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Morita theory for stable derivators

July 2018

July 2018

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Autonomous University of Barcelona

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We give a general construction of realization functors for t-structures on the base of a strong stable derivator. In particular, given such a derivator D, a t-structure t=(D^{≤0},D^{≥0}) on the triangulated category D(e), and letting A=D^{≤0}∩D^{≥0} be its heart, we construct, under mild assumptions, a morphism of prederivators real_{t}: D_{A} → D where D_{A} is the natural prederivator enhancing the derived category of A. Furthermore, we give criteria for this morphism to be fully faithful and essentially surjective. If the t-structure t is induced by a suitably "bounded" co/tilting object, real_{t} is an equivalence. Our construction unifies and extends most of the derived co/tilting equivalences appeared in the literature in the last years.

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arXiv:1807.01505v1 [math.KT] 4 Jul 2018

MORITA THEORY FOR STABLE DERIVATORS

SIMONE VIRILI

Abstract. We give a general construction of realization functors for t-

structures on the base of a strong stable derivator. In particular, given

such a derivator D, a t-structure t“ pDď0,Dě0qon the triangulated

category Dp1q, and letting A“Dď0XDě0be its heart, we construct,

under mild assumptions, a morphism of prederivators

realt:DAÑD

where DAis the natural prederivator enhancing the derived category of

A. Furthermore, we give criteria for this morphism to be fully faithful

and essentially surjective. If the t-structure tis induced by a suitably

“bounded” co/tilting object, realtis an equivalence. Our construction

uniﬁes and extends most of the derived co/tilting equivalences appeared

in the literature in the last years.

Contents

Introduction 1

1. Generalities and preliminaries 6

2. Preliminaries about derived categories 15

3. Lifting t-structures algong diagram functors 22

4. The bounded realization functor 27

5. Half-bounded realization functor 39

6. The unbounded realization functor 48

7. Applications to co/tilting equivalences 53

References 57

Introduction

Tilting theory arose in Representation Theory as an extension of the

classical Morita theory (see [BB80, HR82, Bon81]). In this context, the

equivalences of full module categories were substituted by suitable counter-

equivalences of torsion pairs, induced by a ﬁnite dimensional tilting module.

These results were successively extended to categories of modules over ar-

bitrary rings, but still considering tilting modules with strong ﬁniteness

conditions (see [CF90, Miy86]).

Date: July 5, 2018.

2010 Mathematics Subject Classiﬁcation. 18E30, 18E35, 16E30, 16E35, 14F05.

Key words and phrases. Stable derivator, t-structure, tilting, realization functor.

The author was supported by the Ministerio de Econom´ıa y Competitividad of Spain

via a grant ‘Juan de la Cierva-formaci´on’. He was also supported by the research projects

from the Ministerio de Econom´ıa y Competitividad of Spain (MTM2016-77445-P) and the

Fundaci´on ‘S´eneca’ of Murcia (19880/GERM/15), both with a part of FEDER funds.

2 SIMONE VIRILI

Few years after their introduction, it was noticed that (classical) tilting

modules could be used to construct equivalences between derived categories

(see [CPS86, Hap87, Hap88]). It was ﬁnally shown by Rickard [Ric91] and

Keller [Kel94] that compact tilting complexes guarantee the existence of

(bounded and unbounded) derived equivalences and vice-versa, hence es-

tablishing a derived Morita theory for rings.

Motivated by problems in Approximation Theory of modules, large (i.e.,

non-compact) counterparts of co/tilting modules and complexes over rings

were introduced (see [HC01, CGM07, CT95, ˇ

Sˇ

to14, HMV15, Wei13]). How-

ever, contrary to the compact case, these non-compact objects cannot be

immediately substituted to their classical counterpart to construct derived

equivalences, since their endomorphism rings are not derived equivalent to

the original ring. A solution to this diﬃculty was suggested in [ˇ

Sˇ

to14]: in-

stead of considering endomorphism rings, one should consider the hearts of

the naturally associated t-structures.

t-Structures in triangulated categories were introduced by Beilinson, Bern-

stein and Deligne [BBD82] in their study of perverse sheaves on an algebraic

or analytic variety. A t-structure in a triangulated category Dis a pair of

full subcategories satisfying a suitable set of axioms (see the precise def-

inition in the next section) which guarantee that their intersection is an

Abelian category A, called the heart. t-Structures have found applications

in many branches of Mathematics, such that Algebraic Geometry, Algebraic

Topology and Representation Theory of Algebras.

Already in the ﬁrst paper about t-structures, Beilinson, Bernstein and

Deligne, were concerned in ﬁnding ways to compare the original category

Dwith the bounded derived category DbpAqof the heart Aof a given t-

structure t. When Dis the derived category of a ring R, they were able to

construct, via the ﬁltered derived category of R, a realization functor

realb

t:DbpAq Ñ DbpModpRqq

and to give necessary and suﬃcient conditions for this functor to be an

equivalence. This technique was then generalized by Beilinson [Bei87] with

the introduction of f-categories, an abstraction of the concept of ﬁltered

derived category.

Beilinson’s abstract construction of realization functors via f-categories

is at the base of the recent work of Psaroudakis and Vitoria [PV15], where

they use realization functors to establish a derived Morita theory for Abelian

categories. The picture they obtain is extremely clear but their results do

not extend to unbounded derived categories, due to the lack of general tech-

niques to construct realization functors between unbounded derived cate-

gories. The initial motivation for this paper is to extend Beilinson’s tech-

niques to unbounded derived categories and, in this way, complete the results

in [PV15].

General setting. Beilinson’s f-categories can be viewed, informally, as

“Z-ﬁltered” triangulated categories. What we need for our construction of

unbounded realization functors is an “NˆNop ˆZ-ﬁltered” triangulated

category. It is certainly possible to ﬁnd a suitable set of axioms for such

MORITA THEORY FOR STABLE DERIVATORS 3

gadgets but, in this paper, we have made the choice to adopt instead the

setting of derivators. Recall that a strong stable derivator is just a 2-functor

D: Catop ÑCAT

which satisﬁes certain axioms, where Cat is the 2-category of small cate-

gories and CAT is the 2-“category” of all categories. The axioms in par-

ticular imply that the natural range of this 2-functor is the 2-“category”

of triangulated categories. One usually views Das an enhancement of the

triangulated category Dp1q(where 1is the one-point category), the base of

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 [Gro13]). A prototypical example is the assignment

DG:IÞÑ DpGIqfor a Grothendieck category G.

Localization theory for derivators and, in the stable context, the relation

between t-structures and derivators has recently received a lot of attention

by several researchers (see [Lor18, LV17, Sˇ

SV18, Col18, Lak18]). Our ﬁrst

result in this paper is to reformulate one of the main results of [Sˇ

SV18] in

the following form, which is analogous to a result proved by Lurie [Lur06]

in the setting of p8,1q-categories:

Theorem A (Theorem 3.5).Let D: Catop ÑCAT be a strong and stable

derivator. There are bijections among the following classes:

(1) t-structures in Dp1q;

(2) extension-closed reﬂective sub-derivators of D;

(3) extension-closed coreﬂective sub-derivators of D.

Consider now a Grothendieck category Gand the natural derivator DG

enhancing the derived category of G. It is an exercise on the deﬁnitions

to prove that the (bounded) ﬁltered derived category of Gis a suitable

full subcategory of DGpZq “ DpGZq. For a general strong stable derivator

D, the analogous subcategory of DpZqis an f-category. In view of this

analogy, the NˆNop ˆZ-ﬁltered category we need in order to construct the

unbounded realization functors will be naturally deﬁned as a full subcategory

of DpNˆNop ˆZq, so the language of derivators is extremely appropriate

for the kind of constructions we are introducing.

Main results. Let us now brieﬂy list the main results of this paper. The

main abstract statement about unbounded realization functors is the follow-

ing:

Theorem B (Theorems 6.7 and 6.8).Let D: Catop ÑCAT be a strong and

stable derivator, let t“ pDď0,Dě0qbe a t-structure on its base Dp1q, and let

A:“Dď0XDě0be the heart. Suppose that Ahas enough injectives and it is

(Ab.4˚)-kfor some kPN, then there is an exact morphism of prederivators

realt:DAÑD

that restricts to bounded levels. If the following conditions hold:

(1) tis (co)eﬀa¸cable;

(2) tis non-degenerate (right and left);

(3) tis k-cosmashing for some kPN;

4 SIMONE VIRILI

(4) tis (0-)smashing;

then, realt:DAÑDcommutes with coproducts and it is fully faithful.

Of course a dual statement holds for cosmashing k-smashing t-structures

whose heart has enough projectives. The conditions appearing in the above

general theorem are naturally satisﬁed by certain co/tilting t-structures.

Indeed, recall from [NSZ15, PV15] that an object Xin a triangulated

category is said to be tilting provided pTKą0, T Kă0qis a t-structure and

AddpTq Ď TK‰0;cotilting objects are deﬁned dually. A tilting object is

classical if it is compact. Furthermore, two t-structures ti“ pDď0

i,Dě0

(i“1,2) are said to be at ﬁnite distance if there exists nPNsuch that

Dď´n

1ĎDď0

2ĎDďn

Theorem C (Theorem 7.7).Let D: Catop ÑCAT be a strong, stable

derivator, and let Tbe a tilting object for which the associated t-structure

tT“ pDď0

T,Dě0

Tqis at ﬁnite distance from a classical tilting t-structure.

Letting AT:“Dď0

Tp1q X Dě0

Tp1q, there is an equivalence of prederivators

realtT:DATÑD

that restricts to bounded levels. Hence, DATis a strong, stable derivator.

As a corollary to the above theorem one can deduce one of the main results

of [NSZ15], that is, bounded tilting sets in compactly generated algebraic

triangulated categories induce exact equivalences (see Corollary 7.8). Also in

the dual setting of cotilting t-structures, we can obtain the following general

statement:

Theorem D (Theorem 7.9).Let D: Catop ÑCAT be a strong, stable

derivator, and let Cbe a cotilting object for which the associated t-structure

tC“ pDď0

C,Dě0

Cqis at ﬁnite distance from a classical tilting t-structure.

Letting AC:“Dď0

Cp1q X Dě0

Cp1q, there is an equivalence of prederivators

realtC:DACÑD

that restricts to bounded levels. Hence, DACis a strong, stable derivator.

The setting of the above theorem is inspired to the following situation:

D“DModpRqis the derivator enhancing the derived category of a ring Rand

Cis a big cotilting R-module (see Deﬁnition 7.10). Under these assumptions,

tCis at ﬁnite distance from the canonical t-structure on DModpRq, and this

t-structure is induced by the classical tilting object R. In this way, it is

easy to recover as a corollary of the above theorem, one of the main results

of [ˇ

Sˇ

to14] (see Corollary 7.11).

As another consequence of our general theory of realization functors,

one can obtain a “Derived Morita Theory” for Abelian categories, com-

pleting [PV15, Thm. A]:

Theorem E (Theorem 7.12).Let Abe a Grothendieck category (resp.,

an (Ab.4˚)-hGrothendieck category for some hPN), denote by tA“

pDď0pAq,Dě0pAqq the canonical t-structure on DpAq, and let Bbe an Abelian

category. The following are equivalent:

MORITA THEORY FOR STABLE DERIVATORS 5

(1) Bhas a projective generator (resp., an injective cogenerator) and

there is an exact equivalence DpBq Ñ DpAqthat restricts to bounded

derived categories;

(2) Bhas a projective generator (resp., an injective cogenerator) and

there is an exact equivalence of prederivators DBÑDAthat restricts

to an equivalence Db

BÑDb

(3) there is a tilting (resp., cotilting) object Xin DpAq, whose heart

XK‰0(resp., K‰0X) is equivalent to Band such that the associated

t-structure tX“ pDď0

X,Dě0

Xqhas ﬁnite distance from tA.

Acknowledgement. It is a pleasure for me to thank Dolors Herbera, Fosco

Loregian, Moritz Groth, Chrysostomos Psaroudakis, Manuel Saor´ın, Jan

Sˇ

tov´ıˇcek, and Jorge Vitoria for several discussions and suggestions.

6 SIMONE VIRILI

1. Generalities and preliminaries

1.1. Preliminaries and notation. Given a category Cand two objects

x, y PObpCq, we denote by Cpx, yq:“homCpx, yqthe set of morphism from

xto yin C.

Ordinals. Any ordinal λcan be viewed as a category in the following way:

the objects of λare the ordinals αăλand, given α,βăλ, the hom-set

λpα, βqis a point if αďβ, while it is empty otherwise. Following this

convention,

‚1“ t0uis the category with one object and no non-identity mor-

phisms;

‚2“ t0Ñ1uis the category with exactly two diﬀerent objects and

one non-identity morphism between them;

‚in general, n“ t0Ñ1Ñ ¨ ¨ ¨ Ñ pn´1qu, for any nPNą0.

Functor categories, limits and colimits. A category Iis said to be

(skeletally)small when (the isomorphism classes of) its objects form a

set. If Cand Iare an arbitrary and a small category, respectively, a functor

IÑCis said to be a diagram on Cof shape I. The category of diagrams

on Cof shape I, and natural transformations among them, will be denoted

by CI. A diagram Xof shape I, will be also denoted as pXiqiPI, where

Xi:“Xpiqfor each iPObpIq. When any diagram of shape Ihas a limit

(resp. colimit), we say that Chas all I-limits (resp., colimits). In this case,

limI:CIÑC(resp., colimI:CIÑC) will denote the (I-)limit (resp., (I-

)colimit) functor and it is right (resp., left) adjoint to the constant diagram

functor ∆I:CÑCI. The category Cis said to be complete (resp., co-

complete, bicomplete) when I-limits (resp., I-colimits, both) exist in C,

for any small category I. When Iis a directed set, viewed as a small cate-

gory in the usual way, the corresponding colimit functor is the (I-)directed

colimit functor lim

ÝÑI:CIÑC. The I-diagrams on Care usually called

directed systems of shape Iin C.

Triangulated categories. We refer to [Nee01] for the precise deﬁnition

of triangulated category. In particular, given a triangulated category D, we

will always denote by Σ : DÑDthe suspension functor, and we will

denote (distinguished)triangles in Deither by XÑYÑZ`

Ñor by

XÑYÑZÑΣX. A set SĎObpDqis called a set of generators

of Dif an object Xof Dis zero whenever DpΣkS, X q “ 0, for all SPS

and kPZ. In case Dhas coproducts, we shall say that an object Xis a

compact object when the functor DpX, ´q :DÑAb preserves coproducts.

We will say that Dis compactly generated when it has a set of compact

generators. Given a set Xof objects in Dand a subset IĎZ, we let

XKI:“ tYPD:DpX, ΣiYq “ 0, for all XPXand iPIu

KIX:“ tZPD:DpZ, ΣiXq “ 0, for all XPXand iPIu.

If I“ tiufor some iPZ, then we let XKi:“XKIand KiX:“KIX. If i“0,

we even let XK:“XK0and KX:“K0X.

MORITA THEORY FOR STABLE DERIVATORS 7

D´evissage Let Dbe a triangulated category and SĎDa subclass. we

denote by thickpSqthe smallest subcategory containing Swhich is triangu-

lated and closed under direct summands, and by LocpSq(resp., coLocpSq)

the smallest subcategory containing Swhich is triangulated and closed un-

der small coproducts (resp., products). We recall from [Nic08], that D

is said to satisfy the principle of d´evissage (resp., inﬁnite d´evissage,

dual inﬁnite d´evissage) with respect to Sprovided thickpSq “ D(resp.,

LocpSq “ D, coLocpSq “ D).

Cohomological functors and t-structures. Given a triangulated cate-

gory Dand an Abelian category C, an additive functor H0:DÑCis said to

be a cohomological functor if, for any given triangle XÑYÑZÑΣX,

the sequence H0pXq Ñ H0pYq Ñ H0pZqis exact in C. In particular, one

obtains a long exact sequence as follows:

¨ ¨ ¨ Ñ Hn´1pZq Ñ HnpXq Ñ HnpYq Ñ HnpZq Ñ Hn`1pXq Ñ ¨ ¨ ¨

where Hn:“H0˝Σn, for any nPZ. A t-structure in Dis a pair t“

pDď0,Dě0qof full subcategories, closed under taking direct summands in D,

which satisfy the following properties, where Dďn:“Σ´nDď0, and Děn:“

Σ´nDě0:

(t-S.1) DpX, Y q “ 0, for all XPDď0and YPDě1;

(t-S.2) Dď´1ĎDď0(or, equivalently, Dě1ĎDě0);

(t-S.3) for each XPObpDq, there is a triangle

Xď0ÑXÑXě1`

in D, where Xď0PDď0and Xě1PDě1.

We will call Dď0and Dě0the aisle and the co-aisle of the t-structure, re-

spectively. The objects Xď0and Xě1appearing in the triangle of the above

axiom (t-S.3) are uniquely determined by X, up to a unique isomorphism,

and deﬁne functors p´qď0:DÑDď0and p´qě1:DÑDě1which are right

and left adjoints to the respective inclusion functors. We call them the left

and right truncation functors with respect to the given t-structure t.

Furthermore, the above triangle will be referred to as the truncation tri-

angle of Xwith respect to t. The full subcategory H:“Dď0XDě0is

called the heart of the t-structure.

Model categories and model approximations. For the deﬁnition of

model category we refer to [DS95]. In particular, a model category for

us is just a ﬁnitely bicomplete category Mendowed with a model struc-

ture pW,C,Fq. When needed we will explicitly mention that Mis a Dia-

bicomplete model category to mean that Mhas all limits and colimits of

diagrams of shape I, for any IPDia (where Dia is a suitable class of small

categories).

A rich source of model categories for us is the construction of towers and

telescopes of model categories. We now recall these constructions from [CNPS17]:

Deﬁnition 1.1. Let M‚“ tMn:nPNube a sequence of categories

connected with adjunctions

ln`1:Mn`1//

ooMn:rn.

8 SIMONE VIRILI

The category of towers on M‚,TowpM‚qis deﬁned as follows:

(Tow.1) an object is a pair pa‚, α‚q, where a‚“ tanPMn:nPNuis

a sequence of objects one for each Mn, and α‚“ tαn`1:an`1Ñ

rnpanq:nPNuis a sequence of morphisms;

(Tow.2) amorphism f‚:pa‚, α‚q Ñ pb‚, β‚qis a sequence of morphisms f‚“

tfn:anÑbn:nPNusuch that rnpfnq ˝ αn`1“βn`1˝fn`1, for all

nPN.

The dual of a category of towers is said to be a category of telescopes,

and denoted by TelpM1

‚q.

If each Mnin the above deﬁnition is a Dia-bicomplete category, then one

can construct limits and colimits component-wise in TowpM‚q, so, under

these hypotheses, the category of towers is Dia-bicomplete.

Proposition 1.2. [CNPS17, Prop. 4.3] Let M‚“ tpMn,Wn,Cn,Fnq:nP

Nube a sequence of model categories connected with adjunctions

ln`1:Mn`1//

ooMn:rn

and suppose that each rnpreserves ﬁbrations and acyclic ﬁbrations. Then

there exists a model structure pWTow,CTow,FTowqon the category of towers

TowpM‚q, such that WTow “ tf‚:fnPWn,@nPNuand FTow “ tf‚:fnP

Fn,@nPNu.

The concept of model approximation was introduced by Chach´olski and

Scherer in order to circumvent the diﬃculties in constructing homotopy

limits (see [CS02]):

Deﬁnition 1.3. Let pM1,W1qbe a category with weak equivalences, that

is, W1is a class o morphisms in M1, containing all the isos and with the

3-for-2property. A right model approximation for pM1,W1qis a model

category pM,W,C,Fqand a pair of functors

l:M1//

ooM:r

satisfying the following conditions:

(MA.1) lis left adjoint to r;

(MA.2) if φPW1, then lpφq P W;

(MA.3) if ψis a weak equivalence between ﬁbrant objects, then rpψq P W1;

(MA.4) if lpXq Ñ Yis a weak equivalence in Mwith Xﬁbrant, the adjoint

morphism XÑrpYqis in W1.

One deﬁnes dually left model approximations for pM1,W1q.

In fact, the notion of model approximation is as good as the notion of

model category in order to construct homotopy categories. Indeed, let

l:pM1,W1qÕpM,W,C,Fq:rbe a right model approximation, then

the category M1rW1´1sis locally small as it is equivalent to the category

HopM1q, which is constructed as follows: objects of HopM1qare those of M1

and, given M1, M2PM1,

HopM1qpM1, M2q:“HopMqplpM1q, lpM2qq,

see [CS02, Prop. 5.5]. The category HopM1qis said to be the homotopy

category of the above model approximation. It is easily seen that it is

equivalent to a full subcategory of HopMq.

MORITA THEORY FOR STABLE DERIVATORS 9

1.2. Generalities on (pre)derivators. We denote by Cat the 2-category

of small categories and by Catop the 2-category obtained by reversing the

direction of the functors in Cat (but letting the direction of natural trans-

formations unchanged). Similarly, we denote by CAT the 2-“category” of all

categories. This, when taken literally, may cause some set-theoretical prob-

lems that, for our constructions, can be safely ignored: see the discussion

after [Gro13, Def. 1.1].

Deﬁnition 1.4. Acategory of diagrams is a full 2-subcategory Dia of

Cat, such that

(Dia.1) all ﬁnite posets, considered as categories, belong to Dia;

(Dia.2) given IPDia and iPI, the slices Ii{and I{ibelong to Dia;

(Dia.3) if IPDia, then Iop PDia;

(Dia.4) for every Grothendieck ﬁbration u:IÑJ, if all ﬁbers Ij, for jPJ,

and the base Jbelong to Dia, then so does I.

Example 1.5. A category Iis said to be a ﬁnite directed category if

it has a ﬁnite number of objects and morphisms, and if there is no directed

cycle in the quiver whose vertices are the objects of Iand the arrows are

the non-identity morphisms in I. Equivalently, the nerve of Ihas a ﬁnite

number of non-degenerate simplices. We denote by fd.Cat be the 2-category

of ﬁnite directed categories (categories whose nerve has a ﬁnite number of

non-degenerate simplices). This 2-category is a category of diagrams.

Given a category of diagrams Dia, a prederivator of type Dia is a strict

2-functor

D: Diaop ÑCAT.

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

‚the letter Dwill always denote a (pre)derivator;

‚for a prederivator D: Diaop ÑCAT and a small category IPDia,

we denote by

DI: Diaop ÑCAT

the shifted prederivator such that DIpJq:“DpIˆJq;

‚for any natural transformation α:uÑv:JÑIin Dia, we will

always use the notation α˚:“Dpαq:u˚Ñv˚:DpIq Ñ DpJq. Fur-

thermore, 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 K,U,V,W,X,Y,Z, will be used either for objects in

the base Dp1qor for (incoherent) diagrams on Dp1q, that is, functors

IÑDp1q, for some small category I;

‚the letters K,U,V,W,X,Y,Z, will be used for objects in some

image DpIqof the derivator, for Ia category (possibly) diﬀerent from

1. Such objects will be usually referred to as coherent diagrams

of shape I;

‚given IPDia, consider the unique functor ptI:IÑ1. We usually

denote by hocolimI:DpIq Ñ Dp1qand holimI:DpIq Ñ Dp1qre-

spectively the left and right homotopy Kan extensions of ptI; these

functors are called respectively homotopy colimit and homotopy

limit.

10 SIMONE VIRILI

For a given object iPI, we also denote by ithe inclusion i:1ÑIsuch

that 0 ÞÑ i. We obtain an evaluation functor i˚:DpIq Ñ Dp1q. For an object

XPDpIq, we let Xi:“i˚X. Similarly, for a morphism α:iÑjin I, one

can interpret αas a natural transformation from i:1ÑIto j:1ÑI. In

this way, to any morphism αin I, we can associate α˚:i˚Ñj˚. For an

object XPDpIq, we let Xα:“α˚

X:XiÑXj. For Iin Dia, we denote by

diaI:DpIq Ñ Dp1qI

the diagram functor, such that, given XPDpIq, diaIpXq:IÑDp1qis

deﬁned by diaIpXqpiα

Ñjq “ pXi

Xα

ÑXjq. We will refer to diaIpXqas the

underlying (incoherent) diagram of the coherent diagram X.

Example 1.6. Let D: Diaop ÑCAT be a prederivator. Given IPDia,

consider the unique functor ptI:IÑ1, let XPDp1qand consider X:“

pt˚

IXPDpIq. Then the underlying diagram diaIpXq P Dp1qIis constant,

that is, Xi“Xfor all iPI, and the map Xα:XiÑXjis the identity of

Xfor all pα:iÑjq Ď I.

Deﬁnition 1.7. Aderivator of type Dia, for a given category of diagrams

Dia, is a prederivator D: Diaop ÑCAT satisfying the following axioms:

(Der.1) if šiPIJiis a disjoint union in Dia, then the canonical functor

DpšIJiq Ñ śIDpJiqis an equivalence of categories;

(Der.2) for any IPDia and a morphism f:XÑYin DpIq,fis an

isomorphism if and only if i˚pfq:i˚pXq Ñ i˚pYqis an isomorphism

in Dp1qfor each iPI;

(Der.3) the following conditions hold true:

(L.Der.3) for each functor u:IÑJin Dia, the functor u˚has a left

adjoint u!(i.e. left homotopy Kan extensions are required to

exist);

(R.Der.3) for each functor u:IÑJin Dia, the functor u˚has a right

adjoint u˚(i.e. right homotopy Kan extensions are required to

exist);

(Der.4) the following conditions hold true:

(L.Der.4) the left homotopy Kan extensions can be computed pointwise in

that for each u:IÑJin Dia and jPJ, there is a canonical

isomorphism hocolimu{jp˚pXq – pu!Xqj, where p:u{jÑIis

the canonical functor from the slice category;

(R.Der.4) the right homotopy Kan extensions can be computed pointwise

in that for each u:IÑJin Dia and jPJ, there is a canon-

ical isomorphism pu˚Xqj–holimj{uq˚pXqin Dp1q, where

q:j{uÑIis the canonical functor from the slice category.

If Dia is not explicitly mentioned we just assume that Dia “Cat.

We refer to [Gro13] for a detailed discussion, as well as for the precise

deﬁnitions of pointed derivators (Dp1qhas a zero object), and strong deri-

vators (the partial diagram functors Dp2ˆIq Ñ DpIq2are full and essentially

surjective for each IPCat).

For a ﬁxed category of diagrams Dia, the prederivators of type Dia form

a 2-category, that we denote by PDerDia, where 0-cells are the 2-functors

Diaop ÑCAT (i.e., the prederivators), 1-cells are 2-natural transformations

MORITA THEORY FOR STABLE DERIVATORS 11

among these functors, and 2-cells are modiﬁcations. We will usually refer

to the 1-cells of PDerDia as morphisms of prederivators or, abusing

terminology, functors between prederivators, while we will usually refer to

the 2-cells of PDerDia as natural transformations.

Example 1.8. Given a prederivator D: Diaop ÑCAT and a morphism

u:JÑIin Dia, for any KPDia, one can consider u˚:DIpKq Ñ DJpKq.

These functors can be assembled together to form a morphism of prederiva-

tors that, abusing notation, we denote again by u˚:DIÑDJ.

If Dis a derivator, then one can construct similarly two morphisms of deri-

vators u!, u˚:DJÑDI. Of course there are adjunctions pu!, u˚qand pu˚, u˚q

in the 2-category PDerDia.

1.3. Derivators induced by model categories. In this subsection we

introduce our main source for explicit examples of derivators. Indeed, for a

given Dia-bicomplete model category Mwhose class of weak equivalences

is denoted by W, the following results are proved in [Cis03]:

‚for any IPDia, let WIbe the class of morphisms in CIwhich

belong pointwise to W. Then, [Cis03, Thm. 1] tells us that the cate-

gory of fractions MIrW´1

Iscan always be constructed (in the same

universe);

‚the assignment IÞÑ MIrW´1

Isunderlies a derivator

DpM,Wq: Diaop ÑCAT;

‚the derivator DpM,Wqis strong and it is pointed if Mis pointed.

In fact, there is another approach to the proof of the above facts that uses

model approximations. This is based on the following theorem that collects

several facts proved in [CS02]:

Theorem 1.9. Let pM1,W1qÕpM,W,C,Fqbe a left model approxima-

tion. For any small category I, let pW1qIbe the class of maps in pM1qIthat

belong pointwise to W1, then there exists a model category pMI,WI,CI,FIq

and a left model approximation ppM1qI,pW1qIqÕpMI,WI,CI,FIq. Hence,

the category HoppM1qIqhas small hom-sets and we can deﬁne a prederivator

DpM1,W1q: Catop ÑCAT

IÞÑ HoppM1qIq.

Furthermore, the above approximations are “good for left Kan extensions”,

so they can be used to show that the prederivator DpM1,W1qsatisﬁes (Der.1),

(Der.2), (L.Der.3) and (L.Der.4).

As a consequence, if pM1,W1qadmits both a left and a right model ap-

proximation (e.g., if M1admits a model structure on it for which the class

of weak equivalences is exactly W1), then DpM1,W1qis a derivator.

1.4. Stable derivators. Let :“2ˆ2, with the following labels for

vertices and arrows:

p0,0qN//



p0,1q



p1,0qS

//p1,1q.

12 SIMONE VIRILI

Then we let ι‚:1Ñ(with ‚ P tp0,0q,p0,1q,p1,0q,p1,1qu) and ι‚:2Ñ

(with ‚ P tN, S, W, Eu) be the inclusion of the corresponding vertex and ar-

row, respectively. Furthermore, we consider the inclusion of the two obvious

subcategories ι

ÝÑ ι

ÐÝ ,

p0,0q//



p0,1q p0,0q//



p0,1q



p0,1q



ι//ι

p1,0q p1,0q//p1,1q p1,0q//p1,1q.

Let also ι‚:1Ñ(with ‚ P tp0,0q,p0,1q,p1,0qu), ι‚:2Ñ(with ‚ P

tN, W u), ι‚:1Ñ(with ‚ P tp1,0q,p0,1q,p1,1qu) and ι‚:2Ñ(with ‚ P

tS, Eu) be the inclusion of the corresponding vertex and arrow, respectively.

Let Dia be some category of diagrams and let D: Diaop ÑCAT be a

pointed derivator. The suspension functor Σ : DÑDand the loop func-

tor Ω: DÑDare deﬁned as follows:

Σ :“ι˚

p1,1q˝ pιq!˝ pιp0,0qq˚Ω :“ι˚

p0,0q˝ pιq˚˝ pιp1,1qq!

and they form an adjoint pair pΣ,Ωq:DÕD, which is an equivalence if and

only if Dis a stable derivator. Furthermore, let

C :“ι˚

E˝ pιq!˝ pιNq˚F :“ι˚

W˝ pιq˚˝ pιSq!

this gives an adjoint pair pS,Fq:D2ÕD2which, again, is an equiva-

lence if and only if Dis stable. We deﬁne, respectively, the cone functor

cone: D2ÑDand the ﬁber functor ﬁb : D2ÑDas

cone :“ι˚

p1,1q˝ pιq!˝ pιNq˚ﬁb :“ι˚

p0,0q˝ pιq˚˝ pιSq!.

The key fact about stability, which can be found in [Gro13, Thm. 4.16 and

Cor. 4.19], is that given a strong and stable derivator D, the above structure

can be used to endow each DpIqwith a canonical triangulated structure,

with respect to which, all the u˚and all the Kan extensions u!and u˚are

naturally triangulated functors.

Example 1.10. Let pC,W,B,Fqbe a model category and consider the as-

sociated derivator DpC,Wq: Catop ÑCAT. If pC,W,B,Fqis stable in the

sense of model categories, then DpC,Wqis strong and stable.

1.5. Localization of derivators. In this subsection we recall the deﬁni-

tion and some basic facts about co/reﬂections in PDerDia. This concept is

studied, for example, in [Hel88, Cis08, Lor18, Col18]:

Deﬁnition 1.11. Let L:DÕE:Rbe an adjunction in PDerDia.

‚The adjunction pL, Rqis a reﬂection if Ris fully faithful, i.e. the

counit ε:LR Ñid is invertible.

‚The adjunction pL, Rqis a coreﬂection if Lis fully faithful, i.e.

the unit η: id ÑRL is invertible.

Lemma 1.12. [Lor18, Prop. 3.12] Let L:DÕE:Rbe a reﬂection in

PDerDia. If Dis a (pointed, strong) derivator of type Dia for some category

MORITA THEORY FOR STABLE DERIVATORS 13

of diagrams Dia, so is E. Furthermore, given u:IÑJin Dia, the homo-

topy Kan extensions u!, u˚:EpIq Ñ EpJqcan be computed, respectively, as

follows:

LJu!RI:EpIq Ñ EpJqand LJu˚RI:EpIq Ñ EpJq.

where the u!and u˚appearing in the above equation are the homotopy Kan

extensions in D.

In general a co/reﬂection of a stable derivator need not be stable. How-

ever the following result allows to easily check when a reﬂection of a stable

derivator is again stable:

Lemma 1.13. Let L:DÕE:Rbe a reﬂection in PDerDia and suppose

that Dis strong and stable. Denote by p¯

Σ,¯

Ωqthe suspension-loop adjunction

in E. Then ¯

Σ–LΣRand R¯

Ω–ΩR.

Proof. By [Gro13, Prop. 2.11], left/right exact (in particular right/left ad-

joint) morphisms of pointed derivators commute with loops/suspensions. In

particular, for any IPDia, LIΣRI–¯

ΣLIRI–¯

Σ, while ΩRI–RI¯

Ω. 

If we consider Eas a sub-derivator of D, then the above corollary means

that ¯

Ω is just a restriction of Ω to E, while applying ¯

Σ is the same as ﬁrst

applying Σ and then reﬂecting onto E. In particular, Eis closed under loops

in Dand, being Ω an equivalence, ¯

Ω is always fully faithful, so it is an

equivalence if and only if it is essentially surjective.

1.6. Pseudo-colimits of prederivators. Let Cbe a 2-category (e.g.,

C“CAT or C“PDerDia) and consider a functor F:NÑC, which is

determined by a sequence of (1-)morphisms in C:

Fp0q:“C0

F0//Fp1q:“C1

F1//¨¨¨ //Fpnq:“Cn

Fn//¨¨¨

and let Fn,m :“Fm˝...˝Fn´1for măn. The pseudo-colimit

C:“pcolim F

of this diagram is deﬁned by a universal property expressed by the following

natural isomorphism of categories:

CNpF, ∆Aq – Cppcolim F, Aq

for any APC, where ∆A:NÑCis the functor that takes constantly

the value A. In the following lemma we give an explicit construction of a

pseudo-colimit as above, in the particular case when C“CAT:

Lemma 1.14. Consider a functor F:NÑCAT. The pseudo-colimit C:“

pcolim Fexists and it can be constructed as follows: the class of objects

of Cis the disjoint union of the objects of the Cn, that we can write as

ObpCq “ tpX, nq:XPCnu, while the morphism sets are:

(1.1) CppX, nq,pY, mqq “ colimkěn_mCkpFk,nX, Fk,mYq.

where this direct limit is taken in Set.

14 SIMONE VIRILI

If in the above lemma all the Fnare fully faithful, then the direct limit

(1.1) is eventually constant, that is CppX, nq,pY , mqq – CkpFk,nX, Fk,mYq,

where kěn_mis any positive integer.

The above construction of sequential pseudo-colimits in CAT can be used

to give an explicit construction of sequential pseudo-colimits in PDerDia :

Lemma 1.15. Consider a functor F:NÑPDerDia. The pseudo-colimit

D:“pcolim Fin PDerDia exists and, furthermore, DpIqis the pseudo-colimit

pcolim FIin CAT, where FI:NÑCAT is the functor FIpnq:“ pFpnqqpIq.

Proof. Let us deﬁne a prederivator D: Diaop ÑCAT as follows:

‚for IPDia, let DpIq:“pcolim FI;

‚for a functor u:IÑJ, the functor u˚:DpJq Ñ DpIqis the unique

object in the category on the left-hand side

(1.2) CATpDpJq,DpIqq – CATNpFJ,∆DpIqq

corresponding to an object FJÑ∆DpIqin the category on the

right-hand side, whose n-th component is the composition FJpnq Ñ

FIpnq Ñ DpIq(where FJpnq Ñ FIpnqis u˚for the derivator Fpnq,

while εI

n:FIpnq Ñ DpIqis the canonical map to the pseudo-colimit);

‚for a natural transformation α:uÑv:IÑJ, the natural trans-

formation α˚:u˚Ñv˚:DpJq Ñ DpIqis the unique morphism α˚

in the category on the left-hand side in (1.2), corresponding to a

morphism in CATNpFJ,∆DpIqqwhose n-th component is

FJpnq

u˚

v˚

óα˚FIpnqεI

n//DpIq.

With this explicit deﬁnition, it is not diﬃcult to deduce from the universal

property of pseudo-colimits in CAT that Dis the pseudo-colimit of Fin

PDerDia, that is, that there is an isomorphism of categories

PDerDiapD,Eq – pPDerDia qNpF, ∆Eq,

for any prederivator E.

MORITA THEORY FOR STABLE DERIVATORS 15

2. Preliminaries about derived categories

In this section we start with an Abelian category Aand we study the

standard prederivator whose base is the category of bounded (resp., left-

bounded, right-bounded, unbounded) cochain complexes ChbpAqon A(resp.,

Ch`pAq, Ch´pAq, ChpAq). In particular, we show how to obtain bounded

complexes as a (double) pseudo-colimit of uniformly bounded complexes.

Furthermore, we study the localizations of these prederivators with respect

to weak equivalences, obtaining the natural prederivators enhancing the

bounded (resp., left-bounded, right-bounded, unbounded) derived category

of A. Using model approximations, we give standard assumptions on Afor

the unbounded derived category to have small hom-sets, and to embed it

as a full subcategories of the homotopy category of towers/telescopes over

A. In this way, the concepts of left-complete, right-complete, and two-sided

complete derived category naturally arise.

Setting for Section 2. We ﬁx through this section a category of diagrams

Dia and a Dia-bicomplete Abelian category A. When working with cate-

gories of half-bounded or unbounded complexes we will need Dia to contain

the discrete category on a countable set, that is, we need Ato be countably

bicomplete.

2.1. Categories of bounded complexes. Let us start introducing no-

tations for the following prederivators:

‚ChA: Diaop ÑCAT is the discrete Abelian derivator represented by

the category of unbounded cochain complexes ChpAq;

‚Chb

A: Diaop ÑCAT is the full sub-prederivator of ChAsuch that

Chb

ApIq:“ChbpAIq, the bounded complexes on AI;

‚Chpb,`q

A: Diaop ÑCAT is the full sub-prederivator of Chb

Asuch that

Chpb,`q

ApIq:“Chpb,`qpAIq, the category of those bounded complexes

XPChbpAIqsuch that Xh“0 for all hă0;

‚given nPN, Chn

A: Diaop ÑCAT is the full sub-prederivator of

Chpb,`q

Asuch that Chn

ApIq:“ChnpAIq, the category of those bounded

complexes XPChbpAIqsuch that Xh“0 for all hă0 and all hěn.

Remark 2.1. Notice that Chn

Ais again a discrete Abelian derivator, but

the same is not true for Chb

Aand Chpb,`q

A. In fact, while it is true that

the associated diagram functors are fully faithful, given IPDia, in general

the inclusions ChbpAIq Ď ChbpAqIand Chpb,`qpAIq Ď Chpb,`qpAqImay be

strict.

Consider the following sequence of prederivators, where the maps are the

obvious inclusions:

Ch1

AÑCh2

AÑ ¨ ¨ ¨ Ñ Chn

AÑ ¨ ¨ ¨ .

and let pcolimnChn

Abe the pseudo-colimit in PDerDia, as described in Sub-

section 1.6. Since these inclusions are componentwise fully faithful one can

give a straightforward proof of the following lemma:

16 SIMONE VIRILI

Lemma 2.2. In the above notation, there is an equivalence of prederivators

F: pcolimnChn

AÑChpb,`q

such that, for all nPNą0,IPDia and XPChn

ApIq,pX, nq ÞÑ X.

Let us conclude this subsection showing that the prederivator Chb

Acan

be seen as a pseudo-colimit of copies of Chpb,`q

A. Indeed, deﬁne the following

endofunctors:

Ω: ChAÑChApresp., Σ : ChAÑChAq

such that, given XPChApIq “ ChpAIq, we have pΩXqh:“Xh´1(resp.,

pΣXqh:“Xh`1), with the usual sign conventions for diﬀerentials. Notice

that the above use of the symbols Σ and Ω does not coincide with the general

deﬁnition for pointed derivators given in Subsection 1.4.

Lemma 2.3. In the above notation, consider a diagram F:NÑPDerDia

such that Fpiq “ Chpb,`q

Afor all iPNand Fpiq Ñ Fpi`1qis a suitable

restriction of Ω: ChAÑChA. Then there is an equivalence of prederivators

pcolim F“pcolimnChpb,`q

AÑChb

2.2. Naive and smart truncations of complexes. Let us introduce

the so-called smart truncations of a complex. Indeed, given an integer

kPNand XPChpAq, we let

τďkX:p ¨ ¨ ¨ //Xk´2//Xk´1//kerpdkq//0//¨ ¨ ¨ q;

τěkX:p ¨ ¨ ¨ //0//CoKerpdk´1q//Xk`1//Xk`2//¨ ¨ ¨ q.

Similarly, the naive truncations of Xare deﬁned as follows

Xăk:p ¨ ¨ ¨ //Xk´2//Xk´1//0//¨ ¨ ¨ q;

Xąk:p ¨ ¨ ¨ //0//Xk`1//Xk`2//¨ ¨ ¨ q.

Notice that there is a short exact sequence:

0ÑXąk´1ÑXÑXăkÑ0.

Furthermore, Xis an exact complex if and only if both τďkXand τěk`1X

are exact.

Lemma 2.4. Let Abe an Abelian category and let DbpAq,D`pAq,D´pAq

and DpAqbe its bounded, left-bounded, right-bounded and unbounded derived

category, respectively. Suppose furthermore that these derived categories all

have small hom-sets. Then,

(1) DbpAqsatisﬁes the principle of d´evissage with respect to A, embedded

in ChbpAqas the class of complexes concentrated in degree 0;

(2) if Ahas exact coproducts (resp., products), then D`pAq,D´pAq,

and DpAqsatisfy the principle of inﬁnite (dual) d´evissage with re-

spect to A.

MORITA THEORY FOR STABLE DERIVATORS 17

Proof. (1) is clear, since any bounded complex can be constructed induc-

tively taking extensions of shifted copies of objects in A.

(2) We just deal with the principle of inﬁnite d´evissage since the inﬁnite

dual d´evissage can be proved by dual arguments. Let us start with XP

Ch`pAq, and consider the direct system pτďkXqkPNin ChbpAq, whose direct

limit (in Ch`pAq) is isomorphic to X. There is a short exact sequence of

the form:

0Ñà

kPN

τďkXÑà

kPN

τďkXÑcolimkPNτďkXp– Xq Ñ 0.

Since coproducts are exact in A, the above coproducts are also coprod-

ucts in D`pAqand the above short exact sequence induces a triangle in

D`pAq. Hence, Xbelongs to LocpDbpAqq and, by part (1), LocpDbpAqq “

LocpthickpAqq “ LocpAq.

Let now XPCh´pAq, and consider the direct system pXą´kqkPNin ChbpAq,

whose direct limit (in Ch´pAq) is isomorphic to X. There is a short exact

sequence of the form:

0Ñà

kPN

Xą´kÑà

kPN

Xą´kÑcolimkPNXą´kp– Xq Ñ 0.

Since coproducts are exact in A, the above coproducts are also coprod-

ucts in D´pAqand the above short exact sequence induces a triangle in

D´pAq. Hence, Xbelongs to LocpChbpAqq and, by part (1), LocpChbpAqq “

LocpthickpAqq “ LocpAq.

Finally, suppose XPChpAq. Then Xis an extension of a complex in

Ch´pAqand one in Ch`pAq, hence XPLocpD`pAq Y D´pAqq “ LocpAq,

by what we have just proved for half-bounded complexes. 

2.3. Towers and telescopes of complexes. Let us start introducing the

following sequence of model categories on categories of uniformly left/right

bounded complexes:

Lemma 2.5. [CNPS17, Thm. 5.2] Suppose Ahas enough injectives and let

nPN. Then there is a model category pChě´npAq,Wě´n,Cě´n,Fě´nq,

where:

‚Wě´n“ tφ‚:φ‚is a quasi-isomorphismu;

‚Cě´n“ tφ‚:φkis an epi with injective kernel, for all kě ´nu;

‚Fě´n“ tφ‚:φkis a mono for all kě ´nu.

Furthermore, there is an adjunction

(2.1) ln: Chě´n´1pAq//

ooChě´npAq:rn,

where lnis the obvious inclusion while rnis the (smart) truncation functor.

In this situation, rnpreserves ﬁbrations and acyclic ﬁbrations.

Dually, if Ahas enough projectives, then for any nPNthere is a model

category pChďnpAq,Wďn,Cďn,Fďnq, where:

‚Wďn“ tφ‚:φ‚is a quasi-isomorphismu;

‚Cďn“ tφ‚:φkis an epi for all kďnu;

‚Fďn“ tφ‚:φkis a mono with projective coker, for all kďnu.

18 SIMONE VIRILI

Furthermore, there is an adjunction

(2.2) l1

n: ChďnpAq//

ooChďn`1pAq:r1

where l1

nis the (smart) truncation functor and r1

nis the obvious inclusion.

In this situation, l1

npreserves coﬁbrations and acyclic coﬁbrations.

Lemma 2.5 shows that the adjunctions ln: Chě´n´1pAqÕChě´npAq:rn

give a tower of model categories (see Deﬁnition 1.1), while the adjunctions

n: ChďnpAqÕChďn`1pAq:r1

ngive a telescope. Hence, we can construct a

category of towers TowpAqand a category of telescopes TelpAq. A typical

object X‚

‚of TowpAqcan be thought as a commutative diagram of the form



0//X´2



d´2

2//X´1

t´1



d´1

2//X0



2//X1



2//X2



2//...

0//X´1



d´1

1//X0



1//X1



1//X2



1//...

0//X0

0//X1

0//X2

0//...

where pX‚

n, d‚

nqis a cochain complex for all nPN, and Xm

n“0 for all

mă ´n. An analogous description holds for TelpAq.

Deﬁnition 2.6. Let Abe an Abelian complete category with enough injec-

tives, and let M‚“ tpChě´npAq,Wě´n,Cě´n,Fě´nq:nPNube the tower

of model categories deﬁned in Lemma 2.5. We denote by

pTowpAq,WTow,CTow,FTowq

the induced model category constructed as in Proposition 1.2. Furthermore,

we denote by

tow : ChpAq Ñ TowpAq

the so-called tower functor, which sends a complex Xto the sequence of

its successive truncations ¨ ¨ ¨ Ñ τě´nXÑτě´n`1XÑ ¨ ¨ ¨ Ñ τě0Xand

acts on morphisms in the obvious way (see [CNPS17]).

Dually, let Abe an Abelian cocomplete category with enough projectives,

and let M1

‚“ tpChďnpAq,Wďn,Cďn,Fďnq:nPNube the telescope of model

categories deﬁned in Lemma 2.5. We denote by

pTelpAq,WTow,CTow,FTowq

the induced model category constructed as in Proposition 1.2. Furthermore,

we denote by

tel : ChpAq Ñ TowpAq

the so-called telescope functor, which sends a complex Xto the sequence

of its successive truncations τď0XÑτď1XÑ ¨ ¨ ¨ Ñ τďnXÑ... and acts

on morphisms in the obvious way.

MORITA THEORY FOR STABLE DERIVATORS 19

In the same setting of the above deﬁnition, notice that the category

TowpAqcan be seen as a full subcategory of the category ChpAqNop of func-

tors Nop ÑChpAqand so we can restrict the usual limit functor to obtain

a functor limNop : TowpAq Ñ ChpAqwhich is a right adjoint for the tower

functor. If we endow ChpAqwith the class of weak equivalences given by the

quasi-isomorphisms of complexes, we would like to verify that plimNop,towq

is a right model approximation. A similar question holds for the dual adjunc-

tion ptel,colimNq. In fact, this is not true in general. A suﬃcient condition

for this adjunction to be an approximation is provided by the following

deﬁnition:

Deﬁnition 2.7. [Roo06] If Ahas enough injectives, we say that it is (Ab.4˚)-

kif, for any countable family of objects tBnunPNand any choice of injective

resolutions BnÑEnwith EnPChě0pAq, we have that HhpśnPNEnq “ 0

for all hąk.

Dually, if Ahas enough projectives, it is (Ab.4)-kif, for any countable

family of objects tBnunPNand any choice of projective resolutions BnÑEn

with EnPChď0pAq, we have that HhpšnPNEnq “ 0for all hă ´k.

Proposition 2.8. [CNPS17, Thm. 6.4] Given kPN, suppose that Ahas

enough injectives and it is (Ab.4˚)-k. Then the following adjunction is a

right model approximation,

(2.3) tow : pChpAq,Wq//

oopTowpAq,WTow,CTow,FTowq: lim

Nop .

Dually, if Ahas enough projectives and it is (Ab.4)-k, then the following

adjunction is a left model approximation,

(2.4) colimN:pTelpAq,WTel,CTel,FTelq//

oopChpAq,Wq: tel.

Hence, if Asatisﬁes one of the two following sets of hypotheses

‚enough injectives and (Ab.4˚)-kfor some kPN,

‚enough projectives and (Ab.4)-kfor some kPN,

a consequence of the above Proposition 2.8 and Theorem 1.9 is that the

derived category DpAIqhas small hom-sets for any small category IPDia.

In particular, we can safely deﬁne a prederivator

DA: Diaop ÑCAT, I ÞÑ DpAIq,

enhancing the derived category of A. Notice that all the images of DAare

canonically triangulated but, without more hypotheses on A,DAis just a

pre-derivator. Anyway, one can verify that the homotopy Kan extensions

involved in the deﬁnition of cones, ﬁbers, suspensions and loops, always exist

in DAand that the deﬁnitions given classically coincide with those in Sub-

section 1.4. Let us remark that one can safely deﬁne the above prederivator

under even more general hypotheses, as explained in [San07].

2.4. Two-sided completion of the derived category. Let us deﬁne

the following categories with weak equivalences:

(1) the full subcategory {

ChpAq Ď ChpAqNop spanned by those diagrams

X:Nop ÑChpAqpIqsuch that, for all nPN,

‚Xpnq P Chě´npAq;

20 SIMONE VIRILI

‚the map Xpnq Ð Xpn`1qinduces a q.i. Xpnq Ð τě´nXpn`1q.

Consider also the class of weak equivalences

W:“ tφ:XñX1:φn:Xpnq Ñ X1pnqis a q.i.,@nPNu;

(2) the full subcategory 

ChpAq Ď ChpAqNspanned by those diagrams

Y:NÑChpAqpIqsuch that, for all nPN,

‚Ypnq P ChďnpAq;

‚the map Ypnq Ñ Ypn`1qinduces a q.i. Ypnq Ñ τďnYpn`1q.

Consider also the class of weak equivalences

W:“ tφ:YñY1:φn:Ypnq Ñ Y1pnqis a q.i.,@nPNu;

(3) the full subcategory ChpAq˛ĎChpAqNˆNop spanned by those X:Nˆ

Nop ÑChpAqsuch that, for all nand mPN,

‚Xpm, ´q P {

ChpAq;

‚Xp´, nq P 

ChpAq.

Consider also the class of weak equivalences

W˛:“ tφ:XñX1:φpm,nq:Xpm, nq Ñ X1pm, nqis a q.i.,@n, m PNu.

There are obvious pairs of adjoint morphisms of prederivators:

colimN: Ch˛

ooz

ChA:x

tel and }

tow : ~

ChA//

ooCh˛

A: lim

Nop

where, given YPz

ChApIq,x

telpYqpm, nq:“τďmYpnqwhile, given XP

ChApIq,}

towpXqpm, nq:“τě´nXpmq.

Remember that, given two categories with weak equivalences pC1,W1qand

pC2,W2q, a functor F:C1ÑC2is said to be homotopically invariant if

Fpwq P W2for all wPW1. In this case, the functor Finduces a functor

C1rW´1

1s Ñ C2rW´1

2sthat acts like Fon objects.

Proposition 2.9. Let kPNand suppose that Ahas enough injectives, it is

(Ab.4˚)-kand it has exact coproducts. Then, the localizations {

ChpAqrx

W´1s

and ChpAq˛rW´1

˛sexist (i.e., they are locally small). Furthermore, the func-

tors colimNand x

tel are homotopically invariant and they induce an equiva-

lence DpAq – {

ChpAqrx

W´1s – ChpAq˛rW´1

˛s.

Dually, if Ahas enough projectives, it is (Ab.4)-kand it has exact prod-

ucts, then, the localizations 

ChpAqr|

W´1sand ChpAq˛rW´1

˛sexist (i.e., they

are locally small). Furthermore, the functors limNop and }

tow are homo-

topically invariant and they induce an equivalence DpAq – 

ChpAqr|

W´1s –

ChpAq˛rW´1

˛s.

Proof. We prove just the ﬁrst half of the statement, as the rest of the proof

is completely dual. Consider the right model approximation

(2.5) tow : pChpAq,Wq//

oopTowpAq,WTow,CTow,FTowq: lim

Nop

constructed in Proposition 2.8. Then, DApIqis equivalent to its image via

(the derived functor of) tow in the homotopy category HoWTow pTowpAIqq;

MORITA THEORY FOR STABLE DERIVATORS 21

this essential image is clearly equivalent to {

ChpAIqrx

W´1s, so we get an

equivalence:

tow : DpAIq//

oo{

ChpAIqr x

W´1s: holimNop .

Using the same argument of [Sta18, Tag 05RW] we can embed Ch˛pAIqrW´1

˛s

into HopTowpAqNq, so this category is locally small. Furthermore, we can

consider the following equivalence of categories

tel : {

ChpAq//

ooChpAq˛: colimN

Now notice that x

tel is exact, so it is homotopically invariant and, since A

is (Ab.4), one can use an argument dual to [BN93, Remark 2.3] to show

that the above restriction of the functor colimNis homotopically invariant.

Hence, the above equivalence of categories induces an equivalence of the

localizations. 

22 SIMONE VIRILI

3. Lifting t-structures algong diagram functors

In this section we begin our study of t-structures on a strong and stable

derivator D. Using one of the main results of [Sˇ

SV18], we show that a t-

structure on the base Dp1qof our derivator naturally induces t-structures

on each of the categories DpIq. Furthermore, t-structures in Dp1qare in

bijection with suitable families of co/localizations of D. We also give an

extremely short and self-contained argument for the abelianity of the heart

of a t-structure. In the last subsection we show how a t-structure on Dp1q

induces two special t-structures on DpNqand DpNopqwhich are diﬀerent from

the ones obtained in [Sˇ

SV18].

Setting for Section 3. We ﬁx through this section a category of diagrams

Dia and a strong and stable derivator D: Diaop ÑCAT. In Subsections 3.2

and 3.3 we will need to assume that NPDia.

3.1. t-Structures on stable derivators. Let us start with the following

deﬁnition

Deﬁnition 3.1. At-structure t“ pDp1qď0,Dp1qě0qon Dis, by deﬁnition,

at-structure on the triangulated category Dp1q.

The following proposition is one of the main results of [Sˇ

SV18]:

Proposition 3.2. Let t“ pDp1qď0,Dp1qě0qbe a t-structure on D, and let

IPDia. Letting

DpIqď0:“ tXPDpIq:XiPDp1qď0,@iPIu,

DpIqě0:“ tXPDpIq:XiPDp1qě0,@iPIu,

tI:“ pDpIqď0,DpIqě0qis a t-structure on DpIq. Furthermore,

diaI:DpIq Ñ Dp1qI

induces an equivalence AI–AIbetween the heart AIof tIand the category

AIof diagrams of shape Iin the heart Aof t.

Corollary 3.3. Given a t-structure t“ pDp1qď0,Dp1qě0qon D,

(1) the prederivator Dď0:IÞÑ DpIqď0is a coreﬂection of D;

(2) the prederivator Dě0:IÞÑ DpIqě0is a reﬂection of D;

(3) the prederivator D♥:IÞÑ DpIqď0XDpIqě0can be seen either as a

reﬂection of Dď0, or as a coreﬂection of Dě0.

In particular, Dď0,Dě0, and D♥are pointed, strong derivators of type Dia.

Proof. By Proposition 3.2, tIis a t-structure on DpIqand diaI:DpIq Ñ

Dp1qIinduces an equivalence of categories D♥pIq – D♥p1qI, for all Iin Dia.

The proofs of (1), (2) and (3) are very similar so we just verify (1) and leave

the other two statements to the reader. First of all we need to verify that

Dď0:IÞÑ Dď0pIqand Dě0:IÞÑ Dě0pIqare sub-prederivators of D. For

that one should verify that, given u:IÑJin Dia,

u˚XPDď0pIqfor all XPDď0pJq

u˚YPDě0pIqfor all YPDě0pJq.

MORITA THEORY FOR STABLE DERIVATORS 23

But this is true since XPDď0pJqif and only if XjPDď0p1q, while u˚XP

Dď0pIqif and only if i˚pu˚Xq “ pupiqq˚XPDď0p1qfor all iPI. The proof

of the second statement is analogous.

To conclude the proof of (1) we need to verify that the obvious embedding

Dď0ÑDhas a right adjoint. Since each tIis a t-structure, each component

Dď0pIq Ñ DpIqhas a right adjoint so, by [Gro13, Lem.2.10] there is a unique

way to organize these right adjoints into a lax morphism of prederivators

DÑDď0. Consider a functor u:IÑJin Dia; given XPDď0pIq, let us

show that u!XPDď0pJq, equivalently, we can show that DpJqpu!X,Yq “ 0

for all YPDą0pJq, but this is clear since DpJqpu!X,Yq “ DpIqpX, u˚Yq “

0, as we have already seen that u˚YPDą0pIq. Hence, using the same

argument of the proof of [Gro13, Prop.2.11], one shows that the morphism

DÑDď0is strict.

The last statement is a consequence of Lemma 1.12. 

Recall that a derivator D: Diaop ÑCAT is said to be discrete if, for

any IPDia, the diagram functor diaI:DpIq Ñ Dp1qIis an equivalence.

In particular, this means that the homotopy (co)limits and homotopy Kan

extensions in such a derivator coincide with the classical (i.e., categorical)

(co)limits and Kan extensions. A discrete derivator is Abelian if Dp1qis

an Abelian category.

Corollary 3.4. Given a t-structure t“ pDp1qď0,Dp1qě0qon D,D♥is a

discrete, Abelian derivator.

Proof. D♥is a derivator by Corollary 3.3 and it is discrete by Proposition

3.2. Hence we have just to verify that D♥p1qis an Abelian category. In fact,

we already know that it is an additive and Dia-bicomplete category, so it is

enough to show that any monomorphism (epimorphism) is a (co)kernel.

Let us start with a morphism φ:XÑYin D♥p1q. By Lemma 1.12, the

kernel of φcan be constructed as follows: ﬁrst we notice that such kernel is

the limit (which is the same as “homotopy colimit” since D♥is discrete) of

the following diagram



Xφ//Y.

This limit can be computed ﬁrst as a homotopy limit in Dě0or, equivalently,

in D, and then coreﬂected in D♥(see Lemma 1.12). This shows that Kerpφq

is the coreﬂection of K, where Kis the ﬁber of φ, that is, there is a triangle

in Dp1q:

(3.1) KÑXÑYÑΣK.

This shows that φis a monomorphism if and only if Kerpφq “ 0 if and only

if KPDě1p1q. This means exactly that in the triangle (3.1), X, Y and

ΣKPD♥p1q. By the above discussion, we obtain that φ:XÑYrepresents

a kernel for YÑΣKin D♥p1q. In particular any monomorphism in D♥p1q

is a kernel; one veriﬁes similarly that all the epimorphisms are cokernels. 

We have seen above that a t-structure on the base of a stable derivator can

be used to produce a reﬂection and a coreﬂection. In the following theorem

24 SIMONE VIRILI

we prove that, in fact, all the extension-closed (co)reﬂections arise this way

from a t-structure.

Theorem 3.5. There are bijections among the following classes:

(1) t-structures in Dp1q;

(2) extension-closed reﬂective sub-derivators of D;

(3) extension-closed coreﬂective sub-derivators of D.

Proof. By Proposition 3.2, a t-structure t“ pDp1qď0,Dp1qě0qon Dp1qcan

be lifted to t-structures tI“ pDpIqď0,DpIqě0qin DpIq, for any Iin Dia.

Then, the derivator Dď0: Diaop ÑCAT, such that Dď0pIq:“DpIqď0, is

coreﬂective in Dand, similarly, the derivator Dě0: Diaop ÑCAT, such that

Dě0pIq:“DpIqě0, is reﬂective in D(see Corollary 3.3).

On the other hand, given an extension-closed coreﬂective sub-derivator

UÑD, it is clear that Up1qis an aisle in Dp1q(use [Gro13, Prop. 3.21]). By

Proposition 3.2, we can deﬁne a coreﬂective sub-derivator U1of Dletting

U1pIq:“ tXPDpIq:XiPUp1q,@iPIu.

Thus, Vp1q:“Σ´1pUp1qKqis a co-aisle and, again by Proposition 3.2, we

can deﬁne a reﬂective sub-derivator Vof Dletting

VpIq:“ tXPDpIq:XiPVp1q,@iPIu.

Now, for any IPDia, we have two t-structures: pUpIq,Σ´1pUpIqKqq and

pU1pIq,VpIqq. But clearly, UpIq Ď U1pIqand Σ´1pUpIqKq Ď VpIq, which im-

plies that these two t-structures coincide. In particular, U“U1. Hence, Uis

completely determined by Up1q, so the assignment UÞÑ pUp1q,Σ´1pUp1qKqq

is a bijection between the classes described in parts (3) and (1) of the state-

ment. The bijection between (2) and (1) is proved similarly. 

Due to the above theorem, from now on, we will indiﬀerently denote a t-

structure ton a stable derivator D, either as aisle and coaisle pDp1qď0,Dp1qě0q

in Dp1q, or as a pair of coreﬂective and reﬂective subderivators pDď0,Dě0q

of D.

3.2. Epivalence of certain diagram functors. The following lemma is

a reﬁnement of [Sˇ

SV18, Thm. 3.1] in the case for the special small category N.

We will just give a sketch of the proof since similar statements have already

appeared in [LV17] (for ﬁnite ordinals) and in the appendix of [KN13].

Lemma 3.6. The diagram functor

diaN:DpNq Ñ Dp1qN(resp., diaNop :DpNopq Ñ Dp1qNop )

is full and essentially surjective. Furthermore, given XPDpNq(resp.,

DpNopq), there is a triangle

nPN

pn`1q!XnÑà

nPN

n!XnÑXÑΣà

nPN

pn`1q!Xn

˜resp., YÑź

nPN

n˚YnÑź

nPN

pn`1q˚YnÑΣY¸.

MORITA THEORY FOR STABLE DERIVATORS 25

Finally, given a second object YPDpNqthe diagram functor induces an

isomorphism

DpNqpX,Yq – Dp1qNpdiaNX,diaNYq

presp., DpNopqpX,Yq – Dp1qNoppdiaNop X,diaNop Yqq

provided the following map is surjective (where D:“Dp1q):

śnPNDpΣXn,Ynq Ñ śnPNDpΣXn,Yn`1q

pφnqNÞÑ pdn

Yφn´φn`1Σdn

XqN

ˆśnPNDpΣXn,Ynq Ñ śnPNDpΣXn`1,Ynq

pφnqNÞÑ pdn`1

Yφn`1´φnΣdn`1

XqN˙.

This happens, for example, when the Toda condition Dp1qpΣXn,Yn`1q “ 0

(resp., Dp1qpΣXn`1,Ynq “ 0), is veriﬁed for all nPN.

Sketch of the proof. Let XPDp1qN. For any nPN, consider the following

maps

αn:pn`1q!XnÑn!Xnβn:pn`1q!XnÑ pn`1q!Xpn`1q,

where αnis the image of the identity via the following series of natu-

ral isomorphism Dp1qpXn, Xnq – Dp1qpXn,pn`1q˚n!Xnq – DpNqppn`

1q!Xn, n!Xnq, while βnis induced by the unique natural transformation

bn:p1n

ÑNq ñ p1n`1

ÑNq(hence, βn“ pn`1q!b˚

n). Then deﬁne a mor-

phism in DpNq

ΦX:à

nPN

pn`1q!XnÑà

nPN

n!Xn

where the component of ΦXrelative to pn`1q!Xnis exactly

pβn,´αnqt:pn`1q!XnÑn!Xn‘ pn`1q!Xpn`1q.

Letting Xbe the cone of ΦX, one can verify that diaNpXq – X. Further-

more, given YPDpNq, we can ﬁnd a presentation of Yas the one above and,

applying the functor p´,Zq:“DpNqp´,Zqwe get a long exact sequence

¨¨¨ //śnPNDp1qpΣYn,Znqp˚q //śnPNDp1qpΣYn,Zn`1q//pY,Zq//

//śnPNDp1qpYn,Znqp˚˚q //śiPNDp1qpYn,Zn`1q//¨¨¨

where the kernel of p˚˚q is Dp1qNpdiaNY,diaNZq. This shows that the map

pY,Zq Ñ Dp1qNpdiaNY,diaNZqis always surjective and that, when the

map p˚q is surjective (so for example when Dp1qpΣYn,Zn`1q “ 0 for all

nPN), then it is also injective.

To prove the statement for Nop one uses the same argument with a “dual”

resolution. Indeed, given YPDp1qNop one can construct a map

ΨY:ź

nPN

n˚YnÑź

nPN

pn`1q˚Yn.

Then, calling Ythe ﬁber of the above map, diaNop pYq – Y. Furthermore,

given XPDpNopq, we can ﬁnd a presentation of Yas the one above.

Applying the functor pZ,´q :“DpNop qpZ,´q we get a long exact sequence

with which we can conclude analogously to the ﬁrst half of the proof. 

26 SIMONE VIRILI

3.3. t-structures on DpNqand DpNopq.Let t“ pDď0,Dě0qbe a t-

structure on D. By Proposition 3.2, tN“ pDď0pNq,Dě0pNqq (resp., tNop “

pDď0pNopq,Dě0pNopqq) is a t-structure on the triangulated category DpNq

(resp., DpNopq). In the following lemmas we introduce a diﬀerent kind of

t-structure on the same triangulated categories:

Lemma 3.7. Consider the following full subcategory of DpNopq:

Dď0

tow :“ tXPDpNopq:XnPDď´np1qu,

Dě0

tow :“ tXPDpNopq:XnPDě´np1qu.

Then, ttow :“ pDď0

tow, Dě0

towqis a t-structure in DpNopq.

Proof. It is clear that Dď0

tow and Dě0

tow have the required closure properties.

Let now XPDď0

tow (i.e., XnPDp1qď´n, for all nPN) and YPDě1

tow

(i.e., YnPDp1qě´n`1, for all nPN). Then, Dp1qpΣXn,Yn`1q “ 0 since

ΣXnPΣDp1qď´n“Dp1qď´n´1and Yn`1PDp1qě´nso, by Lemma 3.6,

DpNopqpX,Yq – Dp1qNop pdiaNop X,diaNopYq.

On the other hand,

Dp1qNoppdiaNop X,diaNop Yq Ď ź

iPN

Dp1qpXn,Ynq “ 0,

since XnPDp1qď´nand YnPDp1qě´n`1. This shows that DpNopqpX,Yq “

0. Finally, let XPDpNopqand consider its diagram X:“diaNop XP

Dp1qNop. Then there is an object Xď0PDp1qNop and a morphism φ:Xď0Ñ

Xwith the property that, for any nPN,Xď0pnq P Dď´np1qand the cone of

φnbelongs to Dě´n`1p1q. Using that diaNop is full and essentially surjective,

one can lift φto a morphism Φ: Xď0ÑX. Completing this morphism to

a triangle

Xď0ÑXÑYÑΣXď0

we obtain that, by construction, Xď0PDď0

tow and YPDě1

tow.

We omit the proof of the following lemma since it is completely dual to

that of the above lemma.

Lemma 3.8. Consider the following full subcategory of DpNq:

Dď0

tel :“ tXPDpNq:XnPDďnp1qu,

Dě0

tel :“ tXPDpNq:XnPDěnp1qu.

Then, ttel :“ pDď0

tel , Dě0

tel qis a t-structure in DpNq.

MORITA THEORY FOR STABLE DERIVATORS 27

4. The bounded realization functor

The construction of realization functors (at least in the bounded context)

is classically done using the ﬁltered derived category and, more generally,

f-categories, see [BBD82, Bei87, PV15]. For a given derivator D, one can

deﬁne a “bounded ﬁltered prederivator” FbDsuch that FbDp1qhas a natural

f-structure, whose core is equivalent to Dp1q. As a consequence, any t-

structure on Dp1qcan be lifted to a compatible t-structure on FbDp1q. In

this section we give a direct and detailed proof of this lifting property for

t-structures from Dto FbDin the setting of stable derivators. Moreover, we

give a construction of a morphism of prederivators

realb

t:Db

AÑD,

where Ais the heart of a given t-structure on D. In the ﬁnal part of the

section we show that this morphism is equivalent to the one we obtain via

f-categories.

Setting for Section 4. Fix through this section a category of diagrams

such that NPDia, a strong and stable derivator D: Diaop ÑCAT, a t-

structure t“ pDď0,Dě0qon D, and let A“Dď0p1q X Dě0p1qbe its heart.

Just for the ﬁnal Proposition 4.14, we will need to assume that NPDia.

4.1. The bounded ﬁltered prederivator FbD.Let us start with the

following deﬁnition:

Deﬁnition 4.1. Given nPZ, let rnÑn`1s:2ÑZbe the functor sending

0ÞÑ nand 1ÞÑ n`1. Deﬁne

grn:DZrnÑn`1s˚

//D2cone //D.

Given XPDZpIq

‚Xis bounded above if there exists aPZsuch that Xa´n“0for

all nPN. If Xis bounded above, we let

apXq:“suptaPZ:Xa´n´1“0,@nPNu;

‚Xis bounded below if there exists bPZsuch that grb`nX“0

for all iPN. If Xis bounded below, we let

bpXq:“inftbPZ: grb`nX“0,@nPNu;

‚Xis bounded if it is both bounded above and below. If Xis

bounded, we deﬁne its length ℓpXq “ ´1if X“0and otherwise

ℓpXq “ |bpXq ´ apXq| .

We deﬁne the bounded ﬁltered prederivator

FbD: Diaop ÑCAT,

as a full sub-prederivator of DZ, where FbDpIqis the category of bounded

objects in DZpIq.

In the following remark we try to give some motivation and intuition for

the above deﬁnition:

28 SIMONE VIRILI

Remark 4.2. Let XPDpZqand consider its underlying diagram:

diaZpXq:¨¨¨ //X´2

d´2//X´1

d´1

//X0

//X1

//X2

//¨¨¨

The object Xis then bounded above if Xk“0for all kăă 0and it is

bounded below if dkis an isomorphism for all kąą 0. In particular, when

Xis bounded, we can interpret it as the following (coherent) ﬁnite ﬁltration

of the objects XbpXq:

0“XapXqÑXapXq`1ÑXapXq`2Ñ ¨ ¨ ¨ Ñ XbpXq´1ÑXbpXq.

Hence, the motivation for calling FbDthe bounded ﬁltered derivator of Dis

that its objects are exactly the coherent ﬁnite ﬁltrations of the objects in D.

Another motivation is the following: if we take a Grothendieck category G

and the associated stable derivator DG, the base of the corresponding ﬁltered

derivator FbDGis (equivalent to) what is usually called the ﬁltered derived

category of G.

A ﬁnal remark is that one would expect that the deﬁnition of “bounded

above” object of DpZqis what we call here “bounded below”, and viceversa.

The reason for this counterintuitive choice of words will be clariﬁed in the

proof of Proposition 4.8.

Notice that, in general FbDis not a derivator but, for any IPDia, FbDpIq

is a full triangulated subcategory of DZpIq. In particular, it makes sense

to consider triangles and to speak about t-structures in FbDpIqfor some

IPDia. The following result follows by Lemma 3.6.

Corollary 4.3. For any IPDia, the functor

diaZ:FbDpIq Ñ DpIqpb,Zq

is full and essentially surjective, where DpIqpb,Zqis the full subcategory of

DpIqZspanned by those X:ZÑDpIqsuch that there exists aďbPZ

for which Xa´i“0and Xb`iÑXb`i`1is an isomorphism, for all iPN.

Furthermore, given YPFbDpIq, there is a triangle

YÑź

iďbpYq

i˚YiÑź

iďbpYq

pi´1q˚YiÑΣY

Finally, given a second object XPFbDpIq, the diagram functor induces an

isomorphism

FbDpIqpX,Yq – DpIqZpdiaZX,diaZYq

provided the following map is surjective:

iďb

DpIqpΣXi,Yiq Ñ ź

iďb

DpIqpΣXi´1,Yiq

pφiqiďbÞÑ pdi´1

Yφi´1´φiΣdi´1

Xqiďb,

where b:“maxtbpXq, bpYqu.

Lemma 4.4. Given IPDia and X,YPFbDpIqsuch that apXq ě bpYq,

then

FbDpIqpX,Yq “ DpIqphocolimZpXq,hocolimZpYqq.

MORITA THEORY FOR STABLE DERIVATORS 29

Proof. It is easy to show that the map

iďbpXq

DpIqpΣXi,Yiq Ñ ź

iďbpXq

DpIqpΣXi´1,Yiq

is surjective, so that FbDpIqpX,Yq – DpIqZpdiaZX,diaZYq. Consider now

the underlying diagrams:

¨¨¨ //0//



XapXq//



XapXq`1//



¨¨¨ //XbpXq



–//¨¨¨

¨¨¨ //YapXq´1//YapXq

–//YapXq`1

–//¨¨¨ –//YbpXq

–//¨¨¨

Clearly, hocolimZX–XbpXq, hocolimZY–YbpYq–YbpXqand

DpIqZpdiaZX,diaZYq – DpIqpXbpXq,YbpXqq.

4.2. The Beilinson t-structure.

Proposition 4.5. Consider the following two classes of objects in FbDp1q:

FbDď0p1q “ tXPFbDp1q: grnXPDď´np1q,@nu,

FbDě0p1q “ tXPFbDp1q: grnXPDě´np1q,@nu.

Then tB:“ pFbDď0p1q,FbDě0p1qq is a t-structure on FbDp1q.

Proof. The closure properties of FbDď0p1qand FbDě0p1qfollow by the fact

that all the grnare triangulated functors. Now let XPFbDď0p1q,YP

FbDě1p1q, and let us show that DpZqpX,Yq “ 0. Consider the distinguished

triangle

apXq!XapXqÑXÑX1ÑΣapXq!XapXq,

where ℓpapXq!XapXqq “ 0, ℓpX1q ď ℓpXq ´ 1, and apXq!XapXq,X1P

FbDď0p1q. The following exact sequence

¨ ¨ ¨ Ñ DpZqpX1,Yq Ñ DpZqpX,Yq Ñ DpZqpa!Xa,Yq Ñ ¨ ¨ ¨

shows that DpZqpX,Yq “ 0 provided DpZqpX1,Yq “ 0“DpZqpa!Xa,Yq.

We can use this trick to reduce (by induction on ℓpXq) to the case when

ℓpXq “ 0. Similarly, there is a distinguished triangle apYq!YapYqÑYÑ

Y1ÑΣapYq!YapYq, where ℓpapYq!YapYqq “ 0, ℓpY1q ď ℓpYq ´ 1, and

apYq!YapYq,Y1PDě0

b. Similarly to what we did for X, we can reduce

(by induction on ℓpYq) to the case when ℓpYq “ 0. Hence, suppose that

ℓpXq “ ℓpYq “ 0 and consider the following cases:

‚if apXq ě apYqthen, by adjunction,

DpZqpX,Yq – DpZqpapXq!XapXq, apYq!YapYqq

–Dp1qpXapXq, apXq˚apYq!YapYqq

–Dp1qpXapXq,YapYqq “ 0

where the last equality holds since XapXq–grapXq´1XPDď´apXq`1p1q,

while YapYq–grapYq´1YPDě´apYq`2p1q Ď Dě´apXq`2p1q;

30 SIMONE VIRILI

‚if apXq ă apYqthen, by adjunction,

DpZqpX,Yq – DpZqpapXq!XapXq, apYq!YapYqq

–Dp1qpXapXq, apXq˚apYq!YapYqq

–Dp1qpXapXq,0q “ 0.

Given XPFbDp1q, we need to show that there exists a triangle

Xď0ÑXÑXě1ÑΣpXď0q

such that Xď0PFbDď0p1qand Xě1PFbDě1p1q. If ℓpXq “ ´1 then we can

take everything to be 0. Otherwise, we proceed by induction on ℓpXq P N

to ﬁnd such Xď0and Xě1with the extra conditions that hocolimNXď0P

Dď´apXqp1qand hocolimNXě1PDě´bpXq`1p1q. Indeed:

‚If ℓpXq “ 0, say X“a!Xa, then it is enough to take Xď0:“a!pXď´a

and Xě1:“a!pXě´a`1

aqwith the obvious maps.

‚If ℓpXq ą 0 and a:“apXq, then there is a distinguished triangle in

FbDp1q,a!XaÑXÑX1ÑΣa!Xaand we can consider the following

diagram

Σ´1pX1ď0q//



pa!Xaqď0



//A



//X1ď0



Σ´1X1//



a!Xa



//X



//X1



Σ´1pX1ě1q//



pa!Xaqě1//



B//



X1ě1



X1ď0//Σppa!Xaqď0q//ΣA//ΣpX1ď0q

where all rows and columns are triangles, and all the small squares com-

mute, but the bottom right square that anti-commutes. This diagram is

constructed as follows: ﬁrst we should complete the following diagram to a

morphism of triangles

Σ´1pX1ď0q//



Σ´1X1



//Σ´1pX1ě1q



//X1ď0



pa!Xaqď0//a!Xa//pa!Xaqě1//Σppa!Xaqď0q

where the solid map is given by a rotation of a!XaÑXÑX1ÑΣa!Xa.

In fact, using [BBD82, Prop. 1.1.9], the following two vanishing conditions

imply that the above solid diagram can be completed in a unique way to a

morphism of triangles:

‚using Lemma 4.4, we get that DpZqpΣ´1pX1ď0q,pa!Xaqě1qis iso-

morphic to Dp1qphocolimZΣ´1pX1ď0q,hocolimZpa!Xaqě1qwhich is

trivial since hocolimZΣ´1pX1ď0q P Dď´apX1q`1p1q Ď Dď´apXqp1q,

while hocolimZpa!Xaqě1“Xě´apXq`1

aPDě´apXq`1p1q;

‚using Lemma 4.4, we get that DpZqppX1qď0,pa!Xaqě1qis isomorphic

to Dp1qphocolimZpX1qď0,hocolimZpa!Xaqě1q “ 0, which is trivial

since hocolimZpX1qď0PDď´apX1qp1q Ď Dď´apXq´1p1q, whereas

hocolimZa!Xě1

a“Xě´apXq`1

aPDě´apXq`1p1q.

MORITA THEORY FOR STABLE DERIVATORS 31

Since there exists a unique way to complete the diagram to a morphism

of triangles, this morphism of triangles can be completed to the desired

3ˆ3 diagram (for this use [Nee91, Thms. 1.8 and 2.3], these results say

respectively, in the language of that paper, that any commutative square

can be completed to a “good” morphism of triangles and that any such

morphism gives rise to a 3 ˆ3 diagram as above. Since we have proved

that there is a unique way to complete the above diagram to a morphism

of triangles, this unique way is the “good” one). Now, letting Xď0:“A

and Xě1:“B, it is not diﬃcult to verify that they satisfy the desired

properties. Hence, tBis a t-structure on FbDp1q.

Deﬁnition 4.6. The t-structure tB:“ pFbDď0,FbDě0qon FbDdescribed in

the above proposition is said to be the Beilinson t-structure induced by t.

After the following technical lemma we will show that the heart of the

Beilinson t-structure is precisely the category of bounded complexes over

the heart of the original t-structure t.

Lemma 4.7. Let F:AÑA1be an exact functor between Abelian categories.

If Freﬂects 0-objects (i.e., FpXq “ 0implies X“0), then Fis faithful.

Proof. Consider a morphism φ:AÑBin Asuch that Fpφq “ 0. Let

π:AÑA{kerpφqbe the obvious projection and let ¯

φ:A{kerpφq Ñ Bbe

the unique morphism such that ¯

φπ “φ. Notice that πis an epimorphism

and ¯

φis a monomorphism so, by the exactness of F,Fpπqis an epimorphism

and Fp¯

φqis a monomorphism. One can now conclude as follows: 0 “Fpφq “

Fp¯

φπq “ Fp¯

φqFpπqimplies that Fp¯

φqis 0 as we can cancel the epimorphism

Fpπq, but Fp¯

φqis also a monomorphism, so FpA{Kerpφqq “ 0, which means

that A“Kerpφq, since Freﬂects 0-objects. 

Given an Abelian category A, we denote by ChbpAqthe category of

bounded complexes over A. For a non-trivial complex 0 ‰X‚PChbpAq,

we let:

–bpX‚q:“maxtnPZ:X´n‰0u,apX‚q:“mintnPZ:X´n‰0u;

–ℓpX‚q:“ |apX‚q ´ bpX‚q|.

Let also ℓpXq “ ´1 if X“0.

Proposition 4.8. Let Abe the heart of tand HBĎFbDp1qthe heart of

the Beilinson t-structure tB. Then, there is an equivalence

F:HB

–//ChbpAq

that satisﬁes the following properties for any XPHB:

(1) if ℓpXq “ 0, then FXis a complex concentrated in degree ´a`1,

whose unique non-zero component is Σa´1gra´1X;

(2) if ℓpXq “ 1, then FXis a complex concentrated in degrees ´a`1

and ´a, whose non-zero part is the map Σ´agraXÑΣ´a`1gra´1X;

(3) in general, ℓpFXq “ ℓpXq.

Proof. We start constructing a functor F:HBÑChbpAqas follows: to an

object XPHBwe associate the complex FX‚, where FX´n“Σ´ngrnX

32 SIMONE VIRILI

and with the diﬀerential d´n´1

FX:FX´n´1ÑFX´n, such that Σn`1d´n´1

ﬁts in the following octahedron:

Xn//Xn`1//



grnX



//ΣXn

Xn//Xn`2//



Xrn,n`2s



//ΣXn

grn`1X



grn`1X

Σn`1d´n´1



ΣXn`1//ΣgrnX

With this deﬁnition, the properties (1–3) in the statement are trivially ver-

iﬁed, let us prove that Fis an equivalence:

‚Fis exact. Consider a short exact sequence 0 ÑXÑYÑZÑ

0 in HB. The short exact sequences in HBare the triangles of FbDp1q

that happen to lie in HB, in particular, there exists a map ZÑΣX

such that XÑYÑZÑΣXis a triangle. We have to prove that

0ÑFXÑFYÑFZÑ0 is a short exact sequence in ChbpAq, but

this means exactly that 0 ÑFXnÑFYnÑFZnÑ0 is a short ex-

act sequence in Afor any nPZ. This means that there is a triangle

grnXÑgrnYÑgrnZÑΣgrnXin Dp1q, which is true since grnis a

triangulated functor.

‚Freﬂects 0-objects. Let XPFbDp1qand let us deﬁne a sequence of

objects pX1pnqqnPNin FbDp1q. Indeed, let a:“apXqand

p˚q deﬁne X1p0qto be the cone of the counit a!XaÑX(here Xa–

gra´1X);

p˚q notice that X1p0qa“0 and X1p0qa`1“graX, so there is a natural

map pa`1q!grapXq Ñ X1p0q, and we deﬁne X1p1qto be the cone

of this map;

p˚q we deﬁne X1pn`1qas the cone of the counit pn`a`1q!grn`aXÑ

X1pnq.

One can show that X1pnq “ 0 for any nąℓpXq. This ﬁltration shows that

Xbelongs in the smallest triangulated subcategory of FbDp1qthat contains

the objects of the form pn`1q!grnpXq. Thus, if grnX“0 for all nPZ

(i.e., if FX“0 in ChbpAq), then X“0.

‚Fis fully faithful. By Lemma 4.7 and the properties of Fthat we have

already veriﬁed, it is enough to show that Fis full. Indeed, let X,YP

FbDp1qand let φPChbpAqpFX, F Yq. As in the proof that Freﬂects 0-

objects, one can construct Xand Yas extensions of shifts of objects of the

form pn`1q!grnXand pn`1q!grnY. Of course the map φinduces maps

pn`1q!pΣnφnq:pn`1q!grnXÑ pn`1q!grnY, for any nPZ, and these

maps can be glued together to construct a new map ψ:XÑYsuch that

Fpψq “ φ.

‚Fis essentially surjective. Given a complex C‚PChbpAq, say

¨¨¨ //0//C´bd´b

//C´b`1d´b`1

//¨¨¨ d´a´1

//C´a//0//¨¨¨ ,

MORITA THEORY FOR STABLE DERIVATORS 33

we have to construct and object XPHBsuch that FX–C‚. Suppose,

for simplicity, that a“0, the general case can be handled analogously by

shifting the indices. So our complex becomes:

¨¨¨ //0//C´bd´b

//C´b`1d´b`1

//¨¨¨ d´1

//C0//0//¨¨¨ .

We construct a diagram in Dp1qZ

¨¨¨ //0//Xp0qx0//Xp1qx1//¨¨¨ //Xpb`1q//0//¨¨¨

(one can successively lift it to a coherent diagram Xin FbDp1q, using Corol-

lary 4.3). We deﬁne:

p˚q for nď0, we let Xpnq “ 0;

p˚q Xp1q:“C0PDě0p1qand let Xp0qx0“0//Xp1qidC0“y0

//C0z0:“d0

//0 ;

p˚q Xp2q P Dě´1and x1:Xp1q Ñ Xp2qare deﬁned as part of a triangle:

Xp1qx1//Xp2qy1//ΣC´1z1:“Σd´1

//ΣXp1q

so that Σy0˝z1“idC0˝Σd´1“Σd´1;

p˚q then we proceed inductively. Suppose that we have already con-

structed two triangles with Xpiq P Dě´i`1p1q(so that Xpi`1q P

Dě´ip1qand Xpi`2q P Dě´i´1p1q)

Xpiqxi//Xpi`1qyi//ΣiC´izi//ΣXpiq

Xpi`1qxi`1//Xpi`2qyi`1//Σi`1C´i´1zi`1//ΣXpi`1q

and such that Σyi˝zi`1“Σi`1d´i´1. We deﬁne Xpi`3q P Dě´i´2

and xi`2:Xpi`2q Ñ Xpi`3qas parts of a triangle

Xpi`2qxi`2//Xpi`3qyi`2//Σi`2C´i´2zi`2//ΣXpi`2q

Notice ﬁrst that Σyi˝zi`1˝Σi`1d´i´2“Σi`1pd´i´1d´i´2q “ 0, so

that we have a commutative diagram

0//



Σi`1C´i´2



Σi`1C´i´2//

zi`1Σi`1d´i´2



ΣiC´izi//ΣXpiqΣxi//ΣXpi`1qΣyi//Σi`1C´i.

Now, Σi`1C´i´2PDď´i´1p1qand ΣXpiq P Dě´ip1q, so

Dp1qpΣi`1C´i´2,ΣXpiqq “ 0,

showing that zi`1Σi`1d´i´2“0. Hence, we can deﬁne zi`2in order

to make the following diagram commute

0//



Σi`1C´i´2

Σ´1zi`2



Σi`1C´i´2//

Σi`1d´i´2



Xpi`1qxi`1//Xpi`2qyi`1//Σi`1C´i´1zi`1//ΣXpi`1q.

We deﬁne Xpi`3q,xi`2and yi`2completing zi`2to a triangle. 

34 SIMONE VIRILI

4.3. Totalization of bounded complexes. If ną1, we deﬁne the

following morphism of derivators

trn: Chn

AÑ pChn´1

Aq2

such that, given IPDia and a complex XPChn

ApIq,

ptrnX:Xăn´1ÑΩn´2Xn´1q:“



//0



//0



Xn´2//



Xn´1



0//



‹

‚

We can now introduce the main deﬁnition of this subsection:

Deﬁnition 4.9. Given nPNą0we deﬁne, inductively, a morphism of pre-

derivators, called totalization morphism,

Totn

t: Chn

AÑD

‚for n“1,Ch1

A“ypAq “ D♥so we let Tot1

tbe the inclusion D♥ÑD;

‚in general we deﬁne Totn`1

tto be the following composition

Totn`1

t:“ﬁb ˝ pTotn

tq2˝trn: Chn`1

AÑ pChn

Aq2ÑD2ÑD.

In the following lemma we collect some natural properties of the totaliza-

tion morphisms:

Lemma 4.10. Given nPNą0, the following statements hold true:

(1) Totn

tsends short exact sequences of complexes to triangles;

(2) Totn

tΩn´1A–Ωn´1Afor any APA(where the Ωn´1Aon the left

hand side is the complex concentrated in degree n´1, with Aas the

unique non-zero component);

(3) given XPChnpAq Ď Chn`1pAq,Totn`1

tX–Totn

tX;

(4) given XPChn`1pAq, there is a triangle in Dp1q:

Totn`1

tX//Totn

tXăn//Ωn´1Xn//ΣTotn`1

tX;

(5) given XPChnpAq,Totn`1

tΩX–ΩTotn

tX.

Proof. (1) The proof goes by induction on n. For n“1, short exact se-

quences in ypAq Ď Dare clearly sent to triangles in Dby the inclusion. On

the other hand, Totn`1

tis, by deﬁnition, a composition ﬁb ˝ pTotn

tq2˝trn.

Now, trnis an exact functor, pTotn

tq2sends short exact sequences to tri-

angles by inductive hypothesis, and ﬁb: D2ÑDis an exact morphism of

stable derivators (so it is, component-wise, a triangulated functor).

(2) For n“1 the statement is clear. Suppose we have already proved

our statement for some ną0, then the statement boils down to the usual

MORITA THEORY FOR STABLE DERIVATORS 35

fact that the ﬁber of the morphism 0 ÑΩn´1Ain the triangulated category

Dp1qis isomorphic to ΩnA.

(3) This boils down to the usual fact that, in the triangulated category

Dp1q, the ﬁber of the morphism Totn

tXÑ0 is isomorphic to Totn

tX.

(4) By parts (2) and (3), Totn`1

tXăn–Totn

tXăn, while Totn`1

tΩnXn–

ΩnXn. Consider now the following short exact sequence in Chn`1pAq:

0//ΩnXn//X//Xăn//0

Applying Totn`1

twe get (a rotation of) the triangle in the statement.

(5) For n“1 the statement reduces to part (2). If ną1, let Y:“ΩXP

Chn`1pAqand consider the triangle of part (4),

Totn`1

tY//Totn

tYăn//Ωn´1Yn//ΣTotn`1

tY.

Notice that Yăn“ΩXăn´1and Yn“Xn´1. Hence there is a diagram,

which is commutative by inductive hypothesis, whose ﬁrst line is the above

triangle while the second line is the triangle given by part (4) to which we

have applied Ω:

Totn`1

tΩX//Totn

tΩXăn´1//

–



Ωn´1Xn´1//

–



ΣTotn`1

tΩX

ΩTotn

tX//ΩTotn´1

tXăn´1//Ωn´1Xn´1//Totn`1

tX.

This clearly implies that Totn`1

tΩX–ΩTotn

tX.

Before proceeding further we want to show how the totalization mor-

phisms can be used to detect the exactness of a bounded complex:

Proposition 4.11. Given XPChnpAq,Totn

tXPDďn´1p1q X Dě0p1q. Fur-

thermore, the following are equivalent:

(1) Xis an exact complex;

(2) Totk`1

tpτďkXq “ 0for all kăn.

Proof. The ﬁrst statement follows by induction using the above lemma. In-

deed, if n“1, then Tot1

tXPA“D♥p1q. For ną1, let m:“n´1, there

is a triangle ΩmXmÑTotn

tXÑTotm

tXămÑΩm´1Xmand, by inductive

hypothesis, Totm

tXămPDďm´1p1q X Dě0p1q Ď Dďn´1p1q X Dě0p1q, while

ΩmXmPDďmp1q X Děmp1q Ď Dďn´1p1q X Dě0p1q. Since aisle and co-aisle

are extension-closed, we get that Totn

tXPDďn´1p1q X Dě0p1q.

Let us now show the equivalence of the conditions (1) and (2); we proceed

by induction on n. Indeed, if n“1, then Xis exact if and only if X“0,

if and only if Tot1

tX–X0“0. If n“2, Xis exact if and only if d0is an

isomorphism, if and only if Tot2

tpXq “ ﬁbpd0q “ 0.

For ną2, Xis an exact complex if and only if τďn´2Xand τěn´1Xare

exact. Using the inductive hypothesis we get:

‚τďn´2Xis exact iﬀ Totk`1

tpτďkXq “ 0 for all kăn´1;

‚τěn´1Xis exact iﬀ Totn

tpτěn´1Xq “ Totn

tpτďn´1τěn´1Xq “ 0.

These two conditions together are clearly equivalent to (2). 

36 SIMONE VIRILI

We have constructed a commutative diagram (up to natural iso):

Ch1

Tot1





Ch2

ATot2

❳



Chn

Totn



The above diagram describes an object of pPDerDia qNpF, ∆Dq, where Fpnq “

Chn

A. By the universal property of pseudo-colimits, this corresponds to a

unique object in PDerDiappcolim F, Dq, that is, a morphism of derivators

Totpb,`q

t: Chpb,`q

A–pcolim FÑD.

Similarly, using Lemma 4.10, one shows that the following diagram is com-

mutative up to natural isomorphisms:

Chpb,`q

Totpb,`q



Ω



Chpb,`q

ΣTotpb,`q

❲



Chpb,`q

AΣn´1Totpb,`q

;;



Hence, using that Chb

Ais a sequential pseudo-colimit of copies of Chpb,`q

we can uniquely extend Totpb,`q

tto a unique morphism

Totb

t: Chb

AÑD.

Proposition 4.12. Consider the above morphism Totb

t: Chb

AÑD. For any

IPDia and XPChb

ApIqwe have the following properties:

(1) Totb

tpΩXq – ΩTotb

tpXqand Totb

tpΣXq – ΣTotb

tpXq;

(2) Totb

t: Chb

ApIq Ñ DpIqshort exact sequences to triangles;

(3) Totb

tX“0, provided Xis an exact complex.

Proof. One can check that, given pX, nq P Chpb,`q

ApIq, then Totb

tpX, nq –

Totn

tpXq, so the statement follows by Proposition 4.11. 

4.4. The bounded realization functor. Consider the stable derivator

A: Catop ÑCAT, such that Db

ApIq “ DbpAIqis the bounded derived

category of the Abelian category AI. In the previous section we have con-

structed a morphism Totb

t: Chb

AÑDthat sends short exact sequences to

triangles and that vanishes on exact complexes. Such a morphism factors

MORITA THEORY FOR STABLE DERIVATORS 37

uniquely through the quotient Chb

AÑDb

Agiving us an exact (and t-exact)

morphism of derivators, called (bounded)realization functor,

realb

t:Db

AÑD

extending the inclusion ypAq Ñ D.

Theorem 4.13. The bounded totalization morphism Totb

t: Chb

AÑDfactors

uniquely through the quotient Chb

AÑDb

A, giving rise to an exact and t-exact

morphism of derivators

realb

t:Db

AÑD,

such that realb

tæypAqis naturally isomorphic to the inclusion ypAq “ D♥

tÑD.

We conclude this section about bounded realization functor showing that,

when NPDia and so it makes sense to consider the bounded ﬁltered prede-

rivator FbD, the above construction of realb

tvia totalization of complexes co-

incides with the realization functor constructed via the Beilinson t-structure

on FbD.

Proposition 4.14. Suppose that NPDia. Then, given XPHB(the heart

of the Beilinson t-structure in FbD), there is a natural isomorphism

Totb

tpFpXqq – hocolimZX,

where F:HBÑChbpAqis the equivalence of Proposition 4.8.

Proof. We proceed by induction on ℓpXq ě ´1.

If ℓpXq “ ´1, that is, X“0 then the statement is trivial.

If ℓpXq “ 0, then hocolimZX–Xa–gra´1X, where a:“apXqwhile

Totb

tpFXq – Σ´a`1pΣa´1gra´1Xq – hocolimZX(see Proposition 4.8 (1))

as desired.

If ℓpXq “ 1 then hocolimNX–Xa`1, while Totb

tFXis the ﬁber of the

map graXÑΣgra´1X(see Proposition 4.8); the triangle Xap“ gra´1Xq Ñ

Xa`1ÑgraXÑΣXashows that such a ﬁber is exactly Xa`1.

For ną1, let a:“apXq,b:“bpXq, consider FXPChbpAqand the

following short exact sequence in ChbpAq:

0ÑA:“τď´b`2FXÑFXÑBÑ0.

Then, there are triangles: Totb

tAÑTotb

tFXÑTotb

tBÑΣTotb

tAand

hocolimZF´1AÑhocolimZXÑhocolimZF´1BÑΣ hocolimZF´1A.

Since ℓpF´1Aq ď 1 and ℓpF´1Bq ă ℓpXq, there are isomorphisms Totb

tA–

hocolimZF´1Aand Totb

tB–hocolimZF´1Bby inductive hypothesis. Hence

Totb

tFX–hocolimZX.

Remember that tis said to be cosmashing (resp., smashing) if its aisle

is closed under products (resp., if its coaisle is closed under coproducts).

It is well-known that for a cosmashing t-structure t, the heart has exact

products, while for a smashing t-structure the heart has exact coproducts

(this can be either veriﬁed by hand or it can be deduced from Corollary 3.3

and Lemma 1.12).

38 SIMONE VIRILI

Corollary 4.15. Let tXiuiPIbe a uniformly bounded family of complexes in

ChbpAq, that is, there exist bďaPZsuch that Xn

i“0for all iPIand for

all năbor nąa. The following statements hold true:

(1) if tis smashing, then realb

tpšiXiq – širealb

tXi, where the coprod-

uct on the left-hand side is taken in DbpAq, while the one on the

right-hand side is taken in Dp1q;

(2) if tis cosmashing, then realb

tpśiXiq – śirealb

tXi, where the prod-

uct on the left-hand side is taken in DbpAq, while the one on the

right-hand side is taken in Dp1q.

Proof. (1) Let us start noticing that, if tis smashing, so is tB(use that each

functor of the form grn(nPZ) commutes with coproducts). Hence, coprod-

ucts are the same in HBand in FbDp1q. Furthermore, since coproducts in

Aare exact, the coproduct šiXiin ChbpAqalso represents a coproduct in

DbpAq. Consider the equivalence F:HBÑChbpAq. Then,

realb

t˜ž

Xi¸“hocolimZF´1˜ž

Xi¸“b˚F´1˜ž

Xi¸

“ž

b˚F´1Xi“ž

hocolimZF´1Xi“ž

realb

tXi.

(2) is completely analogous. 

4.5. Eﬀa¸cability of t-structures. In this last subsection we give some

classical conditions on tfor realb

tto be fully faithful.

Deﬁnition 4.16. We say that t“ pDď0,Dě0qis eﬀa¸cable provided the

following condition holds for all IPDia:

‚given objects Xand Yin AI,ną0and φ:XÑΣnYin DpIq,

there is an object Zin AIand an epimorphism ψ:ZÑXin AI,

such that φψ “0.

Dually, tis said to be co-eﬀa¸cable provided the following condition holds

for all IPDia:

‚given objects Xand Yin AI,ną0and φ:XÑΣnYin DpIq,

there is an object Zin AIand a monomorphism ψ:YÑZin A,

such that Σnpψqφ“0.

Proposition 4.17. The following conditions are equivalent:

(1) tis eﬀa¸cable;

(2) tis co-eﬀa¸cable;

(3) realb

tis fully faithful.

If the above equivalent conditions hold true, then the essential image of realb

is exactly Db

t:“ŤnPNDďnXDě´n, so that realb

tinduces an equivalence

realb

t:Db

–

ÝÑ Db

Proof. The equivalence of conditions (1), (2) and (3) is proved in [BBD82,

Prop. 3.1.16], see also [PV15, Thm. 3.10], where some parts of the proof are

explained in better detail. 

MORITA THEORY FOR STABLE DERIVATORS 39

5. Half-bounded realization functor

In this section we introduce the notion of left and right t-complete deri-

vator, analogously to what has been done in the setting of triangulated cate-

gories by Neeman [Nee11] and, in the setting of 8-categories, by Lurie [Lur06,

Chapter 2]. We use these notions to extend the bounded realization functor

realb

t:Db

AÑDto the “half-bounded” realization functors

real´

t:D´

AÑp

Dand real`

t:D`

AÑq

where p

Dand q

Dare the left and right t-completion of D, respectively, while

D´

Aand D`

Aare the natural prederivators enhancing the right- and left-

bounded derived category of A, respectively. We also give suﬃcient condi-

tions on tfor real´

tand real`

tto commute with co/products and to be fully

faithful.

Setting for Section 5. Fix throughout this section a category of diagrams

Dia such that NPDia, a strong and stable derivator D: Diaop ÑCAT, a

t-structure t“ pDď0,Dě0qon D, and let A“D♥p1qbe its heart.

5.1. Left and right t-completeness. Let us start with the following

deﬁnition:

Deﬁnition 5.1. The left t-completion p

D(resp., the right t-completion

D) of Dis the sub-prederivator

D: Diaop ÑCAT (resp., q

D: Diaop ÑCAT)

of DNop (resp., of DN) such that, for any IPDia,p

DpIq Ď DNop pIq(resp.,

DpIq Ď DNpIq) is the full subcategory of those objects Xsuch that

‚XnPDě´npIq(resp., XnPDďnpIq) for all nPN;

‚the canonical map XnÑXn´1(resp., XnÑXn`1) induces an

isomorphism pXnqě´n`1ÑXn´1(resp., XnÑ pXn`1qďn) in DpIq.

In what follows we are going to give conditions under which p

Dis a reﬂec-

tive localization and q

Dis a coreﬂective colocalization of D.

Lemma 5.2. The restriction holimNop :p

DÑD(resp., hocolimN:q

DÑD)

has a left (resp., right) adjoint tow : DÑp

D(resp., tel: DÑq

D).

Proof. We give a complete proof for the left completion, the argument for

the right completion is completely dual. Consider ﬁrst the adjoint pair,

where pt: Nop Ñ1is the obvious functor,

pt˚:DÕDNop : holimNop .

In the notation of Lemma 3.7, the inclusion ιtow :Dě0

tow ÑDNop has a left

adjoint τě0

tow, so we obtain a second pair of adjoints:

τě0

tow :DNop ÕDě0

tow :ιtow.

Putting together the above two adjunctions, we obtain an adjunction

DÕDě0

tow,

where the right adjoint is a restriction of the homotopy limit. It is enough

to notice that the essential image of the left adjoint is contained in p

D.

40 SIMONE VIRILI

Lemma 5.3. Let Db

t:“ŤnPNDďnXDě´n; the restriction tow : Db

tÑp

Dis

fully faithful. Furthermore, consider the full sub-prederivator p

tĎp

Dsuch

that, given IPDia, an object XPp

DpIqbelongs to p

tpIq, if and only if

(1) X0PDb

(2) there exists nPN, such that Xk`1ÑXkis an iso for all kěn.

Then, p

tis exactly the essential image towpDb

tq.

Proof. It it clear that, given YPDb

tpIq, towpYqsatisﬁes the conditions

(1) and (2). Furthermore, holimNop towpYq – ptowYqn–Y(for nas in

condition (2) in the statement). So the unit of the adjunction

tow : Db

tÕp

t: holimNop

is a natural equivalence. To show that also the counit is an equivalence, let

XPp

tpIq. Then, for nas in condition (2), holimNop X“Xnand, clearly,

Xk“ pXnqě´k, for all kPN. Hence, X–tow holimNoppXq.

By a dual argument, we can embed Db

tas a full sub-prederivator q

tof q

Deﬁnition 5.4. We say that tis (countably) k-cosmashing for a non-

negative integer kif, for any (countable) family tXαuof objects in Dď0p1q,

the product śXαbelongs to Dďkp1q. Dually, tis (countably) k-smashing

if, for any (countable) family tXαuof objects in Dě0p1q, the coproduct šXα

belongs to Dě´kp1q.

In particular, a (countably) 0-cosmashing t-structure is a t-structure whose

right truncation functor commutes with (countable) products. Similarly, a

(countably) 0-smashing t-structure is a t-structure whose left truncation

functor commutes with (countable) coproducts.

Lemma 5.5. Suppose that tis (countably) k-(co)smashing for some kPN

and let IPDia, then tIhas the same property.

Proof. Given a (countable) family tXαuof objects in Dď0pIq, consider the

product X:“śXαin DpIq. Then, Xbelongs to DďkpIqif and only if

i˚pXq P Dďkp1qfor all iPI, but this is true since i˚pśXαq “ śi˚pXαq P

Dďkp1qsince tis (countably) k-cosmashing (here we used that, since i˚is

both a left and a right adjoint, it commutes with products and coproducts).

A similar argument can be used in the smashing case. 

Lemma 5.6. Suppose that tis countably k-cosmashing for some kPNand

let XPp

Dp1q; for any nPN, there is a triangle

Kn//holimNop X//Xn//Kn

with KnPDď´n´1p1q.

Proof. Given n1ďn2PN, denote by dn1,n2:Xn2ÑXn1the map in the

underlying diagram of X. Notice that, for any nPN, there is a natural

morphism

πn: holimNop XÑXn,

obtained applying the functor n˚to the counit pt˚

Nop holimNop XÑX

and, by construction, πn1“dn1,n2πn2. Now, given nPN, we apply the

MORITA THEORY FOR STABLE DERIVATORS 41

octahedral axiom to the composition πn“dn,n`k`1πn`k`1, obtaining the

following diagram whose rows and columns are all triangles:

pXn`k`1qď´n´1



pXn`k`1qď´n´1



holimNop X//Xn`k`1//



ΣKn`k`1



//Σ holimNop X

holimNop X//Xn



//ΣKn



//Σ holimNop X

ΣpXn`k`1qď´n´1ΣpXn`k`1qď´n´1

To conclude we should prove that KnPDď´n´1p1qand, for that, it is enough

to verify that Kn`k`1PDď´n´1p1q. Consider now the obvious inclusion

ιn`k`1:pn`k`1qop ÑNop, let Xpďn`k`1q:“ pιn`k`1q˚pιn`k`1q˚X, con-

sider the unit ΦX:XÑXpďn`k`1qand complete it to a triangle in DpNop q:

KÑXÑXpďn`k`1qÑΣK

Applying holimNop we get the following triangle

holimNop KÑXÑXn`k`1ÑΣ holimNop K,

in fact, pXpďn`k`1qqh“Xhprovided hďn`k, while pXpďn`k`1qqh“

Xn`k`1, if hěn`k`1. We obtain an isomorphism holimNop K–Kn`k`1,

from which we deduce that there is a triangle of the form:

Kn`k`1Ñź

hPN

KhÑź

hPN

KhÑΣKn`k`1

Furthermore, for any hPN, there is a distinguished triangle KhÑXhÑ

pXpďn`k`1qqhÑKh, so that Kh“0 if hďn`k`1, and Kh–Xď´n´k´2

otherwise. Since tis countably k-cosmashing, śhPNKhPDď´n´2pIqand

so Kn`k`1PDď´n´1p1q, as desired. 

Theorem 5.7. If tis countably k-cosmashing for some kPN, then p

Dis

a reﬂective localization of Dand so it is itself a pointed, strong derivator.

Furthermore, under these hypotheses, p

Dis stable and, letting

Dď0p1q:“ tXPp

Dp1q:XnPDď0p1q,@nPNuand

Dě0p1q:“ tXPp

Dp1q:XnPDě0p1q,@nPNu,

t:“ pp

Dď0p1q,p

Dě0p1qq is a t-structure on p

Dp1q.

Proof. Denote by ε: tow ˝holimNop Ñidp

Dthe counit of the adjunction

ptow,holimNopq. Given IPDia, we have to show that εIis a natural isomor-

phism. Indeed, given XPp

DpIqand εX: towpholimNop Xq Ñ X, we have

to check that n˚pεXqis an isomorphism for all nPN, that is, the natural

map πn: holimNop XÑXninduces an isomorphism pholimNop Xqě´nÑ

Xn, which is clear by Lemma 5.6.

Denote by pΣ,Ωq(resp., pp

Σ,p

Ωq,pΣNop,ΩNop q) the suspension-loop ad-

junction in Dp1q(resp., p

Dp1q,DpNopq). To show that p

Dis stable we have

to show that p

Ω is an equivalence and we know, by Lemma 1.13, that it is

42 SIMONE VIRILI

fully faithful. Let σ:Nop ÑNop be the map nÞÑ n`1; we claim that

σ˚ΣNopXPp

Dp1qand p

Ωσ˚ΣNopX–X. In fact,

Ωσ˚ΣNop X–tow Ω holimNop σ˚ΣNopX

–tow Ω holimNop ΣNopX

–tow Ω Σ holimNop X

–tow holimNop X

–X.

where holimNop σ˚–holimNop by the dual of [Gro13, Prop. 1.24], using that

σis a left adjoint, while holimNop ΣNop –Σ holimNop since left exact (e.g.,

right adjoint) morphisms of pointed derivators commute with suspensions.

Finally, it is clear that p

Dď0p1qis extension closed and that, given XP

Dď0p1qand nPN,n˚p

ΣX“n˚σ˚ΣNopX“ pn`1q˚ΣNop X“ΣXn`1P

Dď´1p1q Ď Dď0p1q, so p

Σp

Dď0p1q Ď p

Dď0p1q. Furthermore, let XPp

Dď0p1q

and YPp

Dě1p1q, then

Dp1qpX,Yq “ DpNopqpX,Yq

“Dp1qNoppdiaNop X,diaNop Yq

“0

where the second equality holds by Lemma 3.6. Let now XPp

Dp1q. Notice

that the object Xě1:“0˚pX0qě1PDpNop qstill belongs to p

Dp1qand it has

a canonical morphism XÑXě1(just compose the unit XÑ0˚X0with

the image under 0˚of the morphism X0ÑXě1

0); we are ﬁnished if we can

prove that the ﬁber of this morphism is in p

Dď0p1q, but this is clear since,

for any nPN,pXnqě1– pX0qě1.

Deﬁnition 5.8. In the notation of Proposition 5.2, we say that Dis left

t-complete if the adjunction

tow : DÕp

D: holimNop

is an equivalence. Dually, Dis right t-complete if the adjunction

hocolimN:q

DÕD: tel

is an equivalence.

Notice that the above deﬁnition is slightly stronger than that given by

Neeman in [Nee11], in that Neeman requires the unit idDp1qÑholimNop tow

to be an iso, but does not include that the corresponding counit is an

iso in the deﬁnition of left-completeness. Furthermore, in case tis count-

ably k-cosmashing for some kPN, so that p

Dis stable, then tow : DÕ

D: holimNop is t-exact, that is, towpDď0q Ď p

Dď0, towpDě0q Ď p

Dě0and so,

whenever tis also a left-complete, we can deduce: holimNop pp

Dď0q Ď Dď0

and holimNop pp

Dě0q Ď Dě0. Dual observations can be done about right-

completeness.

Proposition 5.9. Given kPN, suppose tis countably k-cosmashing. The

following are equivalent:

(1) Dis left t-complete;

MORITA THEORY FOR STABLE DERIVATORS 43

(2) tis left non-degenerate.

Proof. We prove ﬁrst that (1) implies (2). Indeed, if tis left complete, then

any object XPDp1qﬁts into a triangle XÑśnPNXě´nÑśnPNXě´nÑ

ΣX. Furthermore, given XPŞnPNDď´np1q, clearly Xě´n“0 for all nPN

and so, by the above triangle, also X“0.

To show that (2) implies (1), denote by η: idDp1qÑholimNop tow the unit

of the adjunction ptow,holimNopq, consider XPDp1qand let us show that

ηX:XÑholimNop ptowpXqq is an isomorphism. We have already proved

that towpηXqis an isomorphism, so that ηXinduces isomorphisms Xě´nÑ

pholimNop ptowpXqqqě´nfor all nPN. Thus, if we complete ηXto a triangle

KÑXÑholimNop ptowpXqq Ñ ΣK,

we get that Kě´n“0, for all nPN. As a consequence, KPŞnPNDď´np1q,

so K“0 by (2). 

5.2. The half-bounded ﬁltered prederivators. Let us start with the

following deﬁnition:

Deﬁnition 5.10. Given i, j PN, consider the following functors:

pi, Zq:1ˆZÑNop ˆZpj, Zq:1ˆZÑNˆZ

p0, nq ÞÑ pj, nq p0, nq ÞÑ pj, nq.

(Notice that we are using the same notation for diﬀerent functors, but the

meaning of this notation will always be clear from the context). Given

XPDNopˆZpIqand YPDNˆZpIq, we let Xpi,Zq:“ pi, Zq˚XPDZpIq

and Ypj,Zq:“ pj, Zq˚YPDZpIq. We deﬁne the right-bounded (resp.,

left-bounded)ﬁltered prederivator

F´D: Diaop ÑCAT presp., F`D: Diaop ÑCATq

as a full sub-prederivator of DNopˆZ(resp., of DNˆZ) such that, given IP

Dia,F´DpIq(resp., F`DpIq) is the full category spanned by those Xsuch

that, for any iPN, the object Xpi,Zqbelongs to FbDpIq.

As for FbD, in general, F´Dand F`Dare not derivators but, for any

IPDia, F´DpIqand F`Dare full triangulated subcategories of DNopˆZpIq

and DNˆZpIq, respectively. In particular, it makes sense to consider triangles

and to speak about t-structures in F´DpIqand F`DpIq, for some IPDia.

Corollary 5.11. Given IPDia, the functor diaNop :DNopˆZpIq Ñ pDZpIqqNop

induces a functor

diaNop :F´DpIq Ñ pFbDpIqqNop

which is full and essentially surjective. Furthermore, given X,YPF´DpIq,

the diagram functor induces an isomorphism

F´DpIqpX,Yq – FbDpIqNoppdiaNop X,diaNop Yq

provided FbDpIqpΣXpi`1,Zq,Ypi,Zqq “ 0for all iPN.

Proof. Applying Lemma 3.6 to the shifted stable derivator DZ, one can see

that the functor diaNop :DNopˆZpIq “ pDZqNop pIq Ñ DZpIqNop is full and

essentially surjective. Furthermore, the essentially image of the restriction

44 SIMONE VIRILI

of diaNop to F´DpIqis, by construction, exactly pFbDpIqqNop . This proves

the ﬁrst part of the statement.

To verify the second part, let X,YPF´DpIq Ď DNopˆZpIq, then, again

by Lemma 3.6, the diagram functor induces an isomorphism

F´DpIqpX,Yq “ DNopˆZpIqpX,Yq

–DZpIqNoppdiaNopX,diaNop Yq

“ pFbDpIqqNop pdiaNop X,diaNop Yq

if FbDpIqpΣXpi`1,Zq,Ypi,Zqq “ DZpIqpΣXpi`1,Zq,Ypi,Zqq “ 0 for all iPN.

Proposition 5.12. Consider the following two classes of objects in F´Dp1q:

F´Dď0p1q “ tXPF´Dp1q:Xpi,ZqPFbDď0p1q,@iPNu,

F´Dě0p1q “ tXPF´Dp1q:Xpi,ZqPFbDě0p1q,@iPNu.

Then t´

B:“ pF´Dď0p1q,F´Dě0p1qq is a t-structure on F´Dp1qwhose heart

is equivalent to ChbpAqNop.

Proof. Let us verify the axioms of a t-structure for t´

B. Indeed, the classes

F´Dď0p1qand F´Dě0p1qhave the desired closure properties since all the

pi, Zq˚are triangulated functors, and since FbDď0p1qand FbDě0p1qare an

aisle and a coaisle, respectively. Furthermore, given XPF´Dď0p1qand

YPF´Dě1p1q, then, by Corollary 5.11,

F´DpIqpX,Yq “ FbDpIqNop pdiaNop X,diaNop Yq,

in fact, FbDpIqpΣXpi`1,Zq,Ypi,Zqq “ 0 for all iPN, since ΣXpi`1,ZqPFbDď´1

and Ypi,ZqPFbDě1. Furthermore,

FbDpIqNoppdiaNop X,diaNop Yq Ď ź

iPN

FbDpIqpXpi,Zq,Ypi,Zqq “ 0,

since Xpi,ZqPFbDď0and Ypi,ZqPFbDě1. This shows that F´Dď0p1qis left

orthogonal to F´Dě1p1q. Let now ZPF´DpIqand consider diaNop ZP

FbDpIqNop. Then, there is a map φ:XÑdiaNop Zin FbDpIqNop such that,

for any iPN,φi:Xpiq Ñ Zpi,Zqis the left tb

Btruncation of Zpi,Zqin FbDpIq.

By Corollary 5.11, there is a morphism Φ : XÑZsuch that diaNop pΦq “ φ.

It is now easy to show that the following diagram, obtained by completing

Φ to a triangle:

XΦ

ÝÑ ZÑYÑΣX,

is a truncation triangle of Zwith respect to t´

B. This shows that t´

Bis a

t-structure.

To conclude we should verify that F´Dď0p1q X F´Dě0p1q – ChbpAqNop.

By Proposition 4.8, FbDď0p1q X FbDě0p1q – ChbpAq, so clearly the functor

diaNop induces an essentially surjective and full functor

F´Dď0p1q X F´Dě0p1q Ñ ChbpAqNop .

To show that this functor is faithful notice that, given X,YPF´Dď0p1q X

F´Dě0p1q, then ΣXpi`1,ZqPFbDď´1and Ypi,ZqPFbDě0, so that

FbDp1qpΣXpi`1,Zq,Ypi,Zqq “ 0

MORITA THEORY FOR STABLE DERIVATORS 45

for all iPN, showing that faithfulness follows by Corollary 5.11. 

The above corollary and proposition can be easily dualized to obtain

analogous results for F`D.

Deﬁnition 5.13. The t-structure t´

B:“ pF´Dď0,F´Dě0qon F´D(resp.,

B:“ pF`Dď0,F`Dě0qon F`D) described in the above proposition is said

to be the right-bounded (resp., left-bounded)Beilinson t-structure

induced by t.

5.3. The half-bounded totalization functors. Consider the full sub-

precategory TowAof pCh`

AqNop where, for any IPDia, TowApIqis spanned

by those X:Nop ÑCh`pAIqsuch that Xpnq P Chě´npAIq, for any nPN.

Notice that, without speciﬁc hypotheses on A, we do now have a model

category on TowpAIq(like in Subsection 2.3). On the other hand, we can

still consider the following adjunction of prederivators

tow : ChAÕTowA: lim

Nop .

If we start with a bounded above complex XPCh´pAIq, then the associated

tower towpXqactually belongs to pChbpAIqqNop . Hence, letting Tow´

ApIq:“

pChb

ApIqqNop XTowApIq, we can restrict the above adjunction:

tow´: Ch´

AÕTow´

A: lim

Nop .

Dually, we obtain an adjunction

colimN: Tel`

AÕCh`

A: tel`.

Theorem 5.14. Let t´

B“ pF´Dď0,F´Dě0qand t`

B“ pF`Dď0,F`Dě0qbe

the half-bounded Beilinson t-structures on F´Dand F`D, respectively. Con-

sider the following morphisms of prederivators:

Tot´

t: Ch´

tow´

//pChb

AqNop ĎF´DĎDNopˆZhocolimZ//DNop .

Tot`

t: Ch`

tel`

//pChb

AqNĎF`DĎDNˆZhocolimZ//DN.

where the inclusions pChb

AqNop ĎF´Dand pChb

AqNĎF`Dare possible

because pChb

AqNop and pChb

AqNare equivalent to the heart of t´

Band t`

respectively, by Theorem 4.5. Then the following statements hold:

(1) Tot´

ttakes values in p

DĎDNop, while Tot`

ttakes values in q

DĎDN;

(2) given a quasi-isomorphism φin Ch´pAIq(resp., Ch`pAIq) for some

IPDia, then Tot´

tφ(resp., Tot`

tφ) is an isomorphism.

Proof. We prove our statements just in the right-bounded case since the

left-bounded case is completely dual.

(1) Let IPDia and XPCh´pAIq. Given nPN,

phocolimZtow´Xqn–hocolimZptow´Xqn–hocolimZpτě´nXq,

by [Gro13, Prop. 2.6]. Hence, the underlying diagram of Tot´

tXis given by

Totb

tτě0XÐTotb

tτě´1XÐ ¨ ¨ ¨ Ð Totb

tτě´nXÐ ¨ ¨ ¨ ,

so that Tot´

tXPp

DpIqby the already veriﬁed properties of Totb

46 SIMONE VIRILI

(2) Let IPDia and φ:XÑYbe a quasi-isomorphism in Ch´pAIq. We

have to verify that phocolimZtow´φqnis an isomorphism for any nPN.

By [Gro13, Prop. 2.6],

phocolimZtow´φqn–hocolimZptow´φqn–hocolimZτě´nφ.

To conclude recall that we know by Proposition 4.14 that hocolimZτě´nφ“

Totb

tpτě´nφq, which is an iso since τě´nφis a quasi-isomorphism. 

Suppose now that the Abelian category Ahas enough injectives and it

is (Ab.4˚)-kfor some kPN. Then, we have seen in Subsection 2.3 that is

makes sense to consider the prederivator DAwhere, for any IPDia,

DApIq “ DpAIq “ ChpAIqrW´1

where WIis the class of quasi-isomorphisms in ChpAIq. Consider now the

following sub-prederivator D´

AĎDA

D´

A: Diaop ÑCAT

such that D´

ApIqis spanned by those complexes Xsuch that HnpXq “ 0 for

all nąą 0. One can prove that, for any IPDia,

D´

ApIq “ D´pAIq “ Ch´pAIqrpW´

Iq´1s

where W´

Iis the class of quasi-isomorphisms in Ch´pAIq(for this see, for

example, [Sta18, Tag 05RW]). Hence, a consequence of Theorem 5.14 is that

the corestriction Tot´

t: Ch´

AÑp

Dfactors uniquely through the quotient

Ch´

AÑD´

A, giving rise to a morphism of prederivators

real´

t:D´

AÑp

Applying a dual argument to the case when Ahas enough projectives and

it is (Ab.4)-kfor some kPN, we obtain the following morphism of prederi-

vators

real`

t:D`

AÑp

To conclude this subsection let us explain how the half-bounded realiza-

tion functors can be considered as extensions of the bounded realization

functor. Indeed, in the situation of Theorem 5.14, consider the following

restrictions to bounded complexes

Tot´

t: Chb

tow´

//pChb

AqNop ĎF´DĎDNopˆZhocolimZ//p

Tot`

t: Chb

tel`

//pChb

AqNĎF`DĎDNˆZhocolimZ//q

It is not diﬃcult to show that these restrictions of Tot´

tand Tot`

ttake values

in p

t–Db

tand q

t–Db

t, respectively (see Lemma 5.3). Identifying Db

twith

t(resp., q

t), one can show that the restriction of Tot´

t(resp., Tot`

t) to

bounded complexes is conjugated to Totb

MORITA THEORY FOR STABLE DERIVATORS 47

5.4. Conditions on real´

tand real`

tto be fully faithful. We have

seen in the the previous subsection that, whenever the heart Ahas enough

injectives and it is (Ab.4˚)-kfor some kPN, then we can construct a

suitable morphism of prederivators real´

t:D´

AÑp

Dso, composing it with

holimNop :p

DÑDwe get a morphism of prederivators D´

AÑD. In the

following proposition we give some general conditions to make this morphism

fully faithful.

Proposition 5.15. Suppose that tsatisﬁes the following conditions:

(1) tis (co)eﬀa¸cable;

(2) tis left non-degenerate;

(3) tis k-cosmashing for some kPN;

(4) tis (0-)smashing;

(5) the heart Aof thas enough injectives.

Then, there is an exact and fully faithful morphism of prederivators

real´

t:D´

AÑD

that commutes with coproducts.

Proof. By (the dual of) Proposition 5.9, we know that the hypotheses (2)

and (3) on timply that the morphism holimNop :p

DÑDis an equivalence,

so we just need to prove that real´

t:D´

AÑp

Dis fully faithful. Notice that

the fact that tis k-cosmashing implies that Ais (Ab.4˚)-k, in particular,

Ais a Dia-bicomplete (Ab.4˚)-kAbelian category with enough injectives.

By Proposition 2.8, this implies that the functor tow : ChpAq Ñ TowpAq

induces a fully faithful functor DpAq Ñ HopTowpAqq, that restricts to a

fully faithful functor D´pAq Ñ HopTow´pAqq. Furthermore, notice that the

following functors preserve coproducts (in the sense that, given a family of

objects that admit a coproduct in the source category, their images admit a

coproduct in the target category, and such coproduct is naturally isomorphic

to the image of the coproduct in the source category):

(1) the morphism tow´: Ch´

AÑTow´

AĎ pChb

AqNop , since coproducts

are exact in A;

(2) the inclusion pChb

AqNop ÑF´Dcommutes with coproducts. Again,

this fact is a consequence of the fact that tis smashing;

(3) the morphism hocolimZ:F´DÑp

Dcommutes with coproducts. In

fact, it is enough to show that, for any nPN, hocolimZn˚:F´DÑ

FbDÑDcommutes with coproducts by the same argument of the

proof of Corollary 4.15;

(4) the quotient Ch´

AÑD´

Acommutes with coproducts, since coprod-

ucts are exact in A.

By the above discussion, it is easy to see that real´

t:D´

AÑp

Dcommutes

with coproducts. Furthermore, by Proposition 4.17 and our hypothesis that

tis (co)eﬀa¸cable, we deduce that the restriction of real´

tto Db

AĎD´

Ais

fully faithful. By an inﬁnite d´evissage, we get that real´

tis fully faithful. 

Of course, a dual statement gives conditions on real`

t:D`

AÑDto be

fully faithful.

48 SIMONE VIRILI

6. The unbounded realization functor

In this section we introduce the notion of two-sided t-complete deriva-

tor and we use this notion to extend the half-bounded realization functors

real´

t:D´

AÑp

Dand real`

t:D`

AÑq

D, to the unbounded realization functor

realt:DAÑD˛,

where D˛is the two-sided t-completion of D. When tis smashing, k-

cosmashing for some kPN, non-degenerate (left and right) and (co)eﬀa¸cable,

then we can show that realtcommutes with coproducts and it is fully faith-

ful. With dual hypotheses, realtcommutes with products and it is fully

faithful.

Setting for Section 6. Fix throughout this section a category of diagrams

Dia such that NPDia, a strong and stable derivator D: Diaop ÑCAT, a

t-structure t“ pDď0,Dě0qon D, and let A“D♥p1qbe its heart.

6.1. Bicomplete t-structures. Let us start with the following deﬁnition

that, somehow, puts together the notion of left and right completion of

Deﬁnition 5.1:

Deﬁnition 6.1. We deﬁne the t-bicompletion D˛of Dto be the sub-

prederivator

D˛: Diaop ÑCAT

of DNopˆNsuch that, for any IPCat,D˛pIq Ď DNopˆNpIqis the full subcat-

egory of those objects XPD˛pIqsuch that, for all n, m PN,

(1) Xn,m PDě´npIq;

(2) Xn,m PDďmpIq;

(3) the map Xn,m ÑXn´1,m induces an iso pXn,mqě´n`1ÑXn´1,m;

(4) the map Xn,m ÑXn,m`1induces an iso Xn,m Ñ pXn,m`1qďm.

Suppose now that t“ pDď0,Dě0qis a countably k-smashing and count-

ably k-cosmashing t-structure. Hence, both the left t-completion p

Dand the

right t-completion q

Dare stable derivators, each endowed with a natural t-

structure, which is again countably k-smashing and countably k-cosmashing.

Hence, it actually makes sense to consider the stable derivator q

D, the right

t-completion of the left t-completion, and p

D, the left q

t-completion of the

right t-completion. It is not diﬃcult to check that p

D–D˛–q

Proposition 6.2. If t“ pDď0,Dě0qis a countably k-smashing and count-

ably k-cosmashing t-structure for some kPN, the following diagram com-

mutes up to a natural transformation:

telp

▼

towt

telt

▼

▼D˛

towq

MORITA THEORY FOR STABLE DERIVATORS 49

Furthermore, if tis non-degenerate, then the composition towq

ttelt–telp

ttowt

induces an equivalence of derivators D–D˛.

Proof. The proof boils down to the usual fact that left and right trunca-

tions with respect to a given t-structure commute with each other. For the

equivalence D–D˛use Proposition 5.9 and its dual. 

6.2. The (unbounded) ﬁltered prederivator FD.Let us start with

the following deﬁnition:

Deﬁnition 6.3. Given iPN, let pi, Nop ,Zq:1ˆNop ˆZÑNˆNop ˆZ

be the inclusion such that p0, n, mq ÞÑ pi, n, mq. Given XPDNˆNopˆZpIq

we let Xpi,Nop,Zq:“ pi, Nop,Zq˚XPDNopˆZpIq. We deﬁne the unbounded

ﬁltered prederivator

FD : Diaop ÑCAT,

as a full sub-prederivator of DNˆNopˆZsuch that, given IPDia,FDpIqis

the full subcategory of DNˆNopˆZpIqspanned by those Xsuch that, for any

iPN, the object Xpi,Nop ,Zqbelongs to F´DpIq.

As for FbD,F´D, and F`D, in general, FD is not a derivator but, for any

IPDia, FDpIqis a full triangulated subcategory of DNˆNopˆZpIq. In partic-

ular, it makes sense to consider triangles and to speak about t-structures in

FDpIqfor some IPDia.

Corollary 6.4. Given IPDia,diaN:DNˆNopˆZpIq Ñ pDNopˆZpIqqNinduces

a functor

diaN:FDpIq Ñ pF´DpIqqN

which is full and essentially surjective. Furthermore, given X,YPFDpIq,

the diagram functor induces an isomorphism

FDpIqpX,Yq – F´DpIqNpdiaNX,diaNYq

provided F´DpIqpΣXpi,Nop ,Zq,Ypi`1,Nop,Zqq “ 0for all iPN.

Proof. Applying Lemma 3.6 to the shifted stable derivator DNopˆZ, one can

see that the functor diaN:DNˆNopˆZpIq “ pDNopˆZqNpIq Ñ DNop ˆZpIqNis

full and essentially surjective. Furthermore, the essentially image of the

restriction of diaNto FDpIqis, by construction, exactly pF´DpIqqN. This

proves the ﬁrst part of the statement.

To verify the second part, let X,YPFDpIq Ď DNˆNopˆZpIq, then, again

by Lemma 3.6, the diagram functor induces an isomorphism

FDpIqpX,Yq “ DNˆNopˆZpIqpX,Yq

–DNopˆZpIqNpdiaNX,diaNYq

“ pF´DpIqqNpdiaNX,diaNYq

if F´DpIqpΣXpi,Nop,Zq,Ypi`1,Nop,Zqq “ DNopˆZpIqpΣXpi,Nop,Zq,Ypi`1,Nop,Zqq “

0 for all iPN.

Proposition 6.5. Consider the following two classes of objects in FDp1q:

FDď0p1q “ tXPFDp1q:Xpi,Nop,ZqPF´Dď0p1q,@iPNu,

FDě0p1q “ tXPFDp1q:Xpi,Nop,ZqPF´Dě0p1q,@iPNu.

50 SIMONE VIRILI

Then tB:“ pFDď0p1q,FDě0p1qq is a t-structure on FDp1qwhose heart is

equivalent to ChbpAqNˆNop .

Proof. The proof is completely analogous to that of Proposition 4.5, just

using Corollary 6.4 instead of Corollary 5.11. 

Deﬁnition 6.6. The t-structure tB:“ pFDď0,FDě0qon FD described in

the above proposition is said to be the unbounded Beilinson t-structure

induced by t.

6.3. The unbounded realization functor.

Theorem 6.7. Let tB“ pFDď0,FDě0qbe the Beilinson t-structure in FD

induced by t. Deﬁne a morphism of prederivators

Tott: ChAx

tel tow //pChb

AqNˆNop ĎFD ĎDNˆNopˆZhocolimZ//DNˆNop .

Then the following statements hold true:

(1) Totttakes values in D˛;

(2) given a quasi-iso φin ChpAIqfor some IPCat,Tottφis an iso.

Proof. (1) Let IPCat and XPChpAIq. By [Gro13, Prop. 2.6], for any

pn, mq P NˆNop,

phocolimZx

tel towXqpn,mq–hocolimZpx

tel towXqpn,mq

–hocolimZpτě´mτďnXq,

That is, the underlying diagram of TottXis given by

Totb

tτě0τď0X//Totb

tτě0τď1X//Totb

tτě0τď2X//¨¨¨

Totb

tτě´1τď0X

//Totb

tτě´1τď1X//

Totb

tτě´1τď2X//

¨¨¨

Totb

tτě´2τď0X

//Totb

tτě´2τď1X//

Totb

tτě´2τď2X//

¨¨¨

so that TottXPD˛pIqby the already veriﬁed properties of Totb

(2) Let IPCat and φ:XÑYbe a quasi-isomorphism in ChpAIq.

We have to verify that phocolimZx

tel towφqpn,mqis an isomorphism for any

pn, mq P NˆNop. By [Gro13, Proposition 2.6],

phocolimZx

tel towφqpn,mq–hocolimZpx

tel towφqpn,mq–hocolimZτě´mτďnφ.

To conclude recall that we know by Proposition 4.14 that

hocolimZτě´mτďnφ“Totb

tpτě´mτďnφq,

which is an iso since τě´mτďnφis a quasi-isomorphism. 

MORITA THEORY FOR STABLE DERIVATORS 51

6.7 Suppose now that the Abelian category Ahas enough injectives and

it is (Ab.4˚)-kfor some kPNor, alternatively, that the Abelian category

Ahas enough projectives and it is (Ab.4)-k. In both cases, we have seen in

Subsection 2.3 that it makes sense to consider the prederivator DAwhere,

for any IPDia,

DApIq “ DpAIq “ ChpAIqrW´1

where WIis the class of quasi-isomorphisms in ChpAIq. Hence, a conse-

quence of Theorem 6.7 is that the corestriction Tott: ChAÑD˛factors

uniquely through the quotient ChAÑDA, giving rise to a morphism of

prederivators

realt:DAÑD˛.

6.4. Conditions on realtto be fully faithful. We have seen in the

the previous subsection that, under suitable hypotheses on A, we can con-

struct a morphism of prederivators realt:DAÑD˛so, composing with

holimNop hocolimN:D˛Ñp

DÑDwe get a morphism of prederivators

DAÑD. In the following proposition we give some general conditions

to make this morphism fully faithful:

Theorem 6.8. Suppose that tsatisﬁes the following conditions:

(1) tis (co)eﬀa¸cable;

(2) tis non-degenerate (right and left);

(3) tis k-cosmashing for some kPN;

(4) tis (0-)smashing;

(5) the heart Aof thas enough injectives.

Then, realt:DAÑDcommutes with coproducts and it is a fully faithful.

Dually, realt:DAÑDcommutes with products and it is a fully faithful

provided the following hypotheses hold:

(1’) tis (co)eﬀa¸cable;

(2’) tis non-degenerate (right and left);

(3’) tis k-smashing for some kPN;

(4’) tis (0-)cosmashing;

(5’) the heart Aof thas enough projectives.

Proof. We give an argument just for the ﬁrst half of the statement since

the proof of the rest is completely dual. Let z

ChAĎ pCh`

AqNop and Ch˛

AĎ

pChb

AqNˆNop be the sub-prederivators such that, for any IPDia, z

ChApIq “

{

ChpAIqand ChpAIq˛, as deﬁned in Subsection 2.4. By Proposition 6.2, we

know that the hypotheses (2), (3) and (4) on timply that the morphism

holimNop hocolimN:D˛Ñp

DÑDis an equivalence, so we just need to prove

that realt:DAÑD˛is fully faithful. Notice that the fact that tis k-

cosmashing implies that Ais (Ab.4˚)-k, in particular, Ais a Dia-bicomplete

(Ab.4˚)-kAbelian category with exact coproducts and enough injectives.

By Proposition 2.9, this implies that the composition ChAÑz

ChAÑCh˛

induces an equivalence DAÑD˛

A. Furthermore, notice that the following

functors preserve coproducts:

(1) the morphism ChpAq Ñ ChpAq˛, since coproducts are exact in A;

52 SIMONE VIRILI

(2) the inclusion pChb

AqNˆNop ÑFD commutes with coproducts. Again,

this fact is a consequence of the fact that tis smashing, so that the

t-structure tBon FD is also smashing and the inclusion of the heart

preserves coproducts;

(3) the morphism hocolimZ:FD ÑD˛commutes with coproducts. In

fact, it is enough to show that, for any pn, mq P NˆNop, the compo-

sition hocolimZpn, mq˚:FD ÑFbDÑDcommutes with coproducts

by the same argument of the proof of Corollary 4.15;

(4) the quotient ChAÑDAcommutes with coproducts, since coprod-

ucts are exact in A.

By the above discussion, it is easy to see that realt:DAÑD˛commutes

with coproducts. Furthermore, by Proposition 4.17 and our hypothesis that

tis (co)eﬀa¸cable, we deduce that the restriction of realtto Db

AĎDAis

fully faithful. By an inﬁnite d´evissage, we get that realtis fully faithful. 

MORITA THEORY FOR STABLE DERIVATORS 53

7. Applications to co/tilting equivalences

In this ﬁnal section we list some applications of the general constructions

developed in the rest of the paper to co/tilting equivalence. In this way we

recover some important results from [Ric91, ˇ

Sˇ

to14, NSZ15, PV15]. Let us

start recalling some deﬁnitions from [NSZ15, PV15]:

Deﬁnition 7.1. A set Sin a triangulated category Dis said to be

‚silting if pSKą0,SKă0qis a t-structure in Dand SĎSKą0;

‚cosilting if pKă0S,Ką0Sqis a t-structure in Dand SĎKą0S;

‚tilting if it is silting and AddpSq Ď SK‰0;

‚cotilting if it is cosilting and ProdpSq Ď K‰0S;

‚classical tilting if it is tilting and any SPSis compact.

We collect in the following lemma some basic properties of co/tilting

objects from [NSZ15, PV15]:

Lemma 7.2. Suppose that Dhas products and coproducts, and let SĎD

be a set. Then,

(1) if Sis tilting, the associated t-structure is (co)eﬀa¸cable, cosmashing,

non-degenerate and Sis a set of generators of D. Furthermore,

products are exact in the heart and Sis a set of projective generators

of the heart;

(2) if Sis cotilting, the associated t-structure is (co)eﬀa¸cable, smashing,

non-degenerate and Sis a set of cogenerators of D. Furthermore,

coproducts are exact in the heart and Sis a set of injective cogener-

ators of the heart;

(3) if Sis a classical tilting set, then the associated t-structure is smash-

ing and its heart is equivalent to a category of modules over a ring

with enough idempotents. If, furthermore, S“ tSuconsists of a

single object, then the heart is equivalent to ModpEndDpSqq.

Proof. Parts (1) and (2) follow similarly [PV15, Prop. 4.3, Lem. 4.5 and

Prop. 5.1]. Part (3) follows by [NSZ15, Coro. 4.3]. 

We will apply the following consequence of Brown representability to

prove the essential surjectivity of realtin several cases:

Lemma 7.3. Let Dbe a triangulated category with coproducts and with a set

of compact generators S. Then Dsatisﬁes the principle of inﬁnite d´evissage

with respect to S.

Proof. Consider the inclusion L: LocpSq Ñ Dand notice that this has a

right adjoint R:DÑLocpSqby Brown representability. Let XPD, then

there is a triangle

LRX ÑXÑCÑΣLRX

with LRX PLocpSqand CPLocpSqK(see, [Kra10, Prop. 5.3.1]). Since

LocpSqKĎ`ŤiPZΣiS˘K“0, that is, Sis a set of generators, then X–

LRX PLocpSq.

As a ﬁrst application of our machinery of realization functors we can prove

the following expected extension to the equivalence induced by a classical

tilting object to the context of stable derivators:

54 SIMONE VIRILI

Theorem 7.4. The following are equivalent for a strong and stable derivator

D: Catop ÑCAT:

(1) there is a classical tilting set SĎDp1q(resp., a classical tilting object

TPDp1q);

(2) there is a small preadditive category Cand an equivalence of de-

rivators F:DModpCqÑD(resp., a ring Rand an equivalence of

derivators F:DModpRqÑD).

Proof. If we assume (2), then the representable C-modules form a classical

tilting set in DModpCqp1q, so S:“ tFpCp´, cqq :cPCuis a classical tilting

set in Dp1q.

On the other hand, if Sis a classical tilting set in Dp1q, then let tS“

pDď0

S,Dě0

Sqbe the associated t-structure and consider

F:“realS:DModpCSqÑD,

where CSis the small preadditive category given by Lemma 7.2 (3). By

Theorem 6.8 and Lemma 7.2, Fis fully faithful. Furthermore the image of

Fis a localizing subcategory containing the set of compact generators S,

hence, by Lemma 7.3, Fis also essentially surjective. 

Before stating our main applications, let us recall the following deﬁnition:

Deﬁnition 7.5. [FMT16] A pair of t-structures pt1“ pDď0

1,Dě0

1q,t2“

pDď0

2,Dě0

2qq is said to have shift kPZand gap nPNif kis the maxi-

mal number such that Dďk

2ĎDď0

1and nis the minimal number such that

Dď´n

1ĎDďk

2. In this case we say that the pair pt1,t2qis of type pn, kq.

We say that t1and t2have ﬁnite distance if there exist kPZand nPN

for which the pair pt1,t2qis of type pn, k q.

Lemma 7.6. Let pt1,t2qbe a pair of t-structures is of type pn, kqon a

triangulated category D. If t2is h-smashing (resp., h-cosmashing) for a

given hPN, then t1is pn`hq-smashing (resp., pn`hq-cosmashing).

Proof. Let tDiuibe a family of objects in Dě0

1ĎDěk

2; if t2is h-smashing,

šiDiPDěk´h

2ĎDě´n´h

1. Similarly, consider a family tEiuiof objects in

Dď0

1ĎDďk`n

2; if t2is h-cosmashing, śiDiPDďk`n`h

2ĎDďn`h

1.

Notice that, by Lemmas 7.3 and 7.6, when a tilting t-structure is at

ﬁnite distance from a classical tilting t-structure, then it is cosmashing and

k-smashing for some kPN. Similarly, when a cotilting t-structure is at

ﬁnite distance from a classical tilting t-structure, then it is smashing and

k-cosmashing for some kPN. This easy observation allows us to apply our

machinery of realization functors to such t-structures:

Theorem 7.7. Let D: Catop ÑCAT be a strong, stable derivator, and

suppose that we have a tilting set Tand a classical tilting set Sin Dp1q.

Let tT“ pDď0

T,Dě0

Tqand tS“ pDď0

S,Dě0

Sqbe the induced t-structures and

suppose that they have ﬁnite distance. Letting AT:“Dď0

Tp1qXDě0

Tp1q, there

is an equivalence of prederivators

realtT:DATÑD

MORITA THEORY FOR STABLE DERIVATORS 55

that restricts to equivalences real´

tT:D´

ATÑD´

tT,realb

tT:Db

ATÑDb

tT, and

ypATq Ñ D♥. In particular, DATis a (strong and stable) derivator.

Proof. By Lemma 7.2, pDď0

Tp1q,Dě0

Tp1qq is (co)eﬀa¸cable, cosmashing, non-

degenerate and it is n-smashing, for some nPN, by Lemma 7.6. On the

other hand, AThas exact products, it has a projective generator Xand it

is (Ab.4)-n. Let us verify that tI:“ pDď0

TpIq,Dě0

TpIqq and AI

T“Dď0

TpIq X

Dě0

TpIqhave the same properties for all IPDia. In fact it is easy to verify

that tIis cosmashing, non-degenerate, n-smashing, and that AI

Thas exact

products, it has a set of projective generators ti!T:iPI, T PTuand

it is (Ab.4)-n. It remains to show that tIis eﬀa¸cable. Let Y1, Y2PAI

ną0 and φ:Y1ÑΣnY2in DpIq. Since ti!T:iPI, T PTuis a family of

generators, there is an epimorphism šiPI, T PTi!TpAi,T qÑY1. Then,

DpIq˜ž

iPI, T PT

pi!TpAi,T q,ΣnY2¸“ź

iPI, T PTź

Ai,T

DpIqpT, Σni˚Y2q “ 0

since Y2PTKą0. Hence, by Theorem 6.8,

realtT:DATÑD

is exact, fully faithful and it commutes with coproducts. Therefore, the

essential image of realtTis a localizing category, furthermore SĎSK‰0Ď

tT“realtTpDb

ATq, thus by Lemma 7.3, realtTpDATpIqq “ DpIqfor all I,

so realtTis also essentially surjective. 

As a corollary we obtain one of the main results of [NSZ15]:

Corollary 7.8. [NSZ15, Thm. 7.4] Let Kbe a commutative ring, let Dbe

a compactly generated algebraic triangulated K-category, let Tbe a bounded

tilting set of Dand let ATbe the heart of the associated t-structure. The

inclusion ATÑDextends to a triangulated equivalence Ψ: DpATq Ñ D.

Proof. Compactly generated algebraic triangulated K-categories are trian-

gle equivalent to the derived categories of small dg-categories, and such

derived categories are the homotopy categories of a suitable (combinatorial)

model category. We refer to [ˇ

SP16, Sections 2.2 and 2.3] for a more detailed

discussion. In particular, one can show that there is a strong stable deri-

vator D: Catop ÑCAT such that Dp1q – D. Furthermore, using that Dis

equivalent to the derived category of a small dg-category, one can consider

the set Sof representable modules as a classical tilting set in D. By the

results in [NSZ15, Section 5], the fact that Sis bounded implies that the

t-structures tTand tSare at ﬁnite distance, so Theorem 7.7 applies to give

an equivalence of derivators realtT:DATÑDthat, evaluated at 1gives an

equivalence DpATq Ñ D.

Let us now pass to the dual setting of cotilting t-structures:

Theorem 7.9. Let D: Catop ÑCAT be a strong, stable derivator, and

suppose that we have a cotilting set Cand a classical tilting set Tin Dp1q.

Let tC“ pDď0

C,Dě0

Cqand tT“ pDď0

T,Dě0

Tqbe the induced t-structures and

56 SIMONE VIRILI

suppose that they have ﬁnite distance. Letting AC:“Dď0

Cp1qXDě0

Cp1q, there

is an equivalence of prederivators

realtC:DACÑD

that restricts to equivalences real`

tC:D`

ACÑD`

t,realb

tC:Db

ACÑDb

tC, and

ypACq Ñ D♥. In particular, DACis a (strong and stable) derivator.

Proof. The proof that realtC:DACÑDis exact, fully faithful and that it

commutes with products is completely dual to that given for Theorem 7.7.

It remains to verify that realtCis essentially surjective. By Theorem 7.4,

there is a small preadditive category CTand an equivalence of derivators

realtT:DModpCTqÑD,

so we can identify these two derivators and consider realtCas a morphism

DACÑDModpCTq. Notice that the image of this morphism is a colocalizing

sub-prederivator containing Db

ModpCTqso, by Lemma 2.4, realtCis essentially

surjective. 

The setting of the above theorem may seem a bit unnatural since we

are asking to a cotilting t-structure to be “close” to a tilting t-structure.

To show that this is not so artiﬁcial, let us recall the following deﬁnition

from [HC01].

Deﬁnition 7.10. Given a ring R, a right R-module Cis said to be big

cotilting if there exists nPNsuch that the following three conditions are

satisﬁed:

(1) Chas injective dimension bounded by n;

(2) Extj

RpCI, Cq “ 0for every ją0and every set I;

(3) there exists an exact sequence 0ÑCrÑ... ÑC1ÑC0ÑQÑ0

in ModpRqsuch that Qis an injective cogenerator of ModpRq,rPN

and CiPProdpCqfor all i.

By [ˇ

Sˇ

to14, Thm. 4.5], a big cotilting R-module C, when considered as

an object of DpModpRqq, is a cotilting object. Furthermore, letting tCbe

the associated t-structure in DpModpRqq, applying [PV15, Prop. 4.17 and

4.14], one gets that tCis at ﬁnite distance from the natural t-structure tRin

DpModpRqq. To conclude notice that tRis induced by the classical tilting

object R, so we are exactly in the setting of Theorem 7.9. In this way we

obtain a new proof for the following important result:

Corollary 7.11. [ˇ

Sˇ

to14, Thm. 5.21] Let Rbe a ring, CPModpRqa big

cotilting module and Gthe corresponding tilted Abelian category. Then the

prederivators DModpRqand DGare equivalent and both are strong stable de-

rivators.

As another consequence of our general theory of realization functors,

one can obtain a “Derived Morita Theory” for Abelian categories, com-

pleting [PV15, Thm. A]. Let us remark that the implication “(3)ñ(1)”, in

the special case when XPAĎDpAq, can be proved alternatively with the

methods of [FMS17].

MORITA THEORY FOR STABLE DERIVATORS 57

Theorem 7.12. Let Abe a Grothendieck category (resp., an (Ab.4˚)-h

Grothendieck category for some hPN), denote by tA“ pDď0

A,Dě0

Aqthe

canonical t-structure on DA, and let Bbe an Abelian category. The following

are equivalent:

(1) Bhas a projective generator (resp., an injective cogenerator) and

there is an exact equivalence DpBq Ñ DpAqthat restricts to bounded

derived categories;

(2) Bhas a projective generator (resp., an injective cogenerator) and

there is an exact equivalence of prederivators DBÑDAthat restricts

to an equivalence Db

BÑDb

(3) there is a tilting (resp., cotilting) object Xin DpAq, whose heart

XK‰0(resp., K‰0X) is equivalent to Band such that the associated

t-structure tX“ pDď0

X,Dě0

Xqhas ﬁnite distance from tA.

Proof. The implication “(1)ñ(3)” is proved in [PV15, Thm. A], while the

implication “(2)ñ(1)” is trivial. Let us give an argument for the implication

“(3)ñ(2)” in the tilting case, the cotilting case is dual. Indeed, by Lemma

7.2, tX:“ pXKą0, XKă0qis (co)eﬀa¸cable, cosmashing and non-degenerate.

Since coproducts are exact in any Grothendieck category, tAis smashing

and so, by Lemma 7.6, tXis n-smashing for some nPN. Similarly, Bhas

exact products, it has a projective generator Xand it is (Ab.4)-n. Exactly

as in the proof of Theorem 7.9, one can show that the lifting of tXto any

DpAIqhas the same properties. Hence, Theorem 6.8 applies to give an exact

fully faithful morphism of prederivators

realtX:DBÑDA

that commutes with coproducts. Therefore, for any IPCat, realtXpDpBIqq

is a localizing subcategory of DpAIqthat contains Aso, by Lemma 2.4,

realtXpDpBIqq “ DpAIq, so realtXis also essentially surjective. 

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Morita theory for stable derivators

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