Tensor-triangulated categories and dualities

TENSOR-TRIANGULA TED CA TEGORIES AND DUALI TIES BAPTISTE CALM ` ES AND JENS HORNBOSTEL Abstract. In a triangulated symmetric m onoidal closed category , there are natural dualities induced by the internal Hom. Giv en a monoidal exact f unc- tor f ∗ betw een tw o such categories and adjoin t couples ( f ∗ , f ∗ ), ( f ∗ , f ! ), we establish the commutat ive diagrams necessary for f ∗ and f ∗ to resp ect certain dualities, for a pro jection formula to hold betw een them (as duality preserving exact functors) and for classical base chang e and comp osition f ormulas to hold when suc h duality preserving functors are composed. This framework all o ws us to deﬁne push-forw ards for Witt groups, for example. Contents Int ro duction 2 1. Adjunctions a nd co nsequences 6 1.1. Notations and conv en tions 6 1.2. Useful prop e rties o f adjunctions 6 1.3. Bifunctors and adjunctions 11 1.4. Susp ended and triangula ted catego ries 12 1.5. Susp ended adjunctions 15 2. Dualities 18 2.1. Categor ies with duality 18 2.2. Duality pr e serving functors and morphisms 19 3. Consequences of the closed mono idal structur e 20 3.1. T enso r pro duct and internal Hom 20 3.2. Bidual is omorphism 22 4. F unctors betw een c losed mono idal ca teg ories 23 4.1. The mono idal functor f ∗ 24 4.2. Adjunctions ( f ∗ , f ∗ ) a nd ( f ∗ , f ! ) a nd the pro jection morphism 25 4.3. When the pro jection mor phism is inv ertible 29 4.4. Pro ducts 31 5. Relations b etw een functor s 32 5.1. Comp osition 33 5.2. Base change 36 5.3. Asso ciativity o f pr o ducts 39 5.4. The mono idal functor f ∗ and pro ducts 40 5.5. Pro jection for m ula 42 6. V arious reformulations 44 6.1. Reformulations us ing f ! of unit ob jects 44 6.2. Reformulations us ing mor phisms to corr e ct the dualities 46 6.3. Reformulations us ing a ﬁnal ob ject 48 Appendix A. Signs in the categor y of complexes 5 0 References 52 1 2 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Introduction Several cohomolo g y theor ies on sc hemes use the concept of a dualit y in their deﬁnition. In fact, these co homology t heories are usua lly d eﬁned at a catego r ical level; a gro up is asso cia ted to a ca tegory with some additional structur e, including a duality which is then a contra v ariant endofunctor of or der tw o (up to a natural isomorphism) on the category . The main example that we ha ve in mind is the Witt group, deﬁned for a tria ngulated c a tegory with dualit y , the (coheren t or locally free ) Witt gr oups of a s cheme then b eing deﬁned by using one of the v a rious bo unded derived catego ries o f a sc heme endow ed with a dualit y coming from the derived functor RHom. Of cour s e, most of the s tructural prop er ties of these coho mology theories should b e prov ed a t a catego r ical level. F or exa mple, Paul Balmer has prov e d lo c a lization for Witt gro ups directly at the level of triangula ted categorie s by using lo calization pro p e rties of such catego ries [ 1 ]. In the rest of the ar ticle, we only dis cuss the example of Witt g roups, but everything works exactly the same wa y for Gro thendieck-Witt groups. The reader s hould also hav e in mind that similar consideratio ns should apply to hermitian K-theo ry and Chow-Witt theor y , etc. W e a re int erested in the functor iality of Witt gro ups along mor phisms of schemes. A t the ca tegorical level, this means that we are given functors b etw een catego ries (such as the derived functors of f ∗ or f ∗ ) a nd that we would like to use them to induce morphisms betw een Witt groups. Let us elabor ate on this theme. The basic ideas a re: • a functor betw een ca tegories resp ecting the dua lit y induces a mo rphism of groups and • t w o such morphisms can b e compared if ther e is a mor phism o f functors betw een tw o such functor s, again resp ecting the duality . By “resp ecting the dualit y”, we actually mean the following. Let ( C , D ,  ) b e a category with duality , whic h means that D : C → C is a co nt rav ariant functor and that  : I d C → D 2 is a morphism of functor s such that for an y ob ject A , we hav e (1) D (  A ) ◦  DA = I d DA . A dua lit y pre serving functor from ( C 1 , D 1 ,  1 ) to ( C 2 , D 2 ,  2 ) is a pair { F, φ } where F : C 1 → C 2 is a functor and φ : F D → DF is a morphism o f functors, such that the dia gram of morphism of functors (2) F F  1 / /  2 F   F D 1 D 1 φD 1   D 2 D 2 F D 2 φ / / D 2 F D 1 commutes. In pra c tice , the functor F is usually given. F or exa mple, in the co nt ext of schemes, it can b e the derived functor o f the pull-ba ck or of the push- forward along a morphism of schemes. It then remains to ﬁnd a n interesting φ and to chec k the co mm utativity of diagra m ( 2 ), which can be very intricate. A morphism of dualit y pr eserving functors { F, φ } → { G, ψ } is a mor phism o f functors ρ : F → G suc h that the diag ram (3) F D 1 ρD 1 / / φ   GD 1 ψ   D 2 F D 2 G D 2 ρ o o commutes. This situatio n with tw o functors aris es when we wan t to compar e what t wo diﬀerent functors (res pec ting the dualities) y ie ld as Witt gr oup morphisms. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 3 F or ex ample, we migh t wan t to co mpare the comp osition of tw o pull-backs and the pull-back of the co mpo sition, in the cas e of mor phisms of schemes. In this kind of situation, usually { F , φ } and { G, ψ } ar e already g iven, ρ has to b e found and the commutativ e diagram ( 3 ) has to be prov ed. It turns out tha t in the context of symmetric mono idal closed categ ories, this t yp e of diag ram commutes for certain dua lities, functors and morphisms of functors arising in a na tural way . The main ob ject o f this article is to prov e it. Our original motiv atio n for de a ling with the q ues tions of this pap er was to deﬁne push-forward mo rphisms for Witt g roups with r esp ect to prop er ma ps of schemes and study their prop erties, which in turn s hould b e useful b oth for genera l theorems and c o ncrete co mputatio ns of Witt groups of schemes. The a rticle [ 3 ] uses the results of this article to esta blis h these push-for wards. W e refer the r eader to lo c . cit. for details. As Witt groups are the main applica tion that we have in mind, we hav e sta ted in co rollarie s of the main theorems what they imply for Witt groups. These cor ollaries are trivial, and just rely on the classical pro p o sitions 2.2.3 and 2.2.4 . This should b e seen as some help for the reader familiar with co homology theories and geometr ic intuit ion. The reader not interested in Witt g roups or other Witt-lik e coho mology theories might just skip the cor ollaries. In a ll o f them, the triangulated ca teg ories are a ssumed to b e Z [1 / 2]-linea r (to b e a ble to deﬁne Witt groups). W e now give a more precise descr iption o f the functor s and morphisms of functors we cons ide r . Dualities. Ass ume we ar e given a symmetric monoidal closed categ ory C w ere the tensor pro duct is deno ted by ⊗ and its “right adjoint”, the internal Hom, is denoted by [ − , − ]. Then, as recalled in Sectio n 3 , ﬁxing an ob ject K and setting D K = [ − , K ], a natural mo r phism of functors  K : I d → D K D K can b e deﬁned using the symmetry a nd the adjunction of the tensor pr o duct and the internal Ho m and the fo rmula ( 1 ) is sa tisﬁed. This is well known and re called here just for the sake of completeness. Pull-bac k. Assume now that we are given a monoidal functor denoted by f ∗ (b y analogy with the a lgebro- geometric cas e) fr o m C 1 to C 2 , then, for any ob ject K , as explained ab ov e, we can consider the catego ries with dua lity ( C 1 , D K ,  K ) and ( C 2 , D f ∗ K ,  f ∗ K ). There is a natura l morphism of functors β K : f ∗ D K → D f ∗ K f ∗ such that diagram ( 2 ) is co mmu tative (see Pro p osition 4.1.1 and Theo r em 4.1 .2 ). In other w ords, { f ∗ , β K } is a dua lit y preserving functor from ( C 1 , D K ,  K ) to ( C 2 , D f ∗ K ,  f ∗ K ). Push-forw ard. Assume furthermore that that we a r e given a right adjoint f ∗ , then there is a natura l morphism of pr o jection π : f ∗ ( − ) ⊗ ∗ → f ∗ ( − ⊗ f ∗ ( ∗ )) (see Prop os ition 4.2 .5 ). When f ∗ also has a rig ht a djoint f ! , we can co ns ider the categorie s with dualit y ( C 1 , D K ,  K ) and ( C 2 , D f ! K ,  f ! K ) and ther e is a na tu- ral isomorphism ζ K : f ∗ D f ! K → D K f ∗ such that diagr am ( 2 ) is commutativ e. In o ther words, { f ∗ , ζ K } is a dua lit y preserving functor fro m ( C 2 , D f ! K ,  f ! K ) to ( C 1 , D K ,  K ). Pro duct. Theorems in volving pro ducts of dualities are stated. Giv en a pair of categorie s with dua lit y ( C 1 , D 1 ,  1 ) and ( C 2 , D 2 ,  2 ), there is a n obvious structure of a catego ry with duality ( C 1 × C 2 , D 1 × D 2 ,  1 ×  2 ). If C is monoidal s ymmetric closed, there is a natura l morphism of (bi)functors τ K,M : D K ⊗ D M → D ( − ⊗ − ) K ⊗ M (see Deﬁnition 4.4.1 ). Prop osition 4 .4.6 r ecalls that {− ⊗ − , τ K,M } is a duality pr eserving functor from ( C × C , D K × D M ,  K ×  M ) to ( C , D K ⊗ M ,  K ⊗ M ). This gives a pro duct on Witt groups. 4 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL W e now expla in r elations betw een the functors { f ∗ , β } , { f ∗ , ζ } a nd {⊗ , τ } in diﬀerent contexts. Before going any further, let us remar k that a mo rphism ι : K → M induces a morphism of functors ˜ ι : D K → D M which is easily shown to yield a duality preserving functor I ι = { I d, ˜ ι } from ( C , D K ,  K ) to C , D M ,  M ). Comp os ition. There is a natural wa y to comp o se duality preserving functors: { F , f } ◦ { G, g } := { F G, g F ◦ f G } . Supp ose that we a re given a pseudo contrav ar iant functor ( − ) ∗ from a catego ry B to a category where the ob jects are s ymmetric monoidal c lo sed categ ories and the mo rphisms a re mono ida l functors. As usual, “pseudo” mea ns that we hav e a lmost a functor, exc e pt that we only have a natural isomorphism a g,f : f ∗ g ∗ ≃ ( g f ) ∗ instead of an e quality . Then an obvious question arises: can we compare { ( g f ) ∗ , β K } with { f ∗ , β g ∗ K } ◦ { g ∗ , β K } for compo sable morphisms f and g in B . The answer is that the natura l isomo rphism a is a morphism of dualit y prese rving functors from I a g,f ,K ◦ { f ∗ , β g ∗ K } ◦ { g ∗ , β K } to { ( g f ) ∗ , β K } , i.e. the diagram ( 3 ) comm utes (se e 5.1.3 ) with ρ = a . The correctio n by I a g,f ,K is necessar y for otherwis e , strictly s p ea king, the target categories of the t wo duality preser ving functors are no t equippe d with the same duality . A simila r compo sition question arises when some f ∈ B ar e such that there a re adjunctions ( f ∗ , f ∗ ) and ( f ∗ , f ! ). O n the s ubca tegory B ′ of such f , there is a wa y of deﬁning natur a l (with resp ect to the adjunctions ) pseudo functor structures b : ( g f ) ∗ ≃ g ∗ f ∗ and c : f ! g ! ≃ ( g f ) ! , and we can compa re { ( g f ) ∗ , ζ K } a nd { g ∗ , ζ g,f ! K } ◦ { f ∗ , ζ f ,K } using the natural morphism o f functor s b up to a s mall corre c tion of the duality using c as ab ov e (see Theorem 5.1.9 ). Base c hange. A “bas e change” question arises when we are given a co mm utative diagram V ¯ g / / ¯ f   Y f   X g / / Z in B such that g and ¯ g are in B ′ . In this situation, there is a natural morphism ε : f ∗ g ∗ → ¯ g ∗ ¯ f ∗ (see Section 5.2 ). When ε is an iso morphism, ther e is a natur al morphism γ : ¯ f ∗ g ! → ¯ g ! f ∗ . Starting from an ob ject K ∈ C Z , the mo rphism o f func- tors γ is used to deﬁne a duality pr eserving functor I γ K from ( C V , D ¯ f ∗ g ! K ,  ¯ f ∗ g ! K ) to ( C V , D ¯ g ! f ∗ K ,  ¯ g ! f ∗ K ). Ho w can we compare { f ∗ , β K } ◦ { g ∗ , ζ K } a nd { ¯ g ∗ , ζ f ∗ K } ◦ I γ K ◦ { ¯ f ∗ , β g ! K } ? Theorem 5.2.1 proves that ε deﬁnes a dua lit y pres erving functor from the ﬁrst to the second. Pull-bac k and pro duct. Given a monoidal functor f ∗ the is omorphism α : f ∗ ( − ) ⊗ f ∗ ( − ) → f ∗ ( −⊗− ) (deﬁning f ∗ as a mo noidal functor) deﬁnes a duality pre- serving functor I α K,M from ( C , D f ∗ K ⊗ f ∗ M ,  f ∗ K ⊗ f ∗ M ) to ( C , D f ∗ ( K ⊗ M ) ,  f ∗ ( K ⊗ M ) ). Prop ositio n 5.4.1 shows that the mo rphism o f functors α then deﬁnes a duality pre- serving morphism o f functors fro m I α K,M ◦ {− ⊗ − , τ f ∗ K,f ∗ M } ◦ { f ∗ × f ∗ , β K × β M } to { f ∗ , β K ⊗ M } ◦ {− ⊗ − , τ K,M } . It essentially means tha t the pull-back is a ring morphism o n Witt groups. Pro jection form ul a. Given a monoidal functor f ∗ with a djunctions ( f ∗ , f ∗ ), ( f ∗ , f ! ) such that π is an isomorphism, ther e is a na tural morphism of functor s θ : f ! ( − ) ⊗ f ∗ ( − ) → f ! ( − ⊗ − ). This deﬁnes a duality pr eserving functor I θ K,M from ( C 2 , D f ! K ⊗ f ∗ M ,  f ! K ⊗ f ∗ M ) to ( C 2 , D f ! ( K ⊗ M ) ,  f ! ( K ⊗ M ) ). Theorem 5.5.1 prov es that π is a duality pr eserving morphism of functors fro m { − ⊗ − , τ K,M } ◦ { f ∗ × I d, ζ K × id } to { f ∗ , ζ K ⊗ M } ◦ I θ K,M ◦ {− ⊗ − , τ f ! K,f ∗ M } ◦ { I d × f ∗ , id × β M } . Susp ensi ons and triangul ated structures. As it is r equired for the example of Witt gro ups, all the results are prov ed as well in the setting of t riangulate d TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 5 symmetric monoidal catego ries, functors and morphisms of functors resp ecting the triangulated structure . This means essentially t wo things. First, the categor ies are endow ed with an a uto equiv alence T (called susp ension in the a rticle) a nd the tensor pro ducts deﬁning monoidal struc tur es resp ect this suspens io n, i.e . suc h a tensor pro duct comes equipp ed with a pair of iso morphism of functor s such that a certain dia gram anticomm utes (see Deﬁnition 1.4.12 ). As proved in the a r ticle, it follows that all the other functors (or bifunctors) mentioned ab ov e can b e endow ed by c onstruction (see Poin t 1 of Prop ositions 1.5.3 and 1.5.8 ) with mor phisms of functors to commute with the susp ensions in a suitable wa y inv olving commutativ e diagrams (similar to ( 2 ) with T instea d of D ). All the morphisms of functors inv olved then also r e sp e ct the susp ension, i.e. satisfy c o mmu tative dia grams s imilar to ( 3 ). This is imp or tant, be c a use in the main applications, chec k ing this kind of things by hand a mo unt s to checking signs inv olved in the deﬁnitions of complexes o r maps of co mplexes, and it has been the source of errors in the literature. Here, w e av o id s uch p o tential err ors by cons truction (there are no sig ns and no complexes ). Second, using these mor phis ms of functor s to deal with the susp ensions, the functor s should resp ect the collection of distinguis he d triangles. This is ea sy and has nothing to do with commutative diagrams. It is again o bta ined by c onstruction (see Poin t 2 of P rop ositions 1.5.3 and 1 .5.8 ), ex cept for the ﬁrst v aria ble of the internal Hom, for which it has to b e assumed. The reader who is not interested in triangulated categorie s, but rather o nly in monoidal structure s can just forget ab o ut all the parts of the statements inv olv ing T . What rema ins is the monoidal structure. These functors and morphisms betw een them are discussed in the ﬁrst ﬁve sec- tions o f this a rticle. Section 6 is devoted to reformulations o f the main results when the ba se categ ory B is e nr iched in or der that the ob jects also sp ecify the duality: they a re pairs ( X , K ) where K ∈ X ∗ is used to form the duality D K . The mo r - phisms can then b e chosen in diﬀerent ways (see Sectio ns 6.1 , 6.2 a nd 6 .3 ). These sections a re dir ected tow ar ds geometric applications . In pa rticular, the ob ject ω f int ro duced a t the b eginning o f 6.1 has a clea r geometric interpretation in the ca se o f morphisms of schemes: it is the relative cano nic a l sheaf. Finally , w e hav e included an app endix to recall symmetric monoidal closed s tructures o n categ o ries inv olv- ing chain complexes and to relate the sign choices inv olved with the ones made by v arious a uthors working with Witt gr oups. Remarks on the pro ofs. Nearly every constr uction of a morphism in the article is based on v ariations on a single lemma on a djunctions, namely Lemma 1.2.6 . Its reﬁnements ar e Theorem 1.2.8 , Lemma 1.3.5 a nd Theorem 1.5 .10 . Similarly , to establish commut ativities of na tural transformatio ns arising from b y Lemma 1.2 .6 the main to ol is Le mma 1.2.7 . The who le Section 1 is devoted to these formal results a bo ut a djunctions. In some applications , certain na tural morphisms c o nsidered in this a rticle hav e to b e iso mo rphisms. F or example, to deﬁne Witt gro ups, one requires that the morphism of functors  K is an isomor phis m. This is not imp ortant to pr ov e the commutativit y of the diagrams , ther efore it is not pa rt of our ass umptions. Nev- ertheless, in some ca ses, it might b e useful to know that if a par ticula r morphism is an isomorphism, then another one is also automatically an isomorphism. When po ssible, we have included such statements, for e x ample in Prop o s itions 4.3.1 and 4.3.3 . The attribute “ strong” in Deﬁnitions 2.1.1 , 2.2 .1 and 2.2.2 is part o f this phi- losophy . By con trast, tw o iso mo rphism assumptions are imp or tant to the abstr act setting r e gardless of applications. This is the ca s e of the a s sumption on π , since the essential mor phis m θ is deﬁne d using the inv e r se of π , as well as the a ssumption o n ε whose in verse is in volv ed in the deﬁnition of γ . These isomor phism assumptions 6 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL are listed at the b eginning of Sectio n 4 and a re recalle d in every theor em w her e they a re used. Considering po ssible future applications (e.g. motivic homotopy ca tegories or the stable homotopy catego ry), we hav e presented some asp ects in as muc h generality as p ossible. Co nversely , we do not attempt to provide a complete list of articles where parts o f the gener al fra mework w e study ha s already b een co nsidered. See how ever [ 5 ] for a r ecent refere nc e written by homotopy theorists which con tains many further references. Finally , let us mention a question a c a tegory theo r ist might ask when reading such a pa per : ar e there cohere nce theore ms , in the spir it o f [ 8 ], that would prov e systematically the needed commut ative diagrams? As far a s the a uthors know, there ar e no such co herence theorems av ailable for the moment. Although it is certainly an in ter esting ques tio n, it is unclear (to the authors) how to fo r mulate coherence statements. One problem is that, as mentioned ab ov e, par t o f the in ter- esting co mm utative diag rams inv olve morphisms that ar e deﬁned using inverses of natural mor phisms. Without those inv erted natural morphisms , the commutativit y of the diagrams b ecome meaningless. So, the interested re a der might c o nsider this article a s a source of inspiration for future coherence theo rems. 1. Adjunctions and consequences 1.1. Notations and con v en ti ons. The o pp o s ite c a tegory o f a categor y C is de- noted b y C o . When F and G are functors with sa me sour ce and target, we deno te a morphism of functors b etw ee n them as t : F → G . When s : G → H is another one, their comp osition is deno ted b y s ◦ t . When F 1 , F 2 : C → D , G 1 , G 2 : D → E , f : F 1 → F 2 and g : G 1 → G 2 , we denote by g f the mor phism of functors deﬁned by ( f g ) A = G 2 ( f A ) ◦ g F 1 ( A ) = g F 2 ( A ) ◦ G 1 ( f A ) on any ob ject A . When F 1 = F 2 = F and f = id F (resp. G 1 = G 2 = G and g = i d G ), we us ua lly use the notation g F (resp. Gf ). With this conv ention, g f = G 2 f ◦ g F 1 = g F 2 ◦ G 1 f . When a commutativ e diag ram is o btained by this eq uality or other pr op erties immediate from the deﬁnition of a morphism of functors, we just put an mf lab el o n it and av o id further justiﬁcation. T o save space, it ma y happ en that when lab eling maps in diagrams, we drop the functors from the notation, a nd just keep the imp ortant part, that is the morphis m o f functor s (thus F g H might be reduced to g ). Many of the commutativ e diagra ms in the article will b e lab eled by a symbol in a b ox (letters or num ber s, such as in H or 3 ). When they are used in ano ther commutativ e diagram, even tually after applying a functor to them, we just lab el them with the same sy m bo l, so that the reader recognizes them, but witho ut further comment. 1.2. Useful prop erties of adjunctions. This section is devoted to easy facts and theorems ab out a djunctions, tha t a r e repe atedly used thro ughout this article. All these facts are ob vious, and w e o nly prove the ones that a re not completely classical. A go o d reference for the bac kground on categor ies and adjunctions a s discussed her e is [ 9 ]. Deﬁnition 1.2.1. An adjoint co uple ( L, R ) is the data consisting of tw o functors L : C → D and R : D → C and equiv alently: • a bijection Ho m( LA, B ) ≃ Hom( A, R B ), functoria l in A ∈ C and B ∈ D , or • tw o morphism of functors η : I d C → RL a nd ǫ : LR → I d D , called resp ec- tively unit a nd counit, such that the resulting co mpo sitions R ηR → RL R Rǫ → R and L Lη → LR L ǫL → L ar e identities. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 7 In the co uple, L is called the left adjoint and R the rig ht adjoint. When we w an t to s pe c ify the unit and counit of the couple and the catego r ies inv olved, we say ( L, R , η , ǫ ) is an adjoint co uple from C to D . When the co mm utativit y of a diagr am follows by o ne of the ab ove co mp o s itions giving the identit y , we lab el it ad j . R emark 1.2 .2 . Adjunctions be t ween functors that a re contrav ar iant ca n be co nsid- ered in tw o diﬀerent wa ys, by tak ing the opp os ite catego ry of the sour ce of L or R . This do es not lea d to the sa me notion, essent ially b eca us e if ( L, R ) is an adjoint couple, then ( R o , L o ) is an a djo int co uple (instead of ( L o , R o )). F or this rea s on, we only use cov ariant functors in adjoint couples. Lemma 1. 2.3. L et ( L, R, η , ǫ ) and ( L ′ , R ′ , η ′ , ǫ ′ ) b e two adjoint c ouples b etwe en the same c ate gories C and D , and let l : L → L ′ (r esp. r : R → R ′ ) b e an isomorph ism. Then, ther e is a unique isomorphism r : R → R ′ (r esp. l : L → L ′ ) such that η ′ = r l ◦ η and ǫ ′ = ǫ ◦ l − 1 r − 1 . In p articular, a right (r esp. left) adjoint is un ique up to unique isomoprhism. Pr o of. The morphism r is given by the comp os ition R ′ ǫ ◦ R ′ l − 1 R ◦ η ′ R a nd its inv erse by the comp osition Rǫ ′ ◦ Rl R ′ ◦ η R ′ .  Lemma 1.2. 4. An e qu ivalenc e of c ate gories is an adjoint c ouple ( F , G, a, b ) for which the unit and c ounit ar e isomorphi sms. In p articular, ( G, F, b, a ) is also an adjoint c ouple. Lemma 1.2.5. L et ( L, R, η , ǫ ) (r esp. ( L ′ , R ′ , η ′ , ǫ ′ ) ) b e an adjoint c ouple fr om C to D (r esp. fr om D t o E ) . Then ( L ′ L, RR ′ , R η ′ L ◦ η , ǫ ′ ◦ L ′ ǫR ′ ) is an adjoint c ouple fr om C to E . W e now turn to a series of less standard r esults, nevertheless very easy . Lemma 1 .2.6. (mates) L et H , H ′ , J 1 , K 1 , J 2 and K 2 b e functors with sour c es and tar gets as on the fol lowing diagr am. C 1 K 1   H / / C 2 K 2   C ′ 1 J 1 O O H ′ / / C ′ 2 J 2 O O Assume ( J i , K i , η i , ǫ i ) , i = 1 , 2 ar e adjoint c ouples. L et a : J 2 H ′ → H J 1 (r esp. b : H ′ K 1 → K 2 H ) b e a morphism of functors. Then ther e exists a unique morphism of functors b : H ′ K 1 → K 2 H (r esp. a : J 2 H ′ → H J 1 ) such that the diagr ams J 2 H ′ K 1 aK 1   J 2 b / / H J 2 K 2 H η 2 H   H J 1 K 1 H ǫ 1 / / H and H ′ η 2 H ′   H ′ η 1 / / H ′ H ′ K 1 J 1 bJ 1   K 2 J 2 H ′ K 2 a / / K 2 H J 1 ar e c ommu tative. F urthermor e, given two morphisms of funct ors a and b , the c om- mutativity of one diagr am is e quivalent to the c ommutativity of the other one. In this situation, we say that a and b ar e mates with r esp e ct to t he r est of the data. Pr o of. W e o nly prove that the existence of a implies the uniqueness and existence of b , the pro of of the o ther case is similar. Assume that b exis ts and makes the 8 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL diagrams c o mmu tative. The commutativ e dia gram H ′ K 1 η 2 H ′ K 1   H ′ η 1 K 1 / / H ′ H ′ K 1 J 1 K 1 H ′ K 1 ǫ 1 / /   mf H ′ K 1 b   K 2 J 2 H ′ K 1 K 2 aK 1 / / K 2 H J 1 K 1 K 2 H ǫ 1 / / K 2 H in which the upper hor izontal c o mpo sition is the iden tit y o f H ′ K 1 (b y adjunction) shows that b has to be given by the comp osition H ′ K 1 η 2 H ′ K 1 / / K 2 J 2 H ′ K 1 K 2 aK 1 / / K 2 H J 1 K 1 K 2 H ǫ 1 / / K 2 H . This prov es uniqueness. Now let b b e g iven b y the ab ov e comp ositio n. The com- m utative diag ram J 2 H ′ K 1 / / GF ED J 2 b   O O O O O O O O O O O O O O O O O O O O O O J 2 K 2 J 2 H ′ K 1   / / mf J 2 K 2 H J 1 K 1   / / mf J 2 K 2 H ǫ 2 H   ad j J 2 H ′ K 1 aK 1 / / H J 1 K 1 H ǫ 1 / / H prov e s H and the commutativ e diagra m H ′ K 1 J 1 / / GF ED bJ 1   mf K 2 J 2 H ′ K 1 J 1 / / mf K 2 H J 1 K 1 J 1 / / ad j K 2 H J 1 H ′ H ′ η 1 O O η 2 H ′ / / K 2 J 2 H ′ O O K 2 a / / K 2 H J 1 O O p p p p p p p p p p p p p p p p p p p p p p prov e s H ′ . The fact that the commutativit y o f one of the diagra ms implies com- m utativity to the other is left to the reader.  Lemma 1.2.7 . L et u s c onsider a cub e of functors and morphisms of functors • / /   ? ? ? ? •   ? ? ? ? • ; C         / / q p O O   ? ? ? ? • P X * * * * * * * * * * * * * * * * * * * * * * * * * * O O / / • [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? O O f ront • / / •   ? ? ? ? q p O O   ? ? ? ? / / • [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? O O   ? ? ? ? • ; C         / / • P X * * * * * * * * * * * * * * * * * * * * * * * * * * O O back that is c ommu tative in the fol lowing sense: The morphism b et we en t he two ou t er c omp ositions of fun ctors fr om p to q given by the c omp osition of the thr e e m orphisms of functors of t he fr ont is e qual to the c omp osition of the thr e e morphism of functors of t he b ack. Assu me t hat t he vertic al maps have right adjoints. Then, by L emma 1.2.6 applie d to t he vertic al squar es, we obtain the fol lowing cub e (the top and TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 9 b ottom squar es have not change d). r   / /   ? ? ? ? •   ? ? ? ? •   ; C         / / •   • ; C           ? ? ? ? • ; C                       / / s f ront r   / / •     ? ? ? ? •   •   ? ? ? ? ; C                       / / •   ? ? ? ? ; C         • ; C         / / s back This cu b e is c ommutative (in the sense just deﬁne d, using r and s inste ad of p and q ). Pr o of. This is stra ightforw a rd, using the co mm utative diagrams of L emma 1.2.6 , and left to the rea der .  W e now use Lemma 1.2.6 to prov e a theor em, which do esn’t contain a lot more than the lemma, but is stated in a conv enient wa y for future reference in the applications w e are interested in. Theorem 1.2.8 . L et L , R , L ′ , R ′ , F 1 , G 1 , F 2 , G 2 b e funct ors whose sour c es and tar gets ar e sp e ciﬁe d by t he diagr am C 1 G 1   L / / C 2 R o o G 2   C ′ 1 F 1 O O L ′ / / C ′ 2 . F 2 O O R ′ o o We wil l study morphisms of functors f L , f ′ L , g L , g ′ L , f R , f ′ R , g R and g ′ R whose sour c es and tar get s wil l b e as fol lows: LF 1 f L / / F 2 L ′ f ′ L o o L ′ G 1 g ′ L / / G 2 L g L o o F 1 R ′ f ′ R / / RF 2 f R o o G 1 R g R / / R ′ G 2 g ′ R o o L et us c onsider the fol lowing diagr ams, in which the maps and their dir e ctions wil l b e the obvious ones induc e d by the eight maps ab ove and the adjunctions ac c or dingly to the diﬀer ent c ases discusse d b elow. F 2 L ′ G 1 L LF 1 G 1 F 2 G 2 L L L ′ L ′ G 2 F 2 L ′ L ′ G 1 F 1 G 2 LF 1 F 1 R ′ G 2 R RF 2 G 2 F 1 G 1 R R R ′ R ′ G 1 F 1 R ′ R ′ G 2 F 2 G 1 RF 2 G 1   / / G 1 G 1 RL R ′ L ′ G 1 R ′ G 2 L L ′ G 1 R G 2 G 2 LR   L ′ R ′ G 2 / / G 2 10 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL F 1   / / F 1 F 1 R ′ L ′ RLF 1 RF 2 L ′ LF 1 R ′ F 2 F 2 L ′ R ′   LRF 2 / / F 2 Then 1. L et ( G i , F i ) , i = 1 , 2 b e adjoint c ouples. L et g L (r esp. f L ) b e given, then ther e is a unique f L (r esp. g L ) such that L and L ′ ar e c ommu tative. L et g R (r esp. f R ) b e given, then ther e is a unique f R (r esp. g R ) such that R and R ′ ar e c ommu tative. 1’. L et ( F i , G i ) , i = 1 , 2 b e adjoint c ouples. L et g ′ L (r esp. f ′ L ) b e given, then ther e is a unique f ′ L (r esp. g ′ L ) such that L and L ′ ar e c ommu tative. L et g ′ R (r esp. f ′ R ) b e given, then ther e is a unique f ′ R (r esp. g ′ R ) such that R and R ′ ar e c ommu tative. 2. L et ( L, R ) and ( L ′ , R ′ ) b e adjoint c ouples. L et f L (r esp. f ′ R ) b e given, then ther e is a unique f ′ R (r esp. f L ) su ch that F 1 and F 2 ar e c ommutative. L et g ′ L (r esp. g R ) b e given, then ther e is a u nique g R (r esp. g ′ L ) such t hat G 1 and G 2 ar e c ommu tative. 3. Assuming ( G i , F i ) , i = 1 , 2 , ( L, R ) and ( L ′ , R ′ ) ar e adjoint c ouples, and g L , g ′ L = g − 1 L ar e given (r esp. f R and f ′ R = f − 1 R ). By 1 and 2, we obtain f L and g R (r esp. g R and f L ). We then may c onstruct f R and f ′ R (r esp. g L and g ′ L ) which ar e inverse to e ach other. 3’. Assuming ( F i , G i ) , i = 1 , 2 , ( L, R ) and ( L ′ , R ′ ) ar e adjoint c ouples, f L and f ′ L = f − 1 L ar e given (r esp. g R and g ′ R = g − 1 R ). By 1’ and 2, we obtain g ′ L and f ′ R (r esp. f ′ R and g ′ L ). We then may c onstruct g ′ R and g R (r esp. f ′ L and f L ) which ar e inverse to e ach other. Pr o of. Point s 1, 1’ and 2 a re obvious trans la tions of the pr e vious lemma. W e only prov e Point 3, since 3’ is dua l to it. Let ( L, R, η , ǫ ), ( L ′ , R ′ , η ′ , ǫ ′ ) and ( G i , F i , η i , ǫ i ), i = 1 , 2, b e the adjoint couples . Using 1 a nd 2, we ﬁrst obtain f L and g R , as well as the commutativ e diagrams L , L ′ (bo th involving g L = ( g ′ L ) − 1 ), G 1 and G 2 (bo th in volving g ′ L = ( g L ) − 1 ). The mor phisms of functors f ′ R and f R are resp ectively deﬁned b y the c omp ositions F 1 R ′ ηF 1 R ′ / / RLF 1 R ′ Rf L R ′ / / RF 2 L ′ R ′ RF 2 ǫ ′ / / RF 2 and RF 2 η 1 RF 2 / / F 1 G 1 RF 2 F 1 g R F 2 / / F 1 R ′ G 2 F 2 F 1 R ′ ǫ 2 / / F 1 R ′ . W e compute f R ◦ f ′ R as the upp er r ight comp osition of the following comm utative diagram F 1 R ′ η 1   η / / mf RLF 1 R ′   f L / / mf RF 2 L ′ R ′   ǫ ′ / / mf RF 2 η 1   F 1 G 1 F 1 R ′ / / G 1 η ′ ' ' F 1 G 1 RLF 1 R ′   / / mf F 1 G 1 RF 2 L ′ R ′   / / mf F 1 G 1 RF 2 g R   F 1 R ′ G 2 LF 1 R ′ ( g ′ L ) − 1 = g L   / / L ′ F 1 R ′ G 2 F 2 L ′ R ′   / / mf F 1 R ′ G 2 F 2 ǫ 2   F 1 R ′ L ′ G 1 F 1 R ′ ǫ 1 / / F 1 R ′ L ′ R ′ ǫ ′ / / F 1 R ′ TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 11 The low er left comp osition in the ab ov e dia gram is the iden tit y b e cause it appe a rs as the upp er right comp osition of the commutativ e diagra m F 1 R ′ η 1 / / O O O O O O O O O O O O O O O O F 1 G 1 F 1 R ′   η ′ / / mf F 1 R ′ L ′ G 1 F 1 R ′ ǫ 1   ad j F 1 R ′ / / S S S S S S S S S S S S S S S S S S S S S S S S S S F 1 R ′ L ′ R ′ ǫ ′   ad j F 1 R ′ . The comp ositio n f ′ R ◦ f R = id is pr ov ed in a simila r wa y , inv o lving the diagra ms L a nd G 2 .  The rea der has ce rtainly no tice d that there is a statement 2 ′ which we didn’t sp ell o ut b ecause we do n’t need it. 1.3. Bifunctors and adjunctions. W e ha v e to deal with co uples of bifunctors that give adjoint couples of usua l functors when one o f the entries in the bifunctors is ﬁxed. W e need to expla in how these adjunctions a re functor ial in this entry . The standard example fo r that is the classical a djunction b etw een tensor pro duct and int ernal Ho m. Deﬁnition 1.3. 1. Let X , C , C ′ be three categor ies, and let L : X × C ′ → C a nd R : X o × C → C ′ be bifunctors. W e say that ( L, R ) form an adjoint co uple of bifunctors (abbr eviated as ACB) from C ′ to C with para meter in X if we are given adjoint couples ( L ( X , − ) , R ( X , − ) , η X , ǫ X ) for every X and if furthermo re η and ǫ are gener alized transforma tions in the sense o f [ 4 ], i.e. for any morphism f : A → B in X , the dia grams L ( A, R ( B , C )) L ( f ,id ) / / L ( id,R ( f ,id ))   gen L ( B , R ( B , C )) ǫ B   L ( A, R ( A, C )) ǫ A / / C C ′ η A / / η B   gen R ( A, L ( A, C ′ )) R ( id,L ( f ,id ))   R ( B , L ( B , C ′ )) R ( f ,id ) / / R ( A, L ( B , C ′ )) commute. W e sometimes use the notation ( L ( ∗ , − ) , R ( ∗ , − )), where the ∗ is the ent ry in X a nd write C L / / X C ′ R o o in dia grams. Example 1.3.2 . Let C = C ′ = X b e the category o f mo dules over a commutative ring. The tenso r pro duct (with the v ar iables switched) a nd the internal Hom for m an ACB, with the usua l unit and c ounit. Lemma 1. 3.3. L et F : X ′ → X b e a fun ctor, and ( L, R ) b e an ACB with p ar ameter in X . Then ( L ( F ( ∗ ) , − ) , R ( F o ( ∗ ) , − )) is again an ACB in the obvious way. Lemma 1.3. 4. L et ( L, R ) b e an ACB fr om C ′ to C with p ar ameter in X and let ( F, G ) b e an adjoint c ouple fr om D to C ′ , then ( L ( ∗ , F ( − )) , GR ( ∗ , − )) is an ACB with unit and c ounit deﬁne d in the obvious and natura l way. Pr o of. Left to the reade r .  W e now give a version of Lemma 1.2.6 for ACBs. 12 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Lemma 1. 3.5. F or i = 1 or 2 , let ( J i , K i , η i , ǫ i ) b e an ACB fr om C ′ i to C i with p a- r ameter in X , let H : C 1 → C 2 and H ′ : C ′ 1 → C ′ 2 b e functors. L et a : J 2 ( ∗ , H ′ ( − )) → H J 1 ( ∗ , − ) (r esp. b : H ′ K 1 ( ∗ , − ) → K 2 ( ∗ , H ( − )) ) b e a morphism of bifunctors. Then ther e exists a u nique morphism of bifunctors b : H ′ K 1 ( ∗ , − ) → K 2 ( ∗ , H ( − )) (r esp. a : J 2 ( ∗ , H ′ ( − )) → H J 1 ( ∗ , − ) ) such that the diagr ams J 2 ( X, H ′ K 1 ( X, − )) aK 1   J 2 b / / H J 2 ( X, K 2 ( X, H ( − ))) η 2 H   H J 1 ( X, K 1 ( X, − )) H ǫ 1 / / H and H ′ η 2 H ′   H ′ η 1 / / H ′ H ′ K 1 ( X, J 1 ( X, − )) bJ 1   K 2 ( X, J 2 ( X, H ′ ( − ))) K 2 a / / K 2 ( X, H J 1 ( X, − )) ar e c ommu tative for every morphism X ∈ X . Pr o of. W e only consider the case when a is given. F or ev ery X ∈ X , w e a pply Lemma 1.2.6 to o btain b : H ′ K 1 ( X, − ) → K 2 ( X, H ( − )). It remains to prove that this is functoria l in X . This is an ea sy exe r cise on commutativ e diagrams which we leav e to the rea der (it inv olves that b oth η 1 and ǫ 2 are generalize d transfor mations as in Deﬁnition 1.3.1 ).  Lemma 1. 3.6. L et ( L, R, η , ǫ ) and ( L ′ , R ′ , η ′ , ǫ ′ ) b e ACBs, and let l : L → L ′ (r esp. r : R → R ′ ) b e an isomorphism of bifunctors. Then, t her e exists a unique isomor- phism of bifunctors r : R → R ′ (r esp. l : L → L ′ ) su ch that η ′ X = r X R ( X , l ) η X and ǫ ′ X = ǫ X L ( X , r − 1 X ) l − 1 X for every X ∈ X . In other wor ds, a right ( re sp. left) bifunctor adjoint is unique u p to unique isomorphism. Pr o of. W e only give the pro of in ca se l is given, the ca se with r given is similar. Apply Lemma 1.3.5 with H and H ′ being identit y functors. Starting with l − 1 , we get r , a nd s tarting with l we get a morphis m r ′ : R ′ → R . The mor phisms r and r ′ are inv erse to each other as in the usual pro of o f the unicity of adjunction (which applies to every parameter X ∈ X ).  1.4. Susp ended and triangul ated categories. W e re c a ll her e w ha t we need ab out tria ngulated catego ries. The reaso n why we ma ke a distinction b etw een susp ended ca tegories and triangula ted ones is b eca use a ll the commutativ e diag rams that we are interested in are just rela ted to the susp ension, a nd no t to the exactness of the functors in volv ed. So when w e need to prov e the commutativit y of those diagrams, we forget ab out the exactness of o ur functors, and just think o f them as susp ended functors, in the sens e descr ib ed below. Deﬁnition 1.4.1. A susp ended categor y is an additive categor y C together with an adjoint couple ( T , T − 1 ) fro m C to C which is an equiv alence of categor y (the unit and counit are isomo rphisms). R emark 1.4.2 . W e assume furthermore in all wha t fo llows that T T − 1 and T − 1 T are the ident it y of C and that the unit and counit are also the identit y . This assumption is not true in so me susp ended (triangulated) categor ies ar ising in s ta ble homotopy theory . Nevertheless, it simpliﬁes the exp os ition whic h is a lready suﬃcien tly tech- nical. When working in an example where this as sumption do es not hold, it is of course p oss ible to make the mo diﬁcations to get this ev en more general ca se. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 13 Betw een susp ended categor ies ( C , T C ) a nd ( D , T D ), we use susp ended functors: Deﬁnition 1.4.3 . A s uspe nded functor ( F, f ) from C to D is a functor F to gether with a n isomorphism o f functor s f : F T C → T D F . W e s o metimes forget ab o ut f in the no tation. Without the as sumption in Remark 1.4.2 , we would need another iso morphism f ′ : F T − 1 → T − 1 F a nd co mpatibility diagra ms analo gous to the ones in Lemma 1.2.6 . Then, we would hav e to carry those co mpa tibilities in our co nstructions. Again, this would no t b e a problem, just making things even more tedious. Suspe nded functors can b e comp osed in a n obvious w ay , a nd ( T , id T 2 ) a nd ( T − 1 , id I d ) a re susp ended endofunctors o f C that we call T and T − 1 for sho rt. Deﬁnition 1.4. 4. T o a s uspe nded functor F , one can asso ciate “shifted” ones, comp osing F by T or T − 1 several times on either s ides. The iso morphisms T i F T j ≃ T k F T l with i + j = k + l constructed using f , f − 1 , T − 1 T = I d and T T − 1 = I d all coincide, so whenever we use one, w e lab el it “ f ” without further mention. Deﬁnition 1. 4.5. The opp osite susp ended ca teg ory C o of a susp ended catego ry C is given the susp ension ( T − 1 C ) o . With this co nven tion, we can deal with contra v ariant susp ended functors in tw o diﬀerent wa ys (dep ending where we put the ”op”), and this yields es sentially the same thing, using the deﬁnition of shifted susp ended functors. Deﬁnition 1.4.6 . A morphism of susp ended functors h : ( F , f ) → ( G, g ) is a morphism o f functor s h : F → G s uch that the dia gram F T hT   f / / sus T F T h   GT g / / T G is commutativ e. Lemma 1.4 .7. The c omp osition of t wo morphisms of sus p ende d functors yields a morphism of s u sp ende d fun ctors. Pr o of. Straig ht forward.  A triangula ted catego ry is a sus pended catego ry with the choice of some e x act triangles, satisfying so me axioms. This can b e found in text bo o ks as [ 10 ](see also the nice introduction in [ 1 , Section 1]). W e include the enrich ed o ctahedron a x iom in the list of r e quired axioms as it is suitable to dea l with Witt groups, as explained in lo c . cit. Deﬁnition 1.4. 8. (see for example [ 7 , § 1.1 ]) Let ( F, f ) : C → D be a cov ariant (resp. co nt rav ariant) susp ended functor. W e say that ( F, f ) is δ -ex act ( δ = ± 1) if for a ny exac t triangle A u − → B v − → C w − → T A the tr ia ngle F A F u − → F B F v − → F C δf A ◦ F w − → T F A resp ectively F C F v − → F B F u − → F A δf C ◦ F T − 1 w − → T F C is exa c t. 14 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL R emark 1.4.9 . With this deﬁnition, T and T − 1 are ( − 1)-exac t functor s, b ecaus e of the s econd axiom of triangula ted catego ries, and the comp osition of exa ct functors m ultiplies their signs. Th us, if F is δ -exact, then T i F T j is ( − 1) i + j δ -exa c t. T o deﬁne morphisms b etw een exact functors F a nd G , the signs δ F and δ G of the functors have to b e taken int o account, so that the morphism of functors induces a mor phism b etw een the triangles obtained by applying F or G to a triangle and making the sign modiﬁca tions. Deﬁnition 1.4.1 0 . W e say that h : F → G is a mo r phism of exa ct functors if the diagram sus in Deﬁnition 1 .4.6 is δ F δ G commutativ e. On the other ha nd, we hav e the following lemma . Lemma 1. 4.11. L et h : F → G b e an isomorphism of susp ende d functors such that sus is ν -c ommutative. Assume F is δ -exact. Then G is δ ν -ex act. Pr o of. F o r any triang le A u − → B v − → C w − → T A the tr ia ngle GA Gu − → GB Gv − → F C δg A ◦ Gw − → T GA is eas ily shown to b e isomorphic to F A F u − → F B F v − → F C ν δ f A ◦ F w − → T F A  W e also need to deal with bifunctors from tw o susp ended categ ories to another one. These ar e just suspended functors in each v ariable, with a co mpatibilit y condition. Exa mples a re the in ternal Hom or the tensor pro duct in triangulated categorie s. Deﬁnition 1 .4.12. Let C 1 , C 2 and D b e susp ended ca tegories. A susp ended bi- functor fro m C 1 × C 2 to D is a triple ( B , b 1 , b 2 ) where B : C 1 × C 2 → D is a functor and tw o mor phisms of functors b 1 : B ( T ( − ) , ∗ ) → T B ( − , ∗ ) and b 2 : B ( − , T ( ∗ )) → T B ( − , ∗ ), such tha t the diagram B ( T A, T C ) b 1 ,A,T C   b 2 ,T A,C / / − 1 T B ( T A, C ) b 1 ,A,C   T B ( A, T C ) b 2 ,A,C / / T 2 B ( A, C ) anti-comm utes for every A and C . R emark 1.4.13 . As in Deﬁnition 1.4.4 , we hav e shifted versions o f b 1 (or b 2 ) which we will sometimes lab el “ b 1 ” (or “ b 2 ”) below. It is impor tant to describ e precisely in which order morphisms ar e applied when co mb ining b 1 and b 2 . Deﬁnition 1.4.1 4. A morphism of susp ended bifunctors from a suspe nded bi- functor ( B , b 1 , b 2 ) to a susp ended bifunctor ( B ′ , b ′ 1 , b ′ 2 ) is a mo rphism of functors f : B → B ′ such that the t wo diag rams B ( T A, C ) f T A,C   b 1 ,A,C / / T B ( A, C ) T f A,C   B ( A, T C ) b 2 ,A,C o o f A,T C   B ′ ( T A, C ) b ′ 1 ,A,C / / T B ′ ( A, C ) B ′ ( A, T C ) b ′ 2 ,A,C o o are comm utative for every A and C . TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 15 By co mpo sing with a usual susp ended functor to C 1 or C 2 or from D , we get other susp ended bifunctors (the veriﬁcation is easy). But, if we do that several times, using diﬀeren t functor s, the order in which the susp ended functors to C 1 or C 2 are used do es matter. F or example, as with usua l s uspe nded functors, it is poss ible to deﬁne s hifted versions by comp osing with the susp ensio ns in e ach ca teg ory as men tioned in Remark 1.4.13 . This can b e useful. Unfortunately , acco rding to the order in which we do this (if we mix functors to C 1 and to C 2 ), we don’t get the same iso morphism of functors, even though we get the same functors in the pair . One has to be careful abo ut that. 1.5. Susp ended adjunctions. As with usual functors, ther e is a no tion of a d- junction well suited for sus p ended functors. Deﬁnition 1.5.1. A suspended adjoint couple ( L, R ) is a adjoint couple in the usual sense in which L a nd R are susp ended functors and the unit and counit ar e morphisms o f susp ended functors. Deﬁnition 1.5 .2. When ( L, R ) is an a djoint couple of susp ended functors , using Lemma 1.2.5 we obtain shifted versions ( T i LT j , T − j RT − i ). Using Deﬁnition 1.4.4 , we obtain iso morphisms exchanging the T ’s. Applying Lemma 1.2.3 to them, we get an adjoint couple of susp ended bifunctors ( T i LT j , T − i RT − j ). The following prop o sition seems to b e well-kno wn. Prop ositio n 1.5.3 . L et ( L, R ) b e an adjoint c ouple fr om C to D (of usual fu n ctors) and let ( L, l ) b e a susp ende d functor. Then (1) ther e is a unique isomorphi sm of functors r : RT → T R that tu rns ( R, r ) into a su s p ende d functor and ( L, R ) into a susp ende d adjoint c ouple. (2) if furthermor e C and D ar e triangulate d and ( L , l ) is δ -exact, then ( R, r ) is also δ -exact (with the same δ ). Pr o of. Point 1 is a direct co rollar y of Point 3 of Theorem 1.2.8 , b y ta k ing L = L ′ , R = R ′ , F 1 = T C , G 1 = T − 1 C , F 2 = T D , G 2 = T − 1 D , g L = ( g ′ L ) − 1 = T − 1 l T − 1 . This gives f R = r . The commutativ e diagr ams F 1 and F 2 exactly tell us that the unit and counit ar e susp ended mo rphisms of functors with this c hoice of r . T o prov e Poin t 2, we hav e to show that the pair ( R, r ) is exact. Let A u − → B v − → C w − → T A be an exact triang le. W e w ant to prov e that the triangle RA u − → RB v − → R C r A ◦ Rw − → T RA is exa c t. W e ﬁrst complete RA u − → RB as a n exact triangle RA u − → R B v ′ − → C ′ w ′ − → T R A and we pr ove tha t this tr ia ngle is in fact isomor phic to the prev ious o ne. T o do so, one co mpletes the inco mplete mo rphism of triangles LRA LRu / /   LRB LRv / /   LC ′ f RA ◦ LRw / / h   T LRA   A u / / B v / / C w / / T A . Lo oking at the adjoint diagra m, we see that ad ( h ) : C ′ → RC is an is o morphism by the ﬁve lemma for triangulated c a tegories .  Theorem 1.5. 4. L emma 1.2.6 , L emma 1.2.7 and The or em 1.2. 8 hold when we r eplac e every functor by a s u sp ende d functor, every adjoint c ouple by a s u sp ende d adjoint c ouple and every morphism of functor by a morphism of susp ende d fu n ctors. 16 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Pr o of. The same pr o ofs hold, since they only re ly on o per ations and prop erties of functors a nd morphism of functor s, such as comp osition or co mmu tative diagr ams, that ex ist and b ehav e the same w ay in the suspended case.  W e now a dapt the notion of an adjoint co uple o f bifunctors to susp ended cate- gories. Deﬁnition 1.5.5. Let ( L, R ) be an ACB from C ′ to C with para meter in X , where C , C ′ and X are susp ended ca tegories. Assume moreover tha t ( L, l 1 , l 2 ) and ( R, r 1 , r 2 ) ar e susp ended bifunctor s. W e say that ( L, R ) is a sus pended adjoint couple of bifunctors if (1) (( L ( X , − ) , l 2 ) , ( R ( X, − ) , r 2 )) is a susp ended adjoint couple of functors for every par ameter X , (2) the follo wing diagra ms commute: T C η X,T C / / T η T X,C   R ( X , L ( X, T C )) “ r 1 ” / / T R ( T X , L ( X , T C )) T R ( T X,l 2 )   T R ( T X , L ( T X , C )) T R ( X,l 1 ) / / T R ( T X , T L ( X , C )) T L ( X , R ( T X , C )) l − 1 1 / / l − 1 2   L ( T X , R ( T X , C )) ǫ T X,C   L ( X , T R ( T X , C )) “ r 1 ” − 1 / / L ( X , R ( X , C )) ǫ X,C / / C R emark 1.5.6 . Note that (2) ensur es the co mpatibilit y of the susp ension functor on the pa rameter with the o ther ones . Lemma 1. 5.7. L et ( F, f ) : X ′ → X b e a su sp ende d functor, and ( L, R ) a susp ende d ACB with p ar ameter in X . Then ( L ( F ( ∗ ) , − ) , R ( F o ( ∗ ) , − )) is again a su sp ende d ACB in the obvious way. Pr o of. Left to the reade r .  The following prop o sition is an analog ue o f Prop ositio n 1.5.3 for susp ended bi- functors. Prop ositio n 1 .5.8. L et ( L, R ) b e an A CB such that ( L, l 1 , l 2 ) (r esp. ( R , r 1 , r 2 ) ) is a susp ende d bifunctor. Then (1) ther e ex ist unique r 1 , r 2 (r esp. l 1 , l 2 ) such that ( R, r 1 , r 2 ) (re sp. ( L, l 1 , l 2 ) ) is a susp ende d bifunctor and ( L , R ) is a susp ende d ACB, (2) if ( L ( X , − ) , l 2 ) is δ -exact for some obje ct X , then so is ( R ( X , − ) , r 2 ) . Pr o of. The existence and uniqueness of r 2 follows (for every par a meter X ) fro m Poin t 1 of Pro p o s ition 1.5.3 . Tha t r 1 is also natural in the ﬁrst v ariable follows from a larg e diagr am of co mm utative squa res o f type mf and gen . This yields in pa r ticular morphisms of sus pended functors η X and ǫ X for ev ery X . F or the existence and uniqueness of r 1 , we use Lemma 1 .3 .5 a pplied to H ′ = T , H = I d , J 1 = L ( ∗ , − ), K 1 = R ( ∗ , − ), J 2 = L ( T − 1 ( ∗ ) , − ), K 2 = R ( T − 1 ( ∗ ) , − ) and a = “ l 1 ”“ l 2 ” (the order is imp o rtant). W e obtain tw o diag rams which are easily seen to b e equiv ale n t to the ones requir e d for ( L, R ) to b e a susp ended A CB (Poin t 2 o f Deﬁnition 1.5.5 ). The anticomm utativity requir ed by Deﬁnition 1 .4.12 for r 2 and r 1 may b e prov ed using the anticomm utativity of l 1 and l 2 and the fac t that η X and ǫ T − 1 X are mo rphisms o f suspended functors. Poin t 2 is prov ed in the same wa y a s Poin t 2 o f Prop osition 1.5.3 .  TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 17 W e need a v ersion of Lemma 1.3 .5 for susp ended bifunctors. Lemma 1.5.9. L emma 1.3.5 holds when al l the functors, bifunctors and adjunc- tions b e c ome susp ende d ones. Pr o of. The pr o of of Le mma 1.3.5 works since it o nly in volv es c o mpo sitions and commutativ e diagrams that exist in the susp e nded ca se.  Finally , there is also a version o f Theor em 1.2.8 for (susp ended) bifunctors , which is our main too l for the applica tions. F or this r eason, we state it in full detail. Theorem 1.5.1 0. L et L , R , L ′ , R ′ , F 1 , G 1 , F 2 , G 2 b e (susp ende d) functors whose sour c es and t ar gets ar e sp e ciﬁe d by the diagr am (r e c al l the n otation of Deﬁnition 1.3.1 ), C 1 G 1   X L / / C 2 G 2   X R o o C ′ 1 F 1 O O L ′ / / C ′ 2 F 2 O O R ′ o o and let f L , f ′ L , g L , g ′ L , f R , f ′ R , g R and g ′ R b e morphisms of bifunctors (r esp. susp ende d bifunctors) whose s ou rc es and tar gets wil l b e as fol lows: LF 1 ( ∗ , − ) f L / / F 2 ( ∗ , L ′ ( − )) f ′ L o o L ′ G 1 ( ∗ , − ) g ′ L / / G 2 ( ∗ , L ( − )) g L o o F 1 ( ∗ , R ′ ( − )) f ′ R / / RF 2 ( ∗ , − ) f R o o G 1 ( ∗ , R ( − )) g R / / R ′ G 2 ( ∗ , − ) g ′ R o o L et us c onsider the fol lowing diagr ams, in which the maps and their dir e ctions wil l b e the only obvious ones in al l the c ases discusse d b elow. F 2 ( X, L ′ G 1 ( X, − )) L LF 1 ( X, G 1 ( X, − )) F 2 ( X, G 2 ( X, L ( − )) L o o O O G 1 ( C, − )   / / G 1 G 1 ( C, R L ( − )) R ′ L ′ G 1 ( C, − ) R ′ G 2 ( C, L ( − ) ) F 1 ( X, R ′ G 2 ( X, − )) R RF 2 ( X, G 2 ( X, − )) F 1 ( X, G 1 ( X, R ( − ))) R o o O O F 1 ( C, − )   / / F 1 F 1 ( C, R ′ L ′ ( − )) RLF 1 ( C, − ) RF 2 ( C, L ′ ( − )) L ′ L ′ G 2 ( X, F 2 ( X, L ′ ( − ))) o o L ′ G 1 ( X, F 1 ( X, − )) O O G 2 ( X, LF 1 ( X, − )) L ′ G 1 ( C, R ( − )) G 2 G 2 ( C, LR ( − ))   L ′ R ′ G 2 ( C, − ) / / G 2 ( C, − ) R ′ R ′ G 1 ( X, F 1 ( X, R ′ ( − ))) o o R ′ G 2 ( X, F 2 ( X, − )) O O G 1 ( X, RF 2 ( X, − )) LF 1 ( C, R ′ ( − )) F 2 F 2 ( C, L ′ R ′ ( − ))   LRF 2 ( C, − ) / / F 2 ( C, − ) Then 1. L et ( G i , F i ) , i = 1 , 2 b e ACBs (re sp. susp ende d ACBs). L et g L (r esp. f L ) b e given, then t her e is a un ique (susp ende d) f L (r esp. g L ) such that L and L ′ ar e c ommu tative for any X . L et g R (r esp. f R ) b e given, then ther e is 18 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL a unique ( s usp ende d) f R (r esp. g R ) such t hat R and R ′ ar e c ommutative for any X . 2. L et ( L , R ) and ( L ′ , R ′ ) b e adjoint c ouples (r esp. susp ende d adjoint c ouples). L et f L (r esp. f ′ R ) b e given, then ther e is a unique (susp ende d) f ′ R (r esp. f L ) such that F 1 and F 2 ar e c ommu tative. L et g ′ L (r esp. g R ) b e given, then ther e is a unique (susp ende d) g R (r esp. g ′ L ) su ch that G 1 and G 2 ar e c ommu tative. 3. Assuming ( G i , F i ) , i = 1 , 2 , ar e ACBs (r esp. susp ende d ACB s), ( L, R ) and ( L ′ , R ′ ) ar e (susp ende d) adjoint c ouples, and g L and g ′ L = g − 1 L ar e given (r esp. f R and f ′ R = f − 1 R ). By 1 and 2, we obtain f L and g R (r esp. g R and f L ). We then may c onst ru ct f R and f ′ R (r esp. g L and g ′ L ) which ar e inverse to e ach other. Pr o of. The pro of is the same a s for Theorem 1.2.8 , but using Lemma 1 .3.5 (or 1.5.9 ) instead of Lemma 1.2 .6 for Points 1 and 2.  R emark 1.5 .11 . W e didn’t state the a nalogues o f Poin ts 1’ a nd 2’ of Theorem 1.2.8 in this context b ecause we don’t need them. 2. Dualities W e now intro duce our main sub ject of in terest: duality . As b efore, we state everything for the susp ended (or triangula ted) case and the usual case in a unifor m wa y . 2.1. Categories with duali t y. Deﬁnition 2.1. 1 . A ca teg ory with dua lit y is a tr iple ( C , D,  ) wher e C is a ca teg ory with a n adjoint couple ( D , D o ,  ,  o ) fr om C to C o . When  is an isomor phism, D is an equiv alence of categor ies and we say that the duality is str ong . A susp ended (resp. triangulated) ca teg ory with duality is deﬁned in the sa me way , but ( D, d ) is a susp ended (resp. δ -exa ct) functor on a s usp ended (r esp. triangula ted) catego r y C , and the a djunction is susp ended. R emark 2.1.2 . Obser ve that the standard co ndition ( D o  o ) ◦ (  o D ) = i d D o is satisﬁed by the deﬁnition of an adjunction. The o nly diﬀerence b etw een our deﬁni- tion and Balmer’s deﬁnition [ 1 , Def. 2.2] is that we don’t require the is omorphism d : D T → T − 1 D to b e an eq uality in the susp ended or tria ng ulated case. As- suming that duality and susp ensio n strictly co mm ute is as bad as assuming that the internal Hom and the susp ension strictly commute, o r (by adjunction) that the tenso r pro duct and the susp ensio n strictly commute. This is deﬁnitely a to o strong co nditio n when checking s tr ict commutativit y of diagr ams in so me derived category . Dro pping all s igns in this setting when deﬁning these iso morphisms just by saying “take the ca no nical o nes” may even lea d to contradictions as the results of the Appendix show. When d = id , w e say that the duality is strict . Deﬁnition 2. 1.3. Let ( C , D ,  ) b e a triangulated catego ry w ith duality for which ( D , d ) is δ -exa c t. By Deﬁnition 1.5.2 we get a shifted adjoint co uple T ( D , D o ) = ( T D , T D o ,  ′ , (  ′ ) o ). W e deﬁne the susp ension o f ( C , D ,  ) as T ( C , D ,  ) = ( C , T D , − δ  ′ ). Note that ( T D , T d ) is ( − δ )-exa ct b ecause T is ( − 1)-exact. R emark 2.1.4 . This is the deﬁnition of [ 1 , Deﬁnition 2.8] adapted to cov er the non strict ca s e, and the next one generalizes [ 1 , Deﬁnition 2.13] to the no n strict case. Deﬁnition 2.1.5. F o r any tria ngulated categor y with strong duality ( C , D,  ), we deﬁne its i-th Witt g roup W i by W i ( C , D ,  ) := W ( T i ( C , D ,  )) (extending [ 1 , 2.4 and Deﬁnitions 2.1 2 and 2.13] in the obvious way). If D and  a re under sto o d, we sometimes a lso wr ite W i ( C ) or W i ( C , D )) for shor t. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 19 R emark 2 .1.6 . In co ncrete terms, this means that the c o ndition of lo c. c it. for an element in W 1 ( C ) represented by some φ to b e sy mmetric is that ( T dT − 1 ) ◦ φ = ( D o T φ ) ◦  wher eas in the s trict case the ( T dT − 1 ) may b e omitted. R emark 2.1.7 . T o deﬁne a Witt gr oup, the duality has to b e strong (that is  has to be an iso morphism), but this is not necessa ry to prov e all sorts of comm utative diagrams, s o there is no reaso n to as s ume it here in general. 2.2. Dualit y preserving functors and morphi sms. Deﬁnition 2.2.1. A duality preser ving functor from a (s us pended, tr iangulated) category with duality ( C 1 , D 1 ,  1 ) to another one ( C 2 , D 2 ,  2 ) is a pair { F , f } wher e F is a (susp ended, δ -exact) functor fro m C 1 to C 2 , and f : F D o 1 → D o 2 F o is a morphism (in C 2 ) o f (susp ended, δ -exact) functors where f and f o are mates as in Lemma 1.2.6 (resp. Theorem 1.5 .4 ) when setting J 1 = D 1 , K 1 = D o 1 , J 2 = D 2 , K 2 = D o 2 , H = F o , H ′ = F , a = f o and b = f . In other words, the diagr am H (equiv alent to H ′ ) o f Lemma 1.2.6 must commute: F  2 F   F  1 / / P F D o 1 D 1 f D 1   D o 2 D 2 F D o 2 f o / / D o 2 F o D 1 W e so metimes denote the functor simply by { F } if f is understo o d. When f is an isomorphism, we say the functor is strong duality pr eserving. When the dualities are b oth strict and strong and the functor is strongly du- ality preser v ing, then this co incides with the usual deﬁnition (see for example [ 6 , Deﬁnition 2 .6] wher e Point 2 corres po nds to the fact that f is a morphism o sus- pended functors in o ur deﬁnition and is o nly used in the susp ended ca se). Duality preserving functor s ar e comp osed in the o bvious wa y by s etting { F ′ , f } { F , f } = { F ′ F, f ′ F o ◦ F ′ f } . Deﬁnition 2.2.2. A morphism of (susp ended, exact) duality preser ving functors from { F , f } to { F ′ , f ′ } (with s ame so ur ce and tar get) is a morphism o f the under- lying (susp ended, exact) functors ρ : F → F ′ such that the diagram F D o 1 ρD o 1   f / / M D o 2 F o D o 2 ρ o   F ′ D o 1 f ′ / / D o 2 ( F ′ ) o commutes. W e say that such a morphism is strong if the underlying mo rphism of functors is an isomorphism. Comp osing tw o mor phisms betw een dualit y pres erving functors obviously gives another one, and c omp osition pr eserves b eing strong. The pro ofs o f following tw o prop os itions are stra ightforw ard (see a lso [ 6 , Theo rem 2.7] for a pr o of of the ﬁrst one in the strict ca se). Prop ositio n 2. 2 .3. A 1 -exact str ong duality pr eserving functor { F , f } b etwe en triangulate d c ate gories with str ong dualities induc es a morphism on Witt gr oups by sending an element r epr esente d by a form ψ : A → D 1 ( A ) to the class of the form f A ◦ F ( ψ ) . Prop ositio n 2.2.4 . A duality pr eserving isomorphism b et we en 1 -exact str ong dual- ity pr eserving functors ensu r es that they induc e t he same morphism on Witt gr oups. 20 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Deﬁnition 2.2 .5. (1) Let ( C , D ,  ) b e a tria ngulated category with duality and let A b e a full tr iangulated sub categ o ry o f C preser ved by D . Then, ( A , D | A ,  | A ) is tr ivially a tria ngulated category with duality , and w e say that the duality of C r estricts to A . (2) Let { F , f } : ( C 1 , D 1 ,  1 ) → ( C 2 , D 2 ,  2 ) b e a duality pr eserving functor b e- t ween triangulated ca tegories with dualities as in Deﬁnition 2.2.1 . Assume that there are full tria ngulated s ubca tegories A i ⊂ C i , i = 1 , 2 such that D i restricts to A i and such that F | A 1 factors through A 2 . Then we say that the dua lit y pre s erving pair { F , f } res tr icts to the sub catego ries A 1 and A 2 . (3) Let ρ b e a morphism betw een tw o such restricting functor s { F , f } a nd { F ′ , f ′ } , as in Deﬁnition 2.2.2 , then the restr iction of ρ automatically de- ﬁnes a morphism of exact duality preserv ing functor s b etw e en the res tricted functors. Lemma 2.2.6. L et { F , f } : ( C 1 , D 1 ,  1 ) → ( C 2 , D 2 ,  2 ) b e a duality pr eserving functor b etwe en triangulate d c ate gories with duality t hat r estricts to the sub c ate- gories A 1 and A 2 . (1) Assume that the re stricte d dualities on A 1 and A 2 ar e st ro ng and that f | F ( A o 1 ) is an isomorphism. Then the r estriction of the duality pr eserving p air { F , f } t o A 1 and A 2 is a a st r ongly duality pr eserving functor b etwe en triangulate d c ate gories with st r ong duality and ther efor e induc es a morphism W( A 1 , D 1 | A 1 ,  1 | A 1 ) → W( A 2 , D 2 | A 2 ,  2 | A 2 ) on Witt gr oups. (2) If a morphism ρ b etwe en two such duality pr eserving functors is str ong when r estricte d to A 1 , then they induc e t he same morphisms on Witt gr oups. Pr o of. One ha s to check that certa in diagrams in A 2 are commutativ e. This follows as they a re a lready co mm utative in C 2 by as s umption. No w Prop ositio ns 2.2.3 and 2.2.4 prove the claims.  3. Consequences of the closed mono idal str ucture W e now reca ll a few notions on tenso r pro ducts and internal Hom functors (de- noted by [ − , ∗ ]) a nd pr ov e very basic facts r elated to the susp ensio n. A catego r y satisfying the axioms of this section deserves to be calle d a “ (susp ended or trian- gulated) symmetr ic monoida l clo sed categor y”. Then we prov e that the dualities deﬁned us ing the internal Hom on such a ca teg ory are naturally equipp e d with the necessary data to de ﬁne (triangula ted) catego ries with dualities. 3.1. T ensor p ro duct and in ternal Ho m . Le t ( C , ⊗ ) b e a symmetric mo noidal category (see [ 9 , Chapter VI I]) with an internal Hom [ − , − ] adjoint to the tensor pro duct. Mo r e precisely , we a ssume that ( − ⊗ ∗ , [ ∗ , − ]) is an ACB. W e denote by s = s − 1 the symmetry iso morphism and call this da tum a “ symmetric monoidal closed ca tegory” . When talking ab out a “susp ended symmetric monoida l clos ed categor y”, we assume that we hav e a susp ended bifunctor ( − ⊗ ∗ , tp 1 , tp 2 ) (see Deﬁnition 1.4.12 ) such that the diagram ( T A ⊗ B ) ⊗ C / / tp 1 ⊗ id   assoc T A ⊗ ( B ⊗ C ) tp 1   T ( A ⊗ B ) ⊗ C tp 1   T (( A ⊗ B ) ⊗ C ) / / T ( A ⊗ ( B ⊗ C )) TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 21 commutes, as w ell a s the tw o similar ones in which the s usp ension starts o n o ne of the o ther v ar iables. W e also as sume that the dia gram T ( − ) ⊗ ∗ tp 1   s / / s ∗ ⊗ T ( − ) tp 2   T ( − ⊗ ∗ ) T s / / T ( ∗ ⊗ − ) commutes. By Pro po sition 1.5.8 , we get morphisms th 1 : [ T − 1 ( ∗ ) , − ] → T [ ∗ , − ] th 2 : [ ∗ , T ( − )] → T [ ∗ , − ] that ma ke ([ ∗ , − ] , th 1 , th 2 ) a susp ended bifunctor and ( − ⊗ ∗ , [ ∗ , − ]) a susp ended A CB (Deﬁnition 1.5.5 ). Using s , we obtain a new sus p ended ACB ( ∗ ⊗ − , [ ∗ , − ]) from the previous one. If C is triangula ted, we furthermore assume that ⊗ is e x act in b oth v a r iables (by symmetry it suﬃces to check this for o ne of them). By Pro p o s ition 1.5.8 , [ ∗ , − ] is susp ended in b oth v ariables and automa tically exact in the second v aria ble. W e assume furthermor e that it is exact in the ﬁrst v a r iable, a nd say that we have a “triangula ted clo sed symmetr ic monoidal category” . The morphisms ev l A,K : [ A, K ] ⊗ A → K co ev l A,K : K → [ A, K ⊗ A ] resp ectively ev r A,K : A ⊗ [ A, K ] → K co ev r A,K : K → [ A, A ⊗ K ] induced by the co unit a nd the unit of the (sus p ended) ACB ( − ⊗ ∗ , [ ∗ , − ]) (resp. ( ∗ ⊗ − , [ ∗ , − ])) are called the left (r esp. right) ev aluation and co ev a luation. Lemma 3.1.1 . The fol lowing diagr ams ar e c ommutative. ( T [ A, K ]) ⊗ A tp 1 / / 1 T ([ A, K ] ⊗ A ) T ev l A,K   [ A, T K ] ⊗ A th 2 ⊗ id O O ev l A,T K / / T K A ⊗ ( T [ A, K ]) tp 2 / / 2 T ( A ⊗ [ A, K ]) T ev r A,K   A ⊗ [ A, T K ] id ⊗ th 2 O O ev r A,T K / / T K T K co ev l A,T K / / T co ev l A,K   3 [ A, T K ⊗ A ] tp 1   T [ A, K ⊗ A ] th − 1 2 / / [ A, T ( A ⊗ K )] T K co ev r A,T K / / T co ev r A,K   4 [ A, A ⊗ T K ] tp 2   T [ A, A ⊗ K ] th − 1 2 / / [ A, T ( K ⊗ A )] ( T − 1 [ A, K ]) ⊗ T A T − 1 th 1 ,T A,K ⊗ id   tp 2 / / 5 T ( T − 1 [ A, K ] ⊗ A ) tp − 1 1 / / [ A, K ] ⊗ A ev l A,K   [ T A, K ] ⊗ T A ev l T A,K / / K ( T A ⊗ T − 1 [ A, K ]) id ⊗ T − 1 th 1 ,T A,K   tp 1 / / 6 T ( A ⊗ T − 1 [ A, K ]) tp − 1 2 / / A ⊗ [ A, K ] ev r A,K   T A ⊗ [ T A, K ] ev r T A,K / / K K co ev l T A,K / / co ev l A,K   7 [ T A, K ⊗ T A ] T − 1 th − 1 1 ,T A,K ⊗ T A   [ A, K ⊗ A ] T − 1 th − 1 2 / / T − 1 [ A, T ( K ⊗ A )] tp − 1 2 / / T − 1 [ A, K ⊗ T A ] 22 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL K co ev r T A,K / / co ev r A,K   8 [ T A, T A ⊗ K ] T − 1 th − 1 1 ,T A,T A ⊗ K   [ A, A ⊗ K ] T − 1 th − 1 2 / / T − 1 [ A, T ( A ⊗ K )] tp − 1 1 / / T − 1 [ A, T A ⊗ K ] Pr o of. This is a s traightforw ard conseq uence of Point 1 of Deﬁnition 1.5.5 for the ﬁrst four diagrams and of Poin t 2 of Deﬁnition 1 .5 .5 for the other four.  3.2. Bidual isomo rphi sm. W e still assume that ( C , ⊗ ) is a monoida l ca tegory with an internal Hom as in the previous section. W e now show that the func- tor D K = [ − , K ] na turally deﬁnes a duality on the catego r y C , a nd that in the susp ended case that the dualities D T K and T D K are naturally iso morphic. T o form the adjo int couple ( D K , D o K ,  K ,  o K ), we deﬁne the bidual mor phism of functors  K : I d → D o K D K as the imag e of the right ev aluatio n b y the a djunction ( − ⊗ ∗ , [ ∗ , − ]) isomor phism Hom( A ⊗ [ A, K ] , K ) ∼ / / Hom( A, [[ A, K ] , K ]) . It is functorial in A and deﬁnes a morphism of functors from I d to D o K D K . Note that its deﬁnition uses the adjunction ( − ⊗ ∗ , [ ∗ , − ]) a nd the right ev alua tion, which is not the counit of this a djunction but of the one obtained from it by using s ; so the fact that the monoidal categ ory is symmetric is es sential, here . One cannot pro ceed with only one of these adjunctions. In the susp ended case, D K bec omes a susp ended functor via T − 1 th − 1 1 , − ,K T : D K T → T − 1 D K . Prop ositio n 3.2.1. In the su sp ende d (or triangulate d) c ase,  K is a morphism of susp ende d (or exact) functors. Pr o of. First note that in the ex act case, wha tever the sign of D K is, D o K D K is 1-exact, so there is no sign inv olved in the diagram T A T  K    K T / / [[ T A, K ] , K ] T − 1 th 1 ,T A,K   T [[ A, K ] , K ] th − 1 1 , [ A,K ] ,K / / [ T − 1 [ A, K ] , K ] that we have to chec k (see Deﬁnitions 1.4.10 a nd 1 .4 .6 ). It is obta ined by (sus- pended) adjunction from Diagram 6 in Lemma 3 .1.1  Deﬁnition 3. 2.2. W e say that K is a dualizing ob ject when  K is an isomorphism of (susp ended) functors. Prop ositio n 3.2.3. The functors [ − , ∗ ] : C × C o → C o and [ − , ∗ ] o : C o × C → C form a ( s u sp ende d) ACB with unit  and c ounit  o . Pr o of. W e ﬁr s t hav e to prove that ( D K , D o K ,  K ,  o K ) is an adjoint couple in the usual sense. W e already know that  K is a susp ended morphism. Consider the following dia gram, in which all vertical maps ar e isomor phisms. W e use the no tation f ♯ : Ho m( F ′ , G ) → Hom( F , G ) and f ♯ : Ho m( G, F ) → Hom( G, F ′ ) for the ma ps induced by f : F → F ′ . The unlab eled morphisms are just a djunction bijections , TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 23 and we set  A,K := (  K ) A . Hom([ A, K ] , [ A, K ]) Hom([ A, K ] , [[[ A, K ] , K ] , K ]) ([  A,K ,I d K ]) ♯ o o Hom([ A, K ] ⊗ A, K ) O O Hom([ A, K ] ⊗ [[ A, K ] , K ] , K ) ( I d [ A,K ] ⊗  A,K ) ♯ o o O O Hom( A ⊗ [ A, K ] , K ) s ♯ A, [ A,K ] O O   Hom([[ A, K ] , K ] ⊗ [ A, K ] , K ) (  A,K ⊗ I d [ A,K ] ) ♯ o o s ♯ [[ A,K ] ,K ] , [ A,K ] O O   Hom( A, [[ A, K ] , K ]) Hom([[ A, K ] , K ] , [[ A, K ] , K ]) (  A,K ) ♯ o o The dia gram commutes by functoriality of s and the adjunction bijections. Now I d [[ A,K ] ,K ] in the lower right set is s ent to  [ A,K ] ,K in the upp er right set, which is in turn sent to [  A,K , K ] ◦  [ A,K ] ,K in the upp er left set. But I d [[ A,K ] ,K ] is also sent to  A,K in the lower left set, which is sent to I d [ A,K ] in the upp er left set by deﬁnition o f  A,K . This prov es the tw o required formulas (see Deﬁnition 1.2.1 ) for the comp osition of the unit and the counit in the adjoint couple (which are identical in this cas e ). W e leav e to the reader the easy fact that  A,K is a gener alized tra nsformation (in K ) (just using that the unit o f the adjunction ( − ⊗ ∗ , [ ∗ , − ]) is o ne). The adjoint couple is then a susp ended a djoint couple by Prop ositio n 3.2.1 . This proves Point 1 of Deﬁnition 1 .5.5 . Poin t 2 is pr ov ed using diagrams 6 and 7 .  Corollary 3.2.4. The triple ( C , D K ,  K ) is a (susp ende d) c ate gory with duality (se e Deﬁnition 2.1.1 ). When C is t riangulate d close d, D K is ex act, and ( C , D K ,  K ) is a triangulate d c ate gory with duality which we often denote by C K for short. When K is a dualizing obje ct, the duality is stro ng. Prop ositio n 3.2 . 5. The funct ors D K and D T K ar e exact. The isomorphism th 2 : [ − , T ( ∗ )] → T [ − , ∗ ] deﬁnes a susp ende d duality pr eserving functor { I d C , th 2 , − ,K } fr om C T K to T ( C , D K ,  K ) . This functor is an isomorphism of t riangulate d c ate- gories with duality and ther efor e induc es an isomorphism on Witt gr oups. Pr o of. W e know by 1.4.9 that T D K is ( − 1)-ex act. Diagr am sus should there- fore b e anti-comm utative (see Deﬁnition 1.4.10 ). It follows from the fact that ([ − , ∗ ] , th 1 , th 2 ) is a s usp e nded bifunctor. A squar e o btained by adjunction from 2 in Lemma 3.1 .1 then implies that th 2 deﬁnes a dua lity preserving functor (see Deﬁnition 2.2.1 ).  W e conclude this section by a trivial lemma for future re ference. Lemma 3. 2 .6. L et ι : K → M b e a morphism. Then I ι = { I d, ˜ ι } , wher e ˜ ι : D K → D M is induc e d by ι , is a duality pr eserving functor. This r esp e cts c omp osition: if κ : M → N is another morphism, t hen I κι = I κ I ι . L et ι b e an isomorphism, then if K is dualizing and D K δ -exact, the same is tru e for M and D M , and I ι induc es an isomorphism on Witt gr oups denote d by I W ι . 4. Functors between closed monoidal ca tegories Assume from now on that all categor ies C (maybe with an index) are sy mmetr ic monoidal and equipp ed with an internal Hom, satisfying the set-up o f the previous section. W e say that a functor f ∗ : C 1 → C 2 is a sy mmetric monoidal (susp ended, exact in b oth v ariables) functor when it co mes equipp ed with an isomor phism of (susp ended) bifunctors α : f ∗ ( − ) ⊗ f ∗ ( ∗ ) → f ∗ ( − ⊗ ∗ ) a nd, when a unit 1 for the tensor pr o duct is co ns idered, a n is omorphism f ∗ ( 1 ) ≃ 1 making the standard 24 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL α f ∗ ( − ) ⊗ f ∗ ( ∗ ) → f ∗ ( − ⊗ ∗ ) ( A f ) Section 4 β f ∗ [ − , ∗ ] → [ f ∗ ( − ) , f ∗ ( ∗ )] ( A f ) Prop 4.1.1 λ f ∗ ( − ) ⊗ f ∗ ( ∗ ) → f ∗ ( − ⊗ ∗ ) ( A f ) ( B f ) Prop. 4.2 .1 µ f ∗ [ − , ∗ ] → [ f ∗ ( − ) , f ∗ ( ∗ )] ( A f ) ( B f ) Prop. 4.2 .2 π f ∗ ( − ) ⊗ ∗ → f ∗ ( − ⊗ f ∗ ( ∗ )) ( A f ) ( B f ) Prop. 4.2 .5 κ [ − , f ∗ ( ∗ )] → f ∗ [ f ∗ ( − ) , ∗ ] ( A f ) ( B f ) Prop. 4.2 .5 ζ f ∗ [ ∗ , f ! ( − )] → [ f ∗ ( ∗ ) , − ] ( A f ) ( B f ) ( C f ) Thm. 4.2.9 θ f ! ( − ⊗ ∗ ) → f ! ( − ) ⊗ f ∗ ( ∗ ) ( A f ) ( B f ) ( C f ) ( D f ) Prop. 4.3 .1 ν f ! [ ∗ , − ] → [ f ∗ ( ∗ ) , f ! ( ∗ )] ( A f ) ( B f ) ( C f ) ( D f ) Prop. 4.3 .1 τ K,M D K ⊗ D M → D K ⊗ M Def. 4.4.1 ξ ¯ g ∗ f ∗ → ¯ f ∗ g ∗ ( A f ) ( A ¯ f ) ( A g ) ( A ¯ g ) ( B g ) ( B ¯ g ) ( C g ) ( C ¯ g ) Section 5.2 ε f ∗ g ∗ → ¯ g ∗ ¯ f ∗ ( A f ) ( A ¯ f ) ( A g ) ( A ¯ g ) ( B g ) ( B ¯ g ) ( C g ) ( C ¯ g ) Section 5.2 γ ¯ f ∗ ¯ g ! → g ! f ∗ ( A f ) ( A ¯ f ) ( A g ) ( A ¯ g ) ( B g ) ( B ¯ g ) ( C g ) ( C ¯ g ) ( E f ,g ) Section 5.2 T able 1. Morphisms of functors diagrams commutativ e (see [ 9 , section XI.2] for the deta ils where s uch functors a re called st r ongly monoidal ) W e will consider the following assumptions (used in the deﬁnition o f some mor- phisms of functors): (A f ) The functor f ∗ : C 1 → C 2 is symmetric mo noidal (sus pended, exact in b oth v ariables). (B f ) W e have a functor f ∗ : C 2 → C 1 that ﬁts int o an adjoint couple ( f ∗ , f ∗ , η ∗ ∗ , ǫ ∗ ∗ ). (C f ) W e have a functor f ! : C 1 → C 2 that ﬁts into an adjoint c o uple ( f ∗ , f ! , η ! ∗ , ǫ ! ∗ ). (D f ) The mo r phism π : f ∗ ( − ) ⊗ ∗ → f ∗ ( − ⊗ f ∗ ( ∗ )) from Pr op osition 4.2.5 is an isomorphism (the “pr o jectio n for mula” isomo rphism). (E f ,g ) The morphism of functor s ε of Sectio n 5.2 is an isomorphism. In the following, we will deﬁne sev eral na tural tra nsformations and es tablish commutativ e diagra ms inv olving them. Since ther e ar e so many of them, for the conv enience of the re a der, we include a T able 1 , which displays (from the left to the right): the name of the natur al tra nsformation, its so urce and targ et functor, the necessary assumptions to deﬁne the natura l trans formation and wher e it is deﬁned. 4.1. The monoidal functor f ∗ . In this s ection, we obtain duality preser ving functors a nd mor phisms related to a monoidal functor f ∗ . Prop ositio n 4.1 . 1. Under Assumption ( A f ) , ther e is a un ique morphism β : f ∗ [ ∗ , − ] → [ f ∗ ( ∗ ) , f ∗ ( − )] of (susp ende d) bifunctors su ch that the diagr ams [ f ∗ X , f ∗ ( − ⊗ X )] 9 f ∗ [ X , − ⊗ X ] β o o [ f ∗ X , f ∗ ( − ) ⊗ f ∗ X ] α O O f ∗ ( − ) o o O O f ∗ ( − ) 10 [ f ∗ X , f ∗ ( − )] ⊗ f ∗ X o o f ∗ ([ X, − ] ⊗ X ) O O f ∗ [ X , − ] ⊗ f ∗ X α o o β O O TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 25 c ommute for every X ∈ C 1 . F or any A and B , β is given by the c omp osition f ∗ [ A, B ] co ev l / / [ f ∗ A, f ∗ [ A, B ] ⊗ f ∗ A ] α / / [ f ∗ A, f ∗ ([ A, B ] ⊗ A )] ev l / / [ f ∗ A, f ∗ B ] . Pr o of. Apply Lemma 1.3.5 to H = H ′ = f ∗ , J 1 = − ⊗ ∗ , K 1 = [ ∗ , − ], J 2 = ( − ⊗ f ∗ ( ∗ )), K 2 = [ f ∗ ( ∗ ) , − ] and a = α (recall that in an ACB, the v ar iable deno ted ∗ is the parameter, as explained in Deﬁnition 1.3.1 ). Then deﬁne β = b .  Theorem 4.1.2. (existenc e of the pul l-b ack) Under Assumption ( A f ) , the mor- phism β K : f ∗ D o K → D o f ∗ K ( f ∗ ) o deﬁnes a duality pr eserving functor { f ∗ , β K } of (s usp ende d, triangulate d) c ate gories with duality fr om ( C 1 ) K to ( C 2 ) f ∗ K . Corollary 4.1. 3. (Pul l-b ack for Witt gr oups) When the dualities and the duality pr eserving fun ctor ar e stro ng (i.e.  K ,  f ∗ K and β K ar e isomorphisms), { f ∗ , β K } induc es a morphism of Witt gr oups f ∗ W : W ∗ ( C 1 , K ) → W ∗ ( C 2 , f ∗ K ) by Pr op osition 2.2.3 . Pr o of of The or em 4.1.2 . W e need to show that the diagram f ∗  K f ∗   f ∗  K / / f ∗ D o K D K β K D K   D o f ∗ K D f ∗ K f ∗ D o f ∗ K β o K / / D o f ∗ K ( f ∗ ) o D K commutes. This follows from the commutativ e diagr a m (for all A in C 1 ) f ∗ A co ev l   co ev l / / 9 f ∗ [[ A, K ] , A ⊗ [ A, K ]] β   ev r / / mf f ∗ [[ A, K ] , K ] β   [ f ∗ [ A, K ] , f ∗ A ⊗ f ∗ [ A, K ]] α / / 10 ′ β   [ f ∗ [ A, K ] , f ∗ ( A ⊗ [ A, K ])] / / [ f ∗ [ A, K ] , f ∗ K ] [ f ∗ [ A, K ] , f ∗ A ⊗ [ f ∗ A, f ∗ K ]] ev r 2 2 [[ f ∗ A, f ∗ K ] , f ∗ K ] β O O [[ f ∗ A, f ∗ K ] , f ∗ A ⊗ [ f ∗ A, f ∗ K ]] β O O ev r 2 2 mf where 10 ′ is obtained from 10 by using the compatibility of α with s . By functoriality of co ev, the co unit of the adjunction o f bifunctor s ( − ⊗ ∗ , [ − , ∗ ]), we can co mplete the left vertical part of the diagra m a s a commutativ e sq uare by a mor phis m co e v l from f ∗ A to the bottom ent ry . The outer part of this bigg er diagram is therefor e the one we are lo oking for . In the susp ended (or tria ngulated) case, β is a morphism of susp ended functors b y P rop osition 4.1.1 .  4.2. Adjunctions ( f ∗ , f ∗ ) and ( f ∗ , f ! ) and the pro jection morphism . In this section, we will assume that we have a djoint co uples ( f ∗ , f ∗ ) a nd ( f ∗ , f ! ), a nd obtain the pro jection mor phism f ∗ ( − ) ⊗ ∗ → f ∗ ( − ⊗ f ∗ ( ∗ )) a s well as s everal related commutativ e diag rams. W e also construct the mor phis m ζ which turns f ∗ int o a duality preserving functor (Theorem 4.2.9 ). Prop ositio n 4.2.1. Assume ( A f ) and ( B f ) . Then ther e is a un ique morphism of (susp ende d) bifunctors λ : f ∗ ( − ) ⊗ f ∗ ( − ) → f ∗ ( − ⊗ − ) 26 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL such that the diagr ams − ⊗ X   / / 11 f ∗ f ∗ ( − ) ⊗ f ∗ f ∗ X λ   f ∗ f ∗ ( − ⊗ X ) α − 1 / / f ∗ ( f ∗ ( − ) ⊗ f ∗ X ) f ∗ ( f ∗ ( − ) ⊗ f ∗ X ) λ   α − 1 / / 12 f ∗ f ∗ ( − ) ⊗ f ∗ f ∗ X   f ∗ f ∗ ( − ⊗ X ) / / − ⊗ X c ommute for every X ∈ C 1 . F or any A and B , λ is given by t he c omp osition f ∗ A ⊗ f ∗ B η ∗ ∗ / / f ∗ f ∗ ( f ∗ A ⊗ f ∗ B ) α − 1 / / f ∗ ( f ∗ f ∗ A ⊗ f ∗ f ∗ B ) ǫ ∗ ∗ ⊗ ǫ ∗ ∗ / / f ∗ ( A ⊗ B ) . Pr o of. Apply Lemma 1.2.6 (resp. Theorem 1.5.4 ) to H = H ′ = ( − ⊗ − ), J 1 = f ∗ × f ∗ K 1 = f ∗ × f ∗ , J 2 = f ∗ , K 2 = f ∗ , and a = α − 1 to obtain a unique λ = b satisfying 11 a nd 12 and given by the ab ov e c o mpo sition. In the sus p ended case, λ is a morphism of susp ended bifunctors b eca use it is given by a co mpo sition of morphisms o f susp ended functors a nd bifunctor s.  Prop ositio n 4.2 . 2. Assu me ( A f ) and ( B f ) . Then, ther e is a un ique morphism µ : f ∗ [ ∗ , − ] → [ f ∗ ( ∗ ) , f ∗ ( − )] of (susp ende d) bifunctors su ch that the diagr ams [ f ∗ X , f ∗ ( − ⊗ X )] 13 f ∗ [ X , − ⊗ X ] µ o o [ f ∗ X , f ∗ ( − ) ⊗ f ∗ X ] λ O O f ∗ ( − ) o o O O f ∗ ( − ) 14 [ f ∗ X , f ∗ ( − )] ⊗ f ∗ X o o f ∗ ([ X, − ] ⊗ X ) O O f ∗ [ X , − ] ⊗ f ∗ X λ o o µ ⊗ id O O c ommute for every X ∈ C 1 . F or any A and B , µ is given by the c omp osition f ∗ [ A, B ] co ev l / / [ f ∗ A, f ∗ [ A, B ] ⊗ f ∗ A ] λ / / [ f ∗ A, f ∗ ([ A, B ] ⊗ A )] ev l / / [ f ∗ A, f ∗ B ] . Pr o of. Apply Lemma 1.3.5 (resp. Lemma 1.5.9 ) to J 1 = ( − ⊗ ∗ ), K 1 = [ ∗ , − ], J 2 = ( − ⊗ f ∗ ( ∗ )), K 2 = [ f ∗ ( ∗ ) , − ], H = H ′ = f ∗ and a = λ .  Lemma 4.2.3 . The diagr am f ∗ A ⊗ f ∗ B s ( f ∗ ⊗ f ∗ )   λ / / 15 f ∗ ( A ⊗ B ) f ∗ s   f ∗ B ⊗ f ∗ A λ / / f ∗ ( B ⊗ A ) is c ommut ative. Pr o of. Apply Lemma 1.2.7 to the cube C 2 × C 2 ⊗ / / x   ? ? ? ? ? ? C 2 I d   ? ? ? ? ? ? C 2 × C 2 c ; C           ⊗ / / C 2 C 1 × C 1 f ∗ × f ∗ O O x   ? ? ? ? ? ? C 1 × C 1 id * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * f ∗ × f ∗ O O ⊗ / / C 1 α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? f ∗ O O C 2 × C 2 ⊗ / / C 2 I d   ? ? ? ? ? ? C 2 C 1 × C 1 f ∗ × f ∗ O O x   ? ? ? ? ? ? / / C 1 α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? f ∗ O O I d   ? ? ? ? ? ? C 1 × C 1 c ; C           ⊗ / / C 1 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * f ∗ O O where x is the functor exchanging the comp onents. Note that the morphism of functors f ∗ ( − ⊗ ∗ ) → f ∗ ( − ) ⊗ f ∗ ( ∗ ) obtained o n the fro nt a nd back squar es indeed coincides with λ b y constructio n.  TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 27 Prop ositio n 4.2.4. In the susp ende d c ase, un der As s u mption ( B f ) t her e is a unique way of turn ing f ∗ into a susp ende d funct or su ch that ( f ∗ , f ∗ ) is a susp ende d adjoint c ouple. If further Assumption ( C f ) holds, then ther e is a unique way of turning f ! into a susp ende d fun ctor su ch that ( f ∗ , f ! ) is a su sp ende d adjoint c ou ple. Pr o of. Both results follow directly from P r op osition 1.5.3 .  Prop ositio n 4.2.5. Under assumptions ( A f ) and ( B f ) , ther e is a unique mor- phism π : f ∗ ( − ) ⊗ ∗ → f ∗ ( − ⊗ f ∗ ( ∗ )) and a unique isomorphism κ : [ ∗ , f ∗ ( − )] → f ∗ [ f ∗ ( ∗ ) , − ] of (susp ende d) bifunctors su ch that the diagr ams [ X , f ∗ ( − ⊗ f ∗ X )] 16 f ∗ [ f ∗ X , − ⊗ f ∗ X ] κ − 1 o o [ X , f ∗ ( − ) ⊗ X ] π O O f ∗ o o O O f ∗ 17 [ X , f ∗ ( − )] ⊗ X o o f ∗ ([ f ∗ X , − ] ⊗ f ∗ X ) O O f ∗ [ f ∗ X , − ] ⊗ X π o o κ − 1 O O − ⊗ X   / / 18 f ∗ f ∗ ( − ) ⊗ X π   f ∗ f ∗ ( − ⊗ X ) α − 1 / / f ∗ ( f ∗ ( − ) ⊗ f ∗ X ) f ∗ ( f ∗ ( − ) ⊗ X ) π   α − 1 / / 19 f ∗ f ∗ ( − ) ⊗ f ∗ X   f ∗ f ∗ ( − ⊗ f ∗ X ) / / − ⊗ f ∗ X [ X , − ]   / / 20 [ X , f ∗ f ∗ ( − )] κ   f ∗ f ∗ [ X , − ] β / / f ∗ [ f ∗ X , f ∗ ( − )] f ∗ [ X , f ∗ ( − )] κ   β / / 21 [ f ∗ X , f ∗ f ∗ ( − )]   f ∗ f ∗ [ f ∗ X , − ] / / [ f ∗ X , − ] c ommute for any X ∈ C 1 . F or any A and B , π is given by the c omp osition f ∗ A ⊗ B η ∗ ∗ / / f ∗ f ∗ ( f ∗ A ⊗ B ) α / / f ∗ ( f ∗ f ∗ A ⊗ f ∗ B ) ǫ ∗ ∗ / / f ∗ ( A ⊗ f ∗ B ) κ by [ A, f ∗ B ] η ∗ ∗ / / f ∗ f ∗ [ A, f ∗ B ] β / / f ∗ [ f ∗ A, f ∗ f ∗ B ] ǫ ∗ ∗ / / f ∗ [ f ∗ A, B ] and κ − 1 by f ∗ [ f ∗ A, B ] co ev l / / [ A, f ∗ [ f ∗ A, B ] ⊗ A ] π / / [ A, f ∗ ([ f ∗ A, B ] ⊗ f ∗ A )] ev l / / [ A, f ∗ B ] . Pr o of. Apply Poin t 3 o f Theo rem 1.5.10 with L = L ′ = f ∗ , R = R ′ = f ∗ , G 1 = − ⊗ ∗ , G 2 = − ⊗ f ∗ ( ∗ ), F 1 = [ ∗ , − ], F 2 = [ f ∗ ( ∗ ) , − ], g L = ( g ′ L ) − 1 = α . Then deﬁne π = g R and κ = f ′ R .  Lemma 4.2.6 . The c omp osition f ∗ ( − ) ⊗ ( − ) id ⊗ η ∗ ∗ / / f ∗ ( − ) ⊗ f ∗ f ∗ ( − ) λ / / f ∗ ( − ⊗ f ∗ ( − )) c oincides with π . 28 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Pr o of. This follows fro m the comm utative diag ram (for any A and B in C 2 ) f ∗ A ⊗ B   π ' ' / / mf f ∗ A ⊗ f ∗ f ∗ B   λ   f ∗ f ∗ ( f ∗ A ⊗ B ) α − 1   / / mf f ∗ f ∗ ( f ∗ A ⊗ f ∗ f ∗ B ) α − 1   f ∗ ( f ∗ f ∗ A ⊗ f ∗ B )   / / mf f ∗ ( f ∗ f ∗ A ⊗ f ∗ f ∗ f ∗ B )   f ∗ ( A ⊗ f ∗ B ) / / I d ad j 5 5 f ∗ ( A ⊗ f ∗ f ∗ f ∗ B ) / / f ∗ ( A ⊗ f ∗ B ) in which the curved maps are indeed λ and π by constr uctio n.  Lemma 4.2.7 . The c omp osition f ∗ ( − ) ⊗ f ∗ ( − ) π / / f ∗ ( − ⊗ f ∗ f ∗ ( − )) f ∗ ( id ⊗ ǫ ∗ ∗ ) / / f ∗ ( − ⊗ − ) c oincides with λ . Pr o of. This follows direc tly fr o m the construction of π and diagram 11 .  Lemma 4.2.8 . The c omp osition f ∗ [ f ∗ ( − ) , ∗ ] µ / / [ f ∗ f ∗ ( − ) , f ∗ ( ∗ )] η ∗ ∗ / / [ − , f ∗ ( ∗ )] c oincides with κ . Pr o of. This follows fro m the comm utative diag ram (for any A in C 1 and B in C 2 ) f ∗ [ f ∗ A, B ] co ev l   µ ( ( co ev l / / gen [ A, f ∗ [ f ∗ A, B ] ⊗ A ] η ∗ ∗   π x x [ f ∗ f ∗ A, f ∗ [ f ∗ A, B ] ⊗ f ∗ f ∗ A ] λ   η ∗ ∗ / / mf [ A, f ∗ [ f ∗ A, B ] ⊗ f ∗ f ∗ A ] λ   [ f ∗ f ∗ A, f ∗ ([ f ∗ A, B ] ⊗ f ∗ A )] ev l   / / mf [ A, f ∗ ([ f ∗ A, B ] ⊗ f ∗ A )] ev l   [ f ∗ f ∗ A, f ∗ B ] η ∗ ∗ / / [ A, f ∗ B ] in which the left curved arrow is µ by constructio n, the right one is π b y Lemma 4.2.6 and the comp os ition from the top left corner to the b ottom rig ht o ne a long the upp er right corner is then κ by constructio n.  Theorem 4. 2.9. (existenc e of the push-forwar d) Under assumptions ( A f ) , ( B f ) , ( C f ) , let ζ b e the (susp ende d, exact) bifunctor deﬁne d by the c omp osition f ∗ [ − , f ! ( − )] µ / / [ f ∗ ( − ) , f ∗ f ! ( − )] ǫ ! ∗ / / [ f ∗ ( − ) , − ] . Then { f ∗ , ζ K } is a (susp ende d, exact) duality pr eserving functor fr om ( C 2 , D f ! K ,  f ! K ) to ( C 1 , D K ,  K ) . TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 29 Pr o of. W e need to prove that the dia gram P in Deﬁnition 2.2 .1 is commutativ e. W e consider the commutativ e diagram f ∗ A co ev l / / co ev l   13 f ∗ [[ A, f ! K ] , A ⊗ [ A, f ! K ]] ev r / / µ   f ∗ [[ A, f ! K ] , f ! K ] µ g g mf [ f ∗ [ A, f ! K ] , f ∗ A ⊗ f ∗ [ A, f ! K ]] / / id ⊗ µ   14 ′ [ f ∗ [ A, f ! K ] , f ∗ ( A ⊗ [ A, f ! K ])] ev r   [ f ∗ [ A, f ! K ] , f ∗ A ⊗ [ f ∗ A, f ∗ f ! K ]] ev r / / id ⊗ ǫ ! ∗   mf [ f ∗ [ A, f ! K ] , f ∗ f ! K ] ǫ ! ∗   [ f ∗ [ A, f ! K ] , f ∗ A ⊗ [ f ∗ A, K ]] ev r / / [ f ∗ [ A, f ! K ] , K ] where 14 ′ is obta ine d from 14 b y using s . The top row is f ∗  f ! K and the comp osition from the top right corner to the b ottom one is ζ , b oth by deﬁnition. The r esult follows if we prove that the compo sition fro m the top left co r ner to the bo ttom right one, going thro ugh the b ottom left co r ner, is  K f ∗ follow ed by D o K ζ o . This follows from the co mmutative diagr am f ∗ A co ev l / / co ev l   gen [ f ∗ [ A, f ! K ] , f ∗ A ⊗ f ∗ [ A, f ! K ]] id ⊗ ζ   [[ f ∗ A, K ] , f ∗ A ⊗ [ f ∗ A, K ]] ζ / / ev r   mf [ f ∗ [ A, f ! K ] , f ∗ A ⊗ [ f ∗ A, K ]] ev r   [[ f ∗ A, K ] , K ] ζ / / [ f ∗ [ A, f ! K ] , K ] in which the left vertical comp osition is  K f ∗ and the co mp os ition along the upp er right co rner is the co mp os ition alo ng the lower left corner of the previo us diagra m.  4.3. When the pro jection morphi s m is in vert ible. In this section, we pr ov e that the pro jection morphism is an is o morphism if and only if the morphism ζ is. In that case, we obtain a new morphism θ tha t will b e used in Section 5.5 to state a pro jection formula. - Prop ositio n 4.3. 1. Under assumptions ( A f ) , ( B f ) and ( C f ) , ther e is a unique morphism ν ′ : [ f ∗ ( ∗ ) , f ! ( − )] → f ! [ ∗ , − ] of (susp ende d) bifunctors su ch that the diagr ams [ f ∗ X , − ] / /   22 [ f ∗ X , f ! f ∗ ( − )] ν ′   f ! f ∗ [ f ∗ X , − ] κ − 1 / / f ! [ X , f ∗ ( − )] f ∗ [ f ∗ X , f ! ( − )] ν ′   κ − 1 / / 23 [ X , f ∗ f ! ( − )]   f ∗ f ! [ X , − ] / / [ X , − ] c ommute for any X ∈ C 1 . The morphism of bifunctors ν ′ is invertible if and only if π is invertible (A ssumption ( D f ) ). In that c ase, we denote by ν the inverse of ν ′ , and ther e is a u nique morphism of (susp ende d) bifunctors θ : f ! ( − ) ⊗ f ∗ ( ∗ ) → f ! ( − ⊗ ∗ ) 30 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL such that the diagr ams [ f ∗ X , f ! ( − ⊗ X )] 24 f ! [ X , − ⊗ X ] ν o o [ f ∗ X , f ! ( − ) ⊗ f ∗ X ] θ O O f ! o o O O f ! 25 [ f ∗ X , f ! ( − )] ⊗ f ∗ X o o f ! ([ X, − ] ⊗ X ) O O f ! [ X , − ] ⊗ f ∗ X θ o o ν O O − ⊗ f ∗ X   / / 26 f ! f ∗ ( − ) ⊗ f ∗ X θ   f ! f ∗ ( − ⊗ f ∗ X ) π − 1 / / f ! ( f ∗ ( − ) ⊗ X ) f ∗ ( f ! ( − ) ⊗ f ∗ X ) θ   π − 1 / / 27 f ∗ f ! ( − ) ⊗ X   f ∗ f ! ( − ⊗ X ) / / − ⊗ X c ommute for any X ∈ C 1 . F or any A and B , ν ′ is given by the c omp osition [ f ∗ A, f ! B ] η ! ∗ / / f ! f ∗ [ f ∗ A, f ! B ] κ − 1 / / f ! [ A, f ∗ f ! B ] ǫ ! ∗ / / f ! [ A, B ] and θ by f ! A ⊗ f ∗ B η ! ∗ / / f ! f ∗ ( f ! A ⊗ f ∗ B ) π − 1 / / f ! ( f ∗ f ! A ⊗ B ) ǫ ! ∗ / / f ! ( A ⊗ B ) or e quivalently by f ! A ⊗ f ∗ B co ev l / / f ! [ B , A ⊗ B ] ⊗ f ∗ B ν / / [ f ∗ B , f ! ( A ⊗ B )] ⊗ f ∗ B ev l / / f ! ( A ⊗ B ) . Pr o of. Apply Point 2 o f Theo rem 1.5.10 with L = L ′ = f ∗ , R = R ′ = f ! , G 1 = − ⊗ f ∗ ( ∗ ), G 2 = − ⊗ ∗ , F 1 = [ f ∗ ( ∗ ) , − ], F 2 = [ ∗ , − ] and f L = κ − 1 to obta in ν ′ = f ′ R and diagra ms 22 and 23 . Then Poin t 3 of the sa me Theorem tells us that ν ′ = f ′ R is in v ertible if and only if π = g L is (Note that π and κ indeed corres po nd to each other thr o ugh Point 1 o f the Theorem by construction). In that case, we can use Poin t 2 to obta in θ = g R from π − 1 = g ′ L and diag rams 24 , 25 , 2 6 and 27 .  In pa rticular, under Assumption ( D f ) , this gives iso morphisms ν K : f ! D o K → D o f ! K ( f ∗ ) o (in C 2 ) for each K . Lemma 4.3.2 . The c omp osition f ∗ [ − , f ! ( − )] ǫ ∗ ∗ / / f ∗ [ f ∗ f ∗ ( − ) , f ! ( − )] ν ′ / / f ∗ f ! [ f ∗ ( − ) , − ] ǫ ! ∗ / / [ f ∗ ( − ) , − ] c oincides with ζ . Pr o of. This follows from the commutativ e diagr am (for every A in C 2 and K in C 1 ) f ∗ [ A, f ! K ] µ   / / mf f ∗ [ f ∗ f ∗ A, f ! K ] µ   ν ′ / / 23 f ∗ f ! [ f ∗ A, K ] ǫ ! ∗   [ f ∗ A, f ∗ f ! K ] / / ad j U U U U U U U U U U U U U U U U U U [ f ∗ f ∗ f ∗ A, f ∗ f ! K ] η ∗ ∗   [ f ∗ A, f ∗ f ! K ] ǫ ! ∗ / / [ f ∗ A, K ] ident ifying the vertical comp ositio n in the middle with κ − 1 by Lemma 4.2.8 to recognize 23 .  Prop ositio n 4.3. 3. Under assumptions ( A f ) , ( B f ) , ( C f ) , ζ is an isomorphi sm if and only if ( D f ) holds. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 31 Corollary 4.3. 4. (Push-forwar d for Witt gr oups) When the dualities D f ! K and D K ar e stro ng and Assu mption ( D f ) is satisﬁe d, then { f ∗ , ζ K } is a st r ong duality pr eserving fun ctor and thus induc es a morphism of Witt gr oups f W ∗ : W ∗ ( C 2 , f ! K ) → W ∗ ( C 1 , K ) . Pr o of. This follows from Deﬁnition 2.1.5 , Theorem 4.2.9 , P rop osition 4.3.3 and Prop ositio n 2.2.3 .  Pr o of of Pr op osition 4.3.3 . Let us ﬁr st reca ll that ([ − , ∗ ] , [ − , ∗ ] o ,  ,  o ) is a (sus- pended) ACB from C 1 to C o 1 with pa rameter in C o 1 by Prop osition 3.2.3 , and thus ([ − , ( f ! ) o ( ∗ )] , [ − , f ! ( ∗ )] o ,  ,  o ) is a (susp ended) ACB from C 2 to C o 2 with para me- ter in C o 1 by Lemma 1.5.7 . W e then apply Theor em 1.5.1 0 with the squar e C 1 [ − , ∗ ]   C o 1 f ∗ / / C 2 [ − , ( f ! ) o ( ∗ )]   C o 1 f ∗ o o C o 1 [ − , ∗ ] o O O ( f ! ) o / / C o 2 [ − ,f ! ( ∗ )] o O O f o ∗ o o starting with g ′ L = ( ν ′ ) o . By Poin t 2 of lo c. cit., we o bta in a unique g R such that diagrams G 1 and G 2 commut e. This g R coincides with ζ o by Lemma 4.3.2 . By Poin t 1 o f lo c. cit., w e then obtain a unique f R such that diagr ams R and R ′ commute. But f R has to coincide with ζ by uniqueness, since thos e commutativ e diagrams were a lready prov en in Theo rem 4.2.9 with ζ and ζ o . By Point 3 of lo c. cit., f R is an isomorphism if a nd o nly if g ′ L = ( ν ′ ) o is, which is an isomo r phism if a nd o nly if ( D f ) ho lds by Prop o s ition 4.3.1 . W e hav e th us prov ed that ζ is an isomorphism if and only if ( D f ) holds.  4.4. Pro ducts. W e now deﬁne one mor e class ical morphism rela ted to pr o ducts for Witt groups. It is ea sily chec ked that since ( − ⊗ ∗ , [ ∗ , − ]) for ms an ACB fr om C to C with par ameter in C , (( − 1 ⊗ ∗ 1 ) × ( − 2 ⊗ ∗ 2 ) , [ ∗ 1 , − 1 ] × [ ∗ 2 , − 2 ]) for ms an ACB from C × C to C × C with para meter in C × C , and ( − ⊗ ( ∗ 1 ⊗ ∗ 2 ) , [ ∗ 1 ⊗ ∗ 2 , − ]) forms an A C B from C to C with parameter in C × C . W e can th us consider the situatio n of Lemma 1.3.5 applied to the squar e C × C ⊗ / / [ ∗ 1 , − 1 ] × [ ∗ 2 , − 2 ]   C ×C C [ ∗ 1 ⊗∗ 2 , − ]   C C × C ⊗ / / ( − 1 ⊗∗ 1 ) × ( − 2 ⊗∗ 2 ) O O C −⊗ ( ∗ 1 ⊗∗ 2 ) O O and to the morphism of bifunctors  : ( − 1 ⊗ − 2 ) ⊗ ( ∗ 1 ⊗ ∗ 2 ) → ( − 1 ⊗ ∗ 1 ) ⊗ ( − 2 ⊗ ∗ 2 ). W e th us o btain its mate, a morphism of bifunctors τ : [ ∗ 1 , − 1 ] ⊗ [ ∗ 2 , − 2 ] → [ ∗ 1 ⊗ ∗ 2 , − 1 ⊗ − 2 ] which, for a ny A, B , K , M is g iven by the compo s ition [ A, K ] ⊗ [ B , M ] co ev l / / [ A ⊗ B , ([ A, K ] ⊗ [ B , M ]) ⊗ ( A ⊗ B )]    [ A ⊗ B , K ⊗ M ] [ A ⊗ B , ([ A, K ] ⊗ A ) ⊗ ([ B , M ] ⊗ B )] ev l ⊗ ev l o o and obtain the following commutativ e diagr ams ([ X 1 , K ] ⊗ [ X 2 , M ]) ⊗ ( X 1 ⊗ X 2 )    τ / / 28 [ X 1 ⊗ X 2 , K ⊗ M ] ⊗ ( X 1 ⊗ X 2 ) ev l   ([ X 1 , K ] ⊗ X 1 ) ⊗ ([ X 2 , M ] ⊗ X 2 ) ev l ⊗ ev l / / K ⊗ M 32 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL K ⊗ M co ev l ⊗ co ev l / / co ev l   29 [ X 1 , K ⊗ X 1 ] ⊗ [ X 2 , M ⊗ X 2 ] τ   [ X 1 ⊗ X 2 , ( K ⊗ M ) ⊗ ( X 1 ⊗ X 2 )]  / / [ X 1 ⊗ X 2 , ( K ⊗ X 1 ) ⊗ ( M ⊗ X 2 )] as well. Deﬁnition 4.4.1. Consider o b jects K, M ∈ C . W e write τ K,M for the morphism of functors D K ( − 1 ) ⊗ D M ( − 2 ) → D K ⊗ M ( − 1 ⊗ − 2 ) deﬁned a bove where − 1 = K and − 2 = M . The pro ofs of the next tw o re sults are left to the rea de r . They are not diﬃcult although they requir e large co mm utative diag rams. Prop ositio n 4.4 . 2. The morphism τ K,M is a morphism of susp ende d bifunctors. The pro o f of the following lemma uses the unit o f the tensor pro duct. Lemma 4.4.3. If K is dualizing, M is dualizing and τ K,M is an isomorphism, then K ⊗ M is dualizing. If β f ,K , β f ,M and τ K,M ar e isomorph isms, then β f ,K ⊗ M is an isomorphism. In [ 7 ], Gille and Nenas hev deﬁne tw o natural pro ducts for Witt gro ups. These pro ducts co incide up to a sig n. W e just choo se one of them (the left pr o duct, fo r example), and refer to it as the pro duct, but everything works ﬁne with the other one too. Let us recall the basic prop erties of the pr o duct, r e phrasing [ 7 ] in our terminology . Theorem 4. 4.4. ( [ 7 , Deﬁnition 1.11 and Theore m 2 .9] ) L et C 1 , C 2 and C 3 b e triangulate d c ate gories with dualities D 1 , D 2 and D 3 . L et ( B , b 1 , b 2 ) : C 1 × C 2 → C 3 b e a susp ende d bifunctor (se e Deﬁnition 1.4.12 ) and d : B ( D o 1 × D o 2 ) → D o 3 B o b e an isomorphism of susp ende d bifunctors (se e Deﬁ nition 1.4.14 ) that makes { B , d } a duality pr eserving functor (se e Deﬁnition 2.2. 1 , her e C 1 × C 2 is endowe d with the duality D 1 × D 2 ). Then { B , d } induc es a pr o duct W( C 1 ) × W( C 2 ) → W( C 3 ) . The fo llowing prop os ition is not stated in [ 7 ], but ea sily follows fro m the con- struction o f the pro duct. Prop ositio n 4.4.5. L et σ : B → B ′ b e an isomorphism of susp ende d bifunctors that is duality pr eserving. Then B and B ′ induc e the same pr o duct on Witt gr oups. Let us now a pply this to our context. Prop ositio n 4.4.6. (pr o duct) The morphism of functors τ K,M turns the functor ⊗ into a duality pr eserving functor {− 1 ⊗ − 2 , τ K,M } fr om ( C × C , D K × D M ,  K ×  M ) to ( C , D K ⊗ M ,  K ⊗ M ) . Corollary 4.4 .7. (pr o duct for Witt gr oups) By The or em 4.4.4 , when the dualities and the functor ar e str ong (i.e.  K ,  M and τ K,M ar e isomorphi sms, and thus  K ⊗ M as wel l by L emma 4.4.3 ), {− 1 ⊗ − 2 , τ K,M } induc es a pr o duct W( C , D K ) × W( C , D M ) . / / W( C , D K ⊗ M ) on Witt gr oups. 5. Rela tions between functors In this section, w e prove the main re sults on compo sition, bas e change, and the pro jection formula. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 33 5.1. Comp os ition. This section s tudies the b ehavior of pull-ba cks and pus h-for- wards with r esp ect to comp osition. Le t K be a category whose ob jects are (sus- pended, triang ula ted) c lo sed ca tegories as in Section 3.1 , and whos e morphisms are (susp ended, exa ct) functors. L e t B b e another category , and let ( − ) ∗ be a con- trav a riant pseudo functor from B to K , i.e. a functor, exc e pt that instead of having equalities f ∗ g ∗ = ( g f ) ∗ when f a nd g ar e comp os a ble, we only have isomor phisms of (susp ended) functors a g,f : f ∗ g ∗ → ( g f ) ∗ . W e a lso requir e that ( − ) ∗ sends the ident it y of an o b ject to the identit y . W e denote X ∗ by C X . When, moreover, the diagram f ∗ g ∗ h ∗ a h,g   a g,f / / 30 ( g f ) ∗ h ∗ a h,gf   f ∗ ( hg ) ∗ a hg,f / / ( hg f ) ∗ is commutativ e, we say that the pseudo functor is asso cia tive. R emark 5.1 .1 . An example for this setting is to take for B the ca tegory o f s chemes (or r egular schemes) and C X = D b ( V ect ( X )) (or C X = D b ( O X − M od )). W e requir e that ( − ) ∗ is a monoida l ass o ciative pseudo functor, which means that each f ∗ is symmetric monoida l (Assumption ( A f ) ) a nd that the diag ram g ∗ f ∗ ⊗ g ∗ f ∗ a ⊗ a   α / / 31 g ∗ ( f ∗ ⊗ f ∗ ) α / / g ∗ f ∗ ( − ⊗ − ) a   ( f g ) ∗ ⊗ ( f g ) ∗ α / / ( f g ) ∗ ( − ⊗ − ) commutes for any tw o co mp os able f a nd g . Notation 5. 1 .2. L et X b e an obje ct in B and K an obje ct in C X We denote by C X,K the (susp ende d, t riangulate d) c ate gory with duality ( C X , D K ,  K ) obtaine d by Cor ol lary 3.2.4 . When K is dualizing, we denote by W i ( X, K ) the i -t h shifte d Witt gr oup of C X,K . W e have alre a dy se e n in Theor em 4.1.2 that under Assumption ( A f ) , the c o uple { f ∗ , β K } is a duality pr e serving functor betw een (susp ended, triangulated) ca te- gories with dualities, under Assumption ( A f ) . Recall the I -nota tion o f Lemma 3.2.6 . Theorem 5.1.3. (c omp osition of pul l-b acks) F or any two c omp osable f and g in B , the isomorphism of functors a g,f : f ∗ g ∗ → ( g f ) ∗ is a morphism of duality pr eserving fun ctors fr om I a g,f ,K { f ∗ , β f ,g ∗ K }{ g ∗ , β g,K } to { ( g f ) ∗ , β gf , K } . Corollary 5. 1.4. (c omp osition of pul l-b acks for Witt gro ups) By Pr op osition 2.2.4 , for c omp osable morphisms f and g in B , t he pul l-b ack on Witt gr oups deﬁne d in Cor ol lary 4.1.3 satisﬁes ( g f ) ∗ W = I W a g,f ,K f ∗ W g ∗ W . Pr o of of The or em 5.1.3 . The cla im amounts to chec king that the diagram f ∗ g ∗ [ − , K ] a   β / / f ∗ [ g ∗ ( − ) , g ∗ K ] β / / [ f ∗ g ∗ ( − ) , f ∗ g ∗ K ] a   ( g f ) ∗ [ − , K ] β / / [( g f ) ∗ ( − ) , ( g f ) ∗ K ] a o / / [ f ∗ g ∗ ( − ) , ( g f ) ∗ K ] 34 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL is commutative. Since by construction o f β (Pro po sition 4 .1.1 ), α and β are mates, this co mmu tativity follows from Lemma 1.2.7 applied to the cub e C Z ( gf ) ∗ / / g ∗   ? ? ? ? ? ? ? C X I d   ? ? ? ? ? ? ? C Y a ; C             f ∗ / / C X C Z −⊗ K O O g ∗   ? ? ? ? ? ? ? C Y α P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * −⊗ g ∗ K O O f ∗ / / C X α [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ f ∗ g ∗ K O O C Z ( gf ) ∗ / / C X I d   ? ? ? ? ? ? ? C X C Z −⊗ K O O g ∗   ? ? ? ? ? ? ? ( gf ) ∗ / / C X α [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ ( gf ) ∗ K O O I d   ? ? ? ? ? ? ? C Y a ; C             f ∗ / / C X id ⊗ a * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * −⊗ f ∗ g ∗ K O O which is commutativ e as Diag ram 31 is.  A cov ariant pseudo functor ( − ) ∗ is deﬁned in the same wa y as a contrav ar iant one, exce pt that we are given isomorphisms in the other direction: b g,f : ( g f ) ∗ → g ∗ f ∗ . Deﬁnition 5.1.5. Let ( − ) ∗ (resp. ( − ) ∗ ) be a (susp ended, exact) contrav aria nt (resp. cov ar iant) pseudo functor from B to K . W e say that (( − ) ∗ , ( − ) ∗ ) is a n adjoint couple of pseudo functors if ( f ∗ , f ∗ ) is a (susp ended, exact) adjoint couple for every f (in particular ( − ) ∗ and ( − ) ∗ coincide on ob jects), and the diagrams I d   / / 32 ( g f ) ∗ ( g f ) ∗ b   g ∗ f ∗ f ∗ g ∗ a / / g ∗ f ∗ ( g f ) ∗ and f ∗ g ∗ ( g f ) ∗ a   b / / 33 f ∗ g ∗ g ∗ f ∗   ( g f ) ∗ ( g f ) ∗ / / I d commute for any co mpo sable f a nd g . Spelling out the symmetr ic notion when the left adjoint pseudo functor is cov ari- ant and the right one contra v ariant is left to the reader. As usual, the right (or le ft) adjoint is unique up to uniq ue isomo rphism. Lemma 5.1 .6. L et ( − ) ∗ b e a c ontra variant pseudo-functor. As s ume that for any f ∗ , we ar e given a right (su sp ende d, exact) adjoint f ∗ which is the identity when f is the identity. Then ther e is a unique c ol le ction of isomorphisms b g,f : ( g f ) ∗ → g ∗ f ∗ such that (( − ) ∗ , ( − ) ∗ ) forms an adjoint c ouple of pseudo funct ors. Pr o of. Apply Theo rem 1.2.8 (or 1.5 .4 in the susp ended case) to ( L, R ) = ( f ∗ , f ∗ ), ( F 1 , G 1 ) = ( g ∗ , g ∗ ), ( F 2 , G 2 ) = ( I d, I d ) and ( L ′ , R ′ ) = (( g f ) ∗ , ( g f ) ∗ ). This g ives the r equired is omorphism ( g f ) ∗ → g ∗ f ∗ and the diagr ams 32 and 3 3 .  Lemma 5.1. 7. The right (r esp. left) adjoint of an asso ciative pseudo functor is asso ciative. Pr o of. Left to the reade r .  Lemma 5.1.8. L et (( − ) ∗ , ( − ) ∗ ) b e an adjoint p air of pseudo-functors (left c on- tr avariant and right c ovariant). If ( − ) ∗ satisﬁes the c ommutativity of 31 , then the diagr am ( g f ) ∗ A ⊗ ( g f ) ∗ B b ⊗ b   λ / / 34 ( g f ) ∗ ( A ⊗ B ) b   g ∗ f ∗ A ⊗ g ∗ f ∗ B λ / / g ∗ ( f ∗ A ⊗ f ∗ B ) λ / / g ∗ f ∗ ( A ⊗ B ) TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 35 is c ommut ative for every A and B . Pr o of. The commutativ e diagram g ∗ ( A ⊗ B ) α − 1 / / η ∗ ∗   mf g ∗ A ⊗ g ∗ B η ∗ ∗ ⊗ η ∗ ∗ / /   11 f ∗ f ∗ g ∗ A ⊗ f ∗ f ∗ g ∗ B λ   f ∗ a ⊗ f ∗ a t t f ∗ f ∗ g ∗ ( A ⊗ B ) / / f ∗ a   31 f ∗ f ∗ ( g ∗ A ⊗ g ∗ B ) / / f ∗ ( f ∗ g ∗ A ⊗ f ∗ g ∗ B ) s s g g g g g g g g g g g mf f ∗ ( g f ) ∗ ( A ⊗ B ) α − 1 / / f ∗ (( g f ) ∗ A ⊗ ( g f ) ∗ B ) f ∗ ( g f ) ∗ A ⊗ f ∗ ( g f ) ∗ B λ o o shows that the cub e C X × C X −⊗− / / f ∗ × f ∗   ? ? ? ? ? ? ? C X f ∗   ? ? ? ? ? ? ? C Y × C Y λ ; C             −⊗− / / C Y C Z × C Z ( gf ) ∗ × ( gf ) ∗ O O I d × I d   ? ? ? ? ? ? ? C Z × C Z ( a × a ) ◦ ( η ∗ ∗ × η ∗ ∗ ) * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * g ∗ × g ∗ O O −⊗− / / C Z α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? g ∗ O O C X × C X −⊗− / / C X f ∗   ? ? ? ? ? ? ? C Y C Z × C Z ( gf ) ∗ × ( gf ) ∗ O O I d × I d   ? ? ? ? ? ? ? −⊗− / / C Z α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( gf ) ∗ O O I d   ? ? ? ? ? ? ? C Z × C Z id ; C             −⊗− / / C Z a P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * g ∗ O O is comm utative. W e apply Lemma 1.2.7 to it and obta in a new commutativ e cub e which is Diagram 34 . Note that the mo rphism of functor s on the left face of our ﬁrst cub e indeed gives b in the resulting cube, by construction of b .  Let us now consider the sub catego ry B ′ of B with the same ob jects, but whose morphisms are only those f : X → Y satisfying assumptions ( A f ) , ( B f ) and ( C f ) . W e then choo se successive right adjoints f ∗ and f ! for each morphism f (they are unique up to unique isomo rphism, and we choos e ( I d X ) ∗ = I d ! X = I d C X for simplicity), and by Lemma 5.1.6 , using ( − ) ∗ , w e tur n them in to (suspended, exact) pseudo functors ( − ) ∗ : B ′ → K ( − ) ! : B ′ → K with str ucture mor phisms b g,f and c g,f . They are ass o ciative by Lemma 5.1.7 . Theorem 5. 1.9. F or morphisms f : X → Y and g : Y → Z in B ′ and K an obje ct of C Z , the isomorphism of funct ors b g,f : ( g f ) ∗ → g ∗ f ∗ deﬁne d ab ove is a morphism of duality pr eserving functors fr om { ( g f ) ∗ , ζ K } I c g,f ,K to { g ∗ , ζ K }{ f ∗ , ζ g ! K } . Corollary 5.1.1 0 . (c omp osition of push-forwar ds) By Pr op osition 2.2.4 , for c om- p osable morphisms f and g in B ′ , t he push-forwar d on W itt gr oups deﬁne d in Cor ol- lary 4.3.4 satisﬁes ( g f ) W ∗ I W c g,f ,K = g W ∗ f W ∗ . Pr o of of The or em 5.1.9 . W e have to pr ov e that the dia gram M of Deﬁnition 2.2.2 which is here ( g f ) ∗ [ A, f ! g ! K ] c / / b   35 ( g f ) ∗ [ A, ( g f ) ! K ] ζ / / [( g f ) ∗ A, K ] g ∗ f ∗ [ A, f ! g ! K ] ζ / / g ∗ [ f ∗ A, g ! K ] ζ / / [ g ∗ f ∗ A, K ] b o O O 36 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL is commutative (for a ny A and B ). The co mm utative diagra m 34 may b e wr itten as the commutativ e cub e C Z I d / / I d   ? ? ? ? ? ? ? C Z I d   ? ? ? ? ? ? ? C Z id ; C             I d / / C Z C X g ∗ f ∗ ( −⊗ B ) O O I d   ? ? ? ? ? ? ? C X b P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ( gf ) ∗ ( −⊗ B ) O O ( gf ) ∗ / / C Z λ [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ ( gf ) ∗ B O O C Z I d / / C Z I d   ? ? ? ? ? ? ? C Z C X g ∗ f ∗ ( −⊗ B ) O O I d   ? ? ? ? ? ? ? f ∗ / / C Y λ [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? g ∗ ( −⊗ f ∗ B ) O O g ∗   ? ? ? ? ? ? ? C X b ; C             ( gf ) ∗ / / C Z λ ◦ ( id ⊗ b ) * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * −⊗ ( gf ) ∗ B O O out of which we get a new co mmutative cub e by Lemma 1.2.7 , whos e commutativit y is equiv a le n t to that o f Diag ram 35 .  5.2. Base cha nge. The next fundamen tal theo rem that we will prove is a base change fo rmula. In this se c tion, we ﬁx a c o mmu tative diag ram in B with g and ¯ g in B ′ . V ¯ f   ¯ g / / 36 Y f   X g / / Z Using Lemma 1.2.6 (or its susp ended version 1.5 .4 ) with J 1 = g ∗ , K 1 = g ∗ , J 2 = ¯ g ∗ , K 2 = ¯ g ∗ , H = ¯ f ∗ , H ′ = f ∗ and the isomorphism of (susp ended) functors ξ : ¯ g ∗ f ∗ → ( f ¯ g ) ∗ = ( g ¯ f ) ∗ → ¯ f ∗ g ∗ for a , we obtain its mate, a mor phis m ε : f ∗ g ∗ → ¯ g ∗ ¯ f ∗ and tw o commutativ e dia grams. Assuming ( E f ,g ) (i.e. ε is an isomorphis m) and applying the sa me lemma to J 1 = g ∗ , K 1 = g ! , J 2 = ¯ g ∗ , K 2 = ¯ g ! , H = f ∗ , H ′ = ¯ f ∗ and a = ε − 1 , we obtain a morphis m γ : ¯ f ∗ g ! → ¯ g ! f ∗ and tw o commutativ e diagrams . ¯ g ∗ ¯ f ∗ g ! γ / / ε − 1   37 ¯ g ∗ ¯ g ! f ∗   f ∗ g ∗ g ! / / f ∗ ¯ f ∗ / /   38 ¯ f ∗ g ! g ∗ γ   ¯ g ! ¯ g ∗ ¯ f ∗ ε − 1 / / ¯ g ! f ∗ g ∗ Theorem 5.2.1. (b ase change) Le t f and ¯ f b e morphisms in B and g and ¯ g b e morphisms in B ′ ﬁtting in the c ommu tative diagr am 36 such that Ass u mption ( E f ,g ) is satisﬁe d. Then, the isomorphi sm of fun ctors ε : f ∗ g ∗ → ¯ g ∗ ¯ f ∗ fr om { f ∗ , β K }{ g ∗ , ζ K } to { ¯ g ∗ , ζ f ∗ K } I γ K { ¯ f ∗ , β g ! K } is duality pr eserving. This toge ther with Prop osition 2.2.4 immediately implie sthe following. Corollary 5.2.2. If in the situ ation of the the or em ab ove γ K is an isomorphism, then t he pul l-b acks and push-forwar ds on Witt gr oups deﬁne d in c or ol laries 4.1.3 and 4.3.4 satisfy f ∗ W g W ∗ = ¯ g W ∗ I W γ K ¯ f ∗ W . TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 37 Pr o of of The or em 5.2.1 . W e hav e to prove that the diagr am f ∗ g ∗ [ A, g ! K ] ζ / / ε   39 f ∗ [ g ∗ A, K ] β / / [ f ∗ g ∗ A, f ∗ K ] ¯ g ∗ ¯ f ∗ [ A, g ! K ] β / / ¯ g ∗ [ ¯ f ∗ A, ¯ f ∗ g ! K ] γ / / ¯ g ∗ [ ¯ f ∗ A, ¯ g ! f ∗ K ] ζ / / [ ¯ g ∗ ¯ f ∗ A, f ∗ K ] ε o O O (corresp onding to Diagram M of Deﬁnition 2.2.2 ) is commutativ e. W e ﬁrst pr ov e the tw o following lemmas. Lemma 5.2.3 . The c omp osition ¯ g ∗ ( ¯ f ∗ ( − ) ⊗ ¯ f ∗ ( ∗ )) α / / ¯ g ∗ ¯ f ∗ ( − ⊗ ∗ ) ε − 1 / / f ∗ g ∗ ( − ⊗ ∗ ) and the c omp osition ¯ f ∗ [ ∗ , g ! ( − )] β / / [ ¯ f ∗ ( ∗ ) , ¯ f ∗ ¯ g ! ( − )] γ / / [ ¯ f ∗ ( ∗ ) , ¯ g ! f ∗ ( − )] ar e mates in L emma 1.3.5 when J 1 = g ∗ ( − ⊗ ∗ ) , K 1 = [ ∗ , g ! ( − )] , J 2 = ¯ g ∗ ( − ⊗ ¯ f ∗ ( ∗ )) , K 2 = [ ¯ f ∗ ( ∗ ) , ¯ g ! ( − )] , H = f ∗ and H ′ = ¯ f ∗ . Pr o of. By uniqueness , this follows from the pr o of of one of the commutativ e dia- grams in Lemma 1 .3.5 . W e choose H whic h is as follows. ¯ g ∗ ( ¯ f ∗ [ X , g ! A ] ⊗ ¯ f ∗ X ) β / / α   10 ¯ g ∗ ([ ¯ f ∗ X , ¯ f ∗ g ! A ] ⊗ ¯ f ∗ X )   γ / / mf ¯ g ∗ ([ ¯ f ∗ X , ¯ g ! f ∗ A ] ⊗ ¯ f ∗ X )   ¯ g ∗ ¯ f ∗ ([ X, g ! A ] ⊗ X ) / / ε − 1   mf ¯ g ∗ ¯ f ∗ g ! A γ / / ε − 1   37 ¯ g ∗ g ! f ∗ A   f ∗ g ∗ ([ X, g ! A ] ⊗ X ) / / f ∗ g ∗ g ! A / / f ∗ A  Lemma 5.2.4 . The c omp osition ¯ g ∗ ( − ) ⊗ f ∗ g ∗ ( ∗ ) id ⊗ ε / / ¯ g ∗ ( − ) ⊗ ¯ g ∗ ¯ f ∗ ( ∗ ) λ / / ¯ g ∗ ( − ⊗ ¯ f ∗ ( ∗ )) and the c omp osition ¯ g ∗ [ ¯ f ∗ ( ∗ ) , ¯ g ! ( − )] ζ / / [ ¯ g ∗ ¯ f ∗ ( ∗ ) , − ] ε o / / [ f ∗ g ∗ ( ∗ ) , − ] ar e mates in L emma 1.3.5 when J 1 = ¯ g ∗ ( − ⊗ ¯ f ∗ ( ∗ )) , K 1 = [ ¯ f ∗ ( ∗ ) , ¯ g ! ( − )] , J 2 = ( − ⊗ f ∗ g ∗ ( ∗ )) , K 2 = [ f ∗ g ∗ ( ∗ ) , − ] , H = I d and H ′ = ¯ g ∗ . Pr o of. Similar to the pro of of the prev ious Lemma, this fo llows fro m the co mmu - tative diag ram ¯ g ∗ [ ¯ f ∗ X , ¯ g ! A ] ⊗ f ∗ g ∗ X id ⊗ ε   ζ ⊗ id / / mf [ ¯ g ∗ ¯ f ∗ X , A ] ⊗ f ∗ g ∗ X ε o / /   gen [ f ∗ g ∗ X , A ] ⊗ f ∗ g ∗ X   ¯ g ∗ [ ¯ f ∗ X , ¯ g ! A ] ⊗ ¯ g ∗ ¯ f ∗ X λ   ζ ⊗ id / / [ ¯ g ∗ ¯ f ∗ X , A ] ⊗ ¯ g ∗ ¯ f ∗ X * * V V V V V V V V V V V V V V V 14 ′′ ¯ g ∗ ([ ¯ f ∗ X , ¯ g ! A ] ⊗ ¯ f ∗ X ) / / ¯ g ∗ ¯ g ! A / / A where 14 ′′ is ea sily obtained fro m 14 ′ by lo o king at the deﬁnition of ζ in Theor em 4.2.9 .  38 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL The commutativ e diagra m 31 ea sily implies (by deﬁnition of ξ ) that the diag ram ¯ g ∗ ( f ∗ A ⊗ f ∗ B ) α   α − 1 / / ¯ g ∗ f ∗ A ⊗ ¯ g ∗ f ∗ B ξ ⊗ ξ / / ¯ f ∗ g ∗ A ⊗ ¯ f ∗ g ∗ B α   ¯ g ∗ f ∗ ( A ⊗ B ) ξ / / ¯ f ∗ g ∗ ( A ⊗ B ) α − 1 / / ¯ f ∗ ( g ∗ A ⊗ g ∗ B ) is commutativ e for any A and B . It can b e written as the comm utative cub e C X × C X −⊗− / / ¯ f ∗ × ¯ f ∗   ? ? ? ? ? ? ? C X ¯ f ∗   ? ? ? ? ? ? ? C V × C V α ; C             −⊗− / / C V C Z × C Z g ∗ × g ∗ O O f ∗ × f ∗   ? ? ? ? ? ? ? C Y × C Y ξ × ξ * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * * * ¯ g ∗ × ¯ g ∗ O O −⊗− / / C Y α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ¯ g ∗ O O C X × C X −⊗− / / C X ¯ f ∗   ? ? ? ? ? ? ? C V C Z × C Z g ∗ × g ∗ O O f ∗ × f ∗   ? ? ? ? ? ? ? −⊗− / / C Z α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? g ∗ O O f ∗   ? ? ? ? ? ? ? C Y × C Y α ; C             −⊗− / / C Y ξ P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ¯ g ∗ O O out of which Lemma 1 .2 .7 gives the commu tative cub e C X × C X g ∗ × g ∗   −⊗− / / ¯ f ∗ × ¯ f ∗   ? ? ? ? ? ? ? C X ¯ f ∗   ? ? ? ? ? ? ? C V × C V ¯ g ∗ × ¯ g ∗   α ; C             −⊗− / / C V ¯ g ∗   C Z × C Z f ∗ × f ∗   ? ? ? ? ? ? ? ε × ε     ; C     C Y × C Y −⊗− / / λ ; C                                   C Y C X × C X g ∗ × g ∗   −⊗− / / C X ¯ f ∗   ? ? ? ? ? ? ? g ∗   C V ¯ g ∗   C Z × C Z λ ; C                                   f ∗ × f ∗   ? ? ? ? ? ? ? −⊗− / / C Z ε ; C             f ∗   ? ? ? ? ? ? ? C Y × C Y α ; C             −⊗− / / C Y The commutativit y of the las t cub e implies the commutativit y of the following cub e for a ny A . C Z f ∗ / / I d   ? ? ? ? ? ? ? C Y I d   ? ? ? ? ? ? ? C Z id ; C             f ∗ / / C Y C X g ∗ ( −⊗ A ) O O g ∗   ? ? ? ? ? ? ? C Z λ P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * −⊗ g ∗ A O O f ∗ / / C Y α [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ f ∗ g ∗ A O O C Z f ∗ / / C Y I d   ? ? ? ? ? ? ? C Y C X g ∗ ( −⊗ A ) O O g ∗   ? ? ? ? ? ? ? ¯ f ∗ / / C V x [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ¯ g ∗ ( −⊗ ¯ f ∗ A ) O O ¯ g ∗   ? ? ? ? ? ? ? C Z ε ; C             f ∗ / / C Y y P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * −⊗ f ∗ g ∗ A O O where x is the ﬁr s t comp osition in Lemma 5.2.3 a nd y is the ﬁrst comp osition in Lemma 5 .2.4 . Applying Lemma 1.2.7 to this new cube gives a cube in which the mates of x a nd y may b e des crib ed using lemma s 5.2 .3 and 5.2.4 , and whose commutativit y is the one o f Diagra m 39 .  Let us now establish tw o more co mm utative diagr ams tha t will b e useful in a ppli- cations. In geo metr ic situatio ns they a llow us to check whether θ is an iso mo rphism by res tricting to op en s ubsets. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 39 Prop ositio n 5.2 . 5. The diagr am f ∗ g ∗ ( − ) ⊗ f ∗ ( ∗ ) α   ε / / 40 ¯ g ∗ ¯ f ∗ ( − ) ⊗ f ∗ ( ∗ ) π / / ¯ g ∗ ( ¯ f ∗ ( − ) ⊗ ¯ g ∗ f ∗ ( ∗ )) ξ   ¯ g ∗ ( ¯ f ∗ ( − ) ⊗ ¯ f ∗ g ∗ ( ∗ )) α   f ∗ ( g ∗ ( − ) ⊗ ∗ ) π / / f ∗ g ∗ ( − ⊗ g ∗ ( ∗ )) ε / / ¯ g ∗ ¯ f ∗ ( − ⊗ g ∗ ( ∗ )) is c ommutative. If fur t hermor e ε is an isomorphism ( A ssumption ( E f ,g ) ), γ is deﬁne d and t he diagr am ¯ f ∗ g ! ( − ) ⊗ ¯ f ∗ g ∗ ( ∗ ) α   γ ⊗ ξ − 1 / / 41 ¯ g ! f ∗ ( − ) ⊗ ¯ g ∗ f ∗ ( ∗ ) θ / / ¯ g ! ( f ∗ ( − ) ⊗ f ∗ ( ∗ )) α   ¯ f ∗ ( g ! ( − ) ⊗ g ∗ ( ∗ )) θ / / ¯ f ∗ g ! ( − ⊗ ∗ ) γ / / ¯ g ! f ∗ ( − ⊗ ∗ ) is c ommut ative. Pr o of. T o get the ﬁr s t diagr am, apply Lemma 1.2.7 to the cube C X × C Z ¯ f ∗ × I d / / I d ⊗ g ∗   ? ? ? ? ? ? C V × C Z I d ⊗ ¯ g ∗ f ∗   ? ? ? ? ? ? C X α − 1 ; C           ¯ f ∗ / / C V C Z × C Z g ∗ × I d O O ⊗   ? ? ? ? ? ? C Z α − 1 * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * g ∗ O O f ∗ / / C Y ξ [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ¯ g ∗ O O C X × C Z ¯ f ∗ × I d / / C V × C Z I d ⊗ ¯ g ∗ f ∗   ? ? ? ? ? ? C V C Z × C Z g ∗ × I d O O ⊗   ? ? ? ? ? ? f ∗ × I d / / C Y × C Z ξ × id [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ¯ g ∗ × I d O O I d ⊗ f ∗   ? ? ? ? ? ? C Z α − 1 ; C           f ∗ / / C Y P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ¯ g ∗ O O which is commutativ e by 31 (using f ¯ g = g ¯ f ). This gives a cub e who se commu- tativity is the one of 40 . Exchanging the top and b ottom faces of this new cub e (th us r eversing the vertical ar rows), inv erting the front ( ε ), back ( ε × I d ) and sides ( π ) morphis ms of functors and a pplying again Lemma 1.2.7 to this new cub e shows that 41 is comm utative to o.  5.3. Asso ciativi ty of pro ducts. W e now establish a few commutativ e diagrams implied by the compatibility of f ∗ with the a sso ciativity of the tensor pro duct. F or simplicity , we hide in diagr ams all asso cia tivity mo rphisms a nd brack eting concerning the tensor pro duct (a s if the tenso r pro duct was str ic tly asso ciativ e). Since f ∗ is monoida l, the diagram (inv olving α ) f ∗ A ⊗ f ∗ B ⊗ f ∗ C / /   42 f ∗ ( A ⊗ B ) ⊗ f ∗ C   f ∗ A ⊗ f ∗ ( B ⊗ C ) / / f ∗ ( A ⊗ B ⊗ C ) is commutativ e. Lemma 5.3.1 . Under assumptions ( A f ) and ( B f ) , the diagr am (involving λ ) f ∗ A ⊗ f ∗ B ⊗ f ∗ C / /   43 f ∗ ( A ⊗ B ) ⊗ f ∗ C   f ∗ A ⊗ f ∗ ( B ⊗ C ) / / f ∗ ( A ⊗ B ⊗ C ) is c ommut ative 40 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Pr o of. Left to the reade r .  Prop ositio n 5.3. 2. U nder assum ptions ( A f ) and ( B f ) , the fol lowing diagr ams ar e c ommu tative. f ∗ A ⊗ B ⊗ C π   π / / 44 f ∗ ( A ⊗ f ∗ B ) ⊗ C π   f ∗ ( A ⊗ f ∗ ( B ⊗ C )) α − 1 / / f ∗ ( A ⊗ f ∗ B ⊗ f ∗ C ) f ∗ A ⊗ f ∗ B ⊗ C id ⊗ π   λ ⊗ id / / 45 f ∗ ( A ⊗ B ) ⊗ C π   f ∗ ( A ⊗ f ∗ ( B ⊗ f ∗ C )) λ / / f ∗ ( A ⊗ B ⊗ f ∗ C ) Pr o of. Using Lemma 4.2.6 to decomp ose π , Diag ram 44 is f ∗ A ⊗ B ⊗ C η ∗ ∗ / / η ∗ ∗   11 f ∗ A ⊗ f ∗ f ∗ B ⊗ C η ∗ ∗   λ / / mf f ∗ ( A ⊗ f ∗ B ) ⊗ C η ∗ ∗   f ∗ A ⊗ f ∗ f ∗ B ⊗ f ∗ f ∗ C / / λ   43 f ∗ ( A ⊗ f ∗ B ) ⊗ f ∗ f ∗ C λ   f ∗ A ⊗ f ∗ f ∗ ( B ⊗ C ) α − 1 / / λ + + V V V V V V V V V V f ∗ A ⊗ f ∗ ( f ∗ B ⊗ f ∗ C ) λ / / mf f ∗ ( A ⊗ f ∗ B ⊗ f ∗ C ) f ∗ ( A ⊗ f ∗ ( B ⊗ C )) α − 1 3 3 g g g g g g g g g g g and Diagr am 45 is f ∗ A ⊗ f ∗ B ⊗ C η ∗ ∗ / / λ   mf f ∗ A ⊗ f ∗ B ⊗ f ∗ f ∗ C λ   λ / / 43 f ∗ A ⊗ f ∗ ( B ⊗ f ∗ C ) λ   f ∗ ( A ⊗ B ) ⊗ C η ∗ ∗ / / f ∗ ( A ⊗ B ) ⊗ f ∗ f ∗ C λ / / f ∗ ( A ⊗ B ⊗ f ∗ C )  5.4. The m onoidal functor f ∗ and pro ducts. W e hav e the following dia gram of duality pres erving functors ( C 1 ) K × ( C 1 ) M { f ∗ × f ∗ ,β K × β M }   {−⊗∗ ,τ K,M } / / ( C 1 ) K ⊗ M { f ∗ ,β K ⊗ M }   ( C 2 ) f ∗ K × ( C 2 ) f ∗ M {−⊗∗ ,τ f ∗ K,f ∗ M } / / ( C 2 ) f ∗ K ⊗ f ∗ M I α K,M / / ( C 2 ) f ∗ ( K ⊗ M ) where I α : ( C 2 ) f ∗ K ⊗ f ∗ M → ( C 2 ) f ∗ ( K ⊗ M ) is the duality pres erving functor induced by α by Lemma 3.2.6 ). Prop ositio n 5.4.1. (the pul l-b ack r esp e ct s the pr o duct) Under Assumption ( A f ) , the isomorphism of susp ende d bifunctors α : f ∗ ( − ) ⊗ f ∗ ( ∗ ) → f ∗ ( − ⊗ ∗ ) is an iso- morphism of duality pr eserving functors b etwe en the t wo duality pr eserving functors deﬁne d by t he c omp ositions ab ove. Corollary 5.4.2 . The pul l-b ack and the pr o duct deﬁn e d in Cor ol lary 4.1.3 and Pr op osition 4.4.6 satisfy f ∗ W ( x.y ) = I W α K,M ( f ∗ W ( x ) .f ∗ W ( y )) for al l x ∈ W(( C 1 ) K ) and y ∈ W(( C 1 ) M ) by Pr op osition 2.2.4 pr ovide d al l ne c essary assumptions ab out b eing s tr ong ar e satisﬁe d. Pr o of of Pr op osition 5.4.1 . W e start with a lemma. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 41 Lemma 5.4.3 . The c omp osition f ∗ ( − ) ⊗ ( f ∗ A ⊗ f ∗ B ) id ⊗ α / / f ∗ ( − ) ⊗ f ∗ ( A ⊗ B ) α / / f ∗ ( − ⊗ ( A ⊗ B )) and the c omp osition f ∗ [ A ⊗ B , − ] β / / [ f ∗ ( A ⊗ B ) , f ∗ ( − )] α o / / [ f ∗ A ⊗ f ∗ B , − ] ar e mates in L emma 1.2.6 when J 1 = − ⊗ ( A ⊗ B ) , K 1 = [ A ⊗ B , − ] , J 2 = − ⊗ ( f ∗ A ⊗ f ∗ B ) , K 2 = [ f ∗ A ⊗ f ∗ B , − ] , H = f ∗ and H ′ = f ∗ . Pr o of. By uniqueness, it suﬃces to es tablish the commut ativity of diagra m H which is f ∗ [ A ⊗ B , − ] ⊗ ( f ∗ A ⊗ f ∗ B ) β ⊗ id / / id ⊗ α   mf [ f ∗ ( A ⊗ B ) , f ∗ ( − )] ⊗ ( f ∗ A ⊗ f ∗ B ) α o v v id ⊗ α   gen f ∗ [ A ⊗ B , − ] ⊗ f ∗ ( A ⊗ B ) / / α   10 [ f ∗ ( A ⊗ B ) , f ∗ ( − )] ⊗ f ∗ ( A ⊗ B ) co ev l t t j j j j j j j j j j j j j j j j j j j j j j j j j j j j f ∗ ([ A ⊗ B , − ] ⊗ ( A ⊗ B )) co ev l   [ f ∗ A ⊗ f ∗ B , f ∗ ( − )] ⊗ ( f ∗ A ⊗ f ∗ B ) co ev l r r d d d d d d d d d d d d d d d d d d d d d d f ∗ ( − )  Now, accor ding to Deﬁnition 2.2.2 , we need to pr ov e tha t the following diagr a m is commutativ e for any A, B , K and M . f ∗ [ A, K ] ⊗ f ∗ [ B , M ] α   β ⊗ β / / 46 [ f ∗ A, f ∗ K ] ⊗ [ f ∗ B , f ∗ M ] τ / / [ f ∗ A ⊗ f ∗ B , f ∗ K ⊗ f ∗ M ] α   [ f ∗ A ⊗ f ∗ B , f ∗ ( K ⊗ M )] f ∗ ([ A, K ] ⊗ [ B , M ]) τ / / f ∗ [ A ⊗ B , K ⊗ M ] β / / [ f ∗ ( A ⊗ B ) , f ∗ ( K ⊗ M )] α o O O W e apply Lemma 1.2.7 to the cube C 1 × C 1 −⊗− / / f ∗ × f ∗   ? ? ? ? ? ? ? C 1 f ∗   ? ? ? ? ? ? ? C 2 × C 2 α ; C             −⊗− / / C 2 C 1 × C 1 ( −⊗ A ) × ( −⊗ B ) O O f ∗ × f ∗   ? ? ? ? ? ? ? C 2 × C 2 α × α * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * ( −⊗ f ∗ A ) × ( −⊗ f ∗ B ) O O −⊗− / / C 2  [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ ( f ∗ A ⊗ f ∗ B ) O O C 1 × C 1 −⊗− / / C 1 f ∗   ? ? ? ? ? ? ? C 2 C 1 × C 1 ( −⊗ A ) × ( −⊗ B ) O O f ∗ × f ∗   ? ? ? ? ? ? ? −⊗− / / C 1  [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ ( A ⊗ B ) O O f ∗   ? ? ? ? ? ? ? C 2 × C 2 α ; C             −⊗− / / C 2 α ◦ ( id ⊗ α ) * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * −⊗ ( f ∗ A ⊗ f ∗ B ) O O which is easily shown to be commutativ e using the compa tibilit y of f ∗ with the symmetry and the as so ciativity o f the tensor pro duct (diagr ams 15 and 42 ). This yields a new cub e whose commutativit y is the one of 46 using Lemma 5.4.3 to reco gnize the right face.  42 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL 5.5. Pro jection formula. The last theo rem we wan t to prov e is a pro jection formula. Theorem 5.5.1. (pr oje ction formula) L et C 1 and C 2 b e close d c ate gories and let f ∗ b e a monoidal functor (Assu mption ( A f ) ) fr om C 1 to C 2 satisfying A ssumptions ( B f ) , ( C f ) and ( D f ) . L et K and M b e obje cts of C 1 . Then, π is a morphism of duality pr eserving functors fr om { I d ⊗ I d, τ K,M }{ f ∗ × I d, ζ K × id } to { f ∗ , ζ K ⊗ M } I θ K,M { I d ⊗ I d, τ f ! K,f ∗ M }{ I d × f ∗ , id × β M } . Corollary 5. 5.2. (pr oje ction formula for Witt gr oups) By Pr op osition 2.2.4 , u n der str ongness assumptions ne c essary for their ex istenc e and when θ K,M is an isomor- phism, the pul l-b ack, push-forwar d and pr o duct on Witt gr oups deﬁne d in c or ol laries 4.1.3 , 4.3.4 and Pr op osition 4.4.6 satisfy the e quality f W ∗ ( I W θ K,M ( x.f ∗ W ( y ))) = f W ∗ ( x ) .y in W(( C 1 ) K ⊗ M ) for al l x ∈ W(( C 2 ) f ! K ) and y ∈ W(( C 1 ) M ) . Pr o of of The or em 5.5.1 . According to Deﬁnition 2 .2.2 , we hav e to prov e that the following diag r am is commut ative for any A , B , K a nd M . f ∗ [ A, f ! K ] ⊗ [ B , M ] π   ζ / / 47 [ f ∗ A, K ] ⊗ [ B , M ] τ / / [ f ∗ A ⊗ B , K ⊗ M ] f ∗ ([ A, f ! K ] ⊗ f ∗ [ B , M ]) β   [ f ∗ ( A ⊗ f ∗ B ) , K ⊗ M ] π o O O f ∗ ([ A, f ! K ] ⊗ [ f ∗ B , f ∗ M ]) τ / / f ∗ [ A ⊗ f ∗ B , f ! K ⊗ f ∗ M ] θ / / f ∗ [ A ⊗ f ∗ B , f ! ( K ⊗ M )] ζ O O Decomp osing ζ a s in its deﬁnition (see Theorem 4.2 .9 ) and using a few mf di- agrams and Diagram 27 , it is ea sy to reduce this to the commut ativity of the following diag r am (with N = f ! K ). f ∗ [ A, N ] ⊗ [ B , M ] µ / / π   48 [ f ∗ A, f ∗ N ] ⊗ [ B , M ] τ / / [ f ∗ A ⊗ B , f ∗ N ⊗ M ] f ∗ ([ A, N ] ⊗ f ∗ [ B , M ]) β   [ f ∗ ( A ⊗ f ∗ B ) , f ∗ N ⊗ M ] π o O O f ∗ ([ A, N ] ⊗ [ f ∗ B , f ∗ M ]) τ / / f ∗ [ A ⊗ f ∗ B , N ⊗ f ∗ M ] µ / / [ f ∗ ( A ⊗ f ∗ B ) , f ∗ ( N ⊗ f ∗ M )] π − 1 O O which requir es the following pr e liminary lemma s. Lemma 5.5.3 . The c omp osition f ∗ ( − 1 ⊗ − 2 ) ⊗ f ∗ ( A ⊗ f ∗ B ) λ / / f ∗ (( − 1 ⊗ − 2 ) ⊗ ( A ⊗ f ∗ B ))    f ∗ (( − 1 ⊗ A ) ⊗ ( − 2 ⊗ f ∗ B )) and the c omp osition f ∗ ([ A, − 1 ] ⊗ [ f ∗ B , − 2 ]) τ / / f ∗ [ A ⊗ f ∗ B , − 1 ⊗ − 2 ] µ / / [ f ∗ ( A ⊗ f ∗ B ) , f ∗ ( − 1 ⊗ − 2 )] ar e mates in L emma 1.2.6 when J 1 = ( − 1 ⊗ A ) × ( − 2 ⊗ f ∗ B ) , K 1 = [ A, − 1 ] × [ f ∗ B , − 2 ] , J 2 = − ⊗ f ∗ ( A ⊗ f ∗ B ) , K 2 = [ f ∗ ( A ⊗ f ∗ B ) , − ] , and H = H ′ = f ∗ ( − 1 ⊗ − 2 ) . TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 43 Pr o of. By uniqueness, it suﬃces to pr ov e Diagra m H whic h is her e f ∗ ([ A, − 1 ] ⊗ [ f ∗ B , − 2 ]) ⊗ f ∗ ( A ⊗ f ∗ B ) τ / / λ   mf f ∗ [ A ⊗ f ∗ B , − 1 ⊗ − 2 ] ⊗ f ∗ ( A ⊗ f ∗ B ) λ   µ w w f ∗ (([ A, − 1 ] ⊗ [ f ∗ B , − 2 ]) ⊗ ( A ⊗ f ∗ B )) τ / /    28 f ∗ ([ A ⊗ f ∗ B , − 1 ⊗ − 2 ] ⊗ ( A ⊗ f ∗ B )) k k k k k k k k k k k ev l u u k k k k k k k k k k k 14 f ∗ (([ A, − 1 ] ⊗ A ) ⊗ ([ f ∗ B , − 2 ] ⊗ f ∗ B )) ev l   [ f ∗ ( A ⊗ f ∗ B ) , f ∗ ( − 1 ⊗ − 2 )] ⊗ f ∗ ( A ⊗ f ∗ B ) ev l r r e e e e e e e e e e e e e e e e e e e e e e f ∗ ( − 1 ⊗ − 2 )  Lemma 5.5.4 . The c omp osition ( − 1 ⊗ − 2 ) ⊗ f ∗ ( A ⊗ f ∗ B ) π − 1 / / ( − 1 ⊗ − 2 ) ⊗ ( f ∗ A ⊗ B )  / / ( − 1 ⊗ f ∗ A ) ⊗ ( − 2 ⊗ B ) and the c omp osition [ f ∗ A, − 1 ] ⊗ [ B , − 2 ] τ / / [ f ∗ A ⊗ B , − 1 ⊗ − 2 ] ( π − 1 ) o / / [ f ∗ ( A ⊗ f ∗ B ) , − 1 ⊗ − 2 ] ar e mates in L emma 1.2 .6 when J 1 = ( − 1 ⊗ f ∗ A ) × ( − 2 ⊗ B ) , K 1 = [ f ∗ A, − 1 ] × [ B , − 2 ] , J 2 = − ⊗ f ∗ ( A ⊗ f ∗ B ) , K 2 = [ f ∗ ( A ⊗ f ∗ B ) , − ] , and H = H ′ = ⊗ . Pr o of. By uniqueness, it suﬃces to pr ov e that diagram H commutes, which is ([ f ∗ A, − 1 ] ⊗ [ B , − 2 ]) ⊗ f ∗ ( A ⊗ f ∗ B ) τ / / π − 1   mf [ f ∗ A ⊗ B , − 1 ⊗ − 2 ] ⊗ f ∗ ( A ⊗ f ∗ B ) π − 1   ( π − 1 ) o y y ([ f ∗ A, − 1 ] ⊗ [ B , − 2 ]) ⊗ ( f ∗ A ⊗ B ) τ / /    28 [ f ∗ A ⊗ B , − 1 ⊗ − 2 ] ⊗ ( f ∗ A ⊗ B ) l l l l l l l l l l l ev l u u l l l l l l l l l l l l l gen ([ f ∗ A, − 1 ] ⊗ f ∗ A ) ⊗ ([ B , − 2 ] ⊗ B ) ev l   [ f ∗ ( A ⊗ f ∗ B ) , − 1 ⊗ − 2 ] ⊗ f ∗ ( A ⊗ f ∗ B ) ev l r r e e e e e e e e e e e e e e e e e e e e e ( − 1 ⊗ − 2 )  The cub e C 2 × C 2 ⊗ / / ( −⊗ f ∗ M ) × ( −⊗ f ∗ B ) ? ?   ? ? ? C 2 −⊗ ( f ∗ M ⊗ f ∗ B )   ? ? ? ? ? ? ? C 2 ⊗ C 2  ; C             ⊗ / / C 2 C 1 × C 1 f ∗ × f ∗ O O ( −⊗ M ) ⊗ ( −⊗ B )   ? ? ? ? ? ? ? C 1 × C 1 α − 1 × α − 1 * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * f ∗ × f ∗ O O ⊗ / / C 1 α − 1 [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? f ∗ O O C 2 × C 2 ⊗ / / C 2 −⊗ ( f ∗ M ⊗ f ∗ B )   ? ? ? ? ? ? ? C 2 C 1 × C 1 f ∗ × f ∗ O O ( −⊗ M ) × ( −⊗ B )   ? ? ? ? ? ? ? ⊗ / / C 1 α − 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? f ∗ O O −⊗ ( M ⊗ B ) ? ? ?   ? ? C 1 × C 1  ; C             ⊗ / / C 1 ( id ⊗ α − 1 ) ◦ α − 1 * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * f ∗ O O is commutativ e by a clas sical exerc is e on mono ida l functors, symmetry and asso - ciativity . Applying Lemma 1.2.7 to it, we get a new cub e, in which the mor phism of functors on the r ight fa ce is ( id ⊗ α − 1 ) ◦ π , s imply beca use π is the mate of α − 1 (b y Prop os ition 4.2.5 ) and ( id ⊗ α − 1 ) is just a change in the para meter. The 44 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL commutativit y of the cub e th us obtained ca n be rewritten as the co mm utativity of the cub e C 2 × C 1 f ∗ × I d / / I d × f ∗   ? ? ? ? ? ? ? C 1 × C 1 ⊗   ? ? ? ? ? ? ? C 2 × C 2 π − 1 ; C             f ∗ ( −⊗− ) / / C 1 C 2 × C 1 ( −⊗ A ) × ( −⊗ B ) O O I d × f ∗   ? ? ? ? ? ? ? C 2 × C 2 id × α * * * * * * * * * * * * * * * * * * * * P X * * * * * * * * * * * * * * * * * * * * ( −⊗ A ) × ( −⊗ B ) O O f ∗ ( −⊗− ) / / C 1 x [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? −⊗ f ∗ ( A ⊗ f ∗ B ) O O C 2 × C 1 f ∗ × I d / / C 1 × C 1 ⊗   ? ? ? ? ? ? ? C 1 C 2 × C 1 ( −⊗ A ) × ( −⊗ B ) O O I d × f ∗   ? ? ? ? ? ? ? f ∗ × I d / / C 1 × C 1 λ × id ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? [ c ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( −⊗ f ∗ A ) × ( −⊗ B ) O O ⊗   ? ? ? ? ? ? ? C 2 × C 2 π − 1 ; C             f ∗ ( −⊗− ) / / C 1 y P X * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * −⊗ f ∗ ( A ⊗ f ∗ B ) O O where x and y are the ﬁrst comp ositions in lemmas 5.5 .3 a nd 5 .5.4 . Applying one more time Lemma 1.2.7 to this new cub e yields another one whose c o mmu tativity is the one of 48 .  6. V arious ref ormula tions This section is dev oted to reformulations of the main theorems in view of appli- cations. The corollarie s on Witt groups show that these refor m ulations are useful. The refor m ulations a re obtained b y c hanging the duality a t the source o r at the target using functors I ι (see Lemma 3.2.6 ) with ι a dapted to the s ituation. F or this reason, der iving thos e reformulations from the or iginal theo rems is very easy and most of the times follows fro m simple commutativ e diagr ams inv olv ing functors of the form I ι . W e therefore le av e a ll pro o fs o f Sections 6.1 , 6 .2 and 6 .3 a s exercises for the reader . 6.1. Reformulations using f ! of unit ob jects. In this sec tion, we use the unit ob jects in the mo noidal catego ries to formulate the main theore ms in a diﬀerent wa y . In the application to Witt gro ups and algebr aic geometry , this will relate f ! to f ∗ using canonica l sheaves. Let B ′′ denote the s ubca tegory of B ′ in which the morphisms f are such that ( D f ) is satisﬁed ( π is an is omorphism). F or any morphism f : X → Y in B ′ , we deﬁne ω ′ f = f ! ( 1 Y ) . T o every morphism f , we ass o ciate a n um ber d f such that d gf = d f + d g for comp osable morphisms f an g . F or example, for every ob ject X w e ma y cho ose a nu mber d X and set d f = d X − d Y . W e deﬁne ω f = T − d f f ! ( 1 Y ) . R emark 6.1.1 . In applica tions to schemes, d f can b e the relative dimension o f a morphism f and d X can b e the relative dimensio n of a smo o th X over a bas e scheme. In that ca se, ω ′ f is isomorphic to a shifted line bundle a nd ω f to a line bundle (the ca no nical sheaf ). This ex pla ins why we introduce d f and ω f . On the other hand, it is always p ossible to set d f = 0 fo r any morphism f , in which case ω f = ω ′ f , thus statements in terms o f ω f also apply to ω ′ f . F or any tw o comp osable morphisms f and g in B ′′ , let us denote by i ′ g,f the comp osition ω ′ f ⊗ f ∗ ( ω ′ g ) = f ! ( 1 Y ) ⊗ f ∗ ( ω ′ g ) θ → f ! ( ω ′ g ) = f ! g ! ( 1 Z ) c ≃ ( g f ) ! ( 1 Z ) = ω ′ gf and i g,f : ω f ⊗ f ∗ ( ω g ) − → ω gf TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 45 the co mpo sition obtained by the sa me chain of mor phisms a nd then desusp e nding . Lemma 6.1.2. F or any c omp osable morphisms X f / / Y g / / Z h / / V in B ′′ , the diagr am of isomorphisms ω f ⊗ f ∗ ( ω hg ) i hg,f r r d d d d d d d d d d d ω f ⊗ f ∗ ( ω g ⊗ g ∗ ( ω h )) id ⊗ f ∗ i h,g o o ≀ ( id ⊗ a ) ◦ ( id ⊗ α − 1 )   ω hgf ω gf ⊗ ( g f ) ∗ ( ω h ) i h,gf m m Z Z Z Z Z Z Z Z Z ω f ⊗ f ∗ ( ω g ) ⊗ ( g f ) ∗ ( ω h ) i g,f ⊗ id o o is c ommut ative. In other wor ds, i g,f (as wel l as i ′ g,f ) satisﬁes a c o cycle c ondition. W e ca n then re fo rmulate the main theor ems 4.2.9 , 5 .1 .9 , 5 .2.1 and 5.5.1 a nd their corolla r ies for Witt groups as follows. Deﬁnition 6.1 .3. (push-for ward) Let f : X → Y b e a mo rphism in B ′′ and K b e an ob ject in C Y . W e deﬁne a dualit y pres erving functor C X,ω ′ f ⊗ f ∗ K → C Y ,K by the comp osition { f ∗ , ζ K } I θ 1 ,K . W e denote it simply by { f ∗ } in the res t of the section ( K is alwa ys understo o d). Deﬁnition 6.1.4. (push-for ward for Witt gr oups) When the dualities ar e stro ng and θ 1 ,K is an isomor phism, { f ∗ } induces a push-forward f W ∗ : W i + d f ( X, ω f ⊗ f ∗ K ) → W i ( Y , K ) on Witt gro ups (reca ll Pr o p osition 3.2.5 to switch from susp ending ω ′ f to the sus- pended duality used in Deﬁnition 2.1.5 of shifted Witt gr oups). Theorem 6.1.5. (c omp osition of push-forwar ds) L et f : X → Y and g : Y → Z b e morphisms in B ′′ and K b e an obje ct of C Z . The morphism b g,f : ( g f ) ∗ → g ∗ f ∗ (se e S e ction 5.1 ) is duality pr eserving b et we en t he c omp ositions of duality pr eserving functors in the diagr am C X,ω ′ f ⊗ f ∗ ( ω ′ g ⊗ g ∗ K ) { f ∗ }   I ι / / C X,ω ′ gf ⊗ ( gf ) ∗ K { ( gf ) ∗ }   b g,f p x i i i i i i i i i i i i i i i i i i C Y ,ω ′ g ⊗ g ∗ K { g ∗ } / / C Z,K wher e ι is the c omp osition ω ′ f ⊗ f ∗ ( ω ′ g ⊗ g ∗ K ) I d ⊗ α − 1 / / ω ′ f ⊗ f ∗ ω ′ g ⊗ f ∗ g ∗ K i ′ g,f ⊗ a g,f / / ω ′ gf ⊗ ( g f ) ∗ K. Corollary 6. 1 .6. (c omp osition of push-forwar ds for Witt gr oups) F or f and g as in the the or em, the push-forwar ds on Witt gr oups of Deﬁnition 6.1.4 s atisfy g W ∗ f W ∗ = ( g f ) W ∗ I W ι . Theorem 6.1. 7. (b ase change) In t he situ ation of The or em 5.2.1 , with furthermor e g and ¯ g in B ′′ , the morphism of functors ε : g ∗ f ∗ → ¯ f ∗ ¯ g ∗ is duality pr eserving b etwe en t he two c omp ositions of duality pr eserving functors in the diagr am C V , ¯ f ∗ ( ω ′ g ⊗ g ∗ K ) I ι / / C V ,ω ′ ¯ g ⊗ ¯ g ∗ f ∗ K { ¯ g ∗ } ( ( Q Q Q Q Q Q Q Q C X,ω ′ g ⊗ g ∗ K { ¯ f ∗ } 5 5 k k k k k k k k { g ∗ } / / C Z,K { f ∗ } / / ε K S C Y ,f ∗ K wher e ι is deﬁne d as the c omp osition ¯ f ∗ ( ω ′ g ⊗ g ∗ K ) α − 1 / / ¯ f ∗ ω ′ g ⊗ ¯ f ∗ g ∗ K I d ⊗ ξ   ¯ f ∗ ω ′ g ⊗ ¯ g ∗ f ∗ K γ 1 ⊗ I d / / ¯ g ! f ∗ 1 ⊗ ¯ g ∗ f ∗ K ≃ ω ′ ¯ g ⊗ ¯ g ∗ f ∗ K. 46 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL Corollary 6 .1.8. (b ase change for Witt gr oups) In the situation of the t he or em, when furt hermor e γ 1 is an isomorphism, the pu s h-forwar ds and pul l-b acks for Witt gr oups of Deﬁnition 6.1.4 and Cor ol lary 4.1.3 satisfy f ∗ W g W ∗ = ¯ g W ∗ I W ι ¯ f ∗ W . Theorem 6.1. 9. (pr oje ction formula) In t he situation of The or em 5.5.1 , π is a morphism of duality pr eserving functors b etwe en the two c omp ositions in the dia- gr am C ω ′ f ⊗ f ∗ K × C f ∗ M {⊗} / / C ω ′ f ⊗ f ∗ K ⊗ f ∗ M I I d ⊗ α K,M / / C ω ′ f ⊗ f ∗ ( K ⊗ M ) { f ∗ }   C ω ′ f ⊗ f ∗ K × C M { I d × f ∗ } O O { f ∗ × I d } / / C K × C M π K S {⊗} / / C K ⊗ M Corollary 6. 1.10. (pr oje ction formula for Witt gr oups) In the situation of t he the or em, the push-forwar d, pul l-b ack and pr o duct on Witt gr oups of Deﬁnition 6.1.4 , Cor ol lary 4.1.3 and Pr op osition 4.4.6 satisfy f W ∗ I W I d ⊗ α K,M ( x.f ∗ W ( y )) = f W ∗ ( x ) .y in W i + j ( Y , K ⊗ M ) for any x ∈ W i + d f ( X, ω f ⊗ f ∗ K ) and y ∈ W j ( Y , M ) . 6.2. Reformulations using morphism s to correct the dualities. W e now r e- formulate the main re s ults using a conv enien t c a tegorica l setting which allows us to display the previo us resultsin an even nicer way . This will b e useful in a pplica tions. W e deﬁne new categories . Deﬁnition 6.2. 1. Let B ∗ be the catego ry whose ob jects are pairs ( X , K ) with X ∈ B , K ∈ C X , and whos e mor phisms fro m ( X , K ) to ( Y , L ) are pairs ( f , φ ) where f : X → Y is a morphism in B and φ : f ∗ L → K is a morphism in C X . The comp osition is deﬁned by ( g , ψ )( f , φ ) = ( g f , φ ◦ f ∗ ( ψ ) ◦ ( a g,f ) − 1 ). The identit y morphism o n ( X , K ) is ( I d X , I d K ) a nd the compo sition is clear ly asso ciative. Deﬁnition 6.2.2 . Let B ! be the categor y whose ob jects are pairs ( X , K ) with X ∈ B ′ , K ∈ C X , and whose mor phis ms fr o m ( X , K ) to ( Y , M ) are pairs ( f , φ ) where f : X → Y is a morphism in B ′ and φ : K → f ! M is a morphism in C X . The comp o sition is deﬁned by ( g , ψ )( f , φ ) = ( g f , c g,f ◦ f ! ( ψ ) ◦ φ ). T he identit y morphism o n ( X , K ) is ( I d X , I d K ) a nd the compo sition is clear ly asso ciative. F or applica tions to Witt groups, we need the ob jects de ﬁning the dualities to be dua lizing and the duality prese rving functor s to be s trong. W e thus deﬁne tw o more ca tegories . Deﬁnition 6.2.3 . Let B ∗ W denote the sub categor y of B ∗ in which the ob jects ( X, K ) are such that K is dualizing (  K is an iso morphism) and the mor phisms ( f , φ ) : ( X , K ) → ( Y , L ) are such that φ and β f ,L are isomo rphisms. Deﬁnition 6.2. 4 . Let B ! W denote the sub catego ry of B ! in which the ob jects ( X , K ) are such that K is dualizing (  K is an isomor phism) and the morphisms ( f , φ ) are such that φ is an isomorphism. Using the ca tegories B ! and B ∗ and their sub c ategories , the main theorems 4.1.2 , 4.2.9 , 5.1.3 , 5.1 .9 , 5 .2 .1 , 5.5.1 and their corolla ries on Witt groups c a n b e rephr ased as follows. Deﬁnition 6.2. 5. (pull-back) F or any morphism ( f , φ ) : ( X , K ) → ( Y , L ) in B ∗ , we deﬁne a dua lity pr eserving functor | f , φ | ∗ : C Y ,L → C X,K by the comp osition C Y ,L { f ∗ ,β L } / / C X,f ∗ L I φ / / C X,K . Deﬁnition 6.2.6 . (pull-back on Witt gr oups) F or any morphism ( f , φ ) : ( X , K ) → ( Y , L ) in B ∗ W , the duality pres erving functor | f , φ | induces a morphism | f , φ | ∗ W : W i ( Y , L ) → W i ( X, K ) on Witt groups. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 47 Theorem 6.2.7. (c omp osition of pul l-b acks) L et ( f , φ ) and ( g , ψ ) b e c omp osable morphisms in B ∗ . The morphism a g,f : f ∗ g ∗ → ( g f ) ∗ is a morphism of duality pr eserving fun ctors as in t he fol lowing diagr am. C Y ,L | f ,φ | ∗ & & M M M M M M a g,f   C Z,M | ( g, ψ )( f ,φ ) | ∗ / / | g, ψ | ∗ 8 8 q q q q q q C X,K Corollary 6.2.8 . (c omp osition of pul l-b acks for Witt gr oups) Le t ( f , φ ) and ( g , ψ ) b e c omp osable m orphisms in B ∗ W . Then the pul l-b acks for Witt gr oups of Deﬁnition 6.2.6 satisfy | ( g , ψ )( f , φ ) | ∗ W = | f , φ | ∗ W | g , ψ | ∗ W . In other wor ds, W ∗ is a c ontr avariant functor fr om B ∗ W to gr ade d ab elian gr oups. Deﬁnition 6.2. 9. (push-forward) F or any morphism ( f , φ ) : ( X , K ) → ( Y , L ) in B ! , we deﬁne a dua lity preserving functor | f , φ | ∗ : C X,K → C Y ,L by the comp ositio n C X,K I φ / / C X,f ! L { f ∗ ,ζ L } / / C Y ,L . Deﬁnition 6.2.1 0. (push-for ward for Witt gr oups) F or any mo rphism ( f , φ ) : ( X, K ) → ( Y , L ) in B ! W , the duality prese rving functor | f , φ | ∗ induces a morphism on Witt gro ups | f , φ | W ∗ : W i ( X, K ) → W i ( X, L ). Theorem 6. 2.11. (c omp osition of push-forwar ds) L et ( f , φ ) and ( g , ψ ) b e c omp os- able morphisms in B ! . The morphism b g,f : ( g f ) ∗ → g ∗ f ∗ is a morphism of duality pr eserving fun ctors as in t he fol lowing diagr am. C Y ,L | g, ψ | ∗ & & M M M M M M C X,K | ( g, ψ )( f ,φ ) | ∗ / / | f ,φ | ∗ 8 8 q q q q q q b g,f K S C Z,M Corollary 6.2 .12. (c omp osition of push-forwar ds for Witt gr oups) L et ( f , φ ) and ( g , ψ ) b e c omp osable morphi sms in B ! W . Then the push-forwar ds for Witt gr oups of Deﬁnition 6.2.10 satisfy | ( g , ψ )( f , φ ) | W ∗ = | g , ψ | W ∗ | f , φ | W ∗ . In other wor ds, W ∗ is a c ovariant fun ct or fr om B ! W to gr ade d ab elian gr oups. Theorem 6. 2.13. (b ase change) L et ( f , φ ) and ( ¯ f , ¯ φ ) b e morphisms in B ∗ , ( g , ψ ) , and ( ¯ g , ¯ ψ ) b e morphisms in B ! ﬁtting in the diagr am ( V , N ) (¯ g , ¯ ψ ) / / ( ¯ f , ¯ φ )   ( Y , L ) ( f ,φ )   ( X, M ) ( g,ψ ) / / ( Z, K ) such t hat f ¯ g = g ¯ f ∈ B , such that Ass u mptions ( E f ,g ) is satisﬁe d and such that t he diagr am ¯ f ∗ M ¯ φ s s f f f f f f f f f ¯ f ∗ ( ψ ) / / ¯ f ∗ g ! K γ K   N ¯ ψ + + X X X X X X X X X X ¯ g ! L ¯ g ! f ∗ K ¯ g ! ( φ ) o o 48 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL is c ommut ative. Then the morphism ε : f ∗ g ∗ → ¯ g ∗ ¯ f ∗ is duality pr eserving b etwe en the c omp ositions of duality pr eserving functors in the diagr am. C V ,N | ¯ g , ¯ ψ | ∗ / / C Y ,L C X,M | ¯ f , ¯ φ | ∗ O O | g, ψ | ∗ / / C Z,K | f ,φ | ∗ O O ε b j M M M M M M M M M M M M Corollary 6.2.14 . (b ase change for Witt gr oups) In the situation of the the or em, assuming furthermor e that ( f , φ ) and ( ¯ f , ¯ φ ) ar e in B ∗ W , ( g , ψ ) and ( ¯ g , ¯ ψ ) ar e in B ! W and γ K is an isomorp hism, the pul l-b acks and push-forwar ds on Witt gr oups of deﬁnitions 6.2.6 and 6.2.10 satisfy | ¯ g , ¯ ψ | W ∗ | ¯ f , ¯ φ | ∗ W = | f , φ | ∗ W | g , ψ | W ∗ . Theorem 6. 2.15. (pr oje ction formula) L et f : X → Y b e a morphism in B ′′ . L et K, M ∈ C X and L, N ∈ C Y b e obje cts, φ : f ∗ L → K , ψ : M → f ! N and χ : M ⊗ K → f ! ( N ⊗ L ) b e morphisms su ch that M ⊗ K χ / / f ! ( N ⊗ L ) M ⊗ f ∗ L I d ⊗ φ O O ψ ⊗ I d / / f ! N ⊗ f ∗ L θ N ,L O O is c ommut ative. Then, the morphism π is duality pr eserving b etwe en the c omp ositions in the diagr am C X,M × C X,K {⊗} / / C X,M ⊗ K | f ,χ | ∗ & & N N N N N N C X,M × C Y ,L I d ×| f ,φ | ∗ 5 5 k k k k k k k k | f ,ψ | ∗ × I d / / C Y ,N × C Y ,L π K S {⊗} / / C Y ,N ⊗ L Corollary 6. 2.16. (pr oje ction formula for Witt gr oups) In the situation of t he the or em, assuming furthermor e t hat ( f , φ ) is in B ∗ W and ( f , ψ ) and ( f , χ ) ar e in B ! W , the push-forwar d, pul l-b ack and pr o duct for Witt gr oups of Deﬁnition 6.2.10 , Deﬁnition 6.2.6 and Pr op osition 4.4.6 satisfy | f , χ | W ∗ ( x. | f , φ | ∗ W ( y )) = | f , ψ | W ∗ ( x ) .y in W i + j ( Y , N ⊗ L ) for any x ∈ W i ( X, M ) and y ∈ W j ( Y , L ) . 6.3. Reformulations usi ng a ﬁnal ob ject. When the categor y B has a ﬁnal ob ject denoted by Pt that is als o a ﬁnal ob ject for the sub categor y B ′′ , ther e is a reformulation of the main r esults using absolute (that is relative to the ﬁna l ob ject) canonical ob jects rather than relative ones as in section 6.1 . F or any ob ject X ∈ B ′ , let p X denote the unique morphism X → Pt. W e deﬁne d X = d p X , ω ′ X = ω ′ p X and ω X = ω p X . Deﬁnition 6. 3.1. Let B ∗ be the ca tegory whose ob jects ar e pairs ( X , K ) with X ∈ B ′′ , K ∈ C X , a nd whos e mo rphisms from ( X , K ) to ( Y , L ) are pairs ( f , φ ) where f : X → Y is a morphism in B ′′ and φ : K → f ∗ L is a morphism in C X . The comp osition is deﬁned by ( g , ψ )( f , φ ) = ( g f , b g,f ) f ∗ ( ψ ) φ . The identit y of ( X , K ) is ( I d X , I d K ) a nd the compo sition is clear ly asso ciative. Deﬁnition 6.3 .2. Let B W ∗ be the s ub ca tegory of B ∗ in which the ob jects ( X , K ) are suc h that ω X ⊗ K is dualizing and the mor phis ms ( f , φ ) : ( X , K ) → ( Y , L ) a re such that ι deﬁned as the co mp os ition ω ′ X ⊗ K ( c f,p Y ) − 1 ⊗ φ / / f ! ω ′ Y ⊗ f ∗ L θ ω ′ Y ,L / / f ! ( ω ′ Y ⊗ L ) . is an isomor phism. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 49 Deﬁnition 6.3. 3. (push-forward) F or any morphism ( f , φ ) : ( X , K ) → ( Y , L ) in B ∗ , we deﬁne a dua lit y preserving functor | f , φ | ∗ : C X,ω ′ X ⊗ K → C Y ,ω ′ Y ⊗ L by the comp osition C X,ω ′ X ⊗ K I ι / / C X,f ! ( ω ′ Y ⊗ L ) { f ∗ ,ζ ω ′ Y ⊗ L } / / C Y ,L . where ι is deﬁned as in Deﬁnition 6.3.2 (but not nec e ssarily a n isomor phism). Deﬁnition 6.3.4 . (push-forward for Witt gro ups) Let ( f , φ ) b e a morphis m in B W ∗ . Then the duality preserving functor | f , φ | ∗ of Deﬁnition 6.3.3 induces a morphism | f , φ | W ∗ : W i + d X ( X, ω X ⊗ K ) → W i + d Y ( Y , ω Y ⊗ L ) on Witt gro ups. Theorem 6.3.5 . (c omp osition of push-forwar ds) L et ( f , φ ) and ( g , ψ ) b e c omp os- able m orphisms in B ∗ . Then the morphism b g,f : ( g f ) ∗ → g ∗ f ∗ is a m orphism of duality pr eserving functors as in the fol lowing diagr am. C Y ,ω ′ Y ⊗ L | g, ψ | ∗ ( ( Q Q Q Q Q Q Q C X,ω ′ X ⊗ K | ( g, ψ )( f ,φ ) | ∗ / / | f ,φ | ∗ 6 6 m m m m m m m b g,f K S C Z,ω ′ Z ⊗ M Corollary 6. 3.6. (c omp osition of push-forwar ds for Witt gr oups) L et ( f , φ ) and ( g , ψ ) b e c omp osable morphisms in B W ∗ . Then t he push-forwar ds of Deﬁnition 6.3.4 satisfy | ( g , ψ )( f , φ ) | W ∗ = | g , ψ | W ∗ | f , φ | W ∗ . In other wor ds, W ∗ is a c ovariant fun ctor fr om B W ∗ to gr ade d ab elian gr oups. T o state a base change theore m, we need the fo llowing. Let ( f , φ ) and ( ¯ f , ¯ φ ) b e morphisms in B ∗ and ( g , ψ ) and ( ¯ g , ¯ ψ ) b e morphisms in B ∗ with s ources and targ ets as o n the diagram ( V , ω ′ V ⊗ N ) ( ¯ f , ¯ φ )   ( V , N ) (¯ g , ¯ ψ ) / / ( Y , L ) ( Y , ω ′ Y ⊗ L ) ( f ,φ )   ( X, ω ′ X ⊗ M ) ( X, M ) ( g,ψ ) / / ( Z, K ) ( Z, ω ′ Z ⊗ K ) such that f ¯ g = g ¯ f ∈ B and such that Assumption ( E f ,g ) is sa tisﬁed. The mor phism φ induces a morphism “ φ ” deﬁned by the compo sition ¯ f ∗ ( ω ′ X ⊗ g ∗ K ) “ φ ”     α − 1 / / ¯ f ∗ ω ′ X ⊗ ¯ f ∗ g ∗ K ¯ f ∗ ( c p Z ,g ) − 1 ⊗ I d / / ¯ f ∗ g ! ω ′ Z ⊗ ¯ f ∗ g ∗ K γ ⊗ ξ − 1   ¯ g ! ( ω ′ Y ⊗ L ) ¯ g ! f ∗ ( ω ′ Z ⊗ K ) ¯ g ! ( φ ) o o ¯ g ! f ∗ ω ′ Z ⊗ ¯ g ∗ f ∗ K α ◦ θ o o The mo rphism ¯ ψ induces a mor phism “ ¯ ψ ” deﬁned by the c o mpo sition ω ′ V ⊗ N ( c p Y , ¯ g ) − 1 ⊗ ¯ ψ / / ¯ g ! ω ′ Y ⊗ ¯ g ∗ L θ / / ¯ g ! ( ω ′ Y ⊗ L ) Theorem 6.3.7. (b ase change) L et ( f , φ ) , ( ¯ f , ¯ φ ) , ( g , ψ ) and ( ¯ g , ¯ ψ ) b e as ab ove. We assume that the diagr am ω ′ V ⊗ N “ ¯ ψ ” / / ¯ g ! ( ω ′ Y ⊗ L ) ¯ f ∗ ( ω ′ X ⊗ M ) ¯ φ O O ψ / / ¯ f ∗ ( ω X ⊗ g ∗ K ) “ φ ” O O 50 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL is c ommutative. Then, ε : f ∗ g ∗ → ¯ g ∗ ¯ f ∗ is a morphism of du ality pr eserving functors as in the fol lowing diagr am. C V ,ω ′ V ⊗ N | ¯ g , ¯ ψ | ∗ / / C Y ,ω ′ Y ⊗ L C X,ω ′ X ⊗ M | ¯ f , ¯ φ | ∗ O O | g, ψ | ∗ / / C Z,ω ′ Z ⊗ K | f ,φ | ∗ O O ε d l Q Q Q Q Q Q Q Q Q Q Q Q Q Q Corollary 6 .3.8. (b ase change for Witt gr oups) In the situation of the t he or em, assuming furthermor e that ( f , φ ) and ( ¯ f , ¯ φ ) ar e in B ∗ W and ( g , ψ ) and ( ¯ g , ¯ ψ ) ar e in B W ∗ and γ ω Z is an isomorp hism. Then t he pul l-b acks and push-forwar ds on Witt gr oups of deﬁnitions 6.2.6 and 6.3.4 satisfy | ¯ g , ¯ ψ | W ∗ | ¯ f , ¯ φ | ∗ W = | f , φ | ∗ W | g , ψ | W ∗ . Theorem 6.3. 9 . (pr oje ction formula) L et f : X → Y b e a morphism in B ′′ . L et K, M ∈ C X and L, N ∈ C Y b e obje cts, φ : f ∗ L → K , ψ : M → f ∗ N and χ : M ⊗ K → f ∗ ( N ⊗ L ) b e morphisms su ch that M ⊗ K χ / / ψ ⊗ I d   f ∗ ( N ⊗ L ) α − 1   f ∗ N ⊗ K I d ⊗ φ / / f ∗ N ⊗ f ∗ L is c ommutative. Then, the morphism π is du ality pr eserving b etwe en the c omp osi- tions in t he diagr am C X,ω ′ X ⊗ M × C X,K {⊗} / / C X,ω ′ X ⊗ M ⊗ K | f ,χ | ∗ ' ' O O O O O O O C X,ω ′ X ⊗ M × C Y ,L I d ×| f ,φ | ∗ 4 4 j j j j j j j j j | f ,ψ | ∗ × I d / / C Y ,N × C Y ,L π K S {⊗} / / C Y ,N ⊗ L Corollary 6. 3.10. (pr oje ction formula for Witt gr oups) In the situation of t he the or em, assuming furthermor e t hat ( f , φ ) is in B ∗ W and ( f , ψ ) and ( f , χ ) ar e in B W ∗ . Then the pul l-b ack, push-forwar ds and pr o duct for Witt gr oups of deﬁnitions 6.2.6 , 6.3.4 and Pr op osition 4.4.6 satisfy | f , χ | W ∗ ( x. | f , φ | ∗ W ( y )) = | f , ψ | W ∗ ( x ) .y . Appendix A. Signs in the ca tegor y of complexes Let E b e an exact categ ory E admitting inﬁnite countable dir e c t s ums and pro d- ucts, with a tenso r pro duct • adjo int to an in ternal Hom (denoted b y h ) in the sense o f Deﬁnition 1.3.1 . In this section, we explain how certa in signs have to b e chosen in o rder to induce a susp ended symmetric monoidal clo s ed structur e on the category of chain complexes of E (r esp. the homoto p y categor y , the der ived ca te- gory). W e use ho mological complexes , as it is the usual conv ention in the articles ab out Witt gro ups , s o the diﬀerential of a complex is d A i : A i → A i − 1 . The s usp ension functor T is ( T A ) n = A n − 1 and the tensor pro duct a nd the in ternal Hom are g iven by ( A ⊗ B ) n = M i + j = n A i • B j and [ A, B ] n = Y j − i = n h ( A i , B j ) . In table 2 , we give a po ssible choice of signs for the translation functor , tensor pro duct, the asso ciativit y mor phism (denoted by asso ), the sy mmetry morphis m TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 51 and the adjunction mo rphism (denoted by ath ), a nd what it induces on the internal Hom using P rop osition 1.5.8 . In table 3 further b elow, we state the compatibility that these signs must s a tisfy to ensure tha t all axioms o f susp ended symmetric monoidal closed categ o ries considered hold. Deﬁnition o f Sign Choice Lo cus T A ǫ T i − 1 d T A i +1 = ǫ T i d i A A ⊗ B ǫ 1 ⊗ i,j 1 ǫ 1 ⊗ i,j d A i • id B j ǫ 2 ⊗ i,j ( − 1) i ǫ 2 ⊗ i,j id A i • d B j tp 1 ,A,B ǫ tp 1 i,j 1 ǫ tp 1 i,j id A i • B j tp 2 ,A,B ǫ tp 2 i,j ( − 1) i ǫ tp 2 i,j id A i • B j asso A,B ,C ǫ asso i,j,k 1 ǫ asso i,j,k (( A i • B j ) • C k → A i • ( B j • C k )) s A,B ǫ s i,j ( − 1) ij ǫ s i,j ( A i • B j → B j • A i ) ath A,B ,C ǫ ath i,j ( − 1) i ( i − 1) / 2 ǫ ath i,j (Hom( A i • B j , C i + j ) → Hom( A i , h ( B j , C i + j ))) [ A, B ] ǫ 1 h i,j 1 ǫ 1 h i,j ( d A i +1 ) ♯ ǫ 2 h i,j ( − 1) i + j +1 ǫ 2 h i,j ( d B j ) ♯ th 1 ,A,B ǫ th 1 i,j 1 ǫ th 1 i,j id h ( A i ,B j ) th 2 ,A,B ǫ th 2 i,j ( − 1) i + j ǫ th 2 i,j id h ( A i ,B j ) T able 2. Sig n deﬁnitions Balmer [ 1 ], [ 2 ], Gille a nd Nenashev ,[ 6 ], [ 7 ] always co nsider strict dua lities, that is ǫ th 1 = 1 . The sig ns chosen in [ 2 , § 2.6] imply that ǫ 1 h i, 0 = 1 . The choices ma de by [ 7 , Example 1.4 ] are ǫ 1 ⊗ i,j = 1 and ǫ 2 ⊗ i,j = ( − 1 ) i . In [ 6 , p. 111] the signs ǫ 1 h i,j = 1 and ǫ 2 h i,j = ( − 1) i + j +1 are chosen. Finally , the sign c hosen for  in [ 6 , p. 112] cor resp onds via our deﬁnition of  (see Section 3.2 ) to the equality ǫ ath j − i,i ǫ ath i,j − i ǫ s j − i,i = ( − 1) j ( j − 1) / 2 . It is po ssible to choose the sig ns in a wa y compa tible with a ll these choices and o ur formalism. It is given in the third column of T able 2 . More precisely , we ha ve the following theor e m. Theorem A.0.11 . L et a, b ∈ { +1 , − 1 } . Then ǫ 1 ⊗ i,j = 1 ǫ tp 1 i,j = a ǫ 2 ⊗ i,j = ( − 1 ) i ǫ tp 2 i,j = a ( − 1 ) i ǫ 1 h i,j = 1 ǫ th 1 i,j = 1 ǫ 2 h i,j = ( − 1 ) i + j +1 ǫ th 2 i,j = a ( − 1 ) i + j ǫ ath i,j = b ( − 1) i ( i − 1) / 2 ǫ s i,j = ( − 1 ) ij ǫ T i = − 1 satisﬁes al l e qualities of T able 3 as wel l as ǫ ath j − i,i ǫ ath i,j − i ǫ s j − i,i = ( − 1 ) j ( j − 1) / 2 . Ther e- for e, for any exact c ate gory E the c ate gory of chain c omplexes C h ( E ) and its b ounde d variant C h b ( E ) m ay b e e qu ipp e d with the entir e stru ctur e of susp ende d symmetric monoidal c ate gory discusse d in Se ction 3.1 . Mor e over, al l signs may b e chosen in a c omp atible way with al l the ab ove sign choic es of Balmer, Gil le and Nenashev. Pr o of. Straig ht forward.  52 BAPTISTE CALM ` ES A ND JENS HORNBOSTEL compatibility r eason 1 ǫ 1 ⊗ i,j ǫ 1 ⊗ i,j − 1 ǫ 2 ⊗ i,j ǫ 2 ⊗ i − 1 ,j = − 1 A ⊗ B is a complex 2 ǫ ⊗ 1 i,j ǫ ⊗ 1 i,j + k ǫ ⊗ 1 i + j,k ǫ asso i,j,k ǫ asso i − 1 ,j,k asso is a mo rphism 3 ǫ ⊗ 2 i,j ǫ ⊗ 1 j,k ǫ ⊗ 1 i + j,k ǫ ⊗ 2 i,j + k ǫ asso i,j,k ǫ asso i,j − 1 ,k 4 ǫ ⊗ 2 i + j,k ǫ ⊗ 2 j,k ǫ ⊗ 2 i,j + k ǫ asso i,j,k ǫ asso i,j,k − 1 5 ǫ asso i,j,k ǫ asso i,j + k,l ǫ asso j,k,l ǫ asso i,j,k + l ǫ asso i + j,k,l = 1 the p entagon o f [ 9 , p. 252] co mm utes 6 ǫ 1 ⊗ i,j ǫ 2 ⊗ j,i ǫ s i,j ǫ s i − 1 ,j = 1 s A,B is a morphism 7 ǫ 1 ⊗ j,i ǫ 2 ⊗ i,j ǫ s i,j ǫ s i,j − 1 = 1 8 ǫ s i,j ǫ s j,i = 1 s is self-inverse 9 ǫ s j,k ǫ s i,k ǫ s i + j,k ǫ asso i,j,k ǫ asso k,i,j ǫ asso i,k,j = 1 the hex a gons of [ 9 , p. 253] commutes 10 ǫ T i ǫ T i + j ǫ 1 ⊗ i,j ǫ 1 ⊗ i +1 ,j ǫ tp 1 i,j ǫ tp 1 i − 1 ,j = 1 tp 1 ,A,B is a morphism 11 ǫ T i + j ǫ 2 ⊗ i,j ǫ 2 ⊗ i +1 ,j ǫ tp 1 i,j ǫ tp 1 i,j − 1 = 1 12 ǫ T j ǫ T i + j ǫ 2 ⊗ i,j ǫ 2 ⊗ i,j +1 ǫ tp 2 i,j ǫ tp 2 i,j − 1 = 1 tp 2 ,A,B is a morphism 13 ǫ T i + j ǫ 1 ⊗ i,j ǫ 1 ⊗ i,j +1 ǫ tp 2 i,j ǫ tp 2 i − 1 ,j = 1 14 ǫ tp 1 i,j ǫ tp 1 i,j +1 ǫ tp 2 i,j ǫ tp 2 i +1 ,j = − 1 the sq uare in Deﬁnition 1.4.12 anti-comm utes 15 ǫ tp 1 i,j ǫ tp 1 i + j,k ǫ tp 1 i,j + k ǫ asso i,j,k ǫ asso i +1 ,j,k = 1 assoc et al. commute 16 ǫ tp 2 i,j ǫ tp 1 i + j,k ǫ tp 2 i,j + k ǫ tp 1 j,k ǫ asso i,j,k ǫ asso i,j +1 ,k = 1 17 ǫ tp 2 i + j,k ǫ tp 2 i,j + k ǫ tp 2 j,k ǫ asso i,j,k ǫ asso i,j,k +1 = 1 18 ǫ tp 1 i,j ǫ tp 2 j,i ǫ s i,j ǫ s i +1 ,j = 1 the sq uare s comm utes 19 ǫ 1 h i,j ǫ 1 h i,j − 1 ǫ 2 h i,j ǫ 2 h i +1 ,j = − 1 [ A, B ] is a complex 20 ǫ 1 ⊗ i,j ǫ 2 ⊗ i,j ǫ 1 h j − 1 ,i + j − 1 ǫ ath i,j − 1 ǫ ath i − 1 ,j = − 1 ath is well deﬁned 21 ǫ 1 ⊗ i,j ǫ 2 h j,i + j ǫ ath i − 1 ,j ǫ ath i,j = 1 T able 3. Sig n deﬁnitions These structures trivially pass to the homoto py categor y . If the exact ca tegory E one consider s has enoug h injective and pr o jectiv e ob jects, one obtains a left der ived functor o f the tenso r pro duct and a r ig ht derived functor of the internal Hom which are exact in b oth v aria bles. References 1. P . Balmer, Triangular Witt Gr oups Part 1: The 12-Term Lo c alization Exact Sequenc e , K - Theory 4 (2000), no. 19, 311–363. 2. , T riangular Wit t gr oups. II. Fr om usual to derive d , Math. Z. 236 (2001), no. 2, 351–382. 3. B. Calm` es and J. Hornbostel, Push-forwar ds for Witt gr oups of schemes , prepri nt , 2008. 4. S. Eilenberg and G. M. Kelly , A gener alization of t he functional c alculus , J. Algebra 3 (1966), 366–375. 5. H . F ausk, P . Hu, and J. P . May , Isomorph isms b etwe en left and right adjoints , Theory Appl. Categ. 11 (2003), No. 4, 107–131 (electronic). 6. S. Gille, On Witt gr oups with supp ort , M ath. A nnalen 322 (2002), 103–137. 7. S. Gille and A. Nenashev, Pairings in triangular Witt theo ry , J. Algebra 2 61 (2003), no. 2, 292–309. TENSOR-TRIANGULA TED CA TEGORIES AND DUALITIES 53 8. G. M . Kell y and S. Mac Lane, Coher e nc e in close d c ategor ies , Journal of Pure and Applied Algebra 1 (1971), no. 1, 97–140. 9. S. Mac Lane, Cate gories for the working mathematician , second ed., Graduate T exts in Math- ematics, vol. 5, Spri nger-V erlag, N ew Y ork, 1998. 10. C. A. W eibel, An intr o duction to homolo g ic al algebr a , Cambridge Studies in Adv anced Math- ematics, vol. 38, Cambridge University Press, Cambridge, 1994.

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