Derivators, pointed derivators, and stable derivators

DERIV A TORS, POINTED D ERIV A TORS, AND ST ABLE DERIV A TORS MORITZ GR OTH Abstract. W e dev elop some aspects of the theory of deriv ators, p oint ed deri v ators, a nd stable deriv ators. As a main result, we sho w that the v alues of a stable deriv ator can b e canonically endo wed with the structure of a tri ang ulated cat egory . Moreov er, the f unct ors belonging to the stable deriv ator can be turned into exact f unct ors wi th respect to these triangulate d struct ures. Along the wa y , we giv e a simpliﬁcation of the axioms of a p oint ed deriv ator and a reformulation of the base c hange axiom in terms of Grothendiec k (op)ﬁbration. F urthermore, we hav e a new proof that a combinato rial mo del category has an underlying deriv ator. Contents 0. Int ro duction and plan 1 1. Deriv ators 5 1.1. Basic deﬁnitions 5 1.2. Homotopy exact squares and some prop erties of homotopy Kan extensions 14 1.3. Examples 20 2. The 2-ca tegory o f deriv ators 26 2.1. Morphisms and natura l transformations 26 2.2. Homotopy (co)limit preserving morphisms 28 3. Poin ted deriv ators 35 3.1. Deﬁnition and ba sic examples 35 3.2. coCar tesian a nd Cartesian squares 38 3.3. Susp ensions, Lo ops, Cones, and Fibers 42 4. Stable deriv a tors 46 4.1. The additivity of stable deriv ator s 46 4.2. The canonical tria ngulated structures 53 4.3. Recollements of triangulated categorie s 58 References 60 0. Introduction and plan The theo r y of stable deriv a tors as initiated by Heller [Hel88, Hel97] and Grothendieck [Gro] and studied, at lea st in similar settings, among others , by F ranke [F ra 96 ], Keller [Kel91 ] and Ma ltsinio- tis [Ma l07], can b e motiv a ted by saying that it pr o vides a n enha ncemen t of triangulated categories. T ria ngulated ca teg ories suﬀer the well-kno wn defect that the co ne co nstruction is not functorial. A c o nsequence of this non-functoria lity of the co ne construction is the fact that there is no go o d Date : Nov ember 27, 2024. 1 2 MORITZ GROTH theory of homotopy (co)limits for triangula ted catego ries. One can still deﬁne these notions, at least in so me situations wher e the functor s ar e deﬁned on categorie s which ar e freely gener ated by a graph. This is the ca se e.g . for the cone constructio n itself, the homotopy pushout, and the homotopy co limit of a sequence of mor phis ms . But in all these situa tions, the ‘univ ersal ob jects’ are only unique up to non -cano nical isomorphism. The slo gan used to describ e this situation is the follo wing one: dia grams in a triangulated categ o ry do not car ry suﬃcient informa tion to deﬁne their homotopy (co)limits in a c anonic al wa y . But in the t ypica l situations, a s in the case of the der iv e d category of an a belian categor y or in the case of the homotopy catego ry of a stable mo del resp. ∞− categor y , the ‘mo del in the background’ allows for suc h constructions in a functor ial manner. So, the passag e from the model to the derived resp. homotopy c ategory truncates the av ailable information to o strongly . T o b e mor e sp eciﬁc, let A be a n a belian categ ory such that the derived catego ries which o ccur in the following discussion exist. Moreov er, let us denote by C ( A ) the catego ry o f chain complexes in A . As usual, let [1 ] b e the ordinal 0 ≤ 1 consider ed as a catego ry (0 − → 1) . Hence, for an arbitrar y categor y C , the functor category C [1] of functors fr om [1] to C is the a rrow ca tegory of C . With this notatio n, the cone functor at the level of a belian categ ories is a functor C : C ( A [1] ) ∼ = C ( A ) [1] − → C ( A ) . But to give a c onstruction of the cone functor in terms of homotopic al algebra only , one has to cons ider more general diagra ms. F o r this pur p ose, let f : X − → Y be a morphism of chain complexes in A . Then the cone C f of f is the homotopy pushout of the following dia gram: X   f / / Y 0 A t the level of derived categories , the cone co nstruction is again functorial when c onsider e d as a functor D ( A [1] ) − → D ( A ) . The imp ortant po in t is that one forms the arrow categories b efor e passage to the derived ca tegories. Said diﬀerently , at the level of derived c ategories, we have, in general, D ( A [1] ) ≇ D ( A ) [1] . Moreov er, as we hav e mentioned, to actua lly give a co nstruction of this functor one needs appar en tly also the derived c a tegory o f diagra ms in A of the above shap e and a homoto p y pushout functor. More sys tematically , one should not only consider the deriv ed category of an ab elian categor y but als o the derived ca tegories of diag ram catego ries a nd r e striction and homotopy Kan extension f unctors betw een them. This is t he basic idea b ehind the no tion of a deriv ator . But the theory of deriv ator s is more than ‘only an enhancement of triangulated ca tegories’. In fact, it gives us an a lternativ e axioma tic approa c h to an abstract homotopy theory (cf. Re - mark 2.14). As in the theory of mo del categories and ∞− categ ories, there is a certain hier a rc h y of such structures: the unp oin ted situatio n, the p oin ted situation, and the stable situa tion. In the classical situation of top ology , this hier arch y cor respo nds to the passa ge from spaces to p ointed spaces a nd then to sp ectra. In classical homolog ical algebra , the passag e from the der ived catego ry of non-negatively gra ded chain complex es to the unbounded derived ca tegory c a n b e seen as a sec- ond example for passing from the p ointed to the s ta ble situation. In the theory of deriv ator s this threefold hierarch y o f structure s is also pres en t, and th e corresp onding notions are then deriv ato rs, po in ted deriv ator s , a nd sta ble deriv a to rs. F ranke has introduced in [F ra 96] a theory of systems of tria ng ulated diagram categor ies which is similar to the notion of a stable deriv ato r . The fact that the theory of der iv a tors admits the men tioned threefold hierar c hy of str uctures is o ne main adv antage o ver the approach of F ranke. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 3 The theory desc r ibed in this pap er is not completely new. In particular, it owes a lot to Maltsin- iotis who exp osed and expa nded the foundations of the theory originating with Gr othendiec k. The ﬁrst t w o chapters ca n b e considere d as a review of these foundatio ns, although our exp osition de- viates somewha t from existing o nes. In particular , we make systema tic us e of the ‘ba se change formalism’ from the v ery b eginning, resulting in a streamlined developmen t of the theo ry . The more orig inal part of the pap er lies in the remaining tw o chapters in which we use a simpliﬁed notion of p oin ted deriv ator s a s w e discus s b elow. W e g iv e a co mplete and self-contained proof that the v alues of a stable deriv ator c a n be c anon- ic al ly endow ed with the s tructure of a tr iangulated categor y . Similarly , we sho w that the functors which a re part of the deriv ator can be c anonic al ly tur ned in to exact functors with resp ect to these structures. This is in a sense the main work and will o ccupy the bulk of this pap er. More over, we build on ideas of F ranke [F ra96] from his theory of systems of triangula ted diagram ca tegories and adapt them to this alternativ e set o f ax io ms. Similar idea s are used in Lurie’s [Lur1 1] on the theo ry of stable ∞− ca tegories. Along the way we g iv e a s impliﬁcation o f the axioms o f a pointed der iv a tor. The usual deﬁnition of a po in ted deriv ato r, here called a strongly p ointed deriv ato r, is for m ula ted using the notion of cosieves and siev e s. O ne usua lly demands that the homotopy left K an e x tension functor i ! along a cosieve i has itself a left a djoin t i ? , a nd similarly th at the homotopy r igh t Kan extension functor j ∗ along a sieve j has a right adjoint j ! . Mo tiv a ted b y alg e braic geo metry , these additional adjoin ts are then calle d exceptional r esp. c oexceptiona l inv er se image functors. W e s ho w that this deﬁnition can be simpliﬁed. It suﬃces to ask that the underlying categ o ry of the deriv ator is p oin ted, i.e., has a zero ob ject. This deﬁnition is more easily motiv ated, more int uitive for top ologists , and, of course, simpler to ch eck in examples. W e give a dire ct pro of o f the equiv ale nc e of thes e t w o notions in Section 3. A seco nd pro of of this is given in the s table setting using the fact that recollements of triangula ted categ ories are o v erdetermined (cf. Subsection 4.3). The author is aw are of the fact that there will b e a written up version of a pro of of the existence of these canonical triangulated structures in a future pap er by Maltsiniotis. In fact, Maltsiniotis presented an alterna tiv e, unpublished v ariant of F ra nk e’s theorem in a seminar in P ar is in 2001. He show ed that this notion of stable deriv ators is equiv a len t to a v ariant thereo f (as used in the thesis of Ay o ub [Ayo07a, Ayo07b]) where t he triangulations are par t of the notion. Nevertheless, w e give this indep enden t account. Mo reov er, the construction of the susp ension functor in [CN08 ] and the axioms in [Ma l0 7] indicate that that pro of will use the (co)exceptional in verse image functor s . But one p o in t here is to show that these functors ar e no t needed for these purp oses. W e no w turn to a s ho rt de s cription of the co n tent o f the paper. In Sec tion 1, w e give the cent ral deﬁnitions a nd deduce some immediate consequences of the a xioms. The e xistence of cer tain very sp ecial (co)limits can b e explained using the s o-called (partial) underlying dia gram functors. W e develop so me aspects of the ‘base change calculus’ (Subsection 1.2) which is the ma in tool in most of the pro ofs in this pap er. Using that calc ulus we ar e able to characterize deriv ators by saying that they satisfy ba se change for Grothendieck (o p)ﬁbr a tions. This in tur n is the key ingre die nt to establish the theoretically important cla ss of examples, that for a der iv ator D the prederiv a tor D M (cf. E xample 1.7) is also a deriv a tor (Theorem 1.3 1 ). As a further class o f examples, we give a simple, i.e., completely forma l, proof that co m binator ial model categ ories have underlying deriv ators. In Section 2, we in tro duce mo rphisms and na tur al transfor mations in the co n text o f deriv a tors which leads to the 2 -category Der of deriv ato rs. W e then tur n to homotopy-colimit pr eserving morphisms and establish some basic facts ab out them. In particula r, ag ain using the fact that 4 MORITZ GROTH deriv ator s satisfy base change for Grothendieck (op)ﬁbrations we show that homotopy Kan exten- sions in a deriv ato r of the for m D M are calculated p oin t wise (Pro p osition 2.6) which will b e o f some impo rtance in Section 4. Moreov er, w e study in some detail the notion of an adjunction b et ween deriv ator s . In Section 3, w e consider p ointed deriv a tors and g iv e the typical examples. W e pr ove that o ur ‘weak er ’ deﬁnition of a p oin ted deriv ator is equiv alent to the ‘stronger’ one using the (co)exceptional inv ers e imag e functors (Corolla ry 3.8). Mo reo ver, in the po in ted cont ext ho motop y rig h t Kan ex - tensions alo ng sieves give ‘extension b y zero functors’ and dually f or co siev es (P r opos itio n 3.6). W e brieﬂy talk ab out (co)Ca rtesian squar es in a der iv ator and deduce some pr operties ab out them. An imp ortant example of this k ind of results is the comp osition and cancellation pr operty of (co)Cartesian squares (P ropos ition 3 .14). Another o ne is a ‘detection result’ for (co)Cartes ia n squares in lar ger diagrams (Prop osition 3.11) whic h is due to F ranke [F ra96]. W e clo se the section by a discussion of the imp ortant suspension, lo op, co ne, a nd ﬁber functors . In the ﬁnal se ction, we stick to sta ble deriv ators fo r which by deﬁnition the class es of coCa rtesian and Cartesia n squar es coincide. Some nice consequences of this ar e that the sus p ension and the lo op morphisms deﬁne inverse equiv alences, that biCar tes ian squares satisfy the 2 -out-of-3 prop erty , a nd that w e are w orking in the additiv e con text (Prop osition 4 .7 and Corollar y 4.14). The main aim of the sectio n is to es tablish the canonical triang ulated structures on the v alues of a stable deriv a tor (Theorem 4.15). These are pr eserv ed by exact morphisms of stable deriv ators (Pr opos itio n 4.18) and, in particular, by the functors belo nging to the stable deriv ato r itself (Corollar y 4.19). In the last subsection, we remark that, given a stable der iv ator and a (co)sieve, we obtain a r ecollemen t of tria ngulated ca tegories. This reproves, in the stable cas e, that p oin ted deriv a tors are ‘strongly po in ted’. There are three more rema rks in o rder b efore we b egin with the pap er. Firs t, we do not dev elop the g eneral theory of der iv a tors for its own sa k e and a ls o not in its br oadest generality . In this pap er, w e o nly dev elop as m uc h of the genera l theor y as is needed to give complete, self-contained pro ofs of the mentioned results. Nevertheless, this pap er may serve a s an introduction to ma n y central ideas in the theory of deriv a tors and no prior knowledge is assumed. The second remark concerns duality . Many of the statemen ts in this pap er hav e dua l statemen ts which also hold true by the dual pro of (the reas o n for this is Example 1.16). In most ca ses, we will not make these statements explicit a nd we will hardly ever g iv e a pro of of b oth statement s. Nevertheless, we allow o ur selv es to refer to a statement also in cases where, strictly sp eaking, the dual statement is needed. The last remar k conce r ns the ter minology emplo yed here. In the existing litera tur e on deriv ators , the term ‘triangulated deriv a tor’ is used instead of ‘stable deriv a tor’. W e preferr ed to use this diﬀerent terminolo gy for tw o rea sons: Fir s t, the terminology ‘triangula ted deriv a to r’ (introduced by Maltsiniotis in [Ma l0 7]) is a bit misleading in that no triangulatio ns a re part of the initial data. One main p oint of this pap er is to give a pr oof tha t these tr iangulations can be cano nically constructed. Thus, from the p ersp ectiv e of the t ypical distinction b et ween stru ctur es and pr op erties the autho r do es not lik e the former terminology to o m uch. Second, in th e r elated theories of mo del categorie s and ∞− ca tegories, cor r esponding notions exist and are called st able model catego ries and stable ∞− catego ries resp ectively . So, the terminology stable der iv a tor r eminds us o f the related theories. 1 1 This research w as supp orted by the De utsc he F orsc h ungsgemeinsc haft within the graduate program ‘H omotopy and Cohomology’ (GRK 1150) DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 5 1. D eriv a tors 1.1. Basi c de ﬁni tions. As we mentioned in t he in tro duction, the basic idea b ehind a der iv ator is to conside r simultaneously derived or homotopy categories of dia gram categories of diﬀerent shap es. So, the most ba s ic notion in this business is the following one. Deﬁnition 1.1. A pr e derivator D is a strict 2-functor D : Cat op − → CA T . Here, Cat denotes the 2 -category of small categorie s , Cat op is obtained from Cat by reversing the direction of the functors, while CA T denotes the ‘2 -category’ of not necessa rily small categ ories. There a re the usual set- theoretical problems with the notion of the ‘2-ca teg ory’ CA T in that this will no t be a categor y enric hed ov er Cat . Since we will never need this no n-fact in this pap er, we use slogans as the ‘2 - category CA T ’ as a co n venient parlance and think instead o f a pr ederiv ator as a function D as we des cribe it now. Given a prederiv ator D and a functor u : J − → K , an application of D to u g iv es us tw o categories D ( J ) , D ( K ) , and a functor D ( u ) = u ∗ : D ( K ) − → D ( J ) . Similarly , g iv en t wo functors u, v : J − → K and a natural transformation α : u − → v , w e obtain a n induced natural tra nsformation α ∗ as depicted in the next diagra m: J u ' ' v 8 8 ✤ ✤ ✤ ✤   α K D ( K ) u ∗ * * v ∗ 4 4 ✤ ✤ ✤ ✤   α ∗ D ( J ) This datum is compatible with compositions and iden tities in a strict sense, i.e., w e hav e equalities of the resp ectiv e expressions and not only coherent natura l iso mo rphisms b et w een them. F or the relev ant basic 2-ca tegorical notions, which were in tro duced by Ehre s mann in [Ehr63], we r e fer to [KS05] or to [Bo r94a, Chapter 7], but nothing deep from that theory is needed here . The following examples give a n idea o f how suc h prederiv ators arise. Among these pr obably the second, third, and fourth one are the exa mples to ha ve in mind in later sections . Example 1.2. Every category C gives r ise to the pr e derivator r epr esente d by C : y ( C ) = C : J 7− → C J Here, C J denotes the functor c a tegory of functors from J to C . An ticipating the fact that w e ha ve a 2- category PDer of prederiv ator s (cf. Section 2) w e want to men tion that this example extends to a (2-c a tegorical) Y o neda embedding y : CA T − → PD er . In this a nd the co mpanion pap ers ([Gro11, Gro 12a ]) we in tro duce many notions for der iv ators which are analogs of well-kno wn no tions from ca tegory theory . Then it will be imp ortant to se e that these notions are extensions of the c la ssical ones in that b oth notions coincide on the repres en ted (pre)deriv ator s. Example 1.3. Let A be a s uﬃcien tly nice ab elian categor y , i.e., s uc h that we can form the derived categorie s o ccurring in this ex a mple witho ut running into set theo retical problems. Recall that, by deﬁnition, the derived categor y D ( A ) is the loca lization of the categ ory o f chain complexes at the class of quas i-isomorphisms. F or a categor y J , the functor categor y A J is again an ab elian category . In the asso ciated categor y of chain complexes C ( A J ) ∼ = C ( A ) J , quasi-iso morphisms are deﬁned p oin t wise, so that restriction of diagra m functors induce functors on the level o f derived categorie s. Thus, w e have th e pr e deriva tor D A asso ciate d to an ab elian c ate gory A : D A : J 7− → D A ( J ) = D ( A J ) 6 MORITZ GROTH The next example assumes so me knowledge of mo del catego ries. The orig inal reference is [Qui67] while a well written, leisurely intro duction to the theory can b e found in [DS95]. Much more materia l is treated in the monogra phs [Hov99] and [Hir03]. Con v ention 1. 4. In this pa per mo del categor ie s ar e assumed to have limits and co limits of all smal l (as oppos ed to only ﬁnite) diag rams. F urthermore, we do not take functorial factorizations as part o f the no tion o f a mo del c ategory . Fir st, this would b e a n additio na l structure on the model categorie s which is anyhow not resp ected by the morphisms, i.e., by Quillen functors. Second, this assumption would be a bit in co nﬂict with the philos o ph y of higher categ o ry theory . The categ ory of -say- coﬁbrant replacements of a g iv e n o b ject in a mo del categ ory is contractible so tha t any choice is equally go od and ther e is no e s sen tia l diﬀ erence once one passes to ho motop y categories. W e r efer to [Hir03, Part 2 ] f or many results along these lines. Example 1.5. Let M b e a coﬁbra n tly ge ne r ated mo del categor y . Recall that one of the go o d things ab out co ﬁbran tly g enerated mo del c ategories is that diagra m catego ries M J can b e endowed with the so- c alled pr oje ctive mo del st r u ctur e . In more deta il, let us call a natural tra nsformation of M − v alued functors a pr oje ctive ﬁbr ation if all compone nts a re ﬁbrations, and s imilarly a pr oje ctive we ak e quivalenc e if all comp onen ts are weak equiv a lences in M . A pr oje ctive c oﬁbr ation is a map which has the left-lifting-pro perty with r espect to all maps which ar e s im ultaneo usly pro jective ﬁbrations and pr o jective w ea k e q uiv a lences. With these deﬁnitio ns, M J is aga in a model categor y and we can th us consider the asso ciated homo top y categor y . Recall that the canonica l functor γ : M − → Ho ( M ) from M to its homotopy category is a 2-lo calization. This means, that γ induces for every category C an isomorphism of c ate gories γ ∗ : C Ho ( M ) − → C ( M ,W ) where the right-hand-side denotes the full sub category of C M spanned b y the functors which send weak equiv alences to isomorphisms. Mor eo v er, since pro jectiv e w eak equiv ale nces are deﬁned as levelwise weak equiv alences, these are pr e s erv ed by restriction of diagr am functors. By the univ ersal prop ert y of the lo calization functors the restriction of diagram functors des cend uniquely to the homotopy categorie s . Th us, g iv en suc h a co ﬁbran tly g e nerated mo del catego ry M , we ca n form the pr e derivator D M asso ciate d to M if we s e t D M : J 7− → D M ( J ) = Ho ( M J ) . A similar example can be given using the theory of ∞− categ ories (ak a. quasi-categ ories, weak Kan complexes ), i.e., of simplicia l sets sa tis fying the inner ho rn extension pro perty . These were originally intro duced by Boar dman and V o gt in their w or k [BV73] on homotopy inv ariant algebra ic structures. Detailed accounts of this theory ar e given in the tomes due to Joyal [Joy08b , Joy08c, Joy08a , Joy ] and Lurie [Lur09, Lur11]. A short exp osition of man y of the cen tral ideas and also of the philosophy of this theory can b e found in [Gro 1 0 ]. Example 1.6 . Let C be an ∞− category and let K ∈ Set ∆ be a simplicial set. Then one can sho w that the simplicial mapping space C K • = ho m Set ∆ (∆ • × K , C ) is a gain an ∞− categ ory (as opp osed to a more gener a l s implicia l set). This follo ws from the fact that the Joyal mo del structure ([Jo y08b]) on the category o f simplicial sets is Ca rtesian. W e can hence v ar y the simplicial set K and consider the asso ciated ho motop y catego ries Ho ( C K ). Using the nerve functor N which g iv es us a fully faithful embedding of the ca tegory Cat in the catego ry Set ∆ of s implicia l sets, w e thus o btain the pr e derivator D C asso ciate d to the ∞− c ate gory C : D C : J 7− → D C ( J ) = Ho  C N ( J )  DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 7 The functoriality of this construction follows from Theorem 5.14 of [Joy08a]. The last example which we are a b out to mention now do es not seem to b e too interesting in its o wn r igh t. B ut as w e will see later it largely reduces the amo unt of work in many pro ofs (cf. Theorem 1.31). Example 1.7. Let D b e a prederiv ator and let M b e a ﬁxed ca teg ory . Then the assignment D M : Cat op − → CA T : J 7− → D M ( J ) = D ( M × J ) is again a prederiv ato r. Similar ly , given a functor u : L − → M we obtain a mo rphism of 2-functors u ∗ : D M − → D L . T he r e is a no tion of morphisms of prederiv a tors (cf. Section 2 and more sp eciﬁcally Example 2.1) and it is eas y to see that the pairing ( M , D ) 7− → D M is ac tua lly functorial in both v ariables. Mo reov er, w e ha ve coherent is omorphisms ( D L ) M ∼ = D L × M and D e ∼ = D for the terminal category e . Remark 1.8 . i) In some s ituations, in particular under certain ﬁniteness c onditions, o ne do es not wish to consider diagrams of arbitra ry shap es but only o f a ce r tain kind (e.g. ﬁnite, ﬁnite- dimensional, p osets). There is a no tion o f a diagr am c ate gory Dia which is a 2-sub category of Cat having certain clos ure prop erties. Corresp ondingly , there is then the a ssocia ted notion of a pr e derivator of typ e Dia . W e preferred to no t g iv e these deﬁnitions at the very b eginning since we w ant ed to start immediately with the developmen t of the theor y . Once the main results are established we c hec k which prop erties have b een used and come back to this point (cf. the discussio n befo re Deﬁnition 4.21) . So , the r eader is in vited to r e place ‘a (pre)deriv ator’ by ‘a (pre)deriv ator of typ e Dia ’ th roughout this pap er. An example of the usefulness of this mor e ﬂexible notion is given by Keller in [Ke l0 7 ] where he shows that there is a stable deriv ator as socia ted to an exact category in the sense of Quillen [Qui73] if one r estricts to ﬁnite directed diagr ams. ii) The r e is a n additional remark c oncerning the deﬁnition o f a pre deriv ator. In our setup a pre- deriv ator is a 2-functor D : Cat op − → CA T a s opp osed to a more general pseudo-functor (which is for example used in [F ra96]). Mor e s p eciﬁcally , w e insisted on the fa ct that D preser ves iden tities and comp ositions in a strict sense a nd not o nly up to coherent natural is o morphisms. Since all examples showing up in nature hav e this strong e r functoriality we a re ﬁne with this notion. How ever, from the per spective of ‘homotopical inv ariance of structures’, a deﬁnition based on pse udo -functors would be b etter: let D b e a prederiv ator and let us be given a categ ory E J for each sma ll category J . Let us mo reo ver assume that we are g iv e n equiv alences o f categor ies D ( J ) − → E J . Then, in g eneral, we cannot use the equiv alences to obtain a prederiv a tor E with E ( J ) = E J such that the equiv a le nces of catego ries ass e m ble to an equiv alence of preder iv ators. This w ould only b e the cas e if the equiv alences are, in fact, isomor phisms which –b y the basic philosophy of category theory– is a to o strong notio n. Nevertheless, for the sake of a s impliﬁcation of the exp o sition we pre ferred to stick to 2-functors but w ant to men tion that everything we do her e ca n als o b e do ne with pseudo-functor s. Let now D be a prederiv a tor a nd let u : J − → K b e a functor. Motiv ated b y the ab ov e examples we call the induced functor D ( u ) = u ∗ : D ( K ) − → D ( J ) a r estriction of diagr am functor or pr e c om- p osition functor . As a sp ecial c a se of this, let J = e b e the terminal ca tegory , i.e., the category with one ob ject and identit y morphism only . F or an o b ject k of K , we denote b y k : e − → K the unique functor sending the unique ob ject of e to k . Giv en a prederiv a tor D , w e obtain, in pa rticular, for each ob ject k ∈ K a n asso ciated functor k ∗ : D ( K ) − → D ( e ) whic h takes v alues in the u n derly ing c ate gory D ( e ). Let us ca ll s uc h a functor an evaluation fu n ctor . F or a morphism f : X − → Y in D ( K ) let us write f k : X k − → Y k for its image under k ∗ . 8 MORITZ GROTH Deﬁnition 1.9. Let D b e a prederiv a tor a nd let u : J − → K b e a functor. i) The prederiv a tor D a dmits homotopy left Kan extensions along u if the induced functor u ∗ has a left adjoint: ( u ! = HoLK an u , u ∗ ) : D ( J ) ⇀ D ( K ) The prederiv ator D admits homotopy c olimits of shap e J if the functor p ∗ J induced by p J : J − → e has a left a djoin t: ( p J ! = Ho colim J , p ∗ J ) : D ( J ) ⇀ D ( e ) ii) The prederiv ator D admits homotopy right Kan extensions along u if the induced functor u ∗ has a right adjoin t: ( u ∗ , u ∗ = HoRKan u ) : D ( K ) ⇀ D ( J ) The prederiv ator D admits homotopy limits of shap e J if the functor p ∗ J induced by p J : J − → e has a right adjoin t: ( p ∗ J , p J ∗ = Holim J ) : D ( e ) ⇀ D ( J ) Recall from classical category theory , that under co completeness assumptions left Kan extensio ns can b e calc ula ted p oin t wise by cer tain c olimits, and simila rly that under completeness assumptions right Kan extensions can be calcula ted p oint wise b y certain limits [ML98, p. 237]. More precisely , consider u : J − → K and F : J − → C wher e C is a complete catego ry: J F / / u   C K RKan u ( F ) ? ? ⑦ ⑦ ⑦ ⑦ Then the right Kan ex tens io n RKan u ( F ) of F along u exists and can b e describ ed us ing Kan ’s formula [Ka n58 ] as RKan u ( F ) k ∼ = lim J k/ pr ∗ ( F ) = lim J k/ F ◦ pr , k ∈ K. In the ab ov e for mula, we ha v e used the following notation. Let u : J − → K b e a functor and let k be an ob ject of K . Then one can form the slic e c ate gory J k/ of obje cts u -under k . An ob ject in this catego ry is a pa ir ( j, f ) co nsisting of an ob ject j ∈ J to gether with a mo rphism f : k − → u ( j ) in K . Given tw o such o b jects ( j 1 , f 1 ) and ( j 2 , f 2 ) , a morphism g : ( j 1 , f 1 ) − → ( j 2 , f 2 ) is a mor phism g : j 1 − → j 2 in J such that the obvious tria ngle in K commutes. Dually , o ne can for m the s lic e c ate gory J /k of obje cts u -over k . In b oth case s, there a re canonical functors pr : J k/ − → J and pr : J /k − → J forgetting the morphism component. W e will not dis tinguish these pro jection mo r phisms notation- ally but it will always b e clear from the context whic h pro jection morphis m we are considering. A dual formula ho lds for left Kan extensio n in t he ca se of a coco mplete tar get categ o ry C and will not be made explicit. The corr esponding prop erty f or homotopy K an extensions holds in the ca se of model categor ies (cf. Subsection 1.3) a nd will b e demanded axioma tica lly for a deriv ator . In order to b e able to formulate this axio m, w e have to talk ab out base ch ange mor phisms. F or this purp ose, let D b e a prederiv ato r and consider a na tural transformation of functor s α : w ◦ u − → u ′ ◦ v . By an a pplication DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 9 of D , we th us have the follo wing t wo squares on the left: J v / / u   J ′ u ′   D ( J ) D ( J ′ ) v ∗ o o D ( K ) D ( J ) u ∗ o o D ( J ′ ) v ∗ o o K w / / K ′ ⑧ ⑧ ⑧ ⑧ ; C D ( K ) u ∗ O O D ( K ′ ) w ∗ o o u ′ ∗ O O ✟ ✟ ✟ ✟ @ H D ( K ) = V V u ∗ O O ✞ ✞ ✞ ✞ ? G D ( K ′ ) w ∗ o o u ′ ∗ O O ✟ ✟ ✟ ✟ @ H D ( J ′ ) = k k u ′ ∗ o o ✟ ✟ ✟ ✟ @ H Let us a s sume that D admits homotopy r igh t Ka n extensions a lo ng u and u ′ . W e denote a n y chosen adjoints and the corresp onding adjunction morphisms by ( u ∗ , u ∗ ) , η : id − → u ∗ ◦ u ∗ , and ǫ : u ∗ ◦ u ∗ − → id in the case of u and s imilarly in the ca se of u ′ . This can be used to extend our squa r e to the upper right diagr a m in which the a dditional natural tra ns formation are given by the res pective adjunction morphisms. W e can thus deﬁne the natura l transformation α ∗ by pa sting this diagra m to a single natural transformation. Spelling this out, α ∗ is given by the following comp osition o f natural transfor mations: w ∗ ◦ u ′ ∗ α ∗ / / η   u ∗ ◦ v ∗ D ( J ) u ∗   D ( J ′ ) v ∗ o o u ′ ∗   ✻ ✻ ✻ ✻ W _ u ∗ ◦ u ∗ ◦ w ∗ ◦ u ′ ∗ α ∗ / / u ∗ ◦ v ∗ ◦ u ′∗ ◦ u ′ ∗ ǫ ′ O O D ( K ) D ( K ′ ) w ∗ o o This natura l transformation α ∗ is calle d ‘the’ Be ck-Cheval ley tra nsforme d 2-c el l asso ciate d to α . Since this constructio n is very imp ortant in this pap er let us mak e explicit the dual co nstruction. So, let us co nsider a natural transfor mation α : u ′ ◦ v − → w ◦ u as in: J ⑧ ⑧ ⑧ ⑧ {  v / / u   J ′ u ′   D ( J ) ✟ ✟ ✟ ✟   D ( J ′ ) v ∗ o o D ( K ) ✞ ✞ ✞ ✞   D ( J ) ✟ ✟ ✟ ✟   u ! o o D ( J ′ ) ✟ ✟ ✟ ✟   v ∗ o o K w / / K ′ D ( K ) u ∗ O O D ( K ′ ) u ′ ∗ O O w ∗ o o D ( K ) = V V u ∗ O O D ( K ′ ) w ∗ o o u ′ ∗ O O D ( J ′ ) = k k u ′ ! o o Under the assumption that the pr ederiv ator a dmits homotopy left Kan extensions along u and u ′ we ca n proceed as a bov e and deﬁne the natural tra nsformation α ! by pasting, i.e., as follows: u ! ◦ v ∗ α ! / / η ′   w ∗ ◦ u ′ ! D ( J ) u !   D ( J ′ ) v ∗ o o u ′ !   u ! ◦ v ∗ ◦ u ′∗ ◦ u ′ ! α ∗ / / u ! ◦ u ∗ ◦ w ∗ ◦ u ′ ! ǫ O O D ( K ) ✻ ✻ ✻ ✻   D ( K ′ ) w ∗ o o This natural transfor mation α ! is ag ain called ‘the’ Be ck-Cheval ley t r ansforme d 2-c el l asso ciate d to α . In b oth ca ses, we co nstructed a ne w 2-cell by choos ing cer tain a djoin t functors a nd then c o m- po sing with the a djunction morphisms . It is immediate that the r esult dep ends on these c hoices only up to na tural isomor phis m. This technique will b e developed a bit more systematically in Subsection 1.2 but see also [Gro12b] where some asp ects from classical ca tegory theory are treated from this p erspe ctiv e. 10 MORITZ GROTH A t the very moment, we are int erested in the following situa tio n. Let u : J − → K be a functor and k ∈ K a n ob ject. Identifying k aga in with the co rrespo nding functor k : e − → K , we ha ve the following t wo natural transformations α in the context of the slice constructions: J k/ pr / / p J k/   J u   J /k pr / / p J /k   ✂ ✂ ✂ ✂ }  J u   e k / / K ✂ ✂ ✂ ✂ = E e k / / K The components of α at ( j, f : k − → u ( j )) resp. ( j, f : u ( j ) − → k ) are f in both cases. Assuming D to b e a prederiv ator admitting the necessary homotopy Kan extensions, we thus obta in the following Beck-Chev alley transformed 2-cells α ∗ and α ! : D ( J k/ ) Holim J k/   D ( J ) pr ∗ o o u ∗   ✻ ✻ ✻ ✻ W _ D ( J /k ) Hoc olim J /k   D ( J ) pr ∗ o o u !   D ( e ) D ( K ) k ∗ o o D ( e ) ✻ ✻ ✻ ✻   D ( K ) k ∗ o o Asking these natura l transformations to be isomorphisms is a conv enien t w ay to a xiomatize Kan’s formulas. With these prepa rations we can give the cen tral deﬁnition of a der iv a tor. Deﬁnition 1.10. A preder iv ator D is called a derivator if it sa tisﬁes the following axioms: (Der1) F or tw o categ o ries J 1 and J 2 , the functor D ( J 1 ⊔ J 2 ) − → D ( J 1 ) × D ( J 2 ) induced by the inclusions is a n equiv alence of catego r ies. Moreover, the categ ory D ( ∅ ) is not the empty category . (Der2) A morphism f : X − → Y in D ( J ) is an isomorphism if and only if f j : X j − → Y j is an isomorphism in D ( e ) for e very ob ject j ∈ J. (Der3) F o r ev ery functor u : J − → K , there ar e homotop y left and right Kan extensions along u : ( u ! , u ∗ ) : D ( J ) ⇀ D ( K ) and ( u ∗ , u ∗ ) : D ( K ) ⇀ D ( J ) . (Der4) F o r ev ery functor u : J − → K and every k ∈ K , the morphis ms Ho colim J /k pr ∗ ( X ) α ! − → u ! ( X ) k and u ∗ ( X ) k α ∗ − → Holim J k/ pr ∗ ( X ) are isomor phisms for all X ∈ D ( J ). A few remarks on the axio ms are in orde r . The ﬁrst ax iom says of cour s e that a diag ram on a disjoint union is completely determined by its r estrictions to the dir ect summands. The second pa rt of the ﬁr st ax iom is included in o rder to exclude the ‘empty deriv ator’ as an example. But it will also imply the existence of initial and ﬁnal ob jects (cf. Propo sition 1.1 2). The se c ond a xiom can be motiv ated fr o m the examples as follows. A natura l transformation is an isomor phism if and only if it is p oint wise an isomorphism. Similarly , in the co n text of an abelian category , ther e is the easy fact that a mor phism of chain complexes in a functor catego r y is a qua si-isomorphism if a nd only if it is a q uasi-isomorphism at ea c h ob ject. Moreover, in the cont ext of model categories , whatever mo del s tr ucture one establishes on a dia gram catego ry with v alues in a mo del category , o ne certainly wan ts the clas s of weak equiv alences to b e deﬁned p oin t wise. Finally , the cor responding r esult for ∞− categor ies is established b y Joyal as Theor em 5.14 in [Joy08a ]. The la st tw o axioms of course enco de a ‘homotopica l bic o mpleteness prop erty’ tog ether with Kan’s for m ulas . One could easily develop a more g eneral theory of preder iv ators which are o nly homotopy (co)complete or even o nly hav e a cer tain class of homotopy (co)limits. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 11 Example 1.11 . Let C b e a categor y . The represented preder iv ator y ( C ) : J 7− → C J is a der iv a tor if and only if C is bicomplete. Thus, the 2-ca tegory of bicomplete categor ies is embedded int o the 2-catego ry of deriv a tors. The idea is of course that the deriv a tor enco des additional structure on its v alues. O ne nice feature of this a pproach is that this structure do es not have to b e chosen but its ex is tence can b e deduced fro m the axioms. Note that all axioms are o f the form that they demand a pr op erty ; the only actual structur e is the given prederiv ator . This is similar to the situation of additive catego r ies where the enrichmen t in ab elian gro ups ca n uniquely be deduced fr om the fact tha t the underlying category has certain exactness pro perties. W e will come back to this p oint later in the co ntext of stable deriv ator s (cf. Remark 4.20). As a ﬁrst example o f this ‘higher structure’ w e give the following example. W e will pursue this more systematically from Subsection 1.2 on. Let J b e a categor y a nd consider the c o product J ⊔ J together with the co diago nal a nd th e inclusion functors: J id J " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ i 1 / / J ⊔ J ∇ J   J id J | | ② ② ② ② ② ② ② ② ② i 2 o o J Prop osition 1.1 2. L et D b e a derivator and let J b e a c ate gory. i) The value of D at the empty c ate gory ∅ is trivial, i.e., D ( ∅ ) is e quivalent to e. ii) The c ate gory D ( J ) admits an initial obje ct ∅ and a terminal obje ct ∗ . iii) The c ate gory D ( J ) admits ﬁnite c opr o ducts and ﬁ nite pr o ducts. Pr o of. i) Considering the disjoin t union ∅ = ∅ ⊔ ∅ , (Der1) implies that w e have an equiv alence giv en by the diagonal functor D ( ∅ ) − → D ( ∅ ) × D ( ∅ ). Th us, all morphism s ets are either empt y sets o r singletons. The ﬁrst ca se would deduce a c o n tr a diction to the fact that the diagona l is a bijection on path comp onent s. Thus, D ( ∅ ) is trivial a nd we will hence denote any ob ject of D ( ∅ ) by 0. ii) Consider the unique empt y functor ∅ J : ∅ − → J a nd apply (Der3) in order to obtain le ft r esp. right adjoin ts ∅ J ! : D ( ∅ ) − → D ( J ) , ∅ J ∗ : D ( ∅ ) − → D ( J ) . Since a left (rig h t) adjoint preserves initial (ﬁnal) ob jects, the image of a n y ob ject 0 under ∅ J ! ( ∅ J ∗ ) is an initial (ter mina l) ob ject in D ( J ) . Let us denote any suc h image by ∅ resp ectiv ely ∗ . iii) By (Der1 ), w e ha v e a n equiv alence of categories ( i ∗ 1 , i ∗ 2 ) : D ( J ⊔ J ) ≃ − → D ( J ) × D ( J ). Cho ose an inv ers e equiv alence k and consider the following diagram: D ( J ) × D ( J ) k / / D ( J ⊔ J ) ∇ J ! / / ( i ∗ 1 ,i ∗ 2 ) o o D ( J ) : ∆ D ( J ) ∇ ∗ J o o Since the right a djoin t is the diagonal functor , ∇ J ! ◦ k gives a co pr oduct. Simila r ly , ∇ J ∗ ◦ k will deﬁne a pro duct functor on D ( J ) .  W e wan t to emphasize that, in general, the v alues o f a deriv ator only hav e very few c ate goric al (co)limits. In order to relate this to the homoto pic a l v ariants and also fo r later purpos es, let us intro duce the under lying diagr am functors and their partia l v a rian ts. W e saw already that an ob ject m ∈ M induces an ev aluation functor m ∗ : D ( M ) − → D ( e ) . Similarly , a morphism 12 MORITZ GROTH α : m 1 − → m 2 in M can be considered as a natural tr a nsformation of the co r respo nding cla ssifying functors and thus giv es rise to e m 1 ( ( m 2 6 6 ✤ ✤ ✤ ✤   α M , D ( M ) m ∗ 1 * * m ∗ 2 4 4 ✤ ✤ ✤ ✤   α ∗ D ( e ) . Under the categ orical exponential la w whic h can b e written in a sugg estiv e form as  D ( e ) D ( M )  M ∼ = D ( e ) M × D ( M ) ∼ =  D ( e ) M  D ( M ) , we hence obtain an underlying diagr am functor dia M : D ( M ) − → D ( e ) M . Similarly , g iv en a product M × J of t wo c ategories a nd m ∈ M , we can consider the c orresp onding functor m × id J : J ∼ = e × J − → M × J. F ollowing the same arguments as above, we obtain a p artial u nderlyi ng diagr am functor dia M ,J : D ( M × J ) − → D ( J ) M . Thu s, the natur al iso morphism M ∼ = M × e induces an identiﬁcation of dia M and dia M ,e . No w, the functor p M : M − → e gives rise to the following diagram D ( M ) dia M / / Hoc olim M   Holim M   D ( e ) M     D ( e ) p ∗ M O O id / / D ( e ) ∆ M O O which co mm utes in the s e nse that w e have dia M ◦ p ∗ M = ∆ M : D ( e ) − → D ( e ) M . If the underlying diagram functor dia M happ ens to be an equiv alence for a certain category M , then also ∆ M has adjoints on b oth sides, i.e., the ca tegory D ( e ) has then (co)limits of shap e M . Similar r emarks apply to the case of the partial underlying diag r am functor dia M ,J where we would then deduce a conclusion ab out the category D ( J ) . Now, axiom (Der1) implies that the par tial underlying diagram functors dia ∅ ,J : D ( ∅ ) − → D ( J ) ∅ = e and dia e ⊔ e,J : D ( J ⊔ J ) − → D ( J ) × D ( J ) are equiv alences. This explains wh y we w ere able to deduce Prop osition 1.1 2 from the axioms but, in genera l, do not hav e other catego rical (co)limits. Although, in general, we do not wan t to assume that also o ther par tial under lying diag r am func- tors are equiv alences , the following deﬁnition is very imp ortant. This deﬁnition a gain emphasiz e s the imp ortance of the distinction betw een the categor ie s D ( K ) and D ( e ) K . Deﬁnition 1.13. A deriv a tor D is called str ong if the par tial underlying diagram functor dia [1] ,J : D ([1] × J ) − → D ( J ) [1] is full and essentially surjective for eac h category J. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 13 Remark 1.1 4. The strong ness prop erty of a der iv ator is a bit harder to mo tiv a te. It can be chec ked that the deriv ato r s asso ciated to mo del categor ies are s tr ong. Moreover, since the par tial underlying diagram functors are isomorphisms for represe n ted der iv ators, these a r e cer tainly also strong. Thus, the strongness prop erty is satisﬁed by the naturally o ccurring deriv ators. In this pap er, the stro ng ness will play a key role in the constructio n o f the tria ngulated structures on the v a lue s of a sta ble deriv ator. The p oin t is that the str o ngness prop ert y allows one to lift morphisms in the underlying ca tegory D ( e ) to ob jects in the catego r y D ([1]) where we can apply certain constr uc tio ns to it. Similarly , it allows us to lift morphisms in D ( e ) [1] to morphisms in D ([1]) or even to ob jects in D ([1] × [1 ]) . But it is no t o nly the case for the stable context that the strongness pro p erty is co nvenien t. Already in the context of p oin ted deriv ato rs it is very helpful. This pro perty allows the construction of ﬁb er and coﬁb er sequences asso ciated to a morphism in the underlying ca tegory of a strong, po in ted deriv ator. Similarly , one migh t expect that in later dev elopments of the theory this pr o perty will also b e useful in the unp ointed co n text. Nev ertheless, we follow Maltsiniotis in not including the strongne s s a s a n axiom of the basic notion of a deriv ator. Moreov er, it mig h t be helpful to consider v a rian ts of the deﬁnition. Given a family F of sma ll categorie s, we deﬁne a der iv ator D to b e F - str ong if the partial underlying diagram functors dia M ,J are full and ess en tially surjective for all M ∈ F and all categor ies J. Heller considered in [Hel88] the case where F c onsists of all ﬁnite, free catego ries. Let us quickly recall the dua lization pro cess fo r deriv ators. As the author was confused for a while a bout the diﬀerent dualizations for 2 -categories we will give some details. The p oin t is that given a 2-categor y C we obtain a new 2 -category C op be inv erting the directio n of the 1-morphisms and we g et a further 2-catego ry C co by in verting the direction of the 2-morphisms. Mor eo ver, these op erations can b e c o m bined s o that given a 2-c a tegory using the v arious dualizations we obtain 4 diﬀerent 2-categories (more generally , an n -category has 2 n diﬀerent dua liz a tions). Let us expla in thes e dualizations more conceptually , i.e., fr om the pers p ective of enriched categor y theory . First, we can co nsider ‘the enr ic hment level’ Cat as a symmetric monoidal category . The formation of opp osite catego ries can b e p erformed in the context of enr ic hed ca tegories as so on as the enrichmen t level is sy mmetr ic monoidal (cf. to [Bo r 94b , Section 6 .2 ]). Thus, we can form the dual of a 2-catego ry C as a ca teg ory enrich ed ov er Cat . The result of this dualiza tion is the 2-catego ry C op in which the 1- morphisms hav e changed direc tion. Alternatively , since the Ca rtesian monoidal structur e on the 1-categor y Cat behav es well with dualizatio n, there is a second wa y of dualizing a gener al 2- category . Mor e pr ecisely , we can consider the dualization of sma ll c ategories as a mono idal functor ( − ) op : Cat − → Ca t with resp ect to the Cartesian structures . Since any monoidal functor induces a base change functor at the level o f enr iched categories (cf. to [Bor 94b , Section 6.4]), we obtain a 2-ca tegory C co . This 2-c a tegory is obtained from C by inv erting the dir ection of 2-morphisms. Fina lly , applying bo th dualizations to C we obtain the 2-ca tegory C co , op = C op , co . Remarking that the dualization 2-functor of categories J 7→ J op inv erts the direc tio n of the natural transformatio ns but keeps the direction of the functor s w e th us can make the following deﬁnition. Deﬁnition 1.1 5 . Let D b e a prederiv ator, then we deﬁne the dual prederiv ator D op by the following diagram: Cat op D op / / ( − ) op   CA T Cat op , co D / / CA T co ( − ) op O O 14 MORITZ GROTH Example 1.16. A prederiv ator D is a deriv ator if and only if its dual D op is a deriv a tor. This result implies that in many ge ne r al statements abo ut deriv ator s and morphisms b et w een deriv ator s we only have to prov e claims ab out – sa y– homoto p y left Kan extensions while the co r- resp onding claim for homotopy right Kan extensions fo llows by dualit y . 1.2. Hom otop y exact squares and some prop erties o f homotop y Kan extensions. In this subsection we w a n t to establish some lemmas ab out the forma tion of Beck-Chev alley transformed natural tr ansformations (as it is used in the deﬁnition of a deriv ator ). A very conv enient fact is the nice b ehavior of this formalism with respe c t to pasting. Since the Beck-Chev alley tra nsformation itself already uses pasting of natura l tra nsformations le t us quickly recall the latter. F or this purp ose, let us consider the following tw o diagrams in CA T : C 1 ✂ ✂ ✂ ✂ }  C 2 ✂ ✂ ✂ ✂ }  v 1 o o C 3 v 2 o o C 1 C 2 v 1 o o C 3 v 2 o o D 1 u 1 O O D 2 u 2 O O w 1 o o D 3 w 2 o o u 3 O O D 1 u 1 O O D 2 u 2 O O w 1 o o ✂ ✂ ✂ ✂ = E D 3 u 3 O O w 2 o o ✂ ✂ ✂ ✂ = E If we ca ll in b oth diagr ams the na tural trans formations α 1 and α 2 resp ectiv ely then we can form the following compo site natural transformations: α 1 ⊙ α 2 = α 1 w 2 · v 1 α 2 and α 2 ⊙ α 1 = v 1 α 2 · α 1 w 2 By deﬁnition, these natur al transforma tions are obta ined by p asting of the resp ective diagrams. The chosen nota tio n r e minds us o f the order the tr ansformations show up in the co mposition. This pro cedure can als o be applied to la r ger diagrams if all natural transforma tio ns ‘point in the sa me direction’. T o give an example one ca n use these pasting dia grams to depict the t riangular iden tities for an adjunction ( L, R, η , ǫ ) : C ⇀ D as follows: D     |  C ⑧ ⑧ ⑧ ⑧ {  L o o = D C D R o o = C D = R R R O O C L o o = m m C L O O C = Q Q L O O ⑧ ⑧ ⑧ ⑧ ; C D = m m R o o     < D D R O O Here, the unlabe le d natural transformations a re the adjunction morphisms and w e simpliﬁed the notation of an identit y transfor mation by only displaying the corresp onding fu nctor. Let us now turn to the forma lism of Beck-Chev alley tr a nsformed natural tra ns formations. F or this purp ose, let us consider tw o natural transfor mations α 1 and α 2 in CA T a s indica ted in: C 1 ✂ ✂ ✂ ✂ }  C 2 v o o C 1 C 2 v o o D 1 u 1 O O D 2 u 2 O O w o o D 1 u 1 O O D 2 u 2 O O w o o ✂ ✂ ✂ ✂ = E Under the assumption that the vertical functor s hav e left adjoin ts u i ! we o bta in Be ck-Cheval ley (BC) tr ansforme d 2-c el ls α 1 ! by pasting as depicted in the following diagram on the left. Similarly , the existence of right adjoin ts u i ∗ allows us to construct Be ck-Cheval ley (BC) tr ansforme d 2-c el ls α 2 ∗ DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 15 by pa sting as shown in the diagra m on the r igh t: D 1 ✂ ✂ ✂ ✂ }  C 1 ✂ ✂ ✂ ✂ }  u 1 ! o o C 2 ✁ ✁ ✁ ✁ |  v o o D 1 C 1 u 1 ∗ o o C 2 v o o D 1 = S S u 1 O O D 2 w o o u 2 O O C 2 = l l u 2 ! o o D 1 = S S u 1 O O ✂ ✂ ✂ ✂ = E D 2 w o o u 2 O O ✂ ✂ ✂ ✂ = E C 2 = l l u 2 ∗ o o ✁ ✁ ✁ ✁ < D As an result of thes e pa stings w e obtain natural tra nsformations α 1 ! and α 2 ∗ as follows: C 1 u 1 !   C 2 v o o u 2 !   C 1 u 1 ∗   C 2 v o o u 2 ∗   ❁ ❁ ❁ ❁ Z b D 1 ❁ ❁ ❁ ❁  " D 2 w o o D 1 D 2 w o o Again, the resulting natura l transforma tions dep end o n some ch oices but only up to natura l iso- morphism. Using this termino lo gy , the natur al transforma tions showing up in axiom (Der4) of Deﬁnition 1.1 0 are given by the Bec k- Chev alley transfo r med 2-cells a ssocia ted to the f ollowing ones resp ectiv ely: D ( J /k ) ✟ ✟ ✟ ✟   D ( J ) pr ∗ o o D ( J k/ ) D ( J ) pr ∗ o o D ( e ) p ∗ O O D ( K ) k ∗ o o u ∗ O O D ( e ) p ∗ O O D ( K ) u ∗ O O k ∗ o o ✟ ✟ ✟ ✟ @ H There is a certain ambiguit y in the deﬁnition of the Beck-Chev alley transformation. If – sa y– in the case of α 1 : v ◦ u 2 − → u 1 ◦ w the horizontal functors admit rig ht adjoin ts we obtain a diﬀerent transformed 2-cell. Howev er , this gives rise to a co njugate natura l transformatio n (s e e Lemma 1.20). W e will c o mmit the following abuse of notation. Let us ass ume w e ar e giv en a deriv ator D and let us as sume we hav e α 1 = D ( β 1 ) a nd α 2 = D ( β 2 ) for co rresp onding na tural transfor mations β i in Cat . Then, for simplicit y , w e will denote t he Beck-Chev alley tra nsformed 2- cell a ssocia ted to α 1 and α 2 by β 1 ! and β 2 ∗ resp ectiv ely . Th us, we agree on the following short-hand-notation: β ! = ( β ∗ ) ! and β ∗ = ( β ∗ ) ∗ Moreov er, let us sa y that the transfor mations β ! and β ∗ are obtained by ‘base c hange’. W e giv e so me impo rtan t clas s es o f examples for this base change for malism to indicate its broad applicability beyond the purpo ses in this pap er. Example 1. 17. The following natural transformations resp ectively formations can be obtained by the base change formalism: i) adjunction units a nd adjunction counits ii) the formatio n of conjugate transformations (cf. [ML98]) iii) the natura l maps ex pressing that a functor preser v e s certain (co)limits or Kan extensions iv) in the context of triangulated categ ories, the exa ct structur e on a functor adjoint to an exact functor v) in the context of a tr iangulated catego ry with a t-s tructure ( T , τ ≥ 0 , τ ≤ 0 ) the natur al isomorphisms τ ≤ m ◦ τ ≥ n − → τ ≥ n ◦ τ ≤ m for m, n ∈ Z vi) in the context of monoidal categor ies, the monoidal structure on a functor right adjoint to a comonoidal functor a nd dually 16 MORITZ GROTH Let us now collect a few key technical lemmas on this for malism. The pro ofs of these results are quite for mal and w e w ill only include the easiest one here (cf. how ever to [Gro12b]) to a dv er tise the conv enience of drawing these pasting diag rams. A key fact for the remainder of this pap er is the go od b ehavior of Beck-Chev alley tr ansformed 2-cells with resp ect to pasting. Th us, let us again consider tw o pasting situa tio ns in CA T C 1 ✂ ✂ ✂ ✂ }  C 2 ✂ ✂ ✂ ✂ }  v 1 o o C 3 v 2 o o C 1 C 2 v 1 o o C 3 v 2 o o D 1 u 1 O O D 2 u 2 O O w 1 o o D 3 w 2 o o u 3 O O D 1 u 1 O O D 2 u 2 O O w 1 o o ✂ ✂ ✂ ✂ = E D 3 u 3 O O w 2 o o ✂ ✂ ✂ ✂ = E with natural trans formations α 1 and α 2 . Lemma 1. 18. L et us c onsider the ab ove diagr ams in CA T and let u s assume that the vert ic al functors have left adjoints re sp. right adjoints. Then ther e ar e the fol lowing r elations among the Be ck-Cheval ley tra nsforme d 2 -c el ls: ( α 1 ⊙ α 2 ) ! = α 2 ! ⊙ α 1 ! r esp e ctively ( α 2 ⊙ α 1 ) ∗ = α 1 ∗ ⊙ α 2 ∗ Pr o of. W e giv e a proo f of the ﬁr st eq uation. Unrav eling deﬁnitions this b oils down to depicting the transformatio n α 2 ! ⊙ α 1 ! as D 1 ✂ ✂ ✂ ✂ }  C 1 ✂ ✂ ✂ ✂ }  u 1 ! o o C 2 ✂ ✂ ✂ ✂ }  v 1 o o D 1 = S S u 1 O O D 2 ✂ ✂ ✂ ✂ }  w 1 o o u 2 O O C 2 ✂ ✂ ✂ ✂ }  = l l u 2 ! o o C 3 ✁ ✁ ✁ ✁ |  v 2 o o D 2 = S S u 2 O O D 3 w 2 o o u 3 O O C 3 = l l u 3 ! o o and then using a tr iangular iden tity in order to obtain ( α 1 ⊙ α 2 ) ! .  Thu s, bas e change is compatible with horizo n tal pasting and there is a similar such r esult for vertical pasting . Note, that we only ha ve a compatibilit y with resp ect to pasting and not ‘a functo- riality’. In particular, it can (and will) b e the case that we start with a commutativ e squar e but that the asso ciated tr ansformed 2 -cells are even no t iso morphisms. Nevertheless, this compatibility with resp ect to pasting com bined with the 2-out-o f-3-prop ert y for isomorphisms will b e a key ingredient in many proo fs of this pap er. It is useful to r emark that the a ssignmen ts α 7→ α ! and α 7→ α ∗ are inv erse to each other in a cer tain precis e sense. So, let us aga in consider a natura l transfo rmation α a s depicted b e lo w. W e can then iterate the Beck-Chev alley transformation as indica ted in the following diagr a ms. By doing so we ﬁrst obtain α ! and then ( α ! ) ∗ : C 1 ✂ ✂ ✂ ✂ }  C 2 v o o 7→ C 1 u 1 !   C 2 v o o u 2 !   7→ C 1 ✂ ✂ ✂ ✂ }  C 2 v o o D 1 u 1 O O D 2 u 2 O O w o o D 1 ❁ ❁ ❁ ❁  " D 2 w o o D 1 u 1 O O D 2 u 2 O O w o o In this situation we hav e the following lemma which also has its obvious dual for m. The proo f is left to the r eader but can also be found in [Gro 12b ]. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 17 Lemma 1.19. In the ab ove situation we have the e quality α = ( α ! ) ∗ : v ◦ u 2 − → u 1 ◦ w . These tw o lemmas can fo r example be com bined to deduce that for t w o conjugate natural trans - formations we hav e that o ne of them is an isomorphism if and only if the other is. W e wan t to collect a last lemma a b out this formalism which prov es to b e helpful in the discussion o f adjunctions (or adjunctions o f tw o v ariables as in [Gr o11]) and equiv alences of deriv ato rs. Lemma 1.20. Le t α : v ◦ u 2 − → u 1 ◦ w b e a n atur al tr ansformation in CA T such that the fu n ctors u i have left adjoints u i ! for i = 1 , 2 and such that the functors v and w have right adjoints v ∗ and w ∗ r esp e ctively. The natu r al tr ansformations α ! : u 1 ! ◦ v − → w ◦ u 2 ! and α ∗ : u 2 ◦ w ∗ − → v ∗ ◦ u 1 ar e c onjugate. In p articular, α ! is an isomorphism if and only if α ∗ is an isomorphism More details on the a b ov e examples a nd also the pro ofs of the las t tw o lemmas can b e found in [Gro12b]. Le t us now apply this formalism in the co n text of a deriv ator. Deﬁnition 1 .21. Let D b e a deriv ator a nd let us consider a natura l tra nsformation α as indicated in the following square in Cat : J 1 v / / u 1   ✂ ✂ ✂ ✂ }  J 2 u 2   K 1 w / / K 2 The square is D - exact if the base change α ! : u 1 ! ◦ v ∗ − → w ∗ ◦ u 2 ! (or, b y Lemma 1.20, equiv alen tly α ∗ : u ∗ 2 ◦ w ∗ − → v ∗ ◦ u ∗ 1 ) is a natur al isomorphism. The s quare is called homotopy exact if it is D -exact for all deriv ato r s D . W e w ill also apply the terminology o f D -exact s quares in the cont ext of a prederiv ator D admitting the nece s sary homotopy Kan extensio ns. F or a deriv ator D it follows immediately from Le mma 1.18 that D -exact squa res are stable under horizontal and v ertica l pasting. W arning 1.22 . W e wan t to include a warning on a certain risk of ambiguit y if the na tural tr ans- formation α under consider ation ha ppens to b e an isomor phis m. In that case it can (a nd will) happ en that t he Bec k-Chev alley transfor mation of α is an isomor phism witho ut this being the case for α − 1 (cf. for e x ample to Subsection 2.2). In pa r ticular, this can happ en for commutativ e square s. Thu s, in ca se there is a r isk of ambiguit y we will alwa ys give a direction to natural is omorphisms and even to iden tity transfor mations (cf. for example to Pro position 1.30). W e will next illustrate the notion of homo top y exact squa res by giv ing some examples whic h are central to the development o f the theory of deriv ator s (for a more systematic discussion w e refer t o [Mal11]). Using the 2-functoria lity of prederiv ator s, the following is immediate. Lemma 1. 23. L et D b e a pr e derivator and let ( L, R ) : J ⇀ K b e an adj unction. Then we obtain an adjunction ( R ∗ , L ∗ ) : D ( J ) ⇀ D ( K ) . Mor e over, if L (r esp. R ) is ful ly faithful ly, then so is R ∗ (r esp. L ∗ ). Pr o of. Every 2-functor of the v ariance of a prederiv a tor sends an adjunction ( L, R , η , ǫ ) to an adjunction ( R ∗ , L ∗ , η ∗ , ǫ ∗ ) .  In the s ta temen t of this lemma we allow ed ourselves the following abuse of no tation. Strictly sp eaking an adjunction is not determined by the tw o functors L and R but one a lso ha s to specify 18 MORITZ GROTH either of t he following: the natural iso morphism φ o f the morphism se ts, the unit η , or the counit ǫ . In or der to simplify the nota tion we nevertheless allow ed ourselves (and a lso will do so in the remainder of the pap er) to write ( L, R ) instea d of ( L, R, φ ) , ( L, R, η ) , or ( L, R, ǫ ) . A related result using the notion o f homotopy exact squares can b e formulated a s follo ws. This result expresses the co ﬁna lit y of right adjoin ts. Prop osition 1. 24. F or a rig ht adjoint functor R : J − → K the fol lowing squar e is homo topy exact: J R / / p J   ✁ ✁ ✁ ✁ |  K p K   e id / / e Pr o of. W e have to show that the natural transfor mation p J ! R ∗ − → p K ! is an isomor phism for an arbitra ry deriv ator. But it can b e chec ked that this natural transfor mation is the co njugate transformatio n of the identit y id : p ∗ K − → L ∗ p ∗ J = ( p J L ) ∗ = p ∗ K concluding the pro of.  Thu s, fo r a der iv ator D , a right adjoint functor R : J − → K, and an ob ject X ∈ D ( K ) we hav e a canonical iso morphism Ho colim J R ∗ ( X ) ∼ = − → Ho colim K X . F or later r eference, let us spe ll out the imp ortant sp ecial cas e where the right adjoint R = t : e − → K just speciﬁes a terminal ob ject in K . The second par t of the lemma follo w s immediately b y passing to the conjugate of the natural transfo r mation sho wing up in the ﬁrst part. Lemma 1.25. L et D b e a derivator and let K b e a c ate gory admitting a t erminal obje ct t. i) F or X ∈ D ( K ) we have a natu ra l isomorphism X t ∼ = − → Ho colim K X . ii) We have a c anonic al isomorphism of functors p ∗ K ∼ = − → t ∗ . The essential image of t ∗ c onsists of pr e cisely those obje cts for which al l structur e maps in the underlying diagr am ar e isomorphisms. Here is another imp ortant r esult ab o ut homotop y Kan extensions. Prop osition 1.26 . L et u : J − → K b e a ful ly faithful functor, then the fol lowing squar e is homotopy exact: J id / / id   J u   J u / / K Thus, t he adjunction morphisms η : id − → u ∗ u ! and ǫ : u ∗ u ∗ − → id ar e isomorphisms, i.e., homo- topy K an exten s io n fun ctors along ful ly faithful functors ar e ful ly faithful. Pr o of. Since isomor phis ms can be detected point wise we can reduce our task to showing that the following pasting is homotopy exact for all j ∈ J : J /j pr / / p       |  J id / / id   ⑦ ⑦ ⑦ ⑦ {  J u   e j / / J u / / K DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 19 But, the fully faithfulness of u implies that we have an isomo rphism J /u ( j ) − → J /j so that it suﬃces (e.g. by Prop osition 1.24 ) to show that the next pasting is homotopy ex a ct: J /u ( j ) / / p   ✝ ✝ ✝ ✝   J /j pr / / p       |  J id / / id   ⑦ ⑦ ⑦ ⑦ {  J u   e / / e j / / J u / / K But this is gua r an teed b y axio m (Der4 ).  Since we now know that, for fully faithful u : J − → K , the homo to p y Kan extension functors u ! , u ∗ : D ( J ) − → D ( K ) ar e fully faithful, we would like to obtain a characterization of the ob jects in the essential images . The p oint of the next lemma is that one only ha s to con trol the adjunction morphisms at arg umen ts k ∈ K − u ( J ) . Lemma 1.27. L et D b e a derivator, u : J − → K a ful ly faithful fun ctor, and X ∈ D ( K ) . i) X lies in the essential image of u ! if and only if the adjunction c ounit ǫ : u ! u ∗ − → id induc es an isomorphi sm ǫ k : u ! u ∗ ( X ) k − → X k for al l k ∈ K − u ( J ) . ii) X lies in the essential image of u ∗ if and only if t he adjunction un it η : id − → u ∗ u ∗ induc es an isomorphi sm η k : X k − → u ∗ u ∗ ( X ) k for al l k ∈ K − u ( J ) . Pr o of. W e g iv e a pro of of ii), so let us consider the adjunction ( u ∗ , u ∗ ) : D ( K ) ⇀ D ( J ) . By Prop o- sition 1.2 6, u ∗ is fully faithful. Thus, X ∈ D ( K ) lies in the essential image of u ∗ if and only if the adjunction unit η : X − → u ∗ u ∗ X is an is o morphism. Since isomo rphisms can be tested p oin twise, this is the case if and only if we hav e an iso morphism η k : X k − → u ∗ u ∗ ( X ) k for a ll k ∈ K . F or the c on verse dire c tion, one of the triang ular iden tities for our adjunction r eads as id = ǫ u ∗ · u ∗ η . Thu s, with ǫ also u ∗ η is an isomo r phism so that it suﬃces to chec k at points which do no t lie in the image.  There are tw o imp ortant classes of fully faithful functors where the essential image of ho motop y Kan extensions ca n b e characterized more easily . So let us g iv e their deﬁnition. Deﬁnition 1.28. Let u : J − → K b e a fully faithful functor which is injective on ob jects. i) The functor u is called a c osieve if whenever we hav e a morphism u ( j ) − → k in K then k lies in the image of u. ii) The functor u is called a sieve if whenever we have a morphis m k − → u ( j ) in K then k lies in the image of u. The following prop osition and a v ariant for the case of p ointed deriv ators (cf. Prop osition 3.6) will b e frequently used throughout this pape r . Prop osition 1.2 9. L et D b e a derivator. i) L et u : J − → K b e a c osieve, then the homotopy left Kan ex tension u ! is ful ly faithful and X ∈ D ( K ) lies in the essential image of u ! if and only if X k ∼ = ∅ for al l k ∈ K − u ( J ) . ii) L et u : J − → K b e a sieve, then the homotop y right Kan extension u ∗ is ful ly faithful and X ∈ D ( K ) lies in the essential image of u ∗ if and only if X k ∼ = ∗ for al l k ∈ K − u ( J ) . Pr o of. W e give a proof of i). T he statement ab out the fully faithfulness of u ! follows from the fully faithfulness o f the cosieve and P ropo sition 1 .26 . T o describ e the essential image we use the criter ion of Lemma 1.27. But for k ∈ K − u ( J ) we hav e u ! u ∗ ( X ) k ∼ = Ho colim J /k pr ∗ u ∗ ( X ) = Ho colim ∅ pr ∗ u ∗ ( X ) = ∅ . 20 MORITZ GROTH In this sequence, the iso mo rphism is given b y (Der4), the ﬁrst equality follows from the de ﬁni- tion of a cos iev e , and the seco nd equality follows from the descr iption of initial ob jects. Thus ǫ k : u ! u ∗ ( X ) k − → X k is an iso morphism for all k ∈ K − u ( J ) if and only if X k ∼ = ∅ for all k ∈ K − u ( J ) .  1.3. Exampl e s. T he ﬁr st aim of this subsec tio n co nsists of establis hing a class of examples which is very imp ortant for theoretical issues. Namely , w e wan t to show that with D a lso the prederiv ator D M is a der iv ator for a n a rbitrary sma ll ca tegory M . W e will then show that combinatorial mo del categorie s have underlying der iv ators. F or the ﬁrst p oint, the hardest part will b e to show that D M again sa tisﬁes axiom (Der4). In order to a c hieve this we include a short detour and establis h some r eform ulations of this axiom which also are of independent interest and will again be used further below. Let us b egin by c o nsidering the following pullb ack diagram in Cat : J 1 v / / u 1   J 2 u 2   K 1 w / / K 2 ✂ ✂ ✂ ✂ = E F or the notion of Grothendieck (op)ﬁbrations w e refer to [Bor 94b , Section 8.1] or [Vis 0 5]. Prop osition 1.30. Using the ab ove notation, a pul lb ack diagr am is homotopy exact, if u 2 is a Gr othendie ck ﬁbr ation or if w is a Gr othendie ck opﬁbr ation. Pr o of. W e give the pro of in the case wher e u 2 is a Gro thendieck ﬁbra tion. F or a deriv ator D we th us have to sho w that the canonical map id ∗ : w ∗ u 2 ∗ − → u 1 ∗ v ∗ is a natural iso morphism. Since isomorphisms can b e tes ted p oin t wise, (Der4) implies that it s uﬃces to show that the following pasting is a ho mo top y exact squa re for all k 1 ∈ K 1 : ( J 1 ) k 1 / pr / / p   J 1 v / / u 1   J 2 u 2   e k 1 / / K 1 w / / ✝ ✝ ✝ ✝ ? G K 2 ✁ ✁ ✁ ✁ < D Since our diag ram in Cat is a pullback diagr am, w e deduce that with u 2 also u 1 is a Grothendieck ﬁbration. T hus, the cano nical functor c : ( J 1 ) k 1 − → ( J 1 ) k 1 / : j 1 7→ ( j 1 , k 1 id − → u 1 ( j 1 )) is a left adjoint functor ([Qui7 3]) . Here, we denote by ( J 1 ) k 1 the ﬁb er of u 1 ov er k 1 , i.e., the sub- category of J 1 consisting of a ll o b jects s e n t to k 1 and all morphisms sent to id k 1 . Now, Lemma 1.18 and Prop osition 1.2 4 im ply that the a bov e pasting is homotopy exact if a nd only if this is the case for the pasting in the following le ft diagr a m: ( J 1 ) k 1 c / / p   ( J 1 ) k 1 / pr / / p   J 1 v / / u 1   J 2 u 2   ( J 1 ) k 1 w / / p   ( J 2 ) w ( k 1 ) c / / p   ( J 2 ) w ( k 1 ) / pr / / p   J 2 u 2   e / / e k 1 / / ✠ ✠ ✠ ✠ @ H K 1 w / / ✝ ✝ ✝ ✝ ? G K 2 ✁ ✁ ✁ ✁ < D e / / e / / ☛ ☛ ☛ ☛ A I e w ( k 1 ) / / ✌ ✌ ✌ ✌ B J K 2 ✠ ✠ ✠ ✠ @ H DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 21 It is ea sy to chec k that the ab ov e tw o pas ting s deﬁne the same natural trans formation. Thus, by exactly the same a r gumen ts again it suﬃces to show that the square ( J 1 ) k 1 w / / p   ( J 2 ) w ( k 1 ) p   e / / e ☞ ☞ ☞ ☞ B J is homotopy exact. But, since w e started with a pullback dia gram, w restricted in this wa y is a n isomorphism of categ ories so t hat our cla im follows (e.g . again b y Prop osition 1.24).  W e will refer to this prop osition by saying that a deriv ato r satisﬁes base change for Grothendieck (op)ﬁbrations. This prop osition allows us to establish the next theor em. Theorem 1. 3 1. L et D b e a derivator and let M b e a sm al l c ate gory. Then the pr e derivator D M : Cat op − → CA T : K 7→ D ( M × K ) is a derivator. Pr o of. The axioms (Der1)-(Der 3 ) are immediate so we only have to es ta blish axiom (Der4 ) for D M . By duality , it suﬃces to give the pro of for the case o f homo top y rig h t Ka n ex tensions. In other words, we hav e to sho w that the square M × J k/ / /   M × J   M × e / / M × K ☞ ☞ ☞ ☞ B J is D -exact. But this 2-cell ca n be obtained as the pasting of the following diagr a m M × J k/ / /   M × J   M × K k/ / /   M × K   ☞ ☞ ☞ ☞ B J M × e / / M × K ☞ ☞ ☞ ☞ B J in which the upp er square is a pullback diagram such that the b ottom horizontal arrow is a Grothendieck opﬁbratio n. Th us , by Pro position 1.3 0 it suﬃces to show that the low er square given by the natural transformation α 2 is D -exact. W e claim that it w ould suﬃce to sho w that the pasting obtained by the following lef t diagra m is D -exact for ev ery m ∈ M : M m/ × K k/ / /   M × K k/ / /   M × K   ( M × K ) ( m,k ) / / /   M × K   e m / / M × e / / ✍ ✍ ✍ ✍ C K M × K ☞ ☞ ☞ ☞ B J e ( m,k ) / / M × K ✎ ✎ ✎ ✎ C K Using this claim it then s uﬃces to observe that this pasting is naturally isomorphic to the square on the r igh t-ha nd-side which is D -exac t by Kan’s formula. So , it remains to establish our claim for 22 MORITZ GROTH which pur p ose we call the natural transfor ma tion o n the left α 1 . The asso ciated diagram obtained by ba se c ha nge looks like D ( M m/ × K k/ )   D ( M × K k/ ) o o   ✲ ✲ ✲ ✲ R Z D ( M × K ) pr ∗ o o   ✴ ✴ ✴ ✴ S [ D ( e ) D ( M × e ) m ∗ o o D ( M × K ) o o in which the 2- cells are g iven by α 1 ∗ and α 2 ∗ resp ectiv ely . The compatibility of base change with resp ect to pasting thus giv es us the following equations: ( α 1 ⊙ α 2 ) ∗ = α 2 ∗ ⊙ α 1 ∗ = α 2 ∗ pr ∗ · m ∗ α 1 ∗ Now, the canonical isomorphism M m/ × K k/ ∼ = ( M × K k/ ) m/ and axio m ( Der4) imply tha t α 2 ∗ is an isomorphisms. Using the fact that isomo rphisms ar e detec ted p oin t wise we can no w conclude as follows: α 1 ∗ is an isomo r phism if and only if m ∗ α 1 ∗ is an isomo r phism for all m ∈ M which is the case if and only if ( α 1 ⊙ α 2 ) ∗ is an isomo rphism for all m ∈ M . Thus it was indeed enough to sho w that the ab ov e pasting on the left is D -exact.  Thu s, whenev er we w an t to establish a general r esult ab out the v alues D ( M ) of a deriv ator D we may assume that we are conside r ing the underlying categor y of a deriv ator since w e can alwa ys pass from D to D M . Let us note that the conclusion of Pr opos itio n 1.30 is a ctually equiv ale n t to (Der4). Moreov er , there is a further reformulation using a ‘symmetric v ariant of Kan’s form ulas’. Mor e sp eciﬁcally , let us consider the following square in Cat : ( u 1 /u 2 ) pr 1 / / pr 2   ✞ ✞ ✞ ✞   J 1 u 1   J 2 u 2 / / K 2 Here, the catego ry ( u 1 /u 2 ) is the c omma c ate gory where an ob ject is a triple ( j 1 , j 2 , α : u 1 ( j 1 ) − → u 2 ( j 2 )) , j 1 ∈ J 1 , j 2 ∈ J 2 and the functor s pr i are the o b vio us pro jectio n functors. The arrow comp onen t o f such an ob ject deﬁnes the natural tra nsformation depicted in the diagram. If w e specialize to J 1 = e or J 2 = e w e get back the diagrams showing up in the p oin t wise calculation of K a n extensio ns. Prop osition 1.32. L et D b e a pr e derivator which satisﬁes the axioms (Der1)-(Der3) . Then the fol lowing thr e e s tatements ar e e quivalent: i) The pr e derivator D is a derivator, i.e., it also satisﬁes (Der4) . ii) The pr e derivator D satisﬁes b ase change for Gr othendie ck (op)ﬁbr ations. iii) The pr e deriva tor D satisﬁes b ase change for c omma c ate gories, i.e., the squar es asso ciate d to c omma c ate gories ar e D -exact. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 23 Pr o of. By Pro position 1.30 we a lready kno w that ii) is implied by i). Let us show that also the conv er se holds, i.e., we w ant to show that the square J k/ pr / / p J k/   J u   e k / / K ✂ ✂ ✂ ✂ = E is D - exact if we assume ii). But the r easoning in this case is a simpliﬁed version of the a rgumen ts in the pro of of Theorem 1.31. So, it remains to show that i) and iii) are equiv a len t. O ne direction is immediate by sp ecializing comma ca teg ories to slice categ ories so we only hav e to prove that (Der4) implies iii). Using similar reduction ar gumen ts as in the last pro of (including the behavior of bas e change with resp ect to pas ting and the fact that isomorphisms ar e detected p oin t w is e) it suﬃces to show that the following pasting is D -exact for all ob jects j 2 ∈ J 2 : ( u 1 /u 2 ) /j 2 pr / / p   ✌ ✌ ✌ ✌   ( u 1 /u 2 ) pr 1 / / pr 2   ✝ ✝ ✝ ✝   J 1 u 1   e j 2 / / J 2 u 2 / / K Now, ther e is a canonical functor R : J 1 /u 2 ( j 2 ) − → ( u 1 /u 2 ) /j 2 which is deﬁned by: ( j 1 , u 1 ( j 1 ) − → u 2 ( j 2 )) 7→  ( j 1 , u 1 ( j 1 ) − → u 2 ( j 2 ) , j 2 ) , j 2 id − → j 2  This functor can b e chec ked to deﬁne a rig ht adjoint so that b y Prop osition 1 .24 it suﬃces to show that the pasting in the following diagram is D -exact: J 1 /u 2 ( j 2 ) R / / p   ✌ ✌ ✌ ✌   ( u 1 /u 2 ) /j 2 pr / / p   ✌ ✌ ✌ ✌   ( u 1 /u 2 ) pr 1 / / pr 2   ✝ ✝ ✝ ✝   J 1 u 1   e / / e j 2 / / J 2 u 2 / / K But this pasting is precise ly the square used to calcula te homotopy Kan extensio ns along u 1 so that we ca n conclude b y (Der4).  Let us now turn to the second imp ortant clas s of examples of deriv a tors, na mely the ones as- so ciated to nice mo del ca tegories. This is included not only for the sa k e of co mpletenes s but als o bec ause our pro of diﬀers from the o ne given in [Cis03]. Our pro of is completely self-dua l a nd is simpler in that it do es not make use of the explicit description of the g enerating (acyclic) pro jective coﬁbrations o f a diagram ca tegory asso ciated to a coﬁbr an tly g enerated mo del category . W e restr ic t attent ion to the following situation. Deﬁnition 1.33. A mo del categ ory M is ca lled c ombinatorial if it is coﬁbrantly generated and if the underlying categ ory is presen table. This class of mo del ca tegories was introduced by Smith and is studied e.g. in [Lur 09 , Ros0 9, Bek00, Dug01a]. F or background on coﬁbrantly generated mo del categories we refer to [Ho v99]. The theory of presentable categories was initiated by Gabriel and Ulmer in [GU71]. F ur ther references to this theor y a re [Bor 94b , AR94]. One basic idea of the presen tabilit y as sumption is the follo wing 24 MORITZ GROTH one. The presentabilit y imp oses b ey ond the bico mpleteness a certain ‘s ma llness condition’ on a category which has at least tw o imp ortant conseq uences. The ﬁrst one is that the us ual set- theoretic problems o ccurring when one considers functor categorie s disapp ear at leas t if one restr icts attent ion to colimit-pr eserving functor s. But this is anyho w the adapted c la ss of morphism to b e studied in this context. Moreover, in the world of presentable categories one can fo cus more on conceptual ideas than on technical p oints of certain arguments: a functor b et w een pres en table categorie s is a left adjoint if a nd only if it is colimit-pr eserving. T he usual ‘solution s e t condition’ of F rey d’s a djoin t functor theor em is automatically fulﬁlled in this context. F or more comments in this direction see Subsectio n 2 .6 in [Gro 10 ], where these ideas are discussed in the context o f presentable ∞− ca teg ories. Imp ortan t examples of presentable categorie s a re the categories of sets, simplicial sets, a ll preshea f ca tegories (and, more ge nerally , all Grothendieck top oses), a lgebraic categorie s as well a s the Grothendieck ab elian categor ies and the ca tegory Cat . A non-ex ample is the category of top ological s paces altho ugh this can b e r epaired if o ne sticks to the ‘re ally convenien t category ’ (Smith) of ∆-generated spaces ([FR08 ]). T he slog an is that ‘presentable ca tegories are small enough so that cer tain set-theoretical problems disapp ear but are still large enough to include many important examples’. An yhow, a ll we need fro m the theor y of combinatorial mo del ca tegories is the v alidity of the next theorem so that we could also work axioma tica lly with the conclusion of this theo r em. The statement ab out the pro jective mo del str uctures is a conseq ue nce of the lifting theor em of coﬁ- brantly g enerated mo del str uctures along a left adjoint while the statement ab out the injective mo del structure w as only pro v ed more recen tly . B oth results are for example established in [Lur09, Prop osition A.2.8.2]. Theorem 1.34. L et M b e a c ombinatoria l mo del c ate gory and let J b e a smal l c ate gory. The c ate gory M J c an b e endowe d with the pr oje ctive and with the inje ctive mo del stru ctur e. Recall that the pr oje ctive mo del structure is determined by the fact that the weak equiv alences and the ﬁbra tions are deﬁned levelwise. In the inje ctive model s tr ucture this is the case fo r the weak e q uiv a lences and the coﬁbrations . W e will denote the functor catego r ies M J endow ed with the corres p onding mo del structures by M J pro j resp. M J inj . In the s pecial case where the combinatorial mo del category we start with is the ca tegory of simplicial sets endowed with the homotopy-theoretic Kan model structure, the pro jectiv e mo del structure on a diagra m categ ory is the Bousﬁeld-Ka n structure o f [BK7 2 ] while the injectiv e mo del structure is the Heller struc tur e o f [Hel88]. O ne p oint of these mo del structures is tha t cer tain adjunctions are now Quillen a djunctions for trivial reasons . Lemma 1. 35. L et M b e a c ombinatorial mo del c ate gory and let u : J − → K b e a functor. Then we have the fol lowing Quil len adjunctions ( u ! , u ∗ ) : M J pro j − → M K pro j and ( u ∗ , u ∗ ) : M K inj − → M J inj . W e now hav e almost everything at our disp osal needed to establish the following result. Prop osition 1.3 6. L et M b e a c ombinatoria l m o del c ate gory. Then the assignment D M : Cat op − → CA T : J 7− → Ho ( M J ) deﬁnes a str ong derivator. Pr o of. The ﬁrst axiom (Der1 ) is immediate. (Der2) holds in this case since the weak equiv alences are precisely the morphisms whic h ar e in verted by th e formation of homotop y categories a nd since the weak equiv alences are deﬁned levelwise. It is th us enough to consider the t wo axioms on ho motop y DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 25 Kan extensions. W e treat only the ca se of homotopy righ t Kan extensions. The other case follows by duality . Axiom (Der3) on the e xistence o f homoto p y Kan extension functors fo llo ws easily from the la st lemma since one only has to consider the as sociated derived adjunctions at the level of homotopy catego ries. So it re ma ins to establish Kan’s for m ula. F or this purp ose, let u : J − → K be a functor a nd let k ∈ K b e an ob ject. Consider the following dia gram, which commutes up to natural isomor phism by the usual base change morphism from class ical category theo ry: M J k/ lim   ∼ = M J pr ∗ o o u ∗   M M K k ∗ o o By the last lemma, the functors lim and u ∗ are right Quillen fun ctors with res pect to the injective mo del str uctures. If w e ca n show that also the functors k ∗ and pr ∗ are right Quillen functors with r espect to the injective mo del str uctures, then we a re done. In fact, in that case the tw o comp ositions of derived right Quille n functors are cano nically isomor phic and this in turn s ho ws that the ba se change morphism is an iso mo rphism. So let us show that k ∗ is a right Quillen functor. By deﬁnition of the injective mo del str uctures, k ∗ preserves weak equiv a lences. Hence it is enough to show that k ∗ preserves ﬁbrations . Using the adjunction ( k ! , k ∗ ) it is eno ug h to show that k ! : M − → M K preserves a cyclic coﬁbr ations. But an easy calculation with left K an e x tension shows that we ha ve k ! ( X ) l ∼ = ` hom K ( k,l ) X . F r om this description it is immediate that k ! preserves acyclic coﬁbrations. Fina lly , we will s ho w in Lemma 1.37 that also pr ∗ is a rig h t Quillen functor with resp ect to the injective model str ucture. The stro ngness is le ft to the reader. It can be deduced by some ‘ma pping cylinder ar gumen ts’ using the pro jective mo del structure on M [1] .  T o conclude the pro of of P ropo sition 1.36 we hav e to show that the functor pr ∗ : M J − → M J k/ is a right Quillen functor with resp ect to the injective mo del structures. It is again immediate that pr ∗ preserves injective weak equiv alences. Hence it suﬃces to show tha t pr ∗ preserves injective ﬁbrations. W e will pr o v e suc h a r esult for arbitra ry Grothendieck opﬁbrations with dis crete ﬁber s which applies, in particular, to our situation. Lemma 1.37. L et u : J − → K b e a Gr othendie ck opﬁbr ation with discr ete ﬁ b ers and let M b e a c ombinatorial mo del c ate gory. Then the functor u ∗ : M K − → M J pr eserves inje ctive ﬁbr ations. Pr o of. By adjointness, it is enough to show that the left adjoint u ! : M J − → M K preserves acyclic injectiv e coﬁbrations. F or this purp ose, let X ∈ M J and let k ∈ K . Then we make the following calculation: u ! ( X ) k ∼ = colim J /k X ◦ pr ∼ = colim J k X ◦ pr ◦ c ∼ = a j ∈ J k X j The ﬁr st iso morphism is ag ain Kan’s for m ula for Kan ex tens io ns. The second isomor phism is giv en by the coﬁnalit y of righ t adjoints (Pr opositio n 1 .2 4 ) applied to the ca nonical functor c : J k − → J /k . Finally , the last iso morphism uses the fact that the Grothendieck o pﬁbration has dis crete ﬁb ers. F rom this explicit description of u ! the claim follows immediately .  The pr oof of the ab o ve theore m actually sho ws a bit more. Given a coﬁbra ntly genera ted mo del category M , the prederiv ator D M is a what could be called c o c omplete pr e derivator (with the obvious meaning). But b y far more is true. The r e is the followin g more gener a l r esult which is due to Cisinski [Cis0 3]. 26 MORITZ GROTH Theorem 1.38. L et M b e a mo del c ate gory and let J b e a smal l c ate gory. Denote by W J the class of levelwise we ak e quivalenc es in M J . Then the assignment D M : Cat op − → CA T : J 7− → M J [ W − 1 J ] deﬁnes a derivator. The basic idea is to reduce the situatio n of an ar bitrary diagr am category using certain coﬁnality arguments t o the situa tio n where the indexing catego r ies ar e s o -called Reedy categories ([Ho v99]). The pro of ca n b e found in [Cis0 3]. F rom the pro of it w ill, in par ticular, follow that the ab ov e lo calizations make sense (i.e., that no change o f univ erse is necessa r y!) although, in genera l, there is no mo del structur e on M J with W J as weak equiv alences. F or mor e comments ab out the relatio ns hip betw een mo del categories and deriv ator s s ee Remark 2.14. Remark 1.39 . A com bina tion of Theo rem 1.38 and Example 1.1 1 thus shows that der iv ators fo r m quite a gener al framework. First, they subsume bico mplete catego r ies. Moreover, they provide an abstra ct des c r iption of the ca lculus of homoto p y Ka n extensio ns a t the level of the v arious homotopy categ ories asso ciated to a mo del ca tegories. Thus, this framework allows us to treat categoric al limits and colimits and the homo topical v a r ian ts on an equal foo ting. This is similar to what ha ppens in the related theor ies. In the theor y of ∞ -ca tegories, the notion of limits and colimits also subsumes b oth v a rian ts. In the cas e of ner v es of c a tegories, the notion r e duce s to the cla ssical notion of (co)limits, w hile when applied to co heren t nerves of (lo cally ﬁbrant) simplicial mo del categories it coincides with the notion of homo to p y (co)limits (cf . [Lur09] or [Gro1 0]). Similarly , every bicomplete categ ory can b e endowed with the discr et e mo del structure wher e the weak equiv alences are the isomorphis ms and a ll morphisms are ( co)ﬁbrations. With resp ect to this mo del structure the theory of homoto p y (co)limits reduces to the theory o f c ategorical (co)limits. 2. The 2 -ca tegor y of deriv a tors 2.1. Mo rphisms and natural transformations . Let D and D ′ be prederiv ato rs. A morphism of pr e derivators F : D − → D ′ is a pseudo- na tural transformation b e tween the 2-functors D and D ′ (cf. to Deﬁnition 7.5.2 of [Bor94a]). Sp elling out this deﬁnition such a morphis m is a pair ( F • , γ F • ) consisting of a collectio n of functors F J : D ( J ) − → D ′ ( J ) , J ∈ Cat , and a family of na tural isomorphisms γ F u : u ∗ ◦ F K − → F J ◦ u ∗ , u : J − → K as indicated in D ( K ) F K / / u ∗   ∼ = D ′ ( K ) u ∗   D ( J ) F J / / D ′ ( J ) . This datum is sub ject to the following coherence prop erties. Given a pair of comp osable functor s J u − → K v − → L and a na tur al tr a nsformation α : u 1 − → u 2 : J − → K, we then ha v e the following DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 27 relation resp. co mm utative diagrams: γ id J = id F J u ∗ v ∗ F γ v / / γ vu + + u ∗ F v ∗ γ u   u ∗ 1 F α ∗ / / γ u 1   u ∗ 2 F γ u 2   F u ∗ v ∗ F u ∗ 1 α ∗ / / F u ∗ 2 Here, we suppressed some indices (as w e will frequently do in the seq uel) to av oid a wkward notation. Given such a mo rphism F : D − → D ′ the functor F e : D ( e ) − → D ′ ( e ) is calle d the underlying functor . As usua l the no tion of a ps eudo-natural transforma tion can b e r elaxed or can b e s trengthened. In the more relax ed situation there w o uld b e t wo versions of s uc h morphisms, the lax ones and the colax ones, but these do not play an impo rtan t role in this pap er (they will brieﬂy show up in the next subsection and they will b e of s ome impo rtance in the context of mo r phisms of tw o v ariables in [Gro11]). Strictly sp eaking, a lso in our situation there are tw o notions dep e nding on the direction of the natural transformations γ . Since one can alw ays pass to the in verse natural transformations these notions ar e equiv alent. In what follo ws, w e will b e a bit sloppy in notation in that we will not distinguish no tationally b etw een the natural isomor phisms γ b elonging to such a morphism and their inv erses γ − 1 . In the case o f a 2-natural trans formation, i.e., if a ll natural transformations γ a r e given by ide ntities, w e sp eak o f a strict morphi sm . The clas s of str ict morphisms is to o narrow in that many examples will only b e pseudo - natural transforma tio n but str ic t morphisms are co nceptually easie r. This becomes manifest, for example, in the 2-categor ical Y oneda lemma a s opp osed to the more general bicatego rical Y o neda lemma. Finally , let F, G : D − → D ′ be morphisms of preder iv a tors. A natur al tr ansformation τ : F − → G is a mo diﬁcation of the pseudo-natur al transformatio ns (see [Bor94a, Deﬁnition 7.5.3]). Thus, such a τ is given by a family of na tur al transfor mations τ J : F J − → G J which are compatible with the coherence isomorphisms b elonging to the functors F a nd G in the sense that for every functor u : J − → K the following diagram comm utes: u ∗ F τ / / γ   u ∗ G γ   F u ∗ τ / / Gu ∗ Given tw o para llel mo rphisms F a nd G of preder iv ators let us deno te by na t ( F, G ) the natural transformatio ns from F to G. Thus, with prederiv ato rs as ob jects, morphisms as 1- cells, and natural transformatio ns as 2-cells we obtain the 2- c ategory PD er of prederiv a tors. In fa c t, this is just a sp ecial cas e of a 2 -category given by 2- functors, pseudo- natural transfor mations, and mo diﬁcations. The full sub-2- category spanned by the deriv ators is deno ted b y Der . Given t wo (pre)deriv ator s D and D ′ let us denote the c ategory of morphisms by Hom ( D , D ′ ) while we will write Hom strict ( D , D ′ ) for the full subc a tegory spanned by the strict morphisms. Corresp ondingly , we hav e t w o sub-2 - categorie s PDer strict − → PDer and Der strict − → Der . With Lemma 1.18 and later applications (e.g. Propo sition 2 .5) in mind let us only mention that three of the ab ov e deﬁning co herence conditions can be interpreted as equalities o f certain pasting diagrams. E xamples of morphisms will b e given after the following tw o comment s. Let us only quic kly mention that the 2 - categories PDer and Der admit ﬁnite 2-products and a re, in fact, Car tesian clo s ed 2-ca tegories in a certain pr e c ise sense. This obs e rv ation pla ys a central role in the de velopment of mo noidal asp ects of the theo r y of deriv ators (cf. to [Gro11]) but will not 28 MORITZ GROTH be needed in this pap er. The 2-pro duct and related notions will b e studied in more detail in that reference. In every 2-categ o ry we hav e the notion of adjo int morphisms and eq uiv alences. It turns out that the r esulting notion of an equiv alence of deriv ato rs can be simpliﬁed. In fact, an equiv alence is already co mpletely determined by giving a single morphism of deriv ator s which is lev elwise an equiv alence of catego ries. How ever, the case of an adjunction is slig htly more subtle and will b e taken up a gain in the next subsection (cf. P ropo s ition 2.1 1). Let us co nclude this subsection by giving some exa mples . Example 2 .1. i) Let C and D b e categor ies and let us consider the asso ciated represe nted pre d- eriv ator s y C and y D respectively . A funct or F : C − → D induces a strict morphism o f preder iv ators y F : y C − → y D in the obvious wa y . This a ssignmen t is faithful and t he morphisms in the ima ge ar e precisely the strict ones, i.e., the 2-natural transfo rmations. Thus, we have a fully f aithful 2-functor y : CA T − → PDer strict whose r estriction to bico mplete categor ies factors over Der strict . This can be seen as a sp ecial case o f the 2-categorical Y o ne da lemma and from no w on w e will frequent ly dr op the Y oneda em b edding y from notation. ii) Let D b e a prederiv a tor and let v : L − → M b e a functor. The functors ( v × id K ) ∗ assemble int o a strict pr e c omp osition morphism o f prederiv ators v ∗ : D M − → D L and similarly for na tural transformatio ns. Th us , every pr ederiv ator D induces a 2-functor D ( − ) : Cat op − → PDer strict . In fac t, this is a pa rtial 2 -functor of the 2-functor : ( − ) ( − ) : Cat op × PDer − → PDer : ( M , D ) 7→ D M iii) Given a prederiv ator D and a small categor y M let us denote by D ( − ) M the pre de r iv ator which sends K to D ( K ) M . The partial under ly ing dia gram functors then a ssem ble into a str ic t p artial un- derlying diagr am morphi sm of prederiv ators dia M , − : D M − → D ( − ) M . Axiom (Der1) implies that dia ∅ , − and dia e ⊔ e, − are equiv alence s in the case of a deriv ator . iv) Let M b e a co mbinatorial mo del categ ory . Since the weak eq uiv a lences in the diagr am cat- egories M K are deﬁned levelwise, all the a ssoc ia ted lo calizatio n functors assemble into a strict morphism of deriv ators: γ : M = y M − → D M 2.2. Hom otop y (co)limit pres erving morphism s. Let F : D − → D ′ be a morphism of der iv a- tors and let u : J − → K be a functor. Let us reca ll that the na tural transformation γ F u and its inv ers e are 2 - cells a s indicated in the following diagrams: D ′ ( J ) ✟ ✟ ✟ ✟   D ( J ) F o o D ′ ( J ) D ( J ) F o o D ′ ( K ) u ∗ O O D ( K ) F o o u ∗ O O D ′ ( K ) u ∗ O O D ( K ) F o o u ∗ O O ✟ ✟ ✟ ✟ @ H DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 29 By passing to the corresp onding Beck-Chev alley transformed 2 -cells we obtain natural transforma- tions γ F u ! resp ectiv ely γ F u ∗ as in: D ′ ( J ) u !   D ( J ) F o o u !   D ′ ( J ) u ∗   D ( J ) F o o u ∗   ✻ ✻ ✻ ✻ W _ D ′ ( K ) ✻ ✻ ✻ ✻   D ( K ) F o o D ′ ( K ) D ( K ) F o o Deﬁnition 2.2. Let F : D − → D ′ be a morphism of deriv ators and let u : J − → K b e a functor. The morphism F pr eserves homotopy left resp ectiv ely homotopy right Kan ex tensions along u if the natural tra ns formation γ F u ! : u ! F − → F u ! resp ectiv ely γ F u ∗ : F u ∗ − → u ∗ F is an isomo r phism. Similarly , we sp eak of a morphism o f deriv ators which preserves homo top y left/right Kan ex- tensions if the ab o v e prop erty ho lds for all functors u and als o of a morphism which pres e r v es homotopy (co)limits (of a particular shap e or a ls o in genera l). T o mo tiv a te the deﬁnition le t us quickly consider the following ex ample. Example 2.3. L e t F : C − → D b e a functor b et ween bicomplete ca teg ories and let us conside r the induced strict morphism of repr esen ted deriv ator s. W e describe the above canonical morphisms in the absolute cas e, i.e., let J b e a ca tegory a nd u = p J : J − → e b e the unique functor to the terminal categor y . The ab ov e canonical mor phisms then take t he form: colim J F η   β / / F colim J F lim J β / / η   lim J F colim J F u ∗ colim J colim J u ∗ F colim J ǫ O O lim J u ∗ F lim J lim J F u ∗ lim J ǫ O O In the left diagram, β ev aluated at X ∈ C J is the canonical map from the c o limit o f F ◦ X to the image of colim X under F . Thus, we recover the usua l notion of a colimit pr e serving functor, i.e., the functor F : C − → D pre s erv es colimits if a nd only if the induced morphism of deriv ators F : C − → D preserves homotopy colimits. Dual c o mmen ts apply to the diagr a m on the r igh t. W e will see later that if a morphism F : D − → D ′ preserves certain ho mo top y Kan extensions then this is also the case for all induced morphisms F M : D M − → D ′ M (cf. Coro llary 2.8). This will b e a co ns equence of the fac t that homotopy Kan extens ions in deriv ator s of the form D M are calculated p oin t wise. But let us ﬁrst establish the following exp ected fact. Prop osition 2.4. A morphism F : D − → D ′ of derivators pr eserves homotopy left Kan extensions if and only if it pr eserves homotopy c olimits. Pr o of. Let us a ssume that F preser v es homotopy colimits and let us consider a functor u : J − → K . W e obtain the following pasting diagra m in which the natural tra nsformation on the right is the 30 MORITZ GROTH one we w ant to show to be an isomo rphism: D ′ ( J /k ) p !   D ′ ( J ) pr ∗ o o u !   D ( J ) F o o u !   D ′ ( e ) ✺ ✺ ✺ ✺   D ′ ( K ) k ∗ o o ✼ ✼ ✼ ✼   D ( K ) F o o Using axiom (Der4) and the fact that isomorphisms are detected p oint wise it suﬃces to show that the pasting is an isomor phis m. The compa tibility of the forma tion of BC transformed 2- c ells with resp ect to pasting implies that we hav e to show that the transformed 2-cell as sociated to the following left diagram is an isomor phism: D ′ ( J /k ) ✠ ✠ ✠ ✠   D ′ ( J ) pr ∗ o o ✞ ✞ ✞ ✞   D ( J ) F o o D ′ ( J /k ) ✠ ✠ ✠ ✠   D ( J /k ) F o o ✟ ✟ ✟ ✟   D ( J ) pr ∗ o o D ′ ( e ) p ∗ O O D ′ ( K ) u ∗ O O k ∗ o o D ( K ) F o o u ∗ O O D ′ ( e ) p ∗ O O D ( e ) p ∗ O O F o o D ( K ) k ∗ o o u ∗ O O But, us ing the isomorphisms γ F pr and γ F k , this is eq uiv alen t to showing that the B C transformed 2-cell asso ciated to the rig h t diagram is an isomorphism. This in turn follows fro m our assumption that F pr eserves homoto py colimits and (Der4).  F or conv enie nc e let us co llect the following imp ortant closure prop erties of homotopy Kan ex- tensions preserv ing mo r phisms. Prop osition 2.5 . L et D , D ′ , and D ′′ b e derivators, let u : I − → J and v : J − → K b e functors. i) The identity morphism id D : D − → D pr eserves homotopy left Kan ext ensions. ii) If F : D − → D ′ and G : D ′ − → D ′′ pr eserve homotopy left Kan extensions along u then so do es the c omp osition G ◦ F : D − → D ′′ . iii) If F : D − → D ′ pr eserves homotopy left Kan extensions along u and v then it pr eserves homotopy left Kan extensions along v ◦ u. iv) If τ : F − → G is a natura l isomorphi sm of m orp hisms of derivators D − → D ′ then F pr eserves homotopy left Kan extensions along u if and only if G do es. Pr o of. The ﬁrst claim follows immedia tely from the triangular iden tities o f adjunctions. The o ther three cla ims can be deduced from the compatibility of the formation of BC tr ansformed 2 -cells with resp ect to hor izon tal and v ertical pasting resp ectively . In the pro of of the las t tw o claims it is conv enient to rewr ite some of the cohere nc e conditions imp osed on morphisms and natura l transformatio ns o f deriv ators as equalities b et ween certain pas ting diagrams.  Given tw o deriv ator s D a nd D ′ let us denote by Hom ! ( D , D ′ ) res pectively Hom ∗ ( D , D ′ ) the full sub categories of Hom ( D , D ′ ) s panned by the morphisms whic h resp ect homoto p y c o limits and homo to p y limits resp ectively . By the ab o v e pr o position, these a re replete subcateg ories to which the comp osition law can be restr icted. Co rresp ondingly , w e obtain the following 2-categories : Der ! and Der ∗ DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 31 Prop osition 2.6. Le t D b e a derivator and let v : L − → M b e a functor. The morphism of derivators v ∗ : D M − → D L pr eserves homotopy Kan extens ions. In p articular, this is the c ase for the evaluation morphisms m ∗ : D M − → D . Pr o of. W e will treat the ca se of homotopy left K an extensions. By Pr opositio n 2.4 it is eno ugh to show that v ∗ : D M − → D L preserves homotopy colimits. Th us, we hav e to show that for a n arbitrar y sma ll category J the fo llowing squar e is D -exa c t: L × J v × i d / / pr   ✡ ✡ ✡ ✡   M × J pr   L v / / M But this follows from an application of Pro position 1.30.  Thu s, this propos ition tells us, in par ticular, that ho motop y Ka n extensions in the deriv a to r D M are calculated p oint wise. F or a functor u : J − → K and an ob ject X ∈ D M ( J ) we have natural isomorphisms: HoLKan u ( X m ) ∼ = − → (HoLKan u X ) m and (HoRK an u X ) m ∼ = − → HoRKan u ( X m ) Similarly , in the absolute case, i.e., in the ca se of u = p J : J − → e, we obtain cano nical isomor phisms: Ho colim J ( X m ) ∼ = − → (Ho colim J X ) m and (Holim J X ) m ∼ = − → Holim J ( X m ) These isomorphisms are well-b eha v ed in the s ense that the following diagram commutes. W e g ive the compatibility in the case of homotopy left Kan extensions: u ! m ∗ u ∗ =   γ u ! ∼ = / / m ∗ u ! u ∗ ǫ   u ! u ∗ m ∗ ǫ / / m ∗ In fact, this is immedia te using the compatibility of base c hange with pasting and the e qualit y : J m × id / / u   M × J id × u   id × u / / M × K id   = J u   u / / K id   m × id / / M × K id   K m × id / / M × K id / / M × K K id / / K m × id / / M × K This compatibility implies, in pa r ticular, that, f or X ∈ D M ( K ), the counit ǫ : u ! u ∗ ( X ) − → X is an isomorphism in D M ( K ) if and o nly if the counit ǫ : u ! u ∗ ( X m ) − → X m is a n isomorphis m in D ( K ) for all ob jects m ∈ M . F o r later reference, w e co llect the fo llowing co n venien t consequence for the case of a fully faithful functor u : J − → K . Corollary 2. 7. L et D b e a derivator, M a c ate gory, and let u : J − → K b e a ful ly faithful functor. An obje ct X ∈ D M ( K ) lies in the essential image of u ! : D M ( J ) − → D M ( K ) if and only if X m lies in the essential image of u ! : D ( J ) − → D ( K ) for al l m ∈ M . The fact that homoto p y Kan extensions in the deriv ator D M are calculated p oint wise (Pro posi- tion 2.6) can als o b e used to es tablish the following c on venient r esult. 32 MORITZ GROTH Corollary 2.8. L et F : D − → D ′ b e a morphism of derivators and let u : J − → K b e a functor. Then F pr eserves homotopy left Kan extensions along u if and only if F M : D M − → D ′ M pr eserves homotopy left Kan extensions along u for al l smal l c ate gories M . Pr o of. W e ha v e to show that with γ F u ! also the natural transforma tion γ F M u ! = γ F id M × u ! is a natural isomorphism. Since iso morphisms can b e detected point wise and since m ∗ preserves homotopy left Kan extensions (by Pr o position 2.6) this is equiv alent to the fact that the pasting in the left diag ram is a natura l iso mo rphism: D ′ ( J ) u !   D ′ M ( J ) m ∗ o o u !   D M ( J ) F M o o u !   D ′ ( J ) u !   D ( J ) F o o u !   D M ( J ) m ∗ o o u !   D ′ ( K ) ✺ ✺ ✺ ✺   D ′ M ( K ) m ∗ o o ✹ ✹ ✹ ✹   D M ( K ) F M o o D ′ ( K ) ✼ ✼ ✼ ✼   D ( K ) F o o ✻ ✻ ✻ ✻   D M ( K ) m ∗ o o By the natural is omorphism m ∗ ◦ F M ∼ = F ◦ m ∗ (and strictly sp eaking Pr opositio n 2.5) this is equiv alent to the fact that the pasting in the right diagr a m is a natur a l isomorphism. But this follows from our a s sumption that F preserves homotopy left K an extensions alo ng u a nd the fact that m ∗ lies in Hom ! ( D M , D ) .  Since the v alues of a deriv ator alwa ys admit initial ob jects and ﬁnite copro ducts (Pr oposi- tion 1.12) let us agree on establishing the following terminology . Deﬁnition 2.9. A morphism of deriv ators pr eserves initial obje cts r esp e ctively ﬁnite c opr o ducts if the underlying functor has the resp ective pro perty . By the last cor o llary it follows immediately that for suc h a morphism the corresp onding statement is also true for the functors at all levels. Let us no w aga in take up the notion of adjunctions betw een deriv ators. W e include this sligh tly lengthy dis cussion s ince this will motiv ate how to deﬁne an adjunction of t w o v ariables for deriv a to rs. And this no tion in turn will play an essential r ole in the theory of monoida l and enriched deriv ator s (see [Gro1 1 , Gro12a]). An adjunction be tw een tw o deriv ator s D and D ′ consists of tw o morphisms L : D − → D ′ and R : D ′ − → D and tw o natur a l transfor ma tions η : id − → RL and ǫ : L R − → id which satisfy the us ual triang ular iden tities. One might wonder if less da ta would alrea dy determine such an adjunction. As a ﬁrst step there is the following result. Lemma 2.10. L et L : D − → D ′ b e a morphism of pr e derivators such that L K : D ( K ) − → D ′ ( K ) has a right adjoint R K for e ach K ∈ Cat . Then, ther e is a unique way to extend the { R K } to a lax morphism of pr e derivators R : D ′ − → D such t ha t t he fol lowing diagr am c ommutes for al l functors u : J − → K , X ∈ D ( K ) , and Y ∈ D ′ ( K ) : hom D ′ ( K ) ( LX , Y ) / / u ∗   hom D ( K ) ( X, RY ) u ∗   hom D ′ ( J ) ( u ∗ LX , u ∗ Y ) γ L   hom D ( J ) ( u ∗ X , u ∗ RY ) γ R   hom D ′ ( J ) ( Lu ∗ X , u ∗ Y ) / / hom D ( J ) ( u ∗ X , Ru ∗ Y ) DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 33 Pr o of. If we choose X = R Y and if we tr ace the adjunction counit ǫ : LR Y − → Y around the diagram we obtain the uniqueness of the natural transformations γ R . In fact, γ R as indicated in the right diagram is the Beck-Chev alley transformed 2-cell of the trans formation on the left: D ′ ( J ) D ′ ( K ) u ∗ o o D ′ ( J ) R   D ′ ( K ) u ∗ o o R   ✻ ✻ ✻ ✻ W _ D ( J ) L O O D ( K ) L O O u ∗ o o ✟ ✟ ✟ ✟ @ H D ( J ) D ( K ) u ∗ o o Thu s, in formulas, we have γ R u = ( γ L u ) − 1 ∗ . It remains to chec k that this deﬁnes a lax mor phism of prederiv ators R : D − → D ′ . W e omit the details but men tio n that the resp ectiv e formulas ar e implied by the triangular identit ies for adjunctions and by the be ha v ior of the formation of B C transformed 2-cells with r espect to ho r izon tal and v er tical pasting (Lemma 1.18).  In g eneral, we canno t deduce that the lax mo rphism R : D ′ − → D is an actual morphism, i.e ., a pseudo -natural tr a nsformation. How ever, in the context of deriv a tors this can b e reformulated using Lemma 1.2 0 which guarantees that the following natur al transformations are conjugate: γ L u ! : u ! ◦ L − → L ◦ u ! and γ R u : u ∗ ◦ R − → R ◦ u ∗ Thu s, since these a r e conjugate transformations, o ne of them is an isomo rphism if and only if this is the case for the other. F ro m this w e obtain the following result. Prop osition 2.11. L et L : D − → D ′ b e a morphism of deriva tors which admits levelwise right adjoints and let R : D ′ − → D b e a lax morphism as in L emma 2.10. The morphism L is a left adjoint morphism of derivators if and only if L pr eserves homotopy left Kan extensions if and only if R is a morphism of derivators. In p articular, a morphism of derivators is an e quivalenc e if and only if it is levelwise an e quivalenc e of c ate gories. Pr o of. The eq uiv a lence of the statement s ab out a left adjoint morphism follows immediately fro m the ab ov e. If the morphism L happ ens to b e a levelwise equiv alence of deriv a tors then the la x morphism R is a n ac tua l mor phism o f deriv ators . In fact, the natur al transformations γ R u are comp ositions of three isomor phisms in this case.  F or later refer ence, let us collect the following important c la ss of homotopy (co)limit pres erving morphisms of deriv ato rs. Corollary 2.12. A left adjoint morphi sm of derivators pr eserves homotopy left K an extensions. Prop osition 2.6 and Pro position 2.11 together give us immediately the following tw o classes of examples of adjunctions. Example 2.13 . i) Let D b e a deriv ator and let v : L − → M b e a functor, then we have tw o ad- junctions of deriv ato rs ( v ! , v ∗ ) : D L ⇀ D M and ( v ∗ , v ∗ ) : D M ⇀ D L . ii) Let ( F , U ) : M − → N be a Quillen adjunction betw een combinatorial mo del ca tegories. Then the formation of derived Quillen functor s gives us tw o (in g eneral no n-strict) morphisms of deriv a- tors L F : D M − → D N and R U : D N − → D M . These are pa rt of an a djunction of der iv ators ( L F, R U ) : D M ⇀ D N . In par ticular, L F pr eserves homotopy left Kan extensio ns and R U pre- serves homotopy right K an extensions (Corollary 2.12 ). If w e start with a Q uillen equiv a lence then we obtain a derived equiv ale nce of deriv ators . T his a lready ma kes more precise the statement that a Quillen equiv a lence is not o nly a Q uillen pa ir inducing an e q uiv a lence of ho motop y ca tegories but 34 MORITZ GROTH that it also resp ects the entire ‘homotopy theo ry’. Remark 2.14. • It can be shown that the above a ssignmen t M 7− → D M suitably restricted deﬁnes a bi-equiv alence of theorie s. Lo osely sp eaking this says that nice mo del c a tegories and nice der iv a tors do the sa me jo b. More precisely , Renaudin has s ho wn such a result in [Ren0 9] by establishing the fo llo w ing tw o steps: Let Mo dQ deno te the 2-catego ry of comb inatorial mo del categories with Quillen a djunctions ( F , U ) : M ⇀ N as morphisms a nd natural tra nsformations o f left adjoints as 2-morphisms. Renaudin sho ws tha t there is a pseudo-lo calization Mo dQ [ W − 1 ] of the com binatorial mo del ca teg ories at the class W of Quillen equiv alences. Mor eo v er, let Der Pr denote the 2-category of der iv ators of small presentation together with adjunctions as mor phis ms. A deriv ato r is s a id to be o f small pres en tatio n if it can b e obtained a s a ‘nice’ lo calization of the deriv ator asso ciated to simplicial presheaves. The as s ignmen t D ( − ) : Mo dQ − → Der Pr : M 7− → D M factors then up to natural isomor phism over the pseudo-lo calization Mo dQ [ W − 1 ] as indicated in: Mo dQ D ( − ) / / γ   ∼ = Der Pr Mo dQ [ W − 1 ] D ( − ) H H Renaudin show ed that the induced 2-functor D ( − ) : Mo dQ [ W − 1 ] − → Der Pr is a biequiv alence, i.e., a 2-functor w hich is biessentially s ur jectiv e and fully faithful in the sens e that it induces e quivalenc es of morphism categor ies (for biequiv ale nc e s cf. e .g . to [Str96, Lac10] and to [Lac02, Lac04] for their more conceptual ro le). • W e w ant to include a remark on diﬀ erent appro ac hes to a theory of ( ∞ , 1) − categories. There a r e by now many diﬀerent wa ys to a xiomatize such a theory . Among these a re the mo del categor ie s, the ∞− categ ories, and the der iv ators. These theories ar e interrelated by v arious constr uctions. F or a simplicial mo del categor y , one can use the co heren t ner v e construction of Cor dier [Cor 82 ] to obtain an underly ing ∞− category . Moreov er, g iven a bicomplete ∞− category or a model c ategory , by for ming sys tematically ho motop y categ ories one obtains an a s socia ted deriv ator . T he s e thr ee theories a re in fact all ‘equiv alent in a cer tain sense’ if one is willing to restrict to nice sub classes. These comparison results rely hea vily on homotopical gene r alizations of t he following ‘tw o- step hi- erarch y’. In clas sical categor y theor y there are the pr esheaf categorie s which can b e c onsidered as universal co completions. Mo re precisely , the fact that every contrav a rian t set-v alued functor o n a small ca tegory is canonically a co limit of representable ones can b e used to prov e such a r esult. Nice lo calizations of these pres heaf categ ories (the s o-called accessible, reﬂective loca lizations) g iv e us precisely the pr e s en table categories (Repr esen ta tion Theorem [AR94 , Theorem 1.46]). These tw o main steps, namely to establis h the universal prop erty of presheaf categor ies a nd to characterize presentable c a tegories as nice lo calizations of pr esheaf categories , can b e r edone for all the diﬀer- ent theo r ies. T o achiev e this one ha s to r e place presheaf ca tegories by simplicial pr eshe af c ate gories which is ﬁne with the ba sic philosophy of higher category theory . More o ver, the classic a l lo calization theory is r eplaced b y a suitable Bous ﬁeld lo calization theory [Bou75, Hir0 3]. F or model categories , this was done by Dugge r in [Dug01b, Dug01a], while the cor respo nding results for ∞− categorie s can be found in Lurie’s [Lur 09 ]. The c haracteriza tion of presentable ∞− categories as being precise ly the accessible , reﬂectiv e lo calizatio ns of simplicial pr esheaf categ ories is therein cr edited to [Sim07 ]. F or deriv ators , the free generation pr operty o f the deriv ator ass ociated to simplicial presheav es ca n DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 35 be found in [Cis08]. Note how ev er tha t the ba sic mo del used in the background is not the cate- gory of simplicial sets but the ca tegory of s mall catego ries. It can be shown that this wa y als o all ‘homotopy types a re mo deled’. Finally , unt il now, the Representation Theorem for deriv ator s of small pr esen tation is only turned into a deﬁnition in [Ren09]. The author plans to come bac k to this point in a later pro ject. Having es tablished these similar theories at all diﬀere nt lev els one can then establish the comparison res ults if one restricts to the sub classes of (simplicial) com binatorial mo del categories, presentable ∞− categories, and deriv ator s o f small pr e sen ta tio n. 3. P ointed deriv a tors 3.1. Deﬁni tion and basic examples. Since we a r e mainly interested in stable der iv a tors, we turn immediately to the nex t richer structure, namely the p oin ted deriv a to rs. There ar e at least tw o wa ys to a xiomatize a notion of a p ointed deriv ator. F rom these t wo notions, w e turn the ‘weaker one’ in to a deﬁnit ion. The ‘str onger one’ will be referred to as a str ongly p oin ted der iv ator, but we will show that these t w o notions actually coincide. Deﬁnition 3.1. A deriv a tor D is p ointe d if the underlying catego ry D ( e ) of D is po in ted, i.e., admits a zero ob ject 0 ∈ D ( e ). Note that the pointedness is ag ain only a pr operty and not an additional structure. F or a prederiv ato r one w ould impose a s ligh tly stronger condition: a prederiv ator is pointed if and only if all of its v alues and all restriction of diagram functors are pointed. In the case of a deriv ator these stronger prop erties follow immedia tely from the deﬁnition. Prop osition 3.2. L et D b e a p ointe d derivator and let u : J − → K a functor in Cat . Then D ( K ) is also p ointe d and the functors u ! , u ∗ , u ∗ ar e p ointe d. In p articular, if a derivator D is p ointe d then this is also the c ase for D M for every M ∈ Cat . Pr o of. The map fro m the initial to the ﬁnal o b ject in D ( K ) is a n iso morphism since this is po in twise the case. Moreov er, eac h of the functors u ! , u ∗ , u ∗ has an adjoint on at least o ne s ide and will hence preser v e the zero ob jects.  Example 3.3. i) Let C be a ca teg ory . Then the repres e n ted preder iv ator C is p oin ted if a nd only if the catego ry C is p oin ted. ii) The deriv a tor D M asso ciated to a p o in ted combinatorial mo del categ o ry M is pointed. iii) A deriv a to r D is p oin ted if and only if its dual D op is po in ted. W e now w ant to g iv e the stronger axiom as used by Maltsiniotis in [Mal07]. Deﬁnition 3.4. A deriv a tor D is str ongly p ointe d if it has the fo llowing tw o pr o perties: i) F or every sie v e j : J − → K , the homotopy r igh t Kan extension functor j ∗ has a rig h t adjoint j ! : ( j ∗ , j ! ) : D ( J ) ⇀ D ( K ) ii) F or every co siev e i : J − → K, the homotopy left Kan extension functor i ! has a left a djoin t i ? : ( i ? , i ! ) : D ( K ) ⇀ D ( J ) It is an immediate cor o llary of the deﬁnition that a strongly p oin ted der iv ator is p oin ted. In fact, one of the t w o additional prop erties is enough to ensure this. Corollary 3.5. If D is a stro ngly p ointe d derivator, then D is p ointe d. 36 MORITZ GROTH Pr o of. It is enough to consider the cosieve ∅ e : ∅ − → e . F or an initial ob ject ∅ e ! (0) ∈ D ( e ) and an arbitrar y X ∈ D ( e ) , we then deduce hom D ( e ) ( X, ∅ e ! (0)) ∼ = hom D ( ∅ ) ( ∅ ? e X , 0) = ∗ , s o that ∅ e ! (0) is also terminal.  The aim o f this subs e ction is to prov e that also t he conv er s e ho lds (cf. Coro llary 3.8). A further pro of of tha t conv er se will b e given in the stable situation, i.e ., for stable deriv ators. That seco nd pro of is quite an indirect one. It relies o n the fact that r ecollemen ts of tr ia ngulated catego r ies are overdetermined and will b e given in Subsection 4 .3. As a prepa ration, for the direct pro of, we men tion the following immediate co nsequence of Prop osition 1.2 9. It states that homoto p y left Kan extension along co siev es and homotop y r igh t Kan e x tension along siev es a re given by ‘extensio n by zero functors’ and will be of constant use in the remainder of this pape r. Prop osition 3.6 . L et D b e a p ointe d derivator. i) L et u : J − → K b e a c osieve, then the homotopy left Kan ex tension u ! is ful ly faithful and X ∈ D ( K ) lies in the essential image of u ! if and only if X k ∼ = 0 for al l k ∈ K − u ( J ) . ii) L et u : J − → K b e a sieve, then the homotop y right Kan extension u ∗ is ful ly faithful and X ∈ D ( K ) lies in the essential image of u ∗ if and only if X k ∼ = 0 for al l k ∈ K − u ( J ) . F ollowing Heller [Hel97], we intro duce the follo wing notation. L e t D be a p oin ted deriv ator and let u : J − → K b e the inclusion of a full subcatego ry . Then w e denote by D ( K, J ) ⊆ D ( K ) the full, replete s ubcategor y spanned b y the ob jects X which v a nish on J , i.e ., such that u ∗ ( X ) = 0 . If u is now a cos iev e resp ectively a sieve, the ab o ve prop osition guara n tees that we have the following equiv alences of categ ories: ( u ! , u ∗ ) : D ( J ) ≃ − → D ( K, K − J ) r espectively ( u ∗ , u ∗ ) : D ( K, K − J ) ≃ − → D ( J ) This prop osition, a lthough easily prov ed, will b e of central impo rtance in all what follows. It will, in particula r, b e of constant use in the study of the imp ortant co ne, ﬁb er, s us pension, and lo op functors and also in the pro of that the v a lue s o f a stable der iv ator can b e canonically endowed with the structure of a triang ulated categ ory . Ho wev er, we ﬁrst hav e to establish s ome pro perties of coCartesia n and Car tesian squares and this will b e done in the next Subsection 3.2. T o conclude this s ubsection we will now give the pro of that pointed der iv ators ar e actually strongly p ointed. The c o nstructions inv o lv ed in the pr oof are motiv a ted b y the paper of Rezk [Rez ] in which he gives a nice construction of the ‘natural mo del s tr ucture’ on Ca t . This model structure is due to Joyal and Tierney [JT91] and the adjective ‘natural’ refers to the fact that the weak equiv alences in tha t model structure are precis ely the equiv alences in the 2-category Cat . W e use a minor mo diﬁcation of the mapping cylinder ca tegories used in [Rez]. Ins tead of forming the pro duct with the g r oupoid genera ted by [1] we us e the ca tegory [1] itself. This leads to tw o ‘diﬀerently oriented v er sions’ of the mapping cylinder and b oth of them will b e needed. Lemma 3.7. i) L et u : J − → K b e a c osieve. Then the sub c ate gory D ( K, J ) ⊆ D ( K ) is c or eﬂe ctive, i.e., the inclusion functor ι admits a right adjoint. ii) L et u : J − → K b e a sieve. Then the sub c ate gory D ( K , J ) ⊆ D ( K ) is r eﬂe ctive, i .e., the inclusion functor ι admits a left adjoi nt. Pr o of. W e will give the details for the pro of of ii) and mention the necessary mo diﬁcations for i). So, let u : J − → K be a sieve a nd let us construct the mapping cylinder catego r y cyl ( u ) . By deﬁnition, cyl ( u ) is the full subcatego ry of K × [1 ] spa nned by the ob jects ( u ( j ) , 1) a nd ( k, 0) . Thus, DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 37 it is deﬁned by the following pushout diagram where i 0 is the inclusion a t 0: J u / / i 0   K   J × [1] / / cyl ( u ) There a re the na tural functors i : J − → cyl ( u ) : j 7− → ( u ( j ) , 1) and s : K − → cyl ( u ) : k 7− → ( k , 0). Moreov er, id : K − → K a nd J × [1] pr − → J u − → K induce a unique functor q : c yl ( u ) − → K . These functors satisfy the rela tions q ◦ i = u, q ◦ s = id K . Consider now an ob ject X ∈ D ( cyl ( u ) , i ( J )) a nd let us ca lculate the v a lue o f q ! ( X ) at some u ( j ) ∈ K . F or this purp ose, we sho w that the following pa sting is homotopy ex act: e i ( j ) , 1 / / id   ☛ ☛ ☛ ☛   i ( J ) /i ( j ) × [1] ∼ = / / p   ✏ ✏ ✏ ✏   cyl ( u ) /u ( j ) pr / / p   ☞ ☞ ☞ ☞   cyl ( u ) q   e / / e / / e u ( j ) / / K Since u is a siev e we have an isomorphism a s depicted in th e diagram. Moreover, i ( J ) /i ( j ) × [1] has a terminal element so that the left t wo squares are homotop y exact by P ropo sition 1 .24 . Thus, w e can conclude by (Der4) that the abov e pasting is homotopy exac t and obtain q ! ( X ) u ( j ) ∼ = X i ( j ) = 0 . The adjunction ( q ! , q ∗ ) restricts to an a djunction ( q ! , q ∗ ) : D ( cyl ( u ) , i ( J )) ⇀ D ( K, u ( J )) . Moreov er, since w e deﬁned the mapping cy linder fo rming the pro duct with [1 ] as oppose d to the group oid generated by it, s : K − → cyl ( u ) is a siev e. Hence, b y Propo sition 3.6, we have an induced equiv alence ( s ∗ , s ∗ ) : D ( K ) ≃ − → D ( cyl ( u ) , cyl ( u ) − s ( K )) = D ( cyl ( u ) , i ( J )) . Putting these tw o a djunctions together w e obtain the adjunction ( q ! ◦ s ∗ , s ∗ ◦ q ∗ ) : D ( K ) ⇀ D ( cyl ( u ) , i ( J )) ⇀ D ( K , u ( J )) . The relatio n q ◦ s = id implies that the r igh t a djoin t of th is a djunction is the inclusion ι a s intended and the reﬂection is given b y r = q ! ◦ s ∗ . The pro of of i) is similar. Instead of using cyl ( u ) one uses this time the mapping cylinder category cyl ′ ( u ) whic h is obtained by a similar pushout but using the inclusio n i 1 instead of i 0 . Let us deno te the corresp onding functors again by i, q , and s. Us ing a similar calcula tion of q ∗ and the fact that s is now a c osiev e, w e ca n construct a cor eﬂection c.  Corollary 3.8. L et D b e a p ointe d derivator, then D is also str ongly p ointe d. Pr o of. Given a sieve u : J − → K w e hav e to s ho w that u ∗ has a right adjoint. The inclusion v : K − u ( J ) − → K of the c omplemen t is a cosieve. The ab ov e lemma applied to v th us gives us a coreﬂection c : D ( K ) ⇀ D ( K , K − u ( J )) . Putting this together with th e equiv alence induced by u ∗ (guaranteed b y Pr oposition 3.6) we obtain the desired adjunction: ( u ∗ , u ! ) : D ( J ) u ∗ / / D ( K, K − u ( J )) u ∗ o o ι / / D ( K ) c o o The pro of in the case of a co siev e is, of course, the dual one.  The pro ofs of the last tw o res ults w ere cons tructiv e . So, for la ter reference, let us give precise formulas for these additional adjoint functors . Let D b e a p ointed deriv ato r and let u : J − → K 38 MORITZ GROTH be a cosieve. Let us denote by v : J ′ = K − u ( J ) − → K the s iev e g iven by the complemen t. The adjunctions ( u ? , u ! ) : D ( K ) ⇀ D ( J ) and ( v ∗ , v ! ) : D ( J ) ⇀ D ( K ) a r e given by the following comp osite adjunctions resp ectively: u ? : D ( K ) s ∗ / / D ( cyl ( v ) , i ( J ′ )) q ! / / s ∗ o o D ( K, v ( J ′ )) u ∗ / / q ∗ o o D ( J ) u ! o o : u ! v ∗ : D ( J ′ ) v ∗ / / D ( K, u ( J )) q ′∗ / / v ∗ o o D ( cyl ′ ( u ) , i ′ ( J )) s ′∗ / / q ′ ∗ o o D ( K ) s ′ ! o o : v ! Here, cyl ( v ) is the mapping cylinder o btained fro m identifying the b ottom J ′ × { 0 } of J ′ × [1] with the image o f v , i is the inclusion in the cylinder, q is the pro jection a nd s is the ca nonical section of q . The nota tion in the sec o nd decomp osition is s imilar wher e the role s of 0 and 1 are in terchanged. 3.2. coCartesian and Cartesian squares. In this subsection, we introduce coCar tesian and Cartesian squares in a deriv a tor and e s tablish some fac ts ab out them which will be needed in Section 4. Some of these prop erties are well-known from cla ssical category theor y a nd will b e reprov ed here for the context of deriv a tors. The main results ar e the b eha vior of (co)Cartesia n squares under canc e lla tion and compo sition (Prop osition 3.14) and a ‘detection result’ (due to F ranke [F ra9 6 ]) for (co)Cartesia n squa res (Propos ition 3.11) . W e denote the categ ory [1] × [1] by  , i.e.,  is the following p oset co nsidered as a categor y where we dra w the ﬁrst co ordinate horizontally: (0 , 0) / /   (1 , 0)   (0 , 1) / / (1 , 1) F or the treatment of Cartesian and coCartesian squares, it is imp ortant to consider the following t wo inclusions of subca tegories i p : p − →  resp. i y : y − →  which are given b y the subp osets: (0 , 0) / /   (1 , 0) resp ectiv ely (1 , 0)   (0 , 1) (0 , 1) / / (1 , 1) Deﬁnition 3.9. Let D b e a deriv ator a nd let X ∈ D (  ) . i) The squa re X is c oCartesian if it lies in the esse ntial image of i p ! : D ( p ) − → D (  ) . ii) The squa re X is Cartesian if it lies in the essential image of i y ∗ : D ( y ) − → D (  ) . It follows immediately from the fully fa ithfu lness o f homotopy K an extensions along fully faithful functors (Pro position 1.26) and Le mma 1.27 that such an X ∈ D (  ) is coCartes ian if and only if the canonica l mo r phism ǫ (1 , 1) : i p ! i p ∗ ( X ) (1 , 1) − → X (1 , 1) is an isomo rphism. Dually , the square X is Cartesian if and only if the canonical morphism η (0 , 0) : X (0 , 0) − → i y ∗ i y ∗ ( X ) (0 , 0) is a n iso morphism. Our ﬁr s t aim in this section is to esta blis h a ‘detection result’ for (co)Cartesian squar es in larger diagrams which will be us e d frequently later on. So, let us quickly giv e the notion of a squar e. Deﬁnition 3. 10. Let J b e a category . A squar e in J is a functor i :  − → J w hich is injectiv e on ob jects. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 39 Here is now the in tended ‘detection result’ for (co)Cartesian squar e s. Prop osition 3.1 1. L et i :  − → J b e a squar e in J and let f : K − → J b e a funct or. i) Assu m e that the induc e d functor p ˜ i − →  J − i (1 , 1 )  /i (1 , 1) has a left adj oint and t ha t i (1 , 1) do es not lie in the image of f . Then for al l X = f ! ( Y ) ∈ D ( J ) , Y ∈ D ( K ) , the induc e d squar e i ∗ ( X ) is c oCartesian. ii) Assu me t ha t the induc e d functor y ˜ i − →  J − i (0 , 0)  i (0 , 0) / has a right adjoint and that i (0 , 0) do es not lie in t he image of f . Then for al l X = f ∗ ( Y ) ∈ D ( J ) , Y ∈ D ( K ) , t he induc e d squar e i ∗ ( X ) is Cartesian. Pr o of. W e give a pro of of i). By as sumption on f , f factor s as K ¯ f − → J − i (1 , 1 ) j − → J s o that our setup can b e summarized by:  J − i (1 , 1 )  /i (1 , 1) pr   p R = ˜ i 8 8 r r r r r r r r r r r r / / i p   J − i (1 , 1) j   K ¯ f o o f w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣  i / / J W e wan t to show that the a djunctions counit ǫ : i p ! i p ∗ − → id is an isomo rphism when applied to i ∗ f ! ( Y ) , Y ∈ D ( K ) . But by Lemma 1.27 and Lemma 1.18 this is equiv alent to showing that the base change morphism asso ciated to the pasting on the left-hand-side is an iso morphism when ev aluated at f ! ( Y ) : p ∼ = p / (1 , 1) / / p   ✠ ✠ ✠ ✠   p i p / / i p   ⑦ ⑦ ⑦ ⑦ {   i / / id   ⑦ ⑦ ⑦ ⑦ {  J id   p R / / p   ☞ ☞ ☞ ☞   J − i (1 , 1) /i (1 , 1) pr / / p   ✑ ✑ ✑ ✑   J − i (1 , 1) j / / j   ✠ ✠ ✠ ✠   J id   e (1 , 1) / /  id / /  i / / J e id / / e i (1 , 1) / / J id / / J Using Lemma 1 .18 again, this is equiv alent to showing that the base ch ange morphism a ssocia ted to the pasting o n the r igh t gives an isomor phism when ev alua ted at f ! ( Y ) . But this is the case by (Der4 ) and Prop osition 1.24, since f ! ( Y ) ∼ = j ! ¯ f ! ( Y ) lies in the essential image of j ! .  Typical applicatio ns of this prop osition will b e given when the ca tegories under cons ide r ation are p osets. Let J and K b e p osets consider ed as catego ries. Recall that a functor u : J − → K is the same as an or der-preserving map. Moreov er, an adjunction ( u, v ) : J ⇀ K is equiv alently given by tw o order -preserving maps u : J − → K and v : K − → J such that j ≤ v u ( j ) , j ∈ J, and uv ( k ) ≤ k , k ∈ K . In fact, in this case the triang ular identities are automatica lly satisﬁed. F or n ≥ 0 , we denote b y [ n ] the ordinal num b er 0 < . . . < n considered as a category . Mo reo ver, let us denote the standard co s implicial face resp. degener acy maps by d i : [ n − 1] − → [ n ] , 0 ≤ i ≤ n, resp. s j : [ n + 1] − → [ n ] , 0 ≤ j ≤ n. Her e, d i is the unique mono to ne injection omitting i while s j is the unique monoto ne surjection hitting j twice. T he images of these co s implicial str ucture maps under a contrav a rian t functor will, as usual, b e written as d i resp. s j . 40 MORITZ GROTH Lemma 3 .12. F or every 0 ≤ i ≤ n − 1 we have an adjunction ( s i , d i ) : [ n ] ⇀ [ n − 1] . In p articular, we thus obtain the adjunctions ( s 0 , d 0 ) : [2] × [1] ⇀ [1] × [1 ] and ( s 1 , d 1 ) : [2] × [1] ⇀ [1] × [1 ] . In the next prop osition, we will co ns ider squar es X ∈ D (  ) in a deriv ato r and so me of its asso ciated sub-diag rams. T o establis h so me short hand notation, let us denote by d i v the face ma ps id × d i : [1] − → [1] × [1 ] =  in the ’vertical direction’ giv ing rise to ‘hor izon tal f aces’ and simila rly in the other ca se. If we apply a contra v ariant functor to these morphisms we will interc hange the indices and thus write d h i and d v i resp ectiv ely . Prop osition 3.1 3. L et D b e a derivator. i) A n obj e ct of D ([1]) is an isomorph ism if and only if it lies in the essential image of the homotop y left Kan extension functor 0 ! : D ( e ) − → D ([1]) . ii) L et X ∈ D (  ) b e a squar e such that d v 1 ( X ) is an isomorphism, i.e., we have X 0 , 0 ∼ = − → X 1 , 0 . The squar e X is c oCartesian if and only if also d v 0 ( X ) is an isomorphism. Pr o of. i) This is a sp ecial ca se o f (the dual of ) Lemma 1.25. ii) B y i) our assumption on X is equiv alent to the adjunction counit 0 ! 0 ∗ d v 1 ( X ) − → d v 1 ( X ) b eing an isomor phism. Using Lemma 1.27 and (Der4), we ca n refor m ula te this by saying that the base change mo r phism asso ciated to the follo wing pasting is an iso morphism when ev aluated on X : e ∼ = e / 1 pr / / p   ✝ ✝ ✝ ✝   e 0 / / 0   ⑧ ⑧ ⑧ ⑧ {  [1] d 1 v / / id   ⑦ ⑦ ⑦ ⑦ {   id   = e (0 , 0) / / id   ⑤ ⑤ ⑤ ⑤ z   id   e 1 / / [1] id / /  d 1 v / /  e (1 , 0) / /  W e wan t to reform ulate this in a wa y whic h is more conv enient for this pro of. F or this purp ose let us consider the following factorizatio n of the horizontal face ma p: d 1 h = i p ◦ j : [1] j − → p i p − →  Now, our assumption that d v 1 ( X ) is an isomorphism is equiv alent to the counit j ! j ∗ i p ∗ X − → i p ∗ X being an isomorphism. In fact, using Lemma 1 .27 and (Der4 ), the claim ab out the counit ca n be equiv alently res tated b y saying that the ba s e change of the following pasting is a n isomorphism when ev aluated at X : e ∼ = [1] / (0 , 1) pr / / p   ✡ ✡ ✡ ✡   [1] j / / j   ⑦ ⑦ ⑦ ⑦ {  p i p / / id   ⑥ ⑥ ⑥ ⑥ z   id   = e (0 , 0) / / id   ⑥ ⑥ ⑥ ⑥ z   id   e (0 , 1) / / p id / / p i p / /  e (1 , 0) / /  DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 41 Thu s, the cla im follows from our previo us reasoning. This in turn can b e used to show that under our assumption the sq uare X is coCar tesian if and only if the base change asso ciated to [1] j / / j       |  p i p / / id   ⑦ ⑦ ⑦ ⑦ {   id   = [1] d 1 h / / d 1 h       |   id   p id / / i p   ✁ ✁ ✁ ✁ |  p i p / / i p       |   id    id / /   id / /  id / /  is an isomorphism at X . By Lemma 1.27 this is the case if and only if it is the case at (1 , 1) whic h in turn is eq uiv alen t (by similar arg umen ts as in the beginning of this pro of ) to the fact that d v 0 ( X ) is an isomo r phism.  W e now discuss the comp osition and cancellatio n pr o perty of (co)Ca r tesian square s . Recall from classical categor y theo ry that for a diagra m in a categ ory of the shape X 0 , 0 / /   X 1 , 0   / / X 2 , 0   X 0 , 1 / / X 1 , 1 / / X 2 , 1 the follo wing holds: if the squar e o n the le ft is a pushout, then the square on the rig h t is a pushout if and o nly if the comp osite squa r e is. The corr e sponding result in the theory of deriv ators is the co n tent of the next prop osition. The metho ds are similar to the ones used in the pro of of Prop osition 3 .13 so we will b e a bit more sketc hy . Moreov er, since w e only use horizon tal face maps this time we again drop the additiona l index. Prop osition 3.1 4. L et D b e a derivator and let X ∈ D ([2] × [1]) . i) If d 2 ( X ) ∈ D (  ) is c oCartesian, then d 0 ( X ) is c oCartesian if and only if d 1 ( X ) is c oCartesian. ii) If d 0 ( X ) ∈ D (  ) is Cartesian, t hen d 2 ( X ) is Cartesian if and only if d 1 ( X ) is Cartesian. Pr o of. W e g iv e a pr oof of i). F or this purp ose, let J 0 resp. J 1 be the p osets (0 , 0) / /   (1 , 0) / / (2 , 0) resp. (0 , 0) / /   (1 , 0) / /   (2 , 0) (0 , 1) (0 , 1) / / (1 , 1) and denote the fully faithful inclusio n functors by i : J 0 i 0 − → J 1 i 1 − → J 2 = [2] × [1] . Our assumption on X a nd Lemma 1.27 guarantee that the ba se change morphism asso ciated to the f ollowing square ev aluated at X is an is omorphism if and only if this is the ca se at (2 , 1) : J 0 i / / i       |  J 2 id   J 2 id / / J 2 42 MORITZ GROTH W e wan t to refor m ulate this proper ty in tw o wa ys. First, using (Der4) and Prop osition 1.24 applied to d 1 : p − → J 0 this ca n b e seen to b e equiv a le nt to the claim that d 1 ( X ) is coC a rtesian. Second, using the factoriza tion i = i 1 ◦ i 0 the prop ert y can a lso b e reformulated b y saying tha t the base change mor phis m as sociated to the following pasting diagram is an isomorphism a t (2 , 1 ) when ev aluated at X : J 0 i 0 / / i 0       |  J 1 i 1 / / id       |  J 2 id   J 1 id / / i 1       |  J 1 i 1 / / i 1       |  J 2 id   J 2 id / / J 2 id / / J 2 But under our assumption on d 2 ( X ) a nd using the coﬁna lit y of d 0 : [1] − → [2] this can b e seen to be equiv alent to the fact that d 0 ( X ) is coCar tesian whic h then concludes o ur proo f.  Now, that w e ha ve established the prop erties of (co)Cartesia n squar e s necessa ry for our purp oses, we will quickly deﬁne left exact, right exact, and exact morphisms of deriv a tors. Deﬁnition 3.15. A morphism of der iv ators pr eserves c oCartesian squ ar es if it prese r v es ho motop y left K a n e x tensions along i p : p − →  . Similarly , a mo r phism of deriv ators pr eserves Cartesian squar es if it pr eserves homotopy righ t Kan extensions along i y : y − →  . As an immediate co nsequence of Corollary 2.8 and Corolla ry 2.7 we hav e the following result. Corollary 3 .16. L et F : D − → D ′ b e a morphism of derivators. Then F pr eserves c oCartesia n squar es if and only if F : D M − → D ′ M pr eserves c oCartesian s qu ar es for al l c ate gories M . Mor e- over, an obje ct X ∈ D M (  ) is c oCartesian if and only if the squar es X m ∈ D (  ) ar e c oCartesian for al l obje cts m ∈ M . Deﬁnition 3.17. Let F : D − → D ′ be a morphism of der iv ators. i) The mor phis m F is left exact if it preserves Cartesian squares and ﬁnal ob jects. ii) The mor phis m F is right exact if it preserves co Cartesian squares and initial ob jects. iii) The mor phis m F is exact if it is left exact and r igh t exact. It follows immediately from this deﬁnition that a left exact mor phism preserves, in particular, ﬁnite pro ducts and dually for a right exa c t morphism. Example 3. 1 8. i) Let ( F , U ) : M − → N b e a Quillen a djunction b et ween combinatorial mo del categorie s. The morphism L F : D M − → D N is r igh t exact and the mo rphism R U : D N − → D M is left exact. This holds more ge ne r ally for an arbitrary adjunction of der iv ators. ii) Let D b e a deriv ator and let u : L − → M be a functor. The induced strict mor phism of deriv a tors u ∗ : D M − → D L is exact. 3.3. Susp ensi ons, Lo ops, Cone s, and Fib ers. Let D b e a p oin ted deriv ato r and let J b e a category . In this subsection we w ant to constr uct the susp ension and lo op functors on D ( J ) and the cone and ﬁb er functors on D ( J × [1]) . B y P ropo sition 3 .2, w e ca n assume J = e. Let us beg in with the susp ension functor Σ a nd the lo op functor Ω . The ‘extension by zer o functors’ as given b y Prop osition 3.6 will again be c r ucial. Let us consider the following sequences DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 43 of functors: e (0 , 0) / / p i p / /  e, (1 , 1) o o e (1 , 1) / / y i y / /  e. (0 , 0) o o Since (0 , 0) : e − → p is a sieve the homotopy right Ka n extension functor (0 , 0) ∗ gives us a n ‘extensio n by zero functor’ by Pro position 3.6, and similarly f or the homoto p y left Kan extension (1 , 1) ! along the cosieve (1 , 1) : e − → y . Deﬁnition 3.19. Let D b e a p oin ted deriv ator. i) The susp ension functor Σ is g iven by Σ : D ( e ) (0 , 0) ∗ / / D ( p ) i p ! / / D (  ) (1 , 1) ∗ / / D ( e ) . ii) The lo op fu n ctor Ω is given b y Ω : D ( e ) (1 , 1) ! / / D ( y ) i y ∗ / / D (  ) (0 , 0) ∗ / / D ( e ) . The motiv ation for these deﬁnitions should b e cle a r from top ology . Recall that given a p oint ed top ological space X , the sus pension Σ X is constructed by ﬁrst tak ing tw o instances of the ca nonical inclusion into the (cont ractible!) co ne C X and then forming the pushout: X / /   C X X / /   C X   C X C X / / Σ X W e ca n co nsider this dia gram as a homotopy pusho ut. The abov e deﬁnition abstracts precisely this construction. Of cour se, we wan t to show that thes e functors deﬁne an adjoint pair (Σ , Ω) : D ( e ) ⇀ D ( e ) . F or this purp ose, let us deno te by M ⊂ D (  ) , M p ⊂ D ( p ) , and M y ⊂ D ( y ) the resp ective full sub c ategories spanned b y the ob jects X with X 1 , 0 ∼ = 0 ∼ = X 0 , 1 . Prop osition 3.2 0. If D is a p ointe d derivator, then we have an adjunction (Σ , Ω) : D ( e ) ⇀ D ( e ) . Pr o of. With the notation es tablished ab o ve, the susp ension and the lo op functor can be factor ed as follows: Σ : D ( e ) (0 , 0) ∗ ≃ / / M p i p ! / / M i y ∗ / / M y (1 , 1) ∗ ≃ / / D ( e ) D ( e ) M p (0 , 0) ∗ ≃ o o M i p ∗ o o M y i y ∗ o o D ( e ) ≃ (1 , 1) ! o o : Ω The existence of the fa c torization is clear a nd the f act that the functors (0 , 0) ∗ and (1 , 1) ! restricted this way are equiv a lences follows from their fully faithfulness and Pro p osition 3.6. F rom this description, o ne sees immediately that we hav e an adjunction (Σ , Ω) which is, in fact, given a s a comp osite adjunction of four adjunctions amo ng whic h t wo ar e equiv alences.  Using simila r constructions, one can int ro duce c one a nd ﬁb er functors for p oin ted der iv ators. Again, the deﬁnition is ea sily motiv ated from top ology . If we consider a map of p oin ted s paces f : X − → Y then the ma pping cone C f of f is constructed in tw o steps b y forming a pushout as 44 MORITZ GROTH indicated in the next diagram: X f / /   Y X f / /   Y   C X C X / / C f T o axiomatize this in the cont ext o f a pointed deriv a tor, let us consider the following mor phisms o f po sets: [1] i / / p i p / /  y i y o o [1] j o o Here, i is the sieve classifying the horizo n tal a rrow while j is the cosieve classifying the vertical arrow. In particular, by Prop osition 3.6, w e ha ve a gain extension b y zer o functors i ∗ and j ! . Deﬁnition 3.21. Let D b e a p oin ted deriv ator. i) The c one fun ctor Co ne : D ([1 ]) − → D ([1]) is deﬁned as the co mposition: Cone : D ([1]) i ∗ − → D ( p ) i p ! − → D (  ) j ∗ − → D ([1 ]) ii) The ﬁb er functor Fib er : D ([1]) − → D ([1]) is deﬁned as the comp osition: Fib er : D ([1]) j ! − → D ( y ) i y ∗ − → D (  ) i ∗ − → D ([1]) Moreov er, let C : D ([1]) − → D ( e ) be the functor obtained from the cone functor b y ev a luation at 1, and s imilarly let F : D ([1 ]) − → D ( e ) be th e functor obtained from the ﬁber functor by ev aluatio n 0. Prop osition 3.13 shows that the cone C f of an isomorphism f is the zero ob ject 0 . In gen- eral, the converse is only true in the s table situation (cf. Pro position 4.5). There is the following counterexample to the conv er se in the unsta ble situation. Coun terexample 3.22 . Let E b e an exact categ ory in the sense of Quillen (cf. [Qui73]). Moreover, let us ass ume E to have enoug h injectives but also that E is not F rob enius, i.e., the cla sses of injectives and pro jectives do not coincide. The stable categor y E which is obtained from E by dividing out the maps factoring ov er injectives is a ‘susp ended categor y’ in the sense of [KV87]. Let now X b e a n ob ject of E of injective dimensio n 1 and let 0 − → X − → I 0 = I − → I 1 = Σ X − → 0 be an injectiv e resolution of X . By deﬁnition of the susp ended structure on E (cf. [KV87] o r [Hap88, Chapter I]) the diagra m X u / / id   I id   v / / Σ X id   X / / I / / Σ X gives rise to the distinguis he d triang le X u − → I v − → Σ X id − → Σ X. Since Σ X is trivial in the stable category E the morphism u is an example of a morphism which is not an isomorphism but still has a v anishing cone. In the stable situation, i.e., in the F ro b enius case, this counterexample cannot exist. In fact, the a bov e resolution of X would split beca use Σ X is by assumption injectiv e, hence pro jectiv e, showing that the injectiv e dimension o f X is zero. This example can b e made into an example ab out po in ted deriv ators by using [Kel07]. As a preparation for t he next pro of, let us denote b y N ⊂ D (  ) , N p ⊂ D ( p ) , and N y ⊂ D ( y ) the resp ectiv e full subca tegories spanned b y the ob jects X with X 0 , 1 ∼ = 0 . DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 45 Prop osition 3.2 3. L et D b e a p ointe d derivator, then we have an adjunction: ( Cone , Fib er ) : D ([1]) ⇀ D ([1]) Pr o of. There are the following factorizations of the c o ne and ﬁber functor s: Cone : D ([1]) i ∗ ≃ / / N p i p ! / / N i y ∗ / / N y j ∗ ≃ / / D ([1]) D ([1]) N p ≃ i ∗ o o N i p ∗ o o N y i y ∗ o o D ([1]) j ! ≃ o o : Fib er The existence o f these factorizations is ag ain obvious a nd the fact that the outer functors are equiv alences follows a gain fro m Pr oposition 3.6. Thus, this sho ws tha t the pair ( Cone , Fib er ) is the comp osition of four adjunctions among which t w o are equiv alences.  In the litera tur e, there is also a n alterna tiv e description of some of the functors we just intro- duced. This alternative descriptio n is, for e x ample, helpful in the understanding of morphisms in D ([1 ]) which induce z e ro morphisms o n underlying diagr ams. Using our explicit co ns truction of the (co)exceptio na l inverse image functors a t the end of Subsection 3.1 , let us quickly show these t w o approaches to be equiv a len t. F or this purp ose, let D again b e a p oin ted deriv ator and let us cons ider the categor y [1] together with the cosie ve 1 : e − → [1] and the sieve 0 : e − → [1] . Corollar y 3.8 implies that we ha ve adjunctions: (1 ? , 1 ! ) : D ([1]) ⇀ D ( e ) and (0 ∗ , 0 ! ) : D ( e ) ⇀ D ([1]) The form ulas via the ma pping cylinder constructions can be made v ery explicit in this case so that we have the follo wing descriptions of the additional a djoin ts 1 ? and 0 ! : 1 ? : D ([1]) j ∗ ≃ / / D ( p , (0 , 1)) pr 1 ! / / D ([1] , 0) 1 ∗ ≃ / / D ( e ) 0 ! : D ([1]) j ! ≃ / / D ( y , (1 , 0)) pr 1 ∗ / / D ([1] , 1) 0 ∗ ≃ / / D ( e ) In both formulas, j deno tes the functor classifying the ho rizon tal arrow and the functors pr 1 are suitable restrictions of the pro jection on the ﬁrst component  − → [1] . It follows fr o m Lemma 1.25 that in both case s the comp osition o f the la st tw o functors is naturally iso morphic to the homo to p y colimit and homotopy limit functor resp ectiv ely . A ﬁna l a pplication of (Der4) then implies the following result. Prop osition 3. 24. L et D b e a p ointe d derivator then we have the fol lowing natur al isomorphisms: C ∼ = 1 ? , Σ ∼ = 1 ? ◦ 0 ∗ , F ∼ = 0 ! , and Ω ∼ = 0 ! ◦ 1 ! In p articular, we have adjunctions ( C , 1 ! ) : D ([1]) ⇀ D ( e ) and (0 ∗ , F ) : D ( e ) ⇀ D ([1]) . The a bov e deﬁnitions ca n easily b e ex tended (using E xample 2.1) to mor phisms at the level of deriv ator s . Thus, given a pointed deriv ator D w e obtain, in par ticular, adjunctions of deriv ators (Σ , Ω) : D ⇀ D and ( Co ne , Fibe r ) : D [1] ⇀ D [1] . Since the construction of the ab o ve functors is based only on certain extensio n b y zero functors and the for mation of some (co)Car tesian squa res the follo wing pr opositio n is immediate. It applies, in particular , to the precomp osition morphisms v ∗ : D M − → D L for a p oin ted deriv ator D . 46 MORITZ GROTH Prop osition 3.2 5. L et G : D − → D ′ b e a morphism of p ointe d derivators. i) If G is left exact then we have c anonic al isomorphisms G ◦ Ω − → Ω ◦ G and G ◦ Fib er − → Fib er ◦ G. ii) If G is right exact then we have c anonic al isomorphisms Σ ◦ G − → G ◦ Σ and Co ne ◦ G − → G ◦ Co ne . 4. St able deriv a tors 4.1. The additivity of stable deriv ators. In this subsection, we come to the central notion of a stable deriv ator. Similar ly to the situation of a stable mo del category or a stable ∞ -categ ory , one a dds a ‘linea rit y condition’ to the pointed situation. This will e ns ure, in particular , that the susp ension and the lo op functor deﬁne a pair of in verse eq uiv a lences (Σ , Ω) : D ( e ) ≃ − → D ( e ) . This notion was introduced by Ma lts inio tis in [Ma l0 7] by fo r ming a combination of the a xioms of Grothendiecks der iv a tors [Gro] a nd F r ank e’s systems of tr iangulated dia gram categ ories [F ra 96]. More details on the histo r y can be found in the pap er [CN08] by Cisinski and Neeman. Deﬁnition 4.1 . A strong deriv ator D is stable if it is po in ted and if an ob ject of D (  ) is coCartesia n if and only if it is Cartesian. The stro ngness prop ert y will b e crucial in tw o situations in the construction of the c anonical triangulated structures. Let us call a square biCartesian if it satisﬁes the equiv a len t conditions of being Cartesia n or coCartesia n. Example 4.2. i) Let M be a stable c om bina torial mo del categor y then the asso ciated deriv a- tor D M is stable. Thus, w e ha ve, in particular , the s table deriv ator assoc iated to unbounded chain complexes, mo dules over a diﬀerent ial graded algebra, spectra based on simplicia l sets and mo dule sp ectra over a given symmetric r ing sp ectrum. These deriv ator s can b e endow ed with some a ddi- tional structure: they a re exa mples o f monoidal deriv ator s res p. der iv ators tensored ov er a monoidal deriv ator as discusse d in [Gro11, Gro12a]. ii) A deriv a to r D is stable if a nd only if the dual deriv ator D op is stable. Let us b egin by the following co n venient r e sult. Prop osition 4.3 . L et D b e a stable derivator and let M b e a c ate gory. Then D M is again stable. Pr o of. It is immediate that a deriv ator D is str ong if and only if D M is strong f or all categories M . Moreov er, we know that D M is pointed by P ropos ition 3 .2. Thus, let us consider the (co)Cartesian squares. F o r a n o b ject X ∈ D M (  ), using Co rollary 3.16, we have that X is coCar tesian if and only if X m ∈ D (  ) is coCartesian for all m ∈ M . Using the stability of D and the cor responding result for Cartesia n squares in D M (  ) we are done.  W e give immediately the exp ected result on the suspe ns ion a nd lo op functors in this stable situation. Recall the deﬁnition of the categ ories M , M y , M p , and the factorization of (Σ , Ω) in the case of a pointed deriv ator. L e t us denote, in addition, by M Σ ⊂ M (resp. M Ω ⊂ M ) the full DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 47 sub c ategory spanned b y the coCartesia n (resp. Cartesia n) squares . With this notation, in the case of a p ointe d deriv a tor, t here is the following additional factor ization of (Σ , Ω) : Σ : D ( e ) (0 , 0) ∗ ≃ / / M p i p ! ≃ / / M Σ i y ∗ / / M y (1 , 1) ∗ ≃ / / D ( e ) D ( e ) M p (0 , 0) ∗ ≃ o o M Ω i p ∗ o o M y i y ∗ ≃ o o D ( e ) ≃ (1 , 1) ! o o : Ω In this diagram, a ll but p ossibly the t wo res triction functors in the middle are equiv ale nc e s . In the case of a st abl e deriv ator, w e hav e M Σ = M Ω and these tw o r estriction functors are also equiv alences: Σ : D ( e ) (0 , 0) ∗ ≃ / / M p i p ! ≃ / / M Σ i y ∗ ≃ / / M y (1 , 1) ∗ ≃ / / D ( e ) D ( e ) M p (0 , 0) ∗ ≃ o o M Ω i p ∗ ≃ o o M y i y ∗ ≃ o o D ( e ) ≃ (1 , 1) ! o o : Ω This proves the ﬁrst ha lf of the next result. The second half ca n be proved in a simila r w ay . Prop osition 4.4 . L et D b e a stable derivator, then we have e quivalenc es of derivators (Σ , Ω) : D ≃ − → D and ( Cone , Fib er ) : D [1] ≃ − → D [1] . Let us mention the follo wing result which shows that in the stable situation isomorphisms can be characterized b y the v anishing of the cone. W e use the same notation as in Prop osition 3.13. Prop osition 4. 5. L et D b e a st able derivator and let X ∈ D (  ) . If two of the thr e e fol lowing statements hold for the squar e X then so do es the thir d one: i) The squar e X is c oCartesian, ii) The arr ow d v 0 X is an isomorphism, iii) The arr ow d v 1 X is an isomorphism. In p articular, an obje ct f ∈ D ([1]) is an isomorphism if and only if the c one C f is z er o. Pr o of. F or the ﬁrst pa rt we can apply Pro position 3.13 to see that w e only hav e to show that i) and ii) imply iii). But this statement follows from the dual of Pro position 3.13 which ca n b e applied bec ause ev er y coCartesia n square is also Ca r tesian in the stable situation. Finally , the second part follows from the ﬁrst part when applied to the spe c ial case of the deﬁning squar e of the cone.  The next aim is to show that, in the s table case, ﬁnite copro ducts and ﬁnite pro ducts in D ( J ) ar e canonically iso morphic. By Prop osition 4 .3, we can assume that J = e. But let us ﬁrst mention the following r esult which is immediate f rom Prop osition 3 .14 on the comp osition and the cancellation prop erties of (co)Cartesian squares. That result is crucial in order to establish the semi-additivit y . Prop osition 4.6. L et D b e a stable derivator and let X ∈ D ([2 ] × [1]) . If two of the squar es d 0 ( X ) , d 1 ( X ) , and d 2 ( X ) ar e biCartesian, then so is the thir d one. W e no w give the res ult on the semi-additivity of the v alues of a stable deriv ator , i.e., w e wan t to sho w that the v alues then admit ﬁnite bipr oducts. W e know a lready fr o m Pr opositio n 1.12 that the v alues of a n arbitrary deriv ator admit ﬁnite co products and ﬁnite pr oducts. Prop osition 4. 7. L et D b e a stable derivator and c onsider a functor u : J − → K . Then ﬁnite c opr o ducts and ﬁnite pr o ducts in D ( J ) ar e c anonic al ly isomorphic. Mor e over, these ar e pr eserve d by u ∗ , u ! , and u ∗ . 48 MORITZ GROTH Pr o of. F or the ﬁrs t part, it is a gain enough to show the r esult for the case J = e. Let us consider the inclusion j 2 : L 2 − → L 3 of the left p oset L 2 in the right poset L 3 : (1 , 0) / / (2 , 0) (0 , 0)   / / (1 , 0) / / (2 , 0) (0 , 1)   (0 , 1)   (0 , 2) (0 , 2) Moreov er, let j 1 : e ⊔ e − → L 2 be the map (1 , 0) ⊔ (0 , 1) a nd le t j 3 : L 3 − → [2] × [2 ] = L b e the obvious inclusion. Since j 1 is a sieve the homo top y Kan extension functor j 1 ∗ is an ‘extensio n by zer o functors’ by P ropos ition 3.6, and similarly for the ho motop y K an extension functor j 2 ! asso ciated to the cosieve j 2 . L e t us consider the functor: D ( e ) × D ( e ) ≃ D ( e ⊔ e ) j 1 ∗ − → D ( L 2 ) j 2 ! − → D ( L 3 ) j 3 ! − → D ( L ) The image Q ∈ D ( L ) of a pair ( X, Y ) ∈ D ( e ) × D ( e ) under this functor has as underlying diagram: 0 / /   X / /   0   dia L ( Q ) : Y / /   B / /   Y ′   0 / / X ′ / / Z Let us denote the four inclusions of the s maller squares in L b y i k , k = 1 , . . . , 4 , i.e., let us se t i 1 = d 2 × d 2 , i 2 = d 0 × d 2 , i 3 = d 2 × d 0 , and i 4 = d 0 × d 0 . An application of Prop osition 3.1 1 to these inclusions i k :  − → L, k = 1 , . . . , 4 , and f = j 3 allows us to deduce that all sq uares a re biCar tesian. In fact, in all four ca ses, i k (1 , 1) / ∈ Im( j 3 ) and we only hav e to chec k that the induce d functors ˜ i k : p − → L − i k (1 , 1) /i k (1 , 1) are right adjoints. F or k = 1, this functor is a n iso morphism while in the other three cases Lemma 3.12 applies. By Prop osition 4.6, also the comp osite s quares ( d 2 × d 1 )( Q ) and ( d 1 × d 2 )( Q ) are biCartesian. Hence, Prop osition 3.1 3 ensures that we have is o morphisms X ∼ = X ′ and Y ∼ = Y ′ . Similarly , the square ( d 1 × d 1 )( Q ) is biCar tesian and w e obtain an iso morphism Z ∼ = 0. Th us, w e see that B is simult aneously a copr oduct of X a nd Y and a pro duct of X ′ ∼ = X and Y ′ ∼ = Y . The fact that these bipro ducts ar e preser v ed by u ∗ , u ! , and u ∗ follows immediately s ince ea c h o f the three functors has a n adjoin t functor on at least one side.  Corollary 4.8 . L et D b e a stable derivator and let J b e a c ate gory. Every obje ct of D ( J ) is c anonic al ly a c ommutative monoid obje ct and a c o c ommutative c omonoid obje ct. In p articular, t he morphism set hom D ( J ) ( X, Y ) , X , Y ∈ D ( J ) , c arries c anonic al ly the structu re of an ab elian monoid. Pr o of. F or X ∈ D ( J ) , the diagona l map ∆ X : X − → X × X ∼ = X ⊔ X is co unital, coas sociative and coco mm utative. Dually , the codiago na l ∇ X : X × X ∼ = X ⊔ X − → X is unital, asso ciative and DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 49 commutativ e. These can be use d to deﬁne the sum of tw o mor phis m f , g : X − → Y using the usua l conv o lution or cup pro duct, i.e., as f + g : X ∆ X − → X × X f × g − → Y × Y ∼ = Y ⊔ Y ∇ Y − → Y and it is immediate that this deﬁnes the str ucture of an abe lian monoid on hom D ( J ) ( X, Y ) .  W e will from now on use the standar d notation ⊕ for the bipro duct. The next a im is to sho w that ob jects of the for m Ω X (resp. Σ X ) ar e ev en ab elian gr oup (res p. c o gr oup ) ob jects. W e g iv e the pro of in the ca se of Ω X in which ca se the constructions can b e motiv a ted b y the pro cess of concatenation of lo ops in top ology . L et us b egin with s ome prepar ations. Since the a im is to ‘mo del categoric ally’ the conca tenation and inv ersion of loo ps we hav e to consider ﬁnite direct sums of ‘lo op ob jects’. F or the construction of the ﬁnite sums o f lo op ob jects there is the following conceptual approach which admits an o b vio us dualization. Let y n be the p oset with ob jects e 0 , . . . , e n and t and with or dering ge nerated b y e i ≤ t, i = 0 , . . . , n . The pictures of y n for n = 1 and n = 2 are: e 1   e 1 ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ e 2   e 0 / / t e 0 / / t Let F i n deno te the categ ory of the ﬁnite sets h n i = { 0 , . . . , n } with all set-theoretic maps as morphisms betw een them. The assignment h n i 7− → y n can be extended to a functor F in − → Cat if we send f : h k i − → h n i to y f : y k − → y n with y f ( e i ) = e f ( i ) and y f ( t ) = t. Since t : e − → y n is a cosieve, t ! : D ( e ) − → D ( y n ) gives us an ‘extension b y zero functor’. Deﬁne P n as P n : D ( e ) t ! / / D ( y n ) Holim y n / / D ( e ) and note that we ha ve a canonical isomorphism P 1 X ∼ = Ω X . This construction can be extended to a functor as des c r ibed in the following lemma. Lemma 4.9. L et D b e a st able derivator. The ab ove c onstruction deﬁnes a bifunctor: P : F in op × D ( e ) − → D ( e ) : ( h n i , X ) 7− → P n X Pr o of. The functoria lit y of P in the s econd v a r iable is obvious so let us assume we ar e given a morphism f : h k i − → h n i . F r om such a mor phis m f we obtain the following diagra m given on the left-hand-side: e t   / / e t   D ( e ) t !   D ( e ) t !   o o y k y f / / p   y n p   D ( y k ) Holim   ✻ ✻ ✻ ✻   D ( y n ) Holim   y ∗ f o o ✻ ✻ ✻ ✻ W _ e / / e D ( e ) D ( e ) o o The formation of the cor responding base c hange mo r phisms gives r ise to the pasting diagram o n the r igh t (no te that w e had to use b oth v ariants here). Using the fact tha t isomor phisms ca n be detected p oin t wise and (Der4) it is easy to chec k that the upp er 2- cell is inv ertible. T hus we can deﬁne P f as the following compositio n: P f : P n = Holim y n ◦ t ! − → Holim y k ◦ y ∗ f ◦ t ! − → Holim y k ◦ t ! = P k 50 MORITZ GROTH The functoriality o f this c o nstruction follows from the nice b eha vior of pasting with respect to both the inv er sion of na tural transformations and base change.  Let us ﬁx notation for some morphisms in F in . Given a ( k + 1)- tuple ( i 0 , i 1 , . . . , i k ) of elements of h n i let us denote by ( i 0 i 1 . . . i k ) the corresp onding morphism h k i − → h n i which sends j to i j . F or n ≥ 1 and 1 ≤ k ≤ n, we hav e thus the morphism ( k − 1 , k ) : h 1 i − → h n i . So, for a stable deriv ator D and an ob ject X ∈ D ( e ) , we obtain by the last lemma induced maps: ( k − 1 , k ) ∗ = P (( k − 1 , k ) , id X ) : P n X − → P 1 X ∼ = Ω X These maps taken toge ther deﬁne the following Se gal maps and satisfy the ‘usua l’ Segal condi- tion ([Seg74]). Lemma 4. 1 0. L et D b e a stable derivator and let X ∈ D ( e ) . F or n ≥ 1 and 1 ≤ k ≤ n, the ( k − 1 , k ) ∗ to gether deﬁne a natur al isomorphism in D ( e ) : s = s n : P n X ∼ = − → n Y k =1 P 1 ( X ) ∼ = n M k =1 Ω X Pr o of. By induction on n and by the functoria lit y o f P • X , it is enough to chec k this for n = 2. Let J b e the p oset obtained fro m y 2 by adding tw o new elements ω 0 and ω 1 such that ω 0 ≤ e 0 , e 1 and ω 1 ≤ e 1 , e 2 . Moreover, let us denote the resulting inclusion b y j : y 2 − → J. Under the o b vio us isomorphism J ∼ = [1] × y , we can co nsider the adjunction ( d 1 × id , s 0 × id) : y ⇀ [1] × y as a n adjunction ( L, R ) : y ⇀ J. By Prop osition 1.24 w e ha ve a natural isomorphism betw een P 2 and D ( e ) t ! / / D ( y 2 ) j ∗ / / D ( J ) L ∗ / / D ( y ) Holim / / D ( e ) . But it is easy to see that the comp osition of the ﬁrs t three functors ev aluated on X yields a diagra m which v a nishes at t and is isomorphic to Ω X at the tw o r emaining argumen ts. It th us follows that we have an isomorphism P 2 ( X ) ∼ = Ω X ⊕ Ω X induced by the Segal map.  Having the functorial co nstruction of ﬁnite direct sums of lo op obje cts at our disp osal, we w ant to s ho w now that Ω X is always canonically an a belian group ob ject. As an intermediate step, let us cons truct a pairing ⋆ : Ω X ⊕ Ω X − → Ω X whic h will be called the c onc atenation map . By the last lemma we can inv er t the Segal maps and hence deﬁne the pair ing by the following c o mposition: ⋆ : Ω X ⊕ Ω X P 2 ( X ) ∼ = s o o (02) ∗ / / Ω X Lemma 4. 11. L et D b e a s table derivator and let X b e an obje ct of D ( e ) . The c onc atenation map ⋆ : Ω X ⊕ Ω X − → Ω X is an asso ciative p airing on Ω X . DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 51 Pr o of. Let U b e a further ob ject of D ( e ) and consider three morphisms f , g , h : U − → Ω X in D ( e ). In the following diagram, all maps lab eled by s are Segal maps: Ω X P 2 X (02) ∗ _ _ s q q U f ,g,h ( ( f ,g⋆h / / f ⋆ ( g⋆h ) 5 5 Ω X ⊕ Ω X P 3 X (013) ∗ Y Y (03) ∗ p p s   s q q Ω X ⊕ P 2 X id ⊕ (02) ∗ m m id ⊕ s   Ω X ⊕ Ω X ⊕ Ω X The tw o quadrilatera ls on the left comm ute b y deﬁnition of the concatenation and the right one commutes by functoriality of P • . W e can th us deduce the rela tion f ⋆ ( g ⋆ h ) = (03 ) ∗ m ( f , g , h ) where m ( f , g , h ) : U − → P 3 X is the unique map such that s ◦ m ( f , g , h ) = ( f , g , h ) . This ‘asso- ciative descr iption’ of f ⋆ ( g ⋆ h ) together with the Y oneda lemma implies the asso ciativity of the concatenation map.  Heading for the additive in v erse of the identit y on lo op ob jects, let us cons ider the only non- trivial automorphism σ : h 1 i − → h 1 i in F in . Then y σ : y − → y is the isomorphism interchanging the vertices (1 , 0) and (0 , 1). There is thus an induce d a utomorphism σ ∗ = (10) ∗ : Ω X − → Ω X which w e ca ll the inversion of lo ops . Prop osition 4.12. L et D b e a st abl e derivator and let X ∈ D ( e ) . The inversion of lo ops map σ ∗ : Ω X − → Ω X is an additive inverse to id Ω X . In p articular, Ω X ∈ D ( e ) is an ab elian gr oup obje ct. Pr o of. By functoriality of the construction P • X , there is a right action of the symmetric group on three letters on P 2 X . W e wan t to describ e the cor responding action on Ω X ⊕ Ω X obtained by conjugation with the Segal ma p s . The strategy of the pro of is then to use this a ction in order to relate the conca tenation pr oduct a nd th e addition o f morphisms. F or diﬀerent elemen ts i , j ∈ h 2 i let us denote by σ ij the asso ciated tra nsposition. One chec k s that the following diagram commutes P 2 X s   σ ∗ 02 / / P 2 X s   Ω X ⊕ Ω X   0 σ ∗ σ ∗ 0   / / Ω X ⊕ Ω X where the arr o ws lab eled by s are again Segal maps. F r o m the equalit y of the maps σ 01 ◦ (01) = (01) ◦ σ : h 1 i − → h 2 i 52 MORITZ GROTH we conclude that the endomorphism of Ω X ⊕ Ω X c orresp onding to σ 01 is a lo w er triangular matrix s ◦ σ ∗ 01 ◦ s − 1 =  σ ∗ 0 α β  : Ω X ⊕ Ω X − → Ω X ⊕ Ω X for some maps α, β : Ω X − → Ω X . The fact that σ 01 is an inv olution implies the rela tio ns: ασ ∗ + β α = 0 and β 2 = id The aim is now to show that bo th ma ps α and β ar e identities whic h w ould in particular imply that σ ∗ is an additive in v erse of id Ω X . F rom the relation (02) = σ 01 ◦ (12 ) we immediately get (02) ∗ = (12) ∗ ◦ σ ∗ 01 : P 2 X − → Ω X . Using the matr ix description of the map induced by σ 01 we see that for tw o maps f , g : U − → Ω X there is the following form ula for the conca tena tion pro duct: f ⋆ g = αf + β g : U − → Ω X By Lemma 4.11 we know that the concatenation pairing is ass o ciative. If we take U = Ω X a nd compare the tw o expressions for (0 ⋆ 0) ⋆ id Ω X and 0 ⋆ (0 ⋆ id Ω X ) we a lr eady obtain the ﬁrst intended relation β = id Ω X . Instead of using (0 2) = σ 01 ◦ (12) , we can also use the relation (0 2) = σ 12 ◦ (01 ) : h 1 i − → h 2 i to obtain a further desc r iption of the concatenation pro duct. Firs t, since σ 12 = σ 02 ◦ σ 01 ◦ σ 02 : h 2 i − → h 2 i we obtain that the endomo r phism o n Ω X ⊕ Ω X induced by σ ∗ 12 has the following matrix descr iption: s ◦ σ ∗ 12 ◦ s − 1 =  σ ∗ β σ ∗ σ ∗ ασ ∗ 0 σ ∗  : Ω X ⊕ Ω X − → Ω X ⊕ Ω X F rom this a nd the form ula (0 2 ) ∗ = (01) ∗ ◦ σ ∗ 12 we see that the concatenation pro duct can a lso b e written as: f ⋆ g = σ ∗ β σ ∗ f + σ ∗ ασ ∗ g : U − → Ω X A compar is on of these tw o descriptions concludes the pro of since we obtain α = σ ∗ β σ ∗ = id Ω X .  Remark 4 . 13. Although w e will no t make us e of this remark w e w ant to emphasize the following. The pro of of the last prop osition shows that the addition on mapping spaces in to lo op ob jects coincides with the pairing induced by the concatenation of lo ops. Simila r ly , additiv e inv erses ar e given b y the inversion o f loo ps. Thus for maps f , g : U − → Ω X we hav e: f + g = f ⋆ g and − f def = σ ∗ f A combination of this prop osition, the res ult on the semi-additivity of D ( J ) (Pr o position 4 .7), and the fact that (Σ , Ω) is a pair of inv erse equiv alence s in the stable situatio n giv es us immediately the following corollary . Corollary 4 .14. If D is a stable derivator then D ( J ) is an additive c ate gory for an arbitr ary J. Mor e over, for an arbitr ary functor u : J − → K , the induc e d functors u ∗ , u ! , and u ∗ ar e additive. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 53 4.2. The canonical triangulated structur es. W e c an now attack the main res ult o f this section, namely , that g iv en a stable der iv ator D then the categories D ( J ) are canonically tria ngulated categorie s. Using Prop osition 4.3, we can again assume witho ut loss of generality that w e are in the ca se J = e . The susp ension functor of the tr iangulated s tr ucture will be the susp ension functor Σ : D ( e ) − → D ( e ) we constructed alr eady . Thu s, let us construc t the class of distinguished triangles. F or this purp ose, let K denote the po set: (0 , 0)   / / (1 , 0) / / (2 , 0) (0 , 1) Moreov er, let i 0 : [1] − → K be the map classifying the left ho r izon tal arr o w and let i 1 : K − → [2] × [1] be the inclusion. Let us denote the compos ition by i : [1] i 0 − → K i 1 − → [2] × [1] . Again, since i 0 is a sieve, i 0 ∗ gives us a n extension b y zero functor. Let us cons ide r the functor : T : D ([1]) i 0 ∗ − → D ( K ) i 1 ! − → D ([2] × [1 ]) W e claim that the squa res d 0 T ( f ) , d 1 T ( f ) , and d 2 T ( f ) ∈ D (  ) are then biCa r tesian for an a r- bitrary f ∈ D ([1]) . Moreover, if the underlying diagram of f is X − → Y then we have canonica l isomorphisms T ( f ) 2 , 1 ∼ = Σ X and T ( f ) 1 , 1 ∼ = C ( f ) . In fact, by Prop osition 4.6, it is enough to show the biCartesianness of d 0 T ( f ) and d 2 T ( f ) . This can b e done by tw o a pplications of the detectio n result Pr opositio n 3.11 to i 1 : K − → J = [2] × [1 ] . It is ea sy to chec k (using Lemma 3.1 2 in one of the cases) that the assumptions of that prop osition are satisﬁed. Since i 0 is a sieve, the underlying diagram of d 1 T ( f ) and d 2 T ( f ) respectively look like: X / /   0   X / /   Y   0 / / T ( f ) 2 , 1 0 / / T ( f ) 1 , 1 Moreov er, b y the pr o of of Pr opos itio n 4.4, d 1 T ( f ) lies in the essential image of D ( e ) (0 , 0) ∗ / / D ( p ) i p ! / / D (  ) . Hence, we hav e a cano nical is o morphism T ( f ) 2 , 1 ∼ = Σ X . Similarly , if we let j : [1] − → p denote the functor class ifying the upper horizo ntal morphism d 2 T ( f ) then lies in the essential image of D ([1]) j ∗ / / D ( p ) i p ! / / D (  ) . Hence, we also hav e a canonical iso mo rphism T ( f ) 1 , 1 ∼ = C ( f ) as int ended. Thu s, for f ∈ D ([1]) , b y ﬁrs t r estricting T ( f ) to [3] in the exp ected wa y and then forming the underlying diagr am in D ( e ), we obtain a triangle ( T f ) in D ( e ) which is of the following form: ( T f ) : X − → Y − → C ( f ) − → Σ X Call a triangle in D ( e ) distinguishe d if it is isomorphic to ( T f ) for s o me f ∈ D ([1]) . W e are now in the p osition to state the following impo r tan t theorem. 54 MORITZ GROTH Theorem 4.15 . Le t D b e a stable derivator and let J b e a c ate gory. Endowe d with the susp ension functor Σ : D ( J ) − → D ( J ) and the ab ove class of distinguishe d triangles, D ( J ) is a triangulate d c ate gory. The fact that this tria ng ulated structure is co mpa tible with the res tr iction and homotopy Ka n extension functors will be discussed in Corollar y 4.19. F or easier reference to the axioms o f a triangulated catego r y we include a deﬁnition. F or more background on this theo ry cf. for exam- ple [Nee01] or to [Sch07]. The form o f the o ctahedron axiom given here is suﬃcient in or der to obtain the usual form of the o ctahedron axiom. This obser v ation w as made in [KV8 7] (for a proo f of it see [Sch07]). Deﬁnition 4 .16. Let T b e an additive categor y with a s e lf- e q uiv a lence Σ : T − → T and a class of so- called distinguished tr iangles X − → Y − → Z − → Σ X . The pair consisting o f Σ and the class of disting uished tria ngles deter mines a t riangulate d structur e o n T if the following four axio ms are satisﬁed. In this case, the triple consisting of the category , the endofunctor, a nd the class of distinguished tria ng les is called a triangulate d c ate gory . (T1) F or every X ∈ T , the triang le X id − → X − → 0 − → Σ X is distinguished. Every morphism in T o ccurs as the ﬁrst morphism in a distinguished triangle and the class of dis tinguished triangles is replete, i.e., is closed under isomorphis ms. (T2) A triangle X f − → Y g − → Z h − → Σ X is distinguished if a nd only if the rota ted triang le Y g − → Z h − → Σ X − f − → Σ Y is. (T3) Given t wo distinguished triang le s and a commutativ e solid a rrow diagra m X / / u   Y / / v   Z / / w   ✤ ✤ ✤ Σ X Σ u   X ′ / / Y ′ / / Z ′ / / Σ X ′ there exists a da shed arrow w : Z − → Z ′ as indicated such that the extended diag ram comm utes. (T4) F o r ev ery pair of compo sable arrows f 3 : X f 1 − → Y f 2 − → Z there is a commutativ e diag ram X f 1 / / Y g 1 / / f 2   C 1 h 1 / /   Σ X X f 3 / / Z g 2   g 3 / / C 3 h 3 / /   Σ X Σ f 1   C 2 h 2   C 2 Σ g 1 ◦ h 2   h 2 / / Σ Y Σ Y Σ g 1 / / Σ C 1 in which the ro ws and columns are distinguished triangles. W e will now give t he pro of of the theor e m. Pr o of. (of Theorem 4.1 5 ) It suﬃces to do this for the ca se J = e . The additivity of D ( e ) is already given b y Corolla ry 4.14. DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 55 Moreov er, in this stable setting, the susp ension functor Σ is an equiv alence. (T1): The ﬁrst par t of axiom (T1) is settled b y Pr oposition 4.5 and the second pa rt is s ettled using the a ssumed stro ngness. The last pa rt of (T1 ) holds by deﬁnition of the clas s of distinguis hed triangles. (T3): Axiom (T3) is settled similar ly by r educing ﬁrst to the situation o f tr iangles of the form ( T f ) for f ∈ D ([1]) and then applying the str ongness again. (T2): Befor e w e give the actual proo f o f axiom (T2) w e re c all that the axioms of a triangula ted category as given here are not in a minimal form. In fac t, if one has a lready esta blis hed axioms (T1) a nd (T3) it suﬃces to g iv e a pro of of one half of the rotation a xiom as indica ted in the next claim (cf. ag ain to [Sch07 ] for this fact). Claim: let X f − → Y g − → Z h − → Σ X be a distinguished triangle in D ( e ), then also the rotated triangle Y g − → Z h − → Σ X − Σ f − → Σ Y is distinguished. W e can again r educe to the case where the giv en disting uis hed tria ng le is ( T f ) for some f ∈ D ([1]) . Let us consider the category J given b y the following full subposet of [2] × [2] (0 , 0) / /   (1 , 0) / /   (2 , 0) (0 , 1) + + (1 , 2) and let i : [1] − → J b e the functor cla s sifying the upp er left horizontal morphism. Then i is a sieve and i ∗ gives us thus an extension by zero functor. Moreover, let us denote b y j the canonical inclusion of J in K = [2] × [2] − { (0 , 2) } . F or a given f ∈ D ([1 ]) let us consider j ! i ∗ ( f ). Again, b y a rep eated application of P ropo s ition 3 .1 1 all squares in j ! i ∗ ( f ) are biCartesia n. If the diag r am of f is f : X − → Y then the underlying diagr am o f j ! i ∗ ( f ) lo oks lik e: X f / /   Y / / g   0   0 / / C f h / /   Σ X   0 / / Σ Y In fact, the inclusion ( d 1 × d 2 ) : p − → K allows us to identify the v a lue a t (2 , 1) with Σ X while the inclusion ( d 0 × d 1 ) : p − → K giv es us a n identiﬁcation of the lo w er r igh t cor ner with Σ Y . How ever, this last inclus ion diﬀers from the us ual one b y the a utomorphism σ : p − → p . By Prop osition 4.12, the induced map σ ∗ : Σ Y − → Σ Y is − id Σ Y . Hence, using mo reo ver the unique natural transfor mation of the tw o inclusions ( d 0 × d 1 ) − → ( d 1 × d 2 ) : p − → K, we c a n iden tify the morphism Σ X − → Σ Y as − Σ f and this s ho ws that the triangle ( T g ) is as stated in the claim. (T4): It remains to give a pro of of the o ctahedron axiom. The pr oof of this will b e split into tw o parts. i) In the ﬁrst pa rt, given an ob ject F ∈ D ([2]) , we cons truct an asso ciated o ctahedro n diagr am 56 MORITZ GROTH in D ( e ) . The pattern of this part of the pro of is b y now quite familiar . Conside r the ca tegory J given b y the following full subp oset of [4] × [2] (0 , 0) / /   (1 , 0) / /   (2 , 0) / / (3 , 0)   (0 , 1) / / + + (4 , 1) (1 , 2) and let i : [2] − → J cla ssify the t w o compo sable upper left mo r phisms. Mor eo v er, let j : J − → K = [4 ] × [2] − { (4 , 0) , (0 , 2) } be the canonical inclusion. Since i is a s iev e, the homotopy right Kan extension functor i ∗ is an extension b y zero functor . F or F ∈ D ([2]) let us consider D = j ! i ∗ ( F ) ∈ D ( K ) . If the underlying diagram of F is X f 1 − → Y f 2 − → Z then the underlying diagra m of D is X f 1 / /   Y f 2 / /   Z   / / 0   0 / / c C 1 / /   c C 3 / /   S X / /   0   0 / / c C 2 / / S Y / / S c C 1 A re p eated application of Propo s ition 3.11 g uarantees that all square s in D are biCar tesian. Hence the same is also true for all comp ound squares o ne can ﬁnd in D . This a llo ws us to ﬁnd canonica l isomorphisms c C k ∼ = C ( f k ) if we set f 3 = f 2 ◦ f 1 . More pre c is ely , the co ne functor C has of course to be applied to f 1 = d 2 ( F ) , f 2 = d 0 ( F ) , a nd f 3 = d 1 ( F ) ∈ D ([1]) . Similarly , we o btain isomorphisms S X ∼ = Σ X , S Y ∼ = Σ Y , and S c C 1 ∼ = Σ c C 1 . Th us, one ca n extract an o ctahedron diag r am in D ( e ) from the ob ject D. ii) In this part, w e show that every ‘ﬁrst half of an o ctahedron diagra m’ comes up to isomorphism from an ob ject F ∈ D ([2]) . Let us restric t attention to the upper left squar e X f 1 / / Y f 2   X f 3 / / Z of such a diagram. The s trongness o f D guara n tees that ther e is an ob ject F 1 ∈ D ([1]) and an isomorphism dia F 1 ∼ = ( f 1 : X − → Y ) . Moreov er, let us cons ider p ∗ Z ∈ D ([1]) , where p : [1] − → e is the unique functor. Then, we obtain a morphism φ : F 1 − → p ∗ Z as the ima ge of f 2 under the tw o natural isomor phisms (we applied Lemma 1.25 to o btain the second one): hom D ( e ) ( Y , Z ) ∼ = hom D ([1]) ( F 1 , 1 ∗ Z ) ∼ = hom D ([1]) ( F 1 , p ∗ Z ) DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 57 Considering this map φ : F 1 − → p ∗ Z as an o b ject of D ([1 ]) [1] , a further a pplica tion of the strong ness guarantees the existence of an ob ject Q ∈ D (  ) such that dia [1] , [1] Q ∼ = ( φ : F 1 − → p ∗ Z ): dia Q : X f 1   φ 0 / / Z   Y φ 1 / / Z If i : [2] − →  clas siﬁes the non-degenerate pair of co mp osable arrows passing through the lo w er left corner (0 , 1 ) then let us set F = i ∗ Q ∈ D ([2 ]) . This F does the job.  F rom now on, whenever we consider the v alues of a stable deriv ator as tria ngulated c a tegories we will alwa ys mean the tria ngulated structure of Theorem 4.15. The nex t aim is to show that the functors b elonging to a stable de r iv ator can b e ca no nically made into exact functors with r espect to these structures . In the stable setting, Co rollary 4.14 induces immediately the following one. Corollary 4.17. L et F : D − → D ′ b e a morphism of stable derivators, then: F is left exact ⇐ ⇒ F is exact ⇐ ⇒ F is right exact In p articular, the c omp onents F J : D ( J ) − → D ′ ( J ) of an exact morphism ar e additive fu n ctors. Exact mor phisms are the ‘correc t’ mo rphisms for stable deriv a tors. So me evidence for this is given b y the next result. Prop osition 4.18. L et F : D − → D ′ b e an exact morphism of stable derivators and let J b e a c ate gory. The fun ctor F J : D ( J ) − → D ′ ( J ) c an b e c anonic al ly endowe d with the structu r e of an exact functor of triangulate d c ate gories. Pr o of. By Prop osition 4.3, we can assume without loss of generality that J = e. Moreov er, by deﬁnition, F pr eserves zero ob jects and coCa rtesian sq ua res. In pa rticular, coCartesian squares suc h that the tw o oﬀ-diagona l e n tries v anish are preserved b y F . This gives us the ca nonical isomorphism F ◦ Σ ∼ = Σ ◦ F . Similarly , F pr eserv es composites of tw o coCa r tesian squa res. In par ticular, among the compo sites tho se whic h v anish at (2 , 0) and (0 , 1) ar e preser ved. These were used to deﬁne the class of distinguished triang les in the canonical triang ulated structures from where it follows that F together with the ca nonical isomorphism F ◦ Σ ∼ = Σ ◦ F is exact.  This result can no w be applied to Example 3.18. In par ticular, w e can deduce that the functors belo nging to a stable deriv a to r r espect the canonical triangulated structur es we just constr ucted. Corollary 4.19. L et D b e a stable derivator and let u : J − → K b e a functor. The ind uc e d functors u ∗ : D ( K ) − → D ( J ) and u ! , u ∗ : D ( J ) − → D ( K ) c an b e c anonic al ly endowe d with the structur e of exact functors. Pr o of. Since we hav e a djunctions ( u ! , u ∗ ) a nd ( u ∗ , u ∗ ), it suﬃces to show that u ∗ can b e ca nonically endow ed with the str ucture of an exact functor (cf. [Mar 83 , p.4 63]). But this functor u ∗ can b e considered a s u ∗ : D K ( e ) − → D J ( e ) and hence the result follows by a combination of the last prop osition and Example 3.18.  Remark 4.20. Theorem 4.15 and Prop osition 4 .18 r ev e a l certain adv a n tag es of the languag e of stable der iv a tors ov er the language of tr iangulated ca tegories. A tr iangulated c ategory T is, by the very deﬁnition, a triple consisting of a categ ory T tog ether with a functor Σ : T − → T and a 58 MORITZ GROTH class of disting uis hed tria ng le as additionally sp e ciﬁe d st ructur e . These are then sub ject to a list of axioms. One adv antage of the stable deriv ators is that this str ucture does not hav e to be sp eciﬁed but instead is canonically av ailable. Once the deriv a to r is stable, i.e., ha s some easily motiv a ted pr op erties , tria ngulated structures can b e cano nically co nstructed. In par ticular, the o ctahedron axiom do es not hav e to b e made explicit. Similarly , the fac t that a mo rphism F of triangula ted catego ries is exact mea ns, by the very deﬁnition, that the functor is endow ed with an additional stru ct ur e g iven by a natura l isomor phism σ : F ◦ Σ − → Σ ◦ F which behav es nicely with resp ect to the tw o chosen classes of dis ting uished triangles. But, in fact, the exactness of such a mo rphism sho uld only b e a pr operty and not a structure. In most applications, the e xact functors under co nsideration are ‘derived functors’ of functors deﬁned ‘on ce r tain mo dels in the background’. And in this situation, the exactness then reﬂects the fact that this functor preserves (certa in) ﬁnite homotopy (co)limits. In the setting of stable deriv ators this is pr ecisely the notio n of an exact morphism. In pa rticular, the exactness o f a morphism is a g ain a prop ert y and not the sp eciﬁcation of an additional str ucture. These same adv antages are also shared by stable ∞− catego ries as studied in deta il in Lurie’s bo ok [Lur1 1]. A short introductio n to t hat theory ca n be found in [Gro10, Section 5]. W e are now basica lly done with the developmen t of the theory of (stable) deriv ato rs. So le t us analyze what co nditions o n a 2-sub category Dia ⊆ Cat hav e to b e imp o sed in order to b e able to a lso deduce the same results for (stable) deriv a to rs of type Dia . By the very deﬁnition of a deriv ator, we need that the empty catego ry and the ter minal category b elong to Dia . Moreover, it has to b e closed under ﬁnite copr oducts to give sense to a xiom (Der1). F urthermore, we frequently reduced situations to the case o f the underlying c a tegory by using the pass a ge from D to D M . Thus, Dia has als o to be closed under products. W e also used v a rious ﬁnite pos ets as a dmissible shap es in the pro ofs of this sec tio n so we should ask axiomatica lly for a suﬃcient supply of them. Finally , Dia has to b e clo sed under th e slice constr uction since we imp ose axio matically Kan’s for m ula. There is the following deﬁnition of a diagram ca tegory which we cite from [CN08]. In particular, this notion has the closur e pro perties we used in the development of the theo r y . Deﬁnition 4.21. A full 2 -subcateg ory Dia ⊆ Cat is calle d a diagr am c ate gory if it satisﬁes the following axioms: • All ﬁnite p osets consider ed as c a tegories belong to Di a . • F or every J ∈ Dia a nd every j ∈ J , the slice constructio ns J j / and J /j belo ng to Dia . • If J ∈ Dia then also J op ∈ Dia . • F o r every Gr othendiec k ﬁbration u : J − → K , if all ﬁb ers J k , k ∈ K, and the base K b elong to Dia then also J lies in Dia . With this notion o ne can no w deﬁne pre de r iv ators and (p oin ted, stable) deriv ato rs of type Di a as 2-functors Dia op − → CA T satisfying the corr esponding axioms. W e leav e it to the reader to chec k that all results we established so far ca n also be proved in that more genera l situation. Example 4.2 2 . The full 2-sub categor y o f ﬁnite p osets is the smallest diagram category , Cat itself is the larges t one. F ur ther exa mples ar e given b y the full 2-sub categorie s s panned by the ﬁnite categorie s or the ﬁnite-dimensional categor ies. Moreover, the intersection of a fa mily o f diag ram categorie s is ag ain a diagr a m ca teg ory . 4.3. Re collemen ts of tri angul ated categories. In this shor t s ubs ection, we mainly mention that s iev es and cosie v es give rise to rec ollemen ts of triangula ted categories in the con text of a stable DERIV A TORS, POINTED DE RIV A TORS, AND ST ABLE DERIV A TORS 59 deriv ator . This ca n b e used to r epro ve (in the stable cas e ) that the (co)exce ptional inv e r se image functors show up for free. W e begin with a very short reca p of the theor y o f recollements o f tr iangulated categor ie s . F or classical examples of recollements in algebraic geometry cf. [BBD82], for a very nice mo dern treat- men t c f. a lso to the thesis of Heider [Hei07]. Recollements capture axiomatically the situation in which w e are given three tria ngulated categories T ′ , T , and T ′′ such that every ob ject of T can b e obtained as a n extension of an ob ject of T ′′ by an ob ject of T ′ and v ice-v ersa. Mor e pr e cisely , th ere is the following deﬁnition. Deﬁnition 4. 23. A r e c ol lement of triangulate d c ate gories is a diag ram of triangulated categor ies and exact functors T ′ i ! / / T j ∗ / / i ∗ d d i ? z z T ′′ j ∗ d d j ! z z such tha t the following properties hold: • the pairs ( i ? , i ! ) , ( i ! , i ∗ ) , ( j ! , j ∗ ) , and ( j ∗ , j ∗ ) are adjunctions • j ∗ i ! = 0 • the functors i ! , j ! , and j ∗ are fully faithful and • every ob ject X ∈ T sits in t wo distinguished triangles of the form i ! i ∗ X / / X / / j ∗ j ∗ X / / Σ i ! i ∗ X , j ! j ∗ X / / X / / i ! i ? X / / Σ j ! j ∗ X where in b oth triangles the ﬁrst t w o arrows are the resp ectiv e adjunction morphisms. One can show that in this situation T ′ = ker j ∗ and that T ′′ is the V erdier quotient T / T ′ ([Hei07]). The la tter follows immediately from the ﬁrs t since by deﬁnition a r ecollemen t gives us a reﬂective lo calization and a co reﬂectiv e colo calization ([Kr a 10 ]). Let us r emark further that this deﬁnition is not g iv en in a minimal form but is ov erdetermined. Recall from class ical catego r y theo ry that if a functor admits an a djoin t on b oth sides then if one of the adjoints is fully faithful then this is also the case for the other one ([Bo r94a, Pro p. 3.4,2]). And, even mor e interesting for us , it s uﬃces to only have the right half of a reco llemen t. More precisely , ther e is the following r esult ([Hei07, Prop osition 1.14]). Prop osition 4.2 4. Consider a diagr am of t riangulate d c ate gories and exact functors T j ∗ / / T ′′ j ∗ d d j ! z z such that ( j ! , j ∗ ) and ( j ∗ , j ∗ ) ar e adjunctions and one of the two functors j ! , j ∗ is ful ly faithful. If we denote by T ′ the kernel of j ∗ and by i ! : T ′ − → T the inclusion then the ab ove diagr am c an b e extende d to a r e c ol lement: T ′ i ! / / T j ∗ / / i ∗ d d i ? z z T ′′ j ∗ d d j ! z z 60 MORITZ GROTH In the context of a stable deriv ato r, ther e is the following cla ss of examples. Example 4.25. Let D b e a stable der iv ator and consider a sieve j : U − → X . Mor eo v er, let Z b e the full sub category of X spanned by the o b jects which are not in the image of j . Then the inclus io n i : Z − → X is a cos iev e . Mo reov er, by the f ully fa ithfulness of homotopy K a n extensions along fully faithful functors and by Prop osition 3.6, the last pro position gives us the following recollemen ts: D ( U ) j ∗ / / D ( X ) i ∗ / / j ! h h j ∗ v v D ( Z ) i ∗ h h i ! v v D ( Z ) i ! / / D ( X ) j ∗ / / i ∗ h h i ? v v D ( U ) j ∗ h h j ! v v This example shows that for a sieve j : U − → X (resp. for a cosieve i : Z − → X ) the additiona l adjoint functor j ! : D ( X ) − → D ( U ) (resp. i ? : D ( X ) − → D ( Z )) shows up for free in the ab o ve recollements. Th us, this reproves, in the s table case, tha t a p oin ted der iv ator a dmits (co)ex c e ptional inv ers e image functors . References [AR94] Ji ˇ r ´ ı Ad´ amek and Ji ˇ r ´ ı Rosick´ y. Lo c al ly pr esentable and ac c essible c ate g ories , volume 189 o f L ondon Ma th- ematic al So ciety L e ct ur e Note Series . Cambridge Universit y Press, Cambridge, 1994. [Ayo0 7a] Joseph Ay oub. 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Moritz Groth, Rad boud Univ ersity, Nijmegen, Netherlands, email: m.groth@ma th.ru.nl

Derivators, pointed derivators, and stable derivators

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