

    @C=32pt@R=9pt
    i^*i_!i^*u_!𝒳_J[rr]^-
    i^*(α)=[ccc
    i^*i_!φ_i
    
    
    
    -i^*ϵ_i
    ]
    
           i^*i_!X(i)      ⊔    i^*u_!𝒳_J,

which is a split monomorphism because -i^*ϵ_i i^*i_!i^*u_!𝒳_J→ i^*u_!𝒳_J is an isomorphism. As a consequence, one sees that _I(𝒳) is the cokernel of the map _I(α). Finally, define the following map of (incoherent) diagrams

    φ:=(φ_i)_i∈ I_I(u_!𝒳_J)⟶ X    in ()^I,
 
where φ_i is the map defined in (<ref>),





and consider the following commutative diagram

    @C=28pt@R=30pt∐_I∖ J_I(i_!i^*u_!𝒳_J)[d]^-[origin=c]90∼_χ[rr]^-_I(α)
          ∐_I∖ J_I(i_!X(i))⊔_I( u_!𝒳_J)[d]^-[origin=c]90∼_χ[rr]^-_I(β)      _I(𝒳)
    ∐_I∖ J(i^*u_!𝒳_J)⊗ i[rr]|-[ccc⋱      0
       φ_i⊗ i   
    0      ⋱
    ⋯   -e_i   ⋯]
    
          ∐_I∖ JX(i)⊗ i⊔_I( u_!𝒳_J)   [rr]_-[[   ⋯ e_i   ⋯   φ ]]
          X

where the vertical maps are induced by the natural isomorphisms χ_I∘ i_!⟹̃-⊗ i (with i varying in I∖ J) and e_i is the counit of the adjunction (-⊗ i)⊣ (-)_ i (see Example <ref>). One now checks easily that the second row is point-wise split, so X is a cokernel of the first map, which is isomorphic to _I(α). Therefore, _I(𝒳)≅ X, as desired.


 §.§ Lifting diagrams of countable length

Let ^→ be a strong and stable derivator, I a small category of countable length and 𝒳,  𝒴∈(I). Then, the canonical map 

    _I(-)(I)(𝒳,𝒴)⟶()^I(_I(𝒳),_I(𝒴))
 
is an isomorphism provided ()(Σ^n 𝒳_i,𝒴_j)=0 for all i,  j∈ I and n>0.

Fix Notation <ref> and consider the following triangle given by the Lemmas <ref> and <ref>, where α and β are the explicit matrices given in the lemma:

    @C=20pt∐_(ι_n)_!ι_n^*(𝒳)[rr]^α      ∐_(ι_n)_!ι_n^*(𝒳)[rr]^-β      𝒳[rr]      Σ(∐_(ι_n)_!ι_n^*(𝒳)).
 
Apply (-,𝒴):=(I)(-,𝒴) to the above triangle to get the following long exact sequence:

    @C=8.5pt⋯[r]   0[r]   (𝒳,𝒴)[r]   (∐_ (ι_n)_!ι_n^*(𝒳),𝒴)[rrr]^(-)∘α         (∐_(ι_n)_!ι_n^*(𝒳),𝒴)[r]   ⋯

where we can write the 0 on the left by the following series of isomorphisms: 

    (Σ(∐_(ι_n)_!ι_n^*(𝒳)),𝒴)   ≅∏_((ι_n)_!Σι_n^*(𝒳),𝒴)
       ≅∏_(I_≤ n)(Σι_n^*(𝒳),ι_n^*(𝒴))
       ≅∏_()^I_≤ n(_I_≤ n(Σι_n^*(𝒳)),_I_≤ n(ι_n^*(𝒴)))=0.

where the first isomorphism holds because Σ is an equivalence (and, as such, it commutes with coproducts) and (ι_n)_! is triangulated, the second isomorphism follows by the adjunction (ι_n)_!⊣ι_n^*, the third one is a consequence of Proposition <ref> (as I_≤ n is of finite length) and the last equality to 0 follows by our orthogonality hypotheses. A consequence of the  long exact sequence above is that (𝒳,𝒴) is the kernel of the map (-)∘α and, therefore, we will have concluded if we could prove that also ()^I(_I(𝒳),_I(𝒴)) is a kernel for this map. We start considering the following commutative diagram:

    @C=48pt@R=30pt(I)((ι_n)_!ι_n^*(𝒳),𝒴)@<-[r]^-(-)∘∂^n[d]^-[origin=c]90∼_-ι_n^*(-)∘η_n   (I)((ι_n+1)_!ι_n+1^*(𝒳),𝒴)[d]_-[origin=c]90∼^-ι_n+1^*(-)∘η_n+1
    (I_≤ n)(ι_n^*(𝒳),ι_n^*(𝒴))@<-[r]^ι_n+1,n^*(-)[d]^-[origin=c]90∼_-_I_≤ n(-)   (I_≤ n+1)(ι_n+1^*(𝒳),ι_n+1^*(𝒴))[d]_-[origin=c]90∼^-_I_≤ n+1(-)
    ()^I_≤ n(_I(𝒳)_ I_≤ n,_I(𝒴)_ I_≤ n)@<-[r]^(-)_ I_≤ n   ()^I_≤ n+1(_I(𝒳)_ I_≤ n+1,_I(𝒴)_ I_≤ n+1)

where the upper square commutes by Proposition <ref>, and the lower vertical maps are isomorphisms because of Proposition <ref> (and the fact that I_≤ n has finite length, for all ).







Hence, identifying (I_≤ n)(∐_(ι_n)_!ι_n^*(𝒳),𝒴) with ∏_(I_≤ n)((ι_n)_!ι_n^*(𝒳),𝒴), and also identifying each (I_≤ n)((ι_n)_!ι_n^*(𝒳),𝒴) with ()^I_≤ n(_I(𝒳)_ I_≤ n,_I(𝒴)_ I_≤ n) via the isomorphism _I_≤ n(ι_n^*(-)∘η_n) (as in the columns of the above diagram), we can describe the action of (-)∘α as follows:

    ∏_n()^I_≤ n(_I(𝒳)_I_≤ n,_I(𝒴)_I_≤ n)   →∏_n()^I_≤ n(_I(𝒳)_I_≤ n,_I(𝒴)_I_≤ n)
    
    
    (ϕ_n)_n∈   ↦ (ϕ_n-(ϕ_n+1)_I_≤ n)_n∈.

Thus, a sequence ϕ:=(ϕ_n)_n∈∈∏_n()^I_≤ n(_I(𝒳)_I_≤ n,_I(𝒴)_I_≤ n) is in the kernel of (-)∘α if, and only if, (ϕ_n+1)_I_≤ n=ϕ_n for all n∈, that is if, and only if, ϕ represents (the sequence of successive truncations of) a morphism _I(𝒳)→_I(𝒴). 


Let ^→ be a strong and stable derivator, I a small category of countable length and X∈()^I. Then, there is an object 𝒳∈(I) such that _I(𝒳)≅ X, provided ()(Σ^n X_i,X_j)=0 for all i,  j∈ I and n>0.

For each n∈, the inclusions ι_n I_≤ n→ I and ι_n+1,n I_≤ n→ I_≤ n+1 are cosieves and, therefore, the associated restrictions 

    (-)_ I_≤ n()^I⟶()^I_≤ n    and    (-)_ I_≤ n+1()^I⟶()^I_≤ n+1
 
 have left adjoints that act as extension by zero functors (recall Remark <ref>). We denote these adjoints by
 
    (-)⊗ι_n()^I_≤ n⟶()^I    and    (-)⊗ι_n+1,n()^I_≤ n⟶()^I_≤ n+1,

respectively. We denote the counits of the above adjunctions by 

    e^n=(e^n_i)_I(-)_ I_≤ n⊗ι_n⇒𝕀_()^I  and   d^n=(d^n_i)_I_≤ n+1(-)_ I_≤ n⊗ι_n+1,n⇒𝕀_()^I_≤ n+1

where e^n_i and d^n_i are identities if i∈ I_≤ n and 0 otherwise.

Consider now the diagram X∈()^I of the statement, let X^n:=(X_ I_≤ n)⊗ι_n, for all n∈, and consider the counit 

    d^n (X_ I_≤ n)⊗ι_n+1,n=(X_ I_≤ n+1)_ I_≤ n⊗ι_n+1,n⟶ X_ I_≤ n+1.

Extending this counit we get a morphism d^n⊗ι_n+1 X^n→ X^n+1 (whose components relative to I_≤ n are  identities and the remaining components are trivial). Then there is a pointwise split-exact sequence in ()^I of the form 

    @C=35pt∐_X^n[rrr]^(.47)α̅:=
    [cccc
    1         
    
    -d^0⊗ι_1   1      0
       -d^1⊗ι_2   1   
    0      ⋱   ⋱]
             ∐_X^n[rrr]^-β̅:=
    [cccc
    e^0    e^1    e^2    ⋯]
             X.
 
By Proposition <ref>, for any n∈_>0 we can find an object 𝒳^n∈(I_≤ n) such that _I_≤ n(𝒳^n)≅ X_ I_≤ n. Furthermore, by Proposition <ref>, for each n∈ there is a unique morphism δ^n(ι_n+1,n)_!𝒳^n→𝒳^n+1 such that _I_≤ n+1(δ^n)=d^n. Consider also a natural isomorphism ξ^n (ι_n)_!⇒ (ι_n+1)_!∘ (ι_n+1,n)_!  (which exists because these functors are both left adjoints to ι_n^*=ι_n+1,n^*∘ι_n^*) and construct an object 𝒳∈(I) as the cone of the map α in the following triangle:

    @C=14pt∐_(ι_n)_!(𝒳^n)[rrrrrrrr]^(.46)α:=
    [ccccc
    1         
    
    -(ι_1)_!δ^0∘ξ^0   1      0
       -(ι_2)_!δ^1∘ξ^1      1   
    0      ⋱      ⋱]
                            ∐_(ι_n)_!(𝒳^n)[r]   𝒳[r]   (∐_(ι_n)_!(𝒳^n)).

Just by construction, _I(α) is isomorphic to α̅, so the above triangle is pointwise split-exact and _I(𝒳) is the cokernel of  _I(α). Since X is the cokernel of α̅, we deduce that _I(𝒳)≅ X.


 §.§ Lifting in general


We finally have all the ingredients to complete the proof of the main result of this section:


As in Subsection <ref>, consider the category of countable length Δ(I) and the homotopical epimorphism uΔ(I)→ I. By the results of Subsection <ref>, we know that the statements (1) and (2) hold in (Δ(I)) and we should try to restrict them along the fully faithful functor u^*(I)→(Δ(I)).  


(1) Let 𝒳, 𝒴∈(I) and consider the following commutative diagram:

    (I)(𝒳,𝒴)[d]|≅[rr]      ()^I(_I𝒳,_I𝒴)[d]|(*)
    (Δ(I))(u^*𝒳,u^*𝒴)[rr]|(.4)(**)      ()^Δ(I)((_I𝒳)∘ u,(_I𝒴)∘ u)

where the leftmost vertical map is an isomorphism since u^* is fully faithful, (**) is an isomorphism by Proposition <ref> and the fact that Δ(I) has countable length, and (*) is an isomorphism by Lemma <ref>. 


(2) Given X as in the statement, by Proposition <ref> there is 𝒳_Δ(I) in (Δ(I)) such that _Δ(I)(𝒳_Δ(I))≅ X∘ u. Now one should just prove that 𝒳_Δ(I) does belong in the essential image of u^*, but this is true since being constant on fibers is a property that just depends on the underlying (incoherent) diagrams (see the discussion before Definition <ref>). 



































§ T-STRUCTURES ON STRONG AND STABLE DERIVATORS


Let us start by fixing a strong and stable derivator

    ^→.

By definition, at-structure inis just at-structure in the base of, so let us also fix such at-structure=(,Σ)on()whose heart we denote by$̋.  Furthermore, for each small category I, we let

    _I:={𝒳∈(I):𝒳_i∈, ∀ i∈ I}
       _I:={𝒳∈(I):𝒳_i∈, ∀ i∈ I}.

The goal of this section is to prove Theorem <ref>, i.e., we will verify that each _I:=(_I,Σ_I) is a t-structure in (I) and, moreover, that the heart _̋I of _I is equivalent to the functor category ^̋I. Our proof will rely heavily on the results of Section <ref>; also, the scheme of the proof will be the same: we will verify our statement for categories of finite length, then for categories of countable length and, finally, in full generality.


We start with the following definition:


A subcategory 𝒞 of 𝔻() is said to be closed under taking homotopy colimits (resp.,  directed homotopy colimits) with respect to 𝔻, when for any small category (resp., directed set) I  and any object 𝒳∈𝔻(I), one has that _I 𝒳∈𝒞 whenever 𝒳_i∈𝒞 for all i∈ I. Closedness under (inverse) homotopy limits is defined dually.


In fact, when the class  in the above definition is either an aisle or a co-aisle, then some closure properties are automatic: 


In the above setting, the aisle 𝒰 is closed under taking homotopy colimits and the co-aisle 𝒱 is closed under taking homotopy limits. 

Using the terminology of <cit.>, it is clear that 𝒰 is closed under coproducts and it follows from <cit.> or <cit.> that it is closed under homotopy pushouts. Then,  by <cit.>, we conclude that 𝒰 is closed under homotopy colimits. The argument for the co-aisle is formally dual.


Now we can prove the lifting property of t-structures for categories of finite length.


In the above setting, _I=(_I,Σ_I) is a t-structure in (I), for any small category of finite length I.

We proceed by induction on ℓ(I). If ℓ(I)=1, then I is a disjoint union of copies of   and, by (Der.1), there is nothing to prove. Hence, let I be a small category of finite length ℓ(I)>1, fix Notation <ref>, and let us verify that _I is a t-structure. Note that, by Theorem <ref>, the axiom (t-S.1) holds, while the axiom (t-S.2) holds  by construction. Hence, we have just to verify (t-S.3).
Indeed, take 𝒴∈(I) and let us construct a suitable truncation triangle with respect to _I. By inductive hypothesis, we can consider the following truncation triangles in (J) and (), respectively, 

    @C=28pt𝒰_u^*𝒴[r]^-φ_u    u^*𝒴[r]^-ψ_u   𝒱_u^*𝒴[r]   Σ𝒰_u^*𝒴    U_𝒴_i[r]^-φ_i   𝒴_i[r]^-ψ_i    V_𝒴_i[r]   Σ U_𝒴_i

with i ranging in I∖ J. Furthermore, since φ_i U_𝒴_i→𝒴_i is a coreflection of 𝒴_i onto , and i^*u_!𝒰_u^*𝒴≅_u/i_i^*𝒰_u^*𝒴∈ (by Proposition <ref>), there exists a unique morphism e_i i^*u_!𝒰_u^*𝒴→ U_𝒴_i that makes the following square commute:

    @C=70pt
    i^*u_!𝒰_u^*𝒴@.>[r]^-∃! e_i[d]_-i^*u_!φ_u   U_𝒴_i[d]^-φ_i
    
    i^*u_!u^*𝒴[r]_-i^*ϵ_u   𝒴_i.

Consider now the following commutative square in (I):

    @C=50pt@R=75pt∐_I∖ Ji_!i^*u_!𝒰_u^*𝒴[rr]|-  α_𝒰:=[ccc⋱      0
       i_!e_i   
    0      ⋱
    ⋯   -ϵ_i   ⋯]
    [d]|-Φ^(1):=[ccc⋱      0
       i_!i^*u_!φ_u   
    0      ⋱]
          ∐_I∖ Ji_!U_𝒴_i⊔ u_!𝒰_u^*𝒴[d]|-    [ccc|c⋱      0   ⋮
       i_!φ_i      0
    0      ⋱   ⋮
    ⋯   0   ⋯   u_!φ_u
    ]=:Φ^(2)
    ∐_I∖ Ji_!i^*u_!u^*𝒴[rr]|-  α:=[ccc⋱      0
       i_!i^*ϵ_u   
    0      ⋱
    ⋯   -ϵ_i   ⋯]
          ∐_I∖ Ji_!𝒴_i⊔ u_!u^*𝒴

where the cone of α is just 𝒴, by Lemma <ref>. 


























































By the 3× 3 Lemma in triangulated categories we can complete the above square to a commutative diagram in (I), where all the rows and columns are distinguished triangles

    @R=15pt∐_I∖ Ji_!i^*u_!𝒰_u^*𝒴[rr]^-α_𝒰[dd]_-Φ^(1)      ∐_I∖ Ji_!U_𝒴_i⊔ u_!𝒰_u^*𝒴[dd]^Φ^(2)[rr]^-β_𝒰      𝒰[dd][r]^(.65)+   
    
    ∐_I∖ Ji_!i^*u_!u^*𝒴[rr]^-α[dd]_-Ψ^(1)      ∐_I∖ Ji_!𝒴_i⊔ u_!u^*𝒴[rr]^-β[dd]^-Ψ^(2)      𝒴[dd][r]^(.65)+   
    
    𝒱^(1)[d]^(.65)+[rr]^-α_𝒱      𝒱^(2)[d]^(.65)+[rr]^-β_𝒱      𝒱[d]^(.65)+[r]^(.65)+

and β=[[   ⋯ ϵ_i   ⋯ ϵ_u ]].
We will have concluded if we can prove that 𝒰∈_I and 𝒱∈_I, that is, k^*𝒰∈ and k^*𝒱∈ for any k∈ I. We start from the case when k∈ J, and we apply k^*  to the above 3× 3 diagram, obtaining the following commutative diagram in (), where all the rows and columns are distinguished triangles:

    @C=30pt@R=10pt
    0[rr][dd]      0⊔ k^*u_!𝒰_u^*𝒴[dd][rr]^-∼      𝒰_k[dd][r]^(.65)+   
    
    
    0[rr][dd]      0⊔ k^*u_!u^*𝒴[rr]^-∼[dd]      𝒴_k[dd][r]^(.65)+   
    
    
    (𝒱^(1))_k[d]^(.65)+[rr]      (𝒱^(2))_k[d]^(.65)+[rr]^-∼      𝒱_k[d]^(.65)+[r]^(.65)+

Since k∈ J, the category u/k has a terminal object and, therefore, using (Der.4) and <cit.>, one shows that there is a natural isomorphism k^*u_!≅ k^*. Using this and the fact that k^* sends triangles to triangles, we can observe that the central column is a truncation triangle of 𝒴_k≅ k^*u_!u^*𝒴 with respect to the t-structure (𝒰, Σ𝒱) on (). Since also k^*𝒱^(1)=0 (see the first column), we have 𝒰_k≅ k^*u_!𝒰_u^*𝒴≅ U_𝒴_k∈ and 𝒱_k≅𝒱^(2)_k≅ V_𝒴_k∈.

On the other hand, if k∈ I∖ J, applying k^* to the above diagram we get the following commutative diagram in (), where rows and columns are distinguished triangles:

    @R=10pt
    k^*k_!k^*u_!𝒰_u^*𝒴[rr][dd]      k^*k_!U_𝒴_k⊔ k^*u_!𝒰_u^*𝒴[dd][rr]      𝒰_k[dd][r]^(.65)+   
    
    
    k^*k_!k^*u_!u^*𝒴[rr][dd]      k^*k_!𝒴_k⊔ k^*u_!u^*𝒴[rr][dd]      𝒴_k[dd][r]^(.65)+   
    
    
    (𝒱^(1))_k[d]^(.65)+[rr]      (𝒱^(2))_k[d]^(.65)+[rr]      𝒱_k[d]^(.65)+[r]^(.65)+

The components k^*k_!k^*u_!𝒰_u^*𝒴→ k^*u_!𝒰_u^*𝒴 and k^*k_!k^*u_!u^*𝒴→ k^*u_!u^*𝒴 are isomorphisms by Lemma <ref> (since k→ I is fully faithful), so the  first maps in each of the first two rows is a split monomorphism. Hence, the first two rows are split triangles and, therefore, the map 𝒰_k→𝒴_k is isomorphic to k^*k_!U_𝒴_k→ k^*k_!𝒴_k. Furthermore, by Example <ref> (and the fact that |I(k,k)|=1), there is a natural isomorphism k^*k_!⇒𝕀_(), therefore k^*k_!U_𝒴_k≅ U_𝒴_k∈ and k^*k_!V_𝒴_k≅ V_𝒴_k∈, showing that k^*k_!U_𝒴_k→ k^*k_!𝒴_k (and, therefore, also 𝒰_k→𝒴_k) is a coreflection onto . Hence, 𝒱_k∈ as desired.

In the same setting as above, _I=(_I,Σ_I) is a t-structure in (I), for any small category of countable length I.

As in the proof of Lemma <ref>, we have just to verify (t-S.3).
Indeed, take 𝒳∈(I) and let us construct a suitable truncation triangle with respect to _I. Fix Notation <ref>, then,




by Lemma <ref>, for any n∈_>0, there is a triangle

    𝒰^n[r]^φ_n   ι_n^*𝒳[r]   𝒱^n[r]   Σ𝒰^n

with 𝒰^n∈_I_≤ n and 𝒱^n∈_I_≤ n
Applying (ι_n)_! we obtain a triangle in (I):

    @C=45pt
    (ι_n)_!𝒰^n[r]^-(ι_n)_!(φ_n)    (ι_n)_!ι_n^*𝒳[r]    (ι_n)_!𝒱^n[r]   Σ (ι_n)_!𝒰^n.

Since (ι_n)_! is an extension by 0, we have that (ι_n)_!𝒰^n∈_I and (ι_n)_!𝒱^n∈_I, so  the triangle in (<ref>) is a truncation triangle with respect to _I and, as a consequence, (ι_n)_!(φ_n) is a coreflection of (ι_n)_!ι_n^*𝒳 onto _I. Consider the following solid diagram:

    @C=50pt@R=35pt
    (ι_1)_!𝒰^1[d]|(ι_1)_!(φ_1)@.>[r]^∃! ∂^1_𝒰   (ι_2)_!𝒰^2[d]|(ι_2)_!(φ_2)@.>[r]^∃! ∂^2_𝒰   …@.>[r]   (ι_n)_!𝒰^n[d]|(ι_n)_!(φ_n)@.>[r]^∃! ∂^n_𝒰   …
    
    (ι_1)_!ι_1^*𝒳[r]^∂^1   (ι_2)_!ι_2^*𝒳[r]^∂^2   …[r]   (ι_n)_!ι_n^*𝒳[r]^∂^n   …

Given n∈_>0, since (ι_n+1)_!(φ_n+1) is a coreflection onto _I and (ι_n)_!𝒰^n∈_I, there is a unique map ∂^n_𝒰 (ι_n)_!𝒰^n→ (ι_n+1)_!𝒰^n+1 such that (ι_n+1)_!(φ_n+1)∘∂^n_𝒰=∂^n∘ (ι_n)_!(φ_n).


Take now k∈ I and note that, by Lemma <ref>, k^*(ϵ_n) is invertible for all n≥ d(k). Furthermore, k^*(ϵ_n+1)∘ k^*( ∂^n)=k^*(ϵ_n) by Lemma <ref> and, therefore, k^*(∂^n) is an isomorphism for all n≥ d(k). Similarly, k^*(∂^n_𝒰) is an isomorphism for all n≥ d(k), since k^*(∂^n_𝒰) is a coreflection onto  of the isomorphism k^*(∂^n). Taking the cones of the vertical maps in the above diagram, we obtain a sequence like the following one:

    @C=40pt
    (ι_1)_!𝒱^1[r]^∂_𝒱^1   (ι_2)_!𝒱^2[r]^∂_𝒱^2   …[r]   (ι_n)_!𝒱^3[r]^∂_𝒱^n   ….

Given k∈ I, also k^*(∂_𝒱^n) is an isomorphism for all n≥ d(k). To see this just consider the morphism of triangles (k^*(∂_𝒰^n),k^*(∂^n),k^*(∂_𝒱^n)): as the first two components are invertible for  n≥ d(k), then so has to be the third. We can now conclude by taking the Milnor colimits of these three sequences
(i.e. the rightmost terms in triangles as in Lemma <ref>)
to get the following triangle:

    𝒰→𝒳→𝒱→Σ𝒰

with 𝒰∈_I, as 𝒰_k≅ (𝒰^d(k))_k∈, and 𝒱∈_I, as 𝒱_k≅ (𝒱^d(k))_k∈, for all k∈ I, by Example <ref>(2).


Now we can prove Theorem <ref> as stated in the Introduction. That is, _I=(_I,Σ_I) is a t-structure in (I) for any small category I, and the functor _I(I)→()^I induces an equivalence of categories _̋I≅^̋I.


As in the proof of Lemmas <ref> and <ref>, we have just to verify (t-S.3). Indeed, take 𝒳∈(I) and let us construct a suitable truncation triangle with respect to _I. 
As in Subsection <ref>, we consider the homotopical epimorphism uΔ(I)→ I. By Lemma  <ref>,  _Δ(I) is a  on (Δ(I)), so we can take the following truncation triangle in (Δ(I)):

    𝒰_u^*𝒳→ u^*𝒳→𝒱_u^*𝒳→Σ𝒰_u^*𝒳.

We have to verify that 𝒰_u^*𝒳 and 𝒱_u^*𝒳 belong to the essential image of u^*. For that, we need to prove that 𝒰_u^*𝒳 and 𝒱_u^*𝒳 are constant on fibers. Indeed, let i∈ I and consider the following diagram in (u^-1(i)):

    ι_i^*𝒰_u^*𝒳[r]   ι_i^*u^*𝒳[r]@=[d]    ι_i^*𝒱_u^*𝒳[r]    Σι_i^*𝒰_u^*𝒳.
    _u^-1(i)^*𝒰_𝒳_i[r]   _u^-1(i)^*𝒳_i[r]   _u^-1(i)^*𝒱_𝒳_i[r]   _u^-1(i)^*Σ𝒰_𝒳_i

Both the first and the second line are truncation triangles for ι_i^*u^*𝒳= _u^-1(i)^*𝒳_i in (u^-1(i)) with respect to the t-structure _u^-1(i). By the uniqueness of  truncations, ι_i^*𝒰_u^*𝒳≅_u^-1(i)^*𝒰_𝒳_i and ι_i^*𝒱_u^*𝒳≅_u^-1(i)^*𝒱_𝒳_i, so 𝒰_u^*𝒳 and 𝒱_u^*𝒳 are constant on fibers. Thus, the triangle (<ref>) lies in the essential image of u^* and, therefore, we can lift it back along u^* to obtain the desired truncation triangle of 𝒳 in (I).

Finally, note that the functor _̋I→^̋I induced by _I is fully faithful and essentially surjective by Theorem <ref>.


§ FINITENESS CONDITIONS FOR OBJECTS IN THE BASE


Let us start by fixing a strong and stable derivator 

    ^→.
 
In the first part of the section we recall what a closed monoidal derivator 𝕍 is, and what it means for  to have a closed action by such a 𝕍. We then introduce the class of homotopically finitely presented objects in (), relative to a specific closed action on . In the second part of the section we introduce an intrinsic version of homotopic finite presentability for objects in (), where by “intrinsic” we mean that this notion does not depend on the choice of a specific action, as it just uses data from  itself. We conclude the section by proving that a given object in () is compact (in the usual sense of triangulated categories with coproducts) if, and only if, it is intrinsically homotopically finitely presented if, and only if, it is homotopically finitely presented with respect to any closed action on  by a closed monoidal derivator 𝕍, whose tensor unit is a compact generator. 






 §.§ Closed actions by a closed monoidal derivator


Following <cit.>, a  monoidal structure on a prederivator 𝕍 is given by

  *  a tensor product ⊗𝕍×𝕍→𝕍 and

  *  a tensor unit 𝕊𝕐_→𝕍 (here 𝕐_ stands for the terminal derivator, i.e. 𝕐_(I)= for each I∈, cf. Examples <ref> and <ref>),

together with the usual unitality, associativity and symmetry constraints in the 2-category of prederivators. One can restrict this definition to derivators and, after some tedious work, arrive at the definition of a  closed monoidal derivator𝕍, see <cit.>. We will further say that 𝕍 is monogenic (cf. <cit.>) when it is further a strong and stable derivator, and the image of the unique object by the functor 𝕊:𝕐_1( )=→𝕍(), that we still denote by 𝕊, compactly generates 𝕍( ).

Given a closed monoidal derivator 𝕍 we have, for each I,  J∈, the functor of  internal hom-objects
    𝕍(I)^×𝕍(J)    ⟶𝕍(I^× J)
    
    (𝒳,𝒴)   ⟼𝒳𝒴.


The situation further generalizes as follows: if 𝕍 is a closed monoidal derivator, we say that  has a closed 𝕍-action if there is a morphism ⊗𝕍×→ together with associativity and unitality constraints in the 2-category of derivators which is at the same time a  adjoint in the sense of <cit.>. In that case, <cit.> provides us again with an  internal hom-functor for each I,  J∈,


    (I)^×(J) ⟶𝕍(I^× J).

Moreover, if we specialize to I=, these bifunctors, for any fixed  C∈(), naturally assemble to give a morphism of derivators


    C-⟶𝕍.

Crucial for our purposes is the fact that, in this last situation, for all V∈𝕍() and X, Y∈(), we have an adjunction isomorphism,  which is functorial in V, X and Y: 

    ()(V⊗ X,Y)≅𝕍()(V,X Y).

Our first example of this situation 
is aimed at algebraically minded readers:


Let  be a Grothendieck Abelian category. Then, the derived category () is enriched over () (we have 𝐑_()^×() →()). Similarly, the canonical derivator _ enhancing () (see Example <ref>) is enriched over _ and from this one gets an internal hom bifunctor 

    --=𝐑__(I)^×_(J) ⟶_(I^× J),

Note that _ is strong and stable, and it is also closed monoidal with respect to usual derived tensor product. Furthermore, the  unit object 𝕊∈_()= () is precisely ℤ, viewed as a stalk complex at zero, so that _ is monogenic. For I= and  C∈_()=(), we have the obvious morphism of derivators


    C -=𝐑_(C,-)_⟶_.


 We next remind the reader that the situation  applies to any strong and stable derivator:
 
 
Let  be a closed monoidal model category of spectra and _ be the corresponding strong stable derivator of spectra. More concretely, there are various mutually Quillen equivalent stable closed monoidal model categories of spectra which we can choose for , see <cit.>, and we pass to the corresponding homotopy strong stable derivator (see Example <ref>).
This will be a closed monoidal derivator by <cit.>.  Moreover, its unit object 𝕊 is the sphere spectrum (see <cit.>), which is  compact in _ ()=Ho() (see our Appendix [sec_spectra]A). 
For the sake of completeness, we also remark that the  Bousfield-Friedlander stable model category of spectra studied in Appendix [sec_spectra]A is not suitable to derive the monoidal structure in _—although it is also Quillen equivalent to the others, it does not have the required structure of a monoidal model category.

Now, given any strong and stable derivator, the discussion in <cit.> shows that there is a canonical closed action ⊗_×→. This in particular implies  that for each C ∈(), we have a morphism of derivators


    C -= F(C,-)⟶_,

called the  function spectrum, which is right adjoint (internal to the 2-category of derivators) to -⊗ C_→.


 §.§ Homotopically finite presentability relative to a closed action


All through this subsection, we assume that  is a strong and stable derivator,  𝕍 a closed monoidal, strong and stable derivator, and we fix a closed action:

    ⊗𝕍×⟶,
 
Given C∈(), we have corresponding internal hom-functor C -→𝕍. By construction, C -  is a morphisms of derivators and so it comes equipped with a natural isomorphism

    γ_u^C:=γ_u^C - u^*∘ (C -) ⟹̃ (C -)∘ u^*,

for each u J→ I in . As in Subsection <ref>, one can then construct the following natural transformation:

    (γ_u^C)_! u_!∘ (C -) ⟹ (C -)∘ u_!.
  
In particular, when I= and u:=_J J→, we get a canonical natural transformation 

    (γ__J^C)_!_J∘ (C -)⟹ (C-)∘_J .

Note that the  homotopy colimit  on the left hand side is taken in 𝕍, while that on the right hand side is taken in . This leads to the following crucial definition:


Given a small category I, an object C ∈𝔻() is said to be I-homotopically finitely presented in 𝔻, relative to 𝕍, when the last given natural transformation is an isomorphism, i.e., when the  
canonical map

    _I (C𝒳)⟶ C(_I 𝒳)

is an isomorphism for all 𝒳∈𝔻(I). 
We say that C is homotopically finitely presented in , relative to 𝕍,  when it is I-homotopically finitely presented, for any directed set I.


In the algebraic case, we have the following very intuitive interpretation.


In the situation of Example <ref>, for each C∈_()=(),  the morphism C - is given by 𝐑_(C,-)_(I)= (^I)→ (^I)=_(I). Saying that C is I-homotopically finitely presented in _, relative to _, amounts to say that the canonical morphism 

    _I  𝐑_(C,𝒳)⟶𝐑_(C, _I𝒳)
 
is an isomorphism, for all 𝒳∈_(I)= (^I).


In the next lemma we  show that, for certain shapes I, the I-homotopical finite presentability comes for free. We recall that I∈ is strictly homotopy finite if its nerve contains only finitely many non-degenerate simplices, <cit.>. In our terminology, this is equivalent to I being of finite length and having finitely many morphisms. A small category is homotopy finite if it is equivalent to one that is strictly homotopy finite.


Every object C∈() is I-homotopically finitely presented in , relative to 𝕍, if the small category I is homotopy finite.

The morphism C-→𝕍 is exact (=left exact, as we are in the stable setting) since it is right adjoint to -⊗ C (see <cit.>). This is enough to conclude since, by <cit.>, every exact morphism between stable derivators preserves homotopy finite colimits.



We can now give a handy characterization of homotopical finite presentability:


For an object C∈(), the following assertions are equivalent:

  *  the morphism of derivators C -:→𝕍 preserves all homotopy colimits;

  *  the object C is homotopically finitely presented in , relative to 𝕍;

  *  the morphism of derivators C -:→𝕍 preserves  coproducts.

The implication “(1)⇒(2)” is clear, and “(2)⇒(3)” follows by Lemma <ref>. Finally, the implication  “(3)⇒(1)” follows since C -:→𝕍 is an exact morphism of derivators, and so it preserves homotopy pushouts. By <cit.>, a morphism of derivators that preserves coproducts and homotopy pushouts preserves all homotopy colimits. 


 §.§ Intrinsic definition of homotopical finite presentability


In this subsection we present an alternative definition of homotopically finitely presented object in (), that depends only on , and not on any action by a closed monoidal derivator. 
Indeed, given a small category I, consider the following adjunction:

    _Iη_Iϵ_I_I^*(I)⟶().
 
Recall also that, by Example <ref>, _I∘_I^*=κ_I()→()^I is the constant diagram functor, and so we get the following natural transformation 

    _I(η_I)_I⟹κ_I∘_I(I)⟶()^I.
 
In turn, this yields the following canonical morphism of Abelian groups: 

    μ_C,𝒳_I()(C,𝒳_i)⟶()(C,_I𝒳),
 
for any C∈() and 𝒳∈(I), which is natural in both variables. 


An object C∈() is intrinsically homotopically finitely presented in  when, for every directed set I, the morphism 

    μ_C,𝒳_I()(C,𝒳_i)⟶()(C,_I𝒳)
 
is an isomorphism, for all 𝒳∈(I).







The following result will be crucial in the proof of Theorem <ref>.


If an object C∈() is homotopically finitely presented in , then it is also compact in (). 

Let I be a set and adopt the notation of the proof of Lemma <ref>, that is, I is a set and P:=𝒫^<ω(I) is the directed set of finite subsets of I, with u I→ P the obvious inclusion.

Given F∈ P, there is an equivalence F≅ u/F and, by (Der.4), we get the following isomorphism

    _F𝒳_ F≅_u/F _F^* 𝒳⟶̃(u_!𝒳)_F

for each 𝒳∈(I). We can conclude by the following series of isomorphisms:

    ∐_I()(C,X_i)   ≅_P ∐_F ()(C,𝒳_i)
       ≅_P ()(C,(u_!𝒳)_F)
       ≅()(C,_P(u_!𝒳))   as P is homo. f.p.;
       ≅()(C,(_I)_!𝒳)   as _I=_P∘ u;
       ≅()(C,∐_IX_i)   as I is discrete.


 §.§ Compactness versus homotopical finite presentability


In this final subsection we show that the two versions (extrinsic and intrinsic) of homotopical finite presentability coincide provided that, in the extrinsic case, the ⊗-unit of the given closed action, is a compact generator. Furthermore, both versions of  homotopical finite presentability are equivalent to compactness, in the usual sense of triangulated categories with coproducts.
This generalizes an analogous result obtained in <cit.> for the special case of =_R, the canonical derivator enhancing ( R) for a ring R (cf. Example <ref>), and 𝕍=_.


Let 𝕍 be a closed monoidal derivator which is monogenic, and  a stable derivator with a closed 𝕍-action. The following assertions are equivalent for C∈():

  * C is intrinsically homotopically finitely presented in ;

  * C is compact in (), i.e.,  the functor ()(C,-)()→ preserves coproducts;

  *  the morphism of derivators C -→𝕍 preserves coproducts;

  * C is homotopically finitely presented in , relative to 𝕍;

  *  the morphism of derivators C -→𝕍 preserves all homotopy colimits. 

The equivalence of assertions (3–5) is Proposition <ref> and the implication “(1)⇒(2)” is Lemma <ref>.  


(2)⇔(3). By (Der.1), assertion (3) holds if the following functor preserves coproducts:

    C -()⟶𝕍().
 
Let then I be a set, 𝒳=(X_i)_I∈(I)≅()^I, and consider the canonical morphism 

    λ =(γ__I^C)_! ∐_I(C X_i)⟶ C (∐_ IX_i)    in 𝕍(1),
  
as in Subsection <ref>. Furthermore, consider the canonical morphism 

    μ=μ_Σ^nC,𝒳∐_I()(Σ^n C,X_i)⟶()(Σ^n C,∐_IX_i)    in ,
 
as in Subsection <ref>. Then, assertion (3) holds if, and only if, λ is an isomorphism for all 𝒳∈(I), and assertion (2) holds if, and only if, μ is an isomorphism, for all 𝒳∈(I) and all n∈ℤ. In turn, since the unit 𝕊 is a compact generator of 𝕍(), we know that λ is an isomorphism if, and only if, the  map 

    λ_*𝕍()(Σ^n𝕊,∐_I(C X_i))⟶𝕍()(Σ^n𝕊,C (∐_IX_i))
 
is an isomorphism, for all n∈ℤ. Now, to see that μ is an isomorphism if, and only if, λ_* is an isomorphism, we consider the following commutative diagram, where all vertical arrows are isomorphisms:

    @R=15pt
    	𝕍(1)(Σ^n𝕊,∐_ I(C X_i)) [r]^-λ_∗   𝕍(1)(Σ^n𝕊,C (∐_IX_i)) ∐_I𝕍(1)(Σ^n𝕊,C X_i) [u]^-≅   𝔻()(Σ^n𝕊⊗ C,∐_IX_i) [u]_-≅∐_I𝔻()(Σ^n𝕊⊗ C,X_i) [u]^-≅   𝔻()(Σ^nC,∐_IX_i) [u]_-≅∐_I𝔻()(Σ^nC,X_i) [u]^-≅[ur]_-μ

For the invertibility of the vertical arrows, use the natural isomorphism (𝕊⊗ -)≅𝕀_, the adjunction (-⊗ C)⊢ (C -) and the compactness of 𝕊 in 𝕍(). Therefore, λ_* is an isomorphism if, and only if, so is μ.


(2)⇒(1). As we now know that assertions (2–5) are all equivalent, for any given closed symmetric monoidal derivator 𝕍 which is monogenic and acts on , we can assume in this implication that 𝕍=_ and that (C -)=F(C,-)=→_ is the function spectrum (see Example <ref>). In particular,  C is homotopically finitely presented, relative to _. Let then I be any directed set and 𝒳∈(I). As in the proof of the implication “(2)⇔(3)”, consider the canonical map λ_I(C𝒳)→ C_IX, and the induced maps

    λ^*𝕍()(Σ^n𝕊,_I(C𝒳))⟶𝕍()(Σ^n𝕊, C_IX)
 
for each n∈ℤ. We are going to verify that λ^* is an isomorphism if, and only if, the following canonical map is an isomorphism for all n∈:

    μ=μ_Σ^n C,𝒳_I𝒟(Σ^n C,𝒳_i)⟶𝒟(Σ^n C,_I𝒳).
 
In fact, arguing as above, we get a commutative diagram whose columns are isomorphisms:

    @R=15pt
    	𝕍(1)(Σ^n𝕊,_I(C𝒳)) [r]^-λ_∗   𝕍(1)(Σ^n𝕊,C_I𝒳) _I𝕍(1)(Σ^n𝕊,C𝒳_i) [u]^-≅   𝔻()(Σ^n𝕊⊗ C,_I 𝒳) [u]_-≅_I𝔻()(Σ^n𝕊⊗ C,𝒳_i) [u]^-≅   𝔻()(Σ^nC,_I𝒳) [u]_-≅_I𝔻()(Σ^nC,𝒳_i) [u]^-≅[ur]_-μ

where the upper left vertical arrow is an isomorphism because the sphere spectrum 𝕊 is intrinsically homotopically finitely presented in _ (see Appendix [sec_spectra]A).


As an immediate consequence (see Example  <ref>), we get


Let ^→ be a strong and stable derivator. The following assertions are equivalent for an object C∈():

  * C is intrinsically homotopically finitely presented in ;

  * C is a compact object of ();

  * C is homotopically finitely presented in , relative to _.


§ FINITENESS CONDITIONS AND DIRECTED COLIMITS IN THE HEART


Let us start by fixing a strong and stable derivator 

    ^→.
 
In this section we  discuss some finiteness conditions on a t-structure 𝐭 = (,Σ) in (). The simplest condition which we can impose is that  is closed under coproducts—this in fact makes sense in any triangulated category. However, it will turn out later in Example <ref> that this is not sufficient for the heart to be (Ab.5) Abelian. A stronger condition which does imply exactness of directed colimits in the heart is that  is closed under directed homotopy colimits. This will establish Theorem <ref>.














































































	
































 §.§ Homotopically smashing t-structures


Let us start introducing the finiteness conditions which we are going to study:


A t-structure =(,Σ) in () is said to be:

  * compactly generated when  is of the form =𝒮^⊥_≤ 0, where S⊆() is a set of compact objects (see Subsection <ref>);

  * homotopically smashing (with respect to 𝔻) when 𝒱 is closed under taking directed homotopy colimits;

  * smashing when 𝒱 is closed under taking coproducts. 


The three notions relate as follows.


Let =(,Σ) be a t-structure in () and consider the following conditions:

  *  compactly generated;

  *  homotopically smashing;

  *  smashing.

Then, the implications “(1)⇒(2)⇒(3)” hold and none of them can be reversed in general.

The implication “(1)⇒(2)” is a direct consequence of the definition of intrinsically homotopically finitely presented object and of Corollary <ref>. As for the implication “(2)⇒(3)”, let us assume that  is homotopically smashing and let 𝒳=(X_i)_i∈ I be a family of objects of 𝒰^⊥. We have seen in the proof of Lemma <ref> that, looking at I as a discrete category,  we can view 𝒳 as an object of 𝔻(I) and then there is a canonical isomorphism ∐_i∈ IX_i≅_P u_!𝒳, with the same notation as in that proof. Then we just need to check that (u_!𝒳)_F∈𝒰^⊥, for all finite subsets F⊆ I. But, in fact, (u_!𝒳)_F≅∐_i∈ FX_i, which is an object of 𝒰^⊥ since co-aisles are closed under finite coproducts.
We refer to  Examples <ref> and <ref> in the next section for explicit counterexamples showing that the implications in the statement cannot be reversed in general.


 §.§ Directed colimits in the heart


Here we prove Theorem <ref>, as stated in the Introduction, i.e. that the heart of a homotopically smashing t-structure is an (Ab.5) Abelian category. We start with the following consequence of Theorem <ref>:


Let =(,Σ) be a t-structure in (). Given I∈ and 𝒳∈_̋I, 

    _I _I 𝒳≅τ^Σ(_I 𝒳),

where the  colimit on the left hand side is taken in the heart =̋∩Σ.

By  Theorem <ref>, _I induces an equivalence F_I_̋I→^̋I, and we fix a quasi-inverse F_I^-1^̋I→_̋I. Now, _I is defined as the left adjoint to κ_I→̋^̋I so, composing the two adjunctions _I⊣κ_I and F_I⊣ F_I^-1, we obtain that _I∘ F_I is left adjoint to F_I^-1∘κ_I. Furthermore, F_I^-1∘κ_I≅_I^*_.

On the other hand, (τ^Σ V)_ is left adjoint to the inclusion →̋ and _I is left adjoint to _I^*()→(I). Composing the restrictions to the corresponding subcategories of the two adjunctions, we see that τ^Σ V∘(_I)__̋I is a left adjoint to (_I^*)_→̋_̋I.
Thus, we deduce the desired natural isomorphism:

    τ^Σ∘(_I)__̋I≅_I∘ (_I)__̋I.


The above lemma has the following immediate consequence: 


If =(,Σ) is a homotopically smashing t-structure in (), I is a directed set and 𝒳∈_̋I, then _I _I 𝒳≅_I 𝒳.


We can now proceed with the proof of Theorem <ref>: 


In view of Proposition <ref>, it remains to prove the (Ab.5) condition for the heart of a homotopically smashing t-structure =(,Σ). That is, given three diagrams X, Y, and Z I→$̋ for some directed setI, together with natural transformationsf X⇒ Yandg Y⇒ Z, such that 

    0⟶ X_if_i⟶ Y_ig_i⟶Z_i⟶ 0

is a short exact sequence in$̋ for any i∈ I, then

    0⟶_ IX_i⟶_IY_i⟶_IZ_i⟶ 0

is short exact. By Theorem <ref>, the short exact sequence 0→ X→ Y→ Z→ 0 in ^̋I, can be identified with a short exact sequence in _̋I⊆(I). Remember that a sequence in the heart of a t-structure is short exact if and only if it represents a triangle of the ambient category whose three first terms happen to lie in the heart. Hence, there is a map Z→Σ X such that

    X⟶ Y⟶ Z⟶Σ X

is a triangle in (I). Taking homotopy colimits we get a triangle in ():

    _I X⟶_I Y⟶_I Z⟶Σ_I X.

As  is homotopically smashing,  _I X, _I Y and _I Z belong to $̋, so the following sequence in$̋ is short exact:

    0⟶_I X⟶_I Y⟶_I Z⟶ 0.

One concludes by Corollary <ref> since _I X≅ (X_i)_i∈ I, and similarly for Y and Z.


§ TILTED T-STRUCTURES AND EXAMPLES


This section is devoted to providing examples of smashing t-structures which are not homotopically smashing and of homotopically smashing t-structures which are not compactly generated. This will complete the proof of Proposition <ref>. The strategy is to start with the canonical t-structure of a suitable Grothendieck Abelian category, tilt it using a torsion pair (see Example <ref>), and relate the properties of the resulting tilted t-structure to the properties of the torsion pair. Throughout the section, 

    ^→

will be a fixed, strong and stable derivator.





 §.§ Homotopically smashing tilts of t-structures


Our first result characterizes the situation when the Happel-Reiten-Smalø tilt _τ of a homotopically smashing t-structure  is homotopically smashing again.


Let  =(,Σ) be a homotopically smashing t-structure in (), and let _τ := (_τ,Σ_τ) be the tilt of this t-structure with respect to a torsion pair τ=(𝒯,ℱ) in the heart :̋=∩Σ of . Then _τ is a smashing t-structure. Moreover, the following conditions are equivalent:

  * _τ is homotopically smashing;

  * ℱ is closed under taking directed colimits in $̋. 

Since the heart $̋ ofis an (Ab.5) Abelian category by Theorem <ref>, the canonical map∐_I X_i →∏_I X_iis a monomorphism for any setIand(X_i)_I⊆$̋. Indeed, the latter map is a directed colimit of the split inclusions ∐_J X_i ≅∏_J X_i →∏_I X_i, where J runs over all finite subsets of I. In particular, F is closed under coproducts both in $̋ and()and, sinceis closed under coproducts in()by Proposition <ref>, so is_τ. Thus,_τis smashing.


(1)⇒(2). LetIbe a directed set and letF=(F_i)_i∈ Ibe a direct system in$̋, that is, an object in ^̋I. By Theorem <ref>, ^̋I≅_̋I⊆(I), so we can identify F with an object in _̋I and, as such, there is an isomorphism

    _I F≅_I F_i

where the colimit on the right-hand side is taken in $̋ (see Corollary <ref>). Now, ifF_i∈ℱfor eachi∈ I, the fact that_τis homotopically smashing  tells us that_I F∈_τand so_I F∈_τ∩=̋ℱ. 


(2)⇒(1). LetIbe a directed set and let𝒴∈(I)be such that𝒴_i∈_τfor anyi∈ I. Consider the truncation triangle of𝒴with respect to the liftedt-structure_Iin(I):

    𝒰⟶𝒴⟶𝒱⟶Σ𝒰,

where𝒰_i∈_and𝒱_i∈_, for anyi∈ I. For anyi∈ Iwe get a triangle in():

    Σ^-1𝒱_i⟶𝒰_i⟶𝒴_i⟶𝒱_i.

SinceΣ^-1𝒱_i∈Σ^-1⊆Σand𝒴_i∈_τ⊆Σ, we get that𝒰_i∈∩Σ=$̋. On the other hand, Σ^-1𝒱_i∈Σ^-1⊆Σ^-1_τ⊆_τ, and so 𝒰_i∈_τ. These two observations together give us that 𝒰_i∈∩̋_τ=ℱ. Taking now the homotopy colimit of the triangle in (<ref>), we get the following triangle in ():

    _I 𝒰⟶_I 𝒴⟶_I 𝒱⟶Σ_I 𝒰.

As we know that 𝒰_i∈ℱ⊆$̋ for anyi∈ I, we have_I 𝒰≅_I𝒰_iand, by our assumptions, this last directed colimit belongs toℱ. We can now conclude by noting that_I 𝒴∈ℱ*=_τ.

Let _^→ be the canonical derivator enhancing the derived category () of a Grothendieck Abelian category  (Example <ref>), and let =(,Σ) be the canonical t-structure in (). If τ=(𝒯,ℱ) is a torsion pair in  that is not of finite type (that is, ℱ≠ℱ), then the Happel-Reiten-Smalø tilt _τ of  with respect to τ is smashing but not homotopically smashing.

Explicitly we can take = and for 𝒯 the class of all divisible Abelian groups. In this case, the heart _̋τ of _τ is not an (Ab.5) Abelian category. Indeed, given any prime number p, we have in _̋τ a chain of monomorphisms


    Σ_p ↪Σ_p^2↪Σ_p^3↪…

whose directed colimit is τ^Σ_τ(Σ_p^∞) = 0 by Lemma <ref>.


 §.§ Compactly generated tilts of t-structures


Now we focus on what conditions a torsion pair must satisfy in order for the corresponding tiltedt-structure to be compactly generated.
We start with a lemma which says that compact objects in the aisle induce finitely presented objects in the heart (cp. <cit.>).


Let =(,Σ) be a homotopically smashing t-structure in () and  C ∈. If C is compact in (), then H_^0(C) is finitely presented in $̋.

As C is homotopically finitely presented by Corollary <ref>, we have a canonical isomorphism _I()(C, 𝒳_i)≅()(C,_I 𝒳) for any directed set I and 𝒳∈(I). Furthermore, if 𝒳∈_̋I, then ℋ(H_^0(C), 𝒳_i)≅()(C, 𝒳_i) for each i∈ I and ℋ(H_^0(C),_I 𝒳)≅()(C,_I 𝒳), so the canonical morphism _Iℋ(H_^0(C), 𝒳_i)≅ℋ(H_^0(C),_I 𝒳) is invertible as well.
The conclusion follows since _I 𝒳≅_I 𝒳_i by Corollary <ref>.


If thet-structure is even compactly generated, a much better description of finitely presented objects in the heart was obtained in <cit.>. The proof is more involved and we are not going to discuss it here.


Let =(,Σ) be a compactly generated t-structure in a triangulated category 𝒟 with coproducts and let X∈=̋∩Σ. Then X is finitely presented in $̋ if, and only if, there exists a compact objectC∈such thatX≅ H_^0(C).


We can now give our criterion for a Happel-Reiten-Smalø tilt of at-structure to be compactly generated.


Let =(,Σ) be a homotopically smashing t-structure in () and let _τ =(𝒰_τ,Σ_τ) be the tilt of  with respect to a torsion pair τ = ( T, F) in the heart =̋∩Σ. Consider the following conditions:

  * _τ is compactly generated,

  *  there is a set 𝒮⊆𝒯 of finitely presented objects of $̋ withℱ=⋂_S∈𝒮(ℋ(S,-)).

Then, (1) implies (2). If, moreover,is compactly generated,

then (1) and (2) are equivalent.

(1)⇒(2). Let 𝒮̂⊆𝒰_τ⊆ be a set of compact objects such that Σ𝒮̂⊆𝒮̂ and which generates _τ.

We put 𝒮:={H_^0(X):X∈𝒮̂},where each H_^0(X) is finitely presented in $̋ by Lemma <ref>. As(ℋ(H_^0(X),-)) = ∩̋(()(H_^0(X),-)) = ∩̋(()(X,-))for eachX∈𝒮̂, we have

    ⋂_S∈𝒮(ℋ(S,-)) =
    ∩̋⋂_X∈𝒮̂(()(X,-)) =
    ∩̋_τ =
    ℱ.

(2)⇒(1),  assuming that  is compactly generated.

It is enough to adapt the proof of <cit.>.
We start with a set of objects which are finitely presented andℱ=⋂_S∈𝒮((̋S,-))in$̋. For each S∈𝒮, we fix an object X_S∈ which is compact in () and H_^0(X_S)≅ S. Such an object exists by Proposition <ref>. If 𝒴 is a set of compact generators of , we shall prove that Σ𝒴∪𝒳 generates _τ, where 𝒳:={X_S:S∈𝒮}.
Bearing in mind that Σ𝒴∪𝒳⊆_τ, our task reduces to prove that 𝒱':=(Σ𝒴)^⊥_≤ 0∩𝒳^⊥_≤ 0=(Σ𝒴∪𝒳)^⊥_≤ 0⊆_τ.
Since (Σ𝒴)^⊥_≤ 0 = Σ, an object V∈() is in 𝒱' if, and only if, V∈Σ and ()(X_S,V)=0, for all S∈𝒮.
However, since X_S∈ for all S∈𝒮, the following series of isomorphisms for each V∈Σ,

    ()(X_S,V)≅()(H_^0(X_S),V)≅(̋H_^0(X_S),H_^0(V))=(̋S,H_^0(V)),

shows that V∈𝒱' if, and only if, V∈Σ and H_^0(V)∈ℱ or, equivalently, V∈𝒱_τ. 

















Let _R^→ be the canonical derivator enhancing the derived category ( R) for a ring R (Example <ref>), and let =(,Σ) be the canonical  on ( R).
Suppose that R admits a non-trivial two-sided idempotent ideal I contained in its Jacobson radical J(R) (see <cit.> for an explicit example and note that such an I must be infinitely generated from either side and, consequently, R must be non-Noetherian, by the Nakayama Lemma). Consider the torsion pair τ =(𝒯_I,ℱ_I), where 

    𝒯_I:={T∈ R:TI=T}  and  ℱ_I:={F∈ R:FI=0}.

Then, the tilted t-structure _τ in ( R) is homotopically smashing but not compactly generated.
Indeed, the torsion-free class ℱ_I is closed under directed colimits in R, so _τ is homotopically smashing by Proposition <ref>. On the other hand, due to the Nakayama Lemma, 𝒯_I does not contain any non-zero finitely generated module. Then _τ cannot be compactly generated as a t-structure because of Proposition <ref>.

Example <ref> gives a negative answer to <cit.>.


§ ON THE EXISTENCE OF A SET OF GENERATORS


We conclude with the discussion of when the heart of at-structure in the base()of a strong and stable derivatoris actually is a Grothendieck Abelian category. Unlike the exactness of directed colimits, the existence of a generator is, to a large extent, a purely technical condition which is usually satisfied in examples “from the nature”. In the world of∞-categories, a condition for the heart of at-structure to be Grothendieck Abelian was given by Lurie in <cit.>. We give a similar (and short) discussion also in our setting, restricting to derivators of the form=_(,), where(,, B, F)is a combinatorial stable model category (see Example <ref>). These derivators are called “derivators of small presentation” and they can be characterized internally to the2-category of derivators, see <cit.>.


Recall that a model category(,, B, F)is  combinatorial if 

  *  is a locally presentable category (see <cit.>);

  *  the model structure (, B, F) is cofibrantly generated. 

These model categories were introduced by J. Smith. Let us recall here the following properties whose proof essentially follows by  <cit.> or <cit.>:


Let (,, B, F) be a combinatorial model category and 𝒮⊆ a  set of objects. Then, there exists an infinite regular cardinal λ such that

  *  the functors (S,-) commute with λ-directed colimits for all S∈𝒮;

  *  there are co/fibrant replacement functors which preserves λ-directed colimits;

  * λ-directed colimits of weak equivalences are again weak equivalences;

  *  given a λ-directed set I and a diagram X∈^I, there is an isomorphism in ()
    L_IX≅ F(_I X),

where F→() is the canonical functor. 

(1) follows since  is locally presentable, while (2,3) are <cit.>.


(4). Given a λ-directed set I, since _I induces a functor _I →, the universal property of F_I^I→(^I)=^I[_I^-1] yields a unique functor completing the following solid diagram to a commutative square:

    @C=50pt^I[d]__I[r]^-F_I   (^I)@.>[d]
    [r]_-F   (C)

Of course, such a functor automatically satisfies the universal property for being the total left derived functor of _I, hence we deduce the isomorphism in the statement.


As remarked in Example <ref>, for derivators arising from model categories, the functor_I(^I)→()is the total left derived functor of_I^I→. Using this identification, one easily deduces the following corollary:


Let (,, B, F) be a combinatorial model category, = _(,) the induced strong derivator as in Example <ref> and λ an infinite regular cardinal satisfying (4) in Lemma <ref>. If I is a λ-directed and X∈^I, then there is an isomorphism in ()
    _IX≅ F(_I X),

where F→() is the canonical functor. 



We can prove the existence of a generator for the following large class oft-structures which includes all homotopically smashing ones (they constitute a special case forλ=ℵ_0):


Let (,, B, F) be a stable and combinatorial model category, and =(,Σ)  a t-structure in (). We say that  is λ-accessibly embedded in (), with λ an infinite regular cardinal, if  is closed under λ-directed homotopy colimits in _(,).

Let (,, B, F) be a stable and combinatorial model category, λ an infinite regular cardinal satisfying (3) and (4) in Lemma <ref>, and =(,Σ) a λ-accessibly embedded t-structure with heart =̋∩Σ in ().
Then, the following composition functor preserves λ-directed colimits:

    F⟶()H^0⟶.̋

Given a λ-directed set I, consider the following diagram:

    @C=50pt^I[r]^-F_I[d]__I   (^I)[r]^-H^0_I[d]|_I   _̋I[d]^_I
    [r]_-F   ()[r]_-H^0   .̋

By Theorem <ref>, _̋I≅^̋I and, identifying these two categories, (_I)__̋I is conjugated to _I^̋I→$̋ (see the proof of Corollary <ref>). This observation tells us that it is enough to show that the external square in the above diagram commutes. We  verify instead that the smaller squares do commute. In fact, the commutativity of the square on the left hand side is given by Corollary <ref>, while the commutativity of the square on the right hand side follows from the fact that both(by Proposition <ref>) and(by assumption) are closed underλ-directed homotopy colimits.


Before we prove, as a main result of the section, Theorem <ref> from the introduction, we give a discussion of its assumptions:


A triangulated category is called algebraic if it is the stable category of a Frobenius exact category <cit.>. Compactly generated algebraic triangulated categories are then triangle equivalent to the derived categories of small dg categories, and such derived categories are the homotopy categories of  combinatorial model categories of dg modules. We refer to <cit.> for a more detailed discussion.

More generally, one can consider algebraic triangulated categories which are well generated in the sense of <cit.>. These are, by <cit.>, simply localizations of compactly generated algebraic triangulated categories D with respect to localizing subcategories L generated by a small set S of objects. Such a localization is also the homotopy category of a combinatorial model category. 
As an upshot, each well generated algebraic triangulated category is the homotopy category of a combinatorial model category. 

An analogous result for well generated topological triangulated categories (i.e.,  homotopy categories of spectral model categories) can be found in <cit.>.

Let (,, B, F) be a stable combinatorial model category and =(,Σ) a t-structure in () generated by a  set S of objects of . Then there exists an infinite regular cardinal λ such that  is λ-accessibly embedded. 

Up to replacing 𝒮 by another set of the same cardinality, we can assume that any object in 𝒮 is cofibrant. Let now λ be an infinite regular cardinal with the properties (1–4) described in Lemma <ref>, and fix a fibrant replacement functor R that commutes with λ-directed colimits. We have to prove that =𝒮^⊥ is closed under taking λ-directed homotopy colimits. Indeed, let I be a λ-directed set and X∈^I such that X_i∈ for all i∈ I. For any C∈𝒮 and i∈ I, there is a commutative diagram as follows:

    (C, RX_i)@->>[d]_(*)[rr]      (C, _IRX_i)@->>[d]^(**)
    
    0=()(C, RX_i)[rr]      ()(C, _IRX_i)

where (*) is surjective since C is cofibrant and RX_i is fibrant, while (**) is surjective since _IRX_i≅_IRX_i≅ R(_IX_i) by Lemma <ref> and the fact that R commutes with λ-directed colimits. Using the universal property of _I in , we obtain the following commutative diagram:

    _I(C, RX_i)@->>[d][rr]^≅      (C, _IRX_i)@->>[d]
    
    0=_I()(C, RX_i)[rr]      ()(C, _IRX_i)

where the top row is invertible by  Lemma <ref>, so that ()(C, _IRX_i)=0, for all C∈𝒮, and so _IX_i≅_IRX_i∈𝒮^⊥=.

We remind the reader that we need to prove the following. If (,, B, F) is a stable combinatorial model category and =(,Σ) is a λ-accessibly embedded t-structure in (), then the heart =̋∩Σ has a generator.
We may also, without loss of generality, assume that λ satisfies conditions (3) and (4) from Lemma <ref>.
Since the category  is locally presentable, there exists a set  of objects of  such that every object C∈ is a λ-directed colimit C = _I_C (Q_i) of a λ-direct system (Q_i)_i∈ I_C in . Consider the following set of objects in $̋:

    𝒬:={H^0(Q):Q∈}.

Now notice that

    H^0(F(C))≅ H^0(F(_I_CQ_i))(*)≅_I_CH^0(F(Q_i)),

where(*)follows by Proposition <ref>. 
Since any object in$̋ is of the form H^0(F(C)) for some C∈, we have just verified that 𝒬 is a set of generators for $̋. 

If the t-structure in the above proof is compactly generated, then it is homotopically smashing by Proposition <ref>. Since the heart is (Ab.5) by Theorem <ref> and it has a generator by Theorem <ref>, it is a Grothendieck Abelian category.

Corollary <ref> is a straightforward consequence of Theorems <ref> and <ref>.
On the other hand, under the same set of hypotheses, one even has that =̋ H^0_(𝒯), where 𝒮 is a set of compact generators for  and 𝒯 is the smallest subcategory of () containing  𝒮 which is closed under extensions, suspension and direct summands.
Indeed, 𝒯 is precisely the class of compact objects of () which are contained in the aisle . We refer to <cit.> or <cit.> for this fact. The conclusion then follows from <cit.>, where it was shown, while this paper was under review, that the heart $̋ is locally finitely presentable andH^0_(𝒯)is precisely the class of finitely presented objects (recall Proposition <ref>).





§ APPENDIX A: DIRECTED HOMOTOPY COLIMITS OF SPECTRA
tocsectionAppendix A: Directed homotopy colimits of spectra

The main point of this appendix is to establish that the stable homotopy groups of spectra (in the topological sense) commute with (homotopy) directed colimits.
In the terminology of Subsection <ref>, one can equivalently say that the sphere spectrum is intrinsically homotopically finitely presented in_.
This can be viewed as a topological analogue of the fact that the cohomology groups of complexes of Abelian groups commute with directed colimits. Although this result seems to be well-known to experts in homotopy theory (see, e.g., <cit.> or <cit.>), we are lacking an adequate reference.

Model structures on categories of diagrams. 
When discussing homotopy colimits in detail, we are confronted with the following question: Given a model categorywith the class of weak equivalencesand a small categoryI, is there a suitable model structure on the diagram category^Iwith pointwise weak equivalences? Although the existence of such model structures is not clear in full generality and for many purposes one can work around this problem with other techniques (cf. <cit.>), they nevertheless exist in many situations and make the discussion easier there. Most notably, one usually considers two “extremal” model structures, provided that they exist:


	Given a model category (, , B, F) and a small category I,  a model structure on ^I is called
	
  *  the projective model structure if the weak equivalences and fibrations are defined pointwise (i.e., a morphism f X → Y in ^I is a weak equivalence or fibration if f_i X_i → Y_i is a weak equivalence or fibration, respectively, in  for each i∈ I);
		
  *  the injective model structure if the weak equivalences and cofibrations are defined pointwise.
	

Note that both the projective and the injective model structures, when they exist, are unique. It is also a standard fact that the class of injective fibrations is included in the class of projective fibrations and dually for cofibrations. To see this, note that for eachi∈ I, the evaluation functor(-)_i^I →,X ↦ X(i)has a left adjoint-⊗ i→^Igiven by(X⊗ i)(k)≅∐_I(i,k)X, for allk∈ I(see Example <ref>). One readily checks that-⊗ ipreserves cofibrations and trivial cofibrations if^Iis equipped with the injective model structure. Thus,(-⊗ i, (-)_i)is a Quillen adjunction and(-)_isends injective fibrations to fibrations infor eachi∈ I. Another fact which we will need is the following observation regarding the structure of fibrant and cofibrant objects.


	Let (,, B, F) be a model category and (I,≤)  a partially ordered set (viewed as a small category). If a projective model structure exists, X∈^I is a projectively cofibrant object and i≤ j are elements of I, then X(i) → X(j) is a cofibration in . Dually, if an injective model structure exists and X is an injectively fibrant object, then X(i) → X(j) is a fibration in . 

	We  only prove the part regarding the injective model structure, the other part is dual. In view of <cit.>, we need to prove that, given any commutative square
	
	
    U [d]_-c[r]^-u    X(i) [d] 
    
    			V [r]_-v@.>[ur]||h     X(j)

	in , where c U → V is a trivial cofibration, the dotted arrow can be filled in so that both triangles commute.	To construct h, we start with the fact that X is injectively fibrant, i.e., (V',X) →(U',X) is surjective for any pointwise trivial cofibration f' U' → V' in ^I. We apply this property to a specially crafted f', based on the morphism f above. We let V' = V ⊗ i, i.e., V'(k) = V if k ≥ i and is the initial object 0∈ otherwise. Let also:
	
	
    U'(k) = 
    	V   if  k ≥ j;
    U   if  k ≥ i  and  k ≱j;
    
    	0     otherwise.

	The morphisms U'(k) → U'(ℓ) are copies of c if k≥ i, k≱j and ℓ≥ j, and the identity morphisms or the morphisms from the initial object, otherwise. There is an obvious morphism c' U' → V' whose components are just identity morphisms and copies of c. Furthermore, the commutative square (<ref>) allows us to define a morphism u' U' → X such that

  * u'(k) is the composition of v with X(j)→ X(k) if k≥ j,

  * u'(k) is the composition of u with X(i)→ X(k) if k≥ i, but k≱j and

  * u'(k) is the morphism from the initial object otherwise.


As mentioned above, our assumption dictates that u' U' → X factors through c' U' → V' via a map h' V' → X. The restriction of the morphism c' and h' to the components i,  j∈ I yields the following commutative diagram in ,
	
	
    @C=50pt
    		U [d]_-c[r]^-c    V @=[d] [r]^-h'(i)    X(i) [d] 
    
    		V @=[r]     V [r]_-h'(j)    X(j),

	where the compositions in the rows are u and v respectively. It follows that v=h'(j) and h = h'(i) fits into (<ref>).


Our main motivation for considering the projective model structure is that, if it exists, the constant diagram functorκ_I→^Ipreserves fibrations and trivial fibrations, so that(_I, κ_I)is a Quillen adjunction. In other words, the  total left derived functor𝐋_I(^I) →()exists and it can be computed using projectively cofibrant resolutions of diagrams. We will denote𝐋_Iby_Iand call it the homotopy colimit functor. Of course, formally dual statements hold for limits and the injective model structure.

Finally, we touch the problem of the existence of projective an injective model structures in the case of combinatorial model structures (see Section <ref>):

	Let (,, B, F) be a combinatorial model category and I a small category. Then the diagram category ^I admits both the projective and the injective model structures.
Model categories of simplicial sets and spectra.
LetΔbe the category with all finite ordinals1, 2, 3, …, where𝐧={0→ 1→…→ (n-1)}, as objects and order-preserving maps as morphisms. Here we keep our convention for ordinals from set-theory rather than from topology (there one often denotes by[n]the ordinal𝐧+1as it indexes then-dimensional simplices). The categoryof simplicial sets is defined as the category of functorsΔ^→. As customary, we denote byΔ[n] := Δ(-,(𝐧+1))the representable functors in.

A simplicial set can be viewed as a combinatorial model for a topological space. More precisely, there is a geometric realization functor|-|→,X ∈↦| X|∈. This is a left adjoint functor which preserves finite limits up to a weak homotopy equivalence. More in detail, recall that a continuous map if topological spacesf X → Yis a weak homotopy equivalence ifπ_n(f)π_n(X,x) →π_n(Y,f(x))is a bijection for alln≥ 0and all base pointsx∈ X(1)(here,π_0(X,x)is just the set of all path components ofX, whileπ_n(X,x)is a group forn≥ 1and an Abelian group forn≥ 2). Now ifI∈is a finite category andX∈^I, the canonical map|lim_i∈ I X_i|→lim_i∈ I| X_i|is a bijection of sets, but the topologies may not agree. Nevertheless, it is always a weak homotopy equivalence—see <cit.> for details.
 The topological spaces of the form| X|have very nice properties as they naturally have a structure of CW-complexes <cit.>. In fact, ifX⊆ Yis an inclusion of simplicial sets, then| X|is naturally a CW-subcomplex of| Y|in the sense of <cit.>. This follows, e.g., directly from the proof of <cit.>.

GivenX ∈,x∈ X(1)andn≥ 0, one definesπ_n(X,x):=π_n(| X|,| x| )with the base point corresponding tox. It is also possible to defineπ_n(X,x)combinatorially directly on the simplicial setX, see <cit.>.
The categorycomes equipped with a standard model structure such that cofibrations are inclusions of simplicial sets, fibrations are the so-called Kan fibrations, andf X → Yis a weak equivalence ifπ_n(f)π_n(X,x) →π_n(Y,f(x))is a bijection for alln≥ 0and all base pointsx∈ X(1)(or, equivalently, by <cit.>, if| f|is a homotopy equivalence of topological spaces).

Since homotopy groups need a choice of a base point, it will be helpful to work with the category_*of pointed simplicial sets, which is simply the slice categoryΔ[0]/. In pedestrian terms, its objects are pairs(X,x), whereX∈andx∈ X(1), and the morphisms are base point-preserving morphisms of simplicial sets. This category also carries a model structure where a morphismf (X,x) → (Y,y)is a weak equivalence, cofibration or fibration if the underlying mapX→ Yis such in(see <cit.>).

IfIis a directed set and(X_i,x_i)_i∈ I∈_*^I, then there is a canonical homomorphism_Iπ_n(X_i,x_i) →π_n(_I(X_i,x_i))for eachn≥ 0. More explicitly, we first replace(X_i,x_i)_i∈ Iby a pointwise weakly equivalent projectively cofibrant diagram(X'_i,x'_i), then applyπ_nto the adjunction unitη (X'_i,x'_i) →κ_I(_I (X'_i,x'_i)) = κ_I(_I (X_i,x_i)), and finally take the colimit map of the resulting cocone of sets or groups.


	In the above setting, the map _Iπ_n(X_i,x_i) →π_n(_I(X_i,x_i)) is a bijection.

	The morphisms X_i → X_j in the projectively cofibrant diagram (X'_i,x'_i)∈_* are inclusions by Lemma <ref>, so we can assume that X'_i are simplicial subsets of _I X'_i and that _I X'_i is the directed union of the X'_i.
	As any compact subset of |_I X'_i | is contained in | X_i|, for some i∈ I, by <cit.>, we can conclude by <cit.>.


Since our main object of interest are stable Grothendieck derivators and any stable derivator is enriched over spectra by <cit.>,  our main goal is the analogue of Lemma <ref> for spectra. To this end, let us quickly recall one model structure for the category of spectra from <cit.>.

First note that the categorycarries a Cartesian (closed) symmetric monoidal structure(,×,Δ[0]). Now the forgetful functor^*→has a left adjoint which sendsX∈to the disjoint unionX_+ := X ∪Δ[0], and this adjunction allows to define a unique monoidal structure(,∧,Δ[0]_+)such that∧preserves colimits in each component andX ↦ X_+is a monoidal functor. The functor∧is called the smash product.

The key object for the definition of a spectrum is a combinatorial model for the topological circle.
We can define𝕊^1∈as the coequalizer ofΔ[0] ⇉Δ[1], where the two maps come from the two morphisms1→2inΔ. As this construction gives a canonical mapΔ[0]→𝕊^1, we can view𝕊^1as an object of_*.


A spectrum is a sequence X = (X^n,σ^n) indexed by the natural numbers such that X^n∈_* and σ^n𝕊^1∧ X^n → X^n+1 is a map of pointed simplicial sets for each n∈. Maps of spectra f (X^n,σ^n) → (Y^n,σ'^n) are defined in the obvious way as collections of maps f^n X^n→ Y^n such that f^n+1σ^n = σ'^n (𝕊^1∧ f^n) for all n. The category of spectra will be denoted by .


For a spectrumX = (X^n,σ^n)andm∈, we can define the m-th stable homotopy group

    π^s_m(X) := _n≫ 0π_m+n(X^n)

(see <cit.> for details). In fact,π_m^sis a functor→and one can define the classof weak equivalences onas those morphismsf X → Yfor whichπ_m^s(f)is an isomorphism for allm∈.

The classis a part of the Bousfield-Friedlander model structure(,, B, F), <cit.>. A morphismf X → Yina cofibration in this model structure iff^0 X^0 → Y^0and the pushout morphisms


    X^n+1⊔_𝕊^1∧ X^n (𝕊^1∧ Y^n) ⟶ Y^n+1

are cofibrations (i.e., inclusions) in_*for alln≥ 0. It follows by induction onn(e.g., using <cit.>) that allf^n X^n→ Y^nare cofibrations in_*. Fibrant spectraXare those for which eachX^nis fibrant in_*(i.e.,X^nis a Kan complex) and the maps of topological spaces| X^n|→| X^n+1|^|𝕊^1|adjoint to|σ^n||𝕊^1|∧| X^n|→| X^n+1|are weak homotopy equivalences (i.e., they induce bijections for allπ_n(-,x)withx ∈| X^n|andn≥ 0).
This model structure is known to be combinatorial (to see  this, either combine <cit.> with <cit.>, or directly apply <cit.>). Now we can prove the desired result (cp. <cit.>):


Let I be a directed set and X ∈^I. Then the canonical map _I π_m^s(X(i)) →π_m^s(_I X) is an isomorphism for each m∈.

The projective model structure on ^I exists by Proposition <ref>. If X' is a projectively cofibrant replacement of X, the maps X'(i)^n → X'(j)^n of pointed simplicial sets are inclusions (by Lemma <ref>) for each i≤ j in I and each n ≥ 0. Using Lemma <ref>,


    π_m^s(_I X') =
    _n≫ 0π_m+n(_I X'(i)^n) ≅_I_n≫ 0π_m+n(X'(i)^n) =
    _Iπ_m^s(X'(i)).

We conclude by combining this with _I X ≅_I X'.

The category  is a stable model category, so _ is a strong, stable derivator and () = _() is a triangulated category. 
There is a privileged object in () which plays the role of ℤ in ()—the sphere spectrum𝕊 = (𝕊^n, 𝕀_𝕊^n+1), where for each n≥0 we define


    𝕊^n := 𝕊^1∧…∧𝕊^1_n times

(we formally put 𝕊^0 = Δ[0]_+, the monoidal unit for the smash product).

The sphere spectrum 𝕊 is intrinsically homotopically finitely presented in  _.

 Since there is a natural equivalence ()(𝕊,-) ≅π_0^s of functors ()→, the result immediately follows by Proposition <ref>.
tocsectionReferencesalpha5cm0.4pt
Manuel Saorín –
Departamento de Matemáticas,
Universidad de Murcia, Aptdo. 4021,
30100 Espinardo, Murcia,
SPAINJan Šťovíček –

Department of Algebra,
Faculty of Mathematics and Physics,
Charles University in Prague,
Sokolovská 83, 18675 Praha 8,
CZECH REPUBLIC
Simone Virili – or 
Departamento de Matemáticas,
Universidad de Murcia,  Aptdo. 4021,
30100 Espinardo, Murcia,
SPAIN