THE AXIOMS FOR TRIANGULATED CATEGORIES

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1 THE AIOMS FOR TRIANGULATED CATEGORIES J. P. MA Contents 1. Trianulated cateories 1 2. Weak pusouts and weak pullbacks 4 3. How to prove Verdier s axiom 6 Reerences 9 Tis is an edited extract rom my paper [11]. We deine trianulated cateories and discuss omotopy pusouts and pullbacks in suc cateories in 1 and 2. We ocus on Verdier s octaedral axiom, since te axiom tat is usually rearded as te most substantive one is redundant: it is implied by Verdier s axiom and te remainin, less substantial, axioms. Stranely, since trianulated cateories ave been in common use or over tirty years, tis observation seemed to be new in [11]. We explain intuitively wat is involved in te veriication o te axioms in Trianulated cateories We recall te deinition o a trianulated cateory rom [15]; see also [2, 7, 10, 16]. Actually, one o te axioms in all o tese treatments is redundant. Te most undamental axiom is called Verdier s axiom, or te octaedral axiom ater one o its possible diarammatic sapes. However, te sape tat I ind most convenient, a braid, does not appear in te literature o trianulated cateories. It does appear in Adams [1, p. 212], wo used te term sine wave diaram or it. We call a diaram o te orm (1.1) a trianle and use te notation (,, ) or it. Z Σ Deinition 1.2. A trianulation on an additive cateory C is an additive selequivalence Σ : C C toeter wit a collection o trianles, called te distinuised trianles, suc tat te ollowin axioms old. Axiom T 1. Let be any obect and : be any map in C. (a) Te trianle id Σ is distinuised. (b) Te map : is part o a distinuised trianle (,, ). (c) Any trianle isomorpic to a distinuised trianle is distinuised. Axiom T 2. I (,, ) is distinuised, ten so is (,, Σ). 1

2 2 J. P. MA Axiom T 3 (Verdier s axiom). Consider te ollowin diaram. Z U V W Σ Σ Σ Σ ΣU Assume tat =, = Σ, and (,, ) and (,, ) are distinuised. I and are iven suc tat (,, ) is distinuised, ten tere are maps and suc tat te diaram commutes and (,, ) is distinuised. We call te diaram a braid o distinuised trianles enerated by = or a braid coenerated by = Σ. We ave labeled our axioms (T?), and we will compare tem wit Verdier s oriinal axioms (TR?). Our (T1) is Verdier s (TR1) [15], our (T2) is a weak orm o Verdier s (TR2), and our (T3) is Verdier s (TR4). We ave omitted Verdier s (TR3), since it is exactly te conclusion o te ollowin result. Lemma 1.3 (TR3). I te rows are distinuised and te let square commutes in te ollowin diaram, ten tere is a map k tat makes te remainin squares commute. i Z Σ Proo. Tis is part o te 3 3 lemma, wic we state and prove below. Te point is tat te construction o te commutative diaram in tat proo requires only (T1), (T2), and (T3), not te conclusion o te present lemma; compare [2, ]. Verdier s (TR2) includes te converse, (T2 ) say, o (T2). Tat too is a consequence o our (T1), (T2), and (T3). A standard arument usin only (T1), (T2), (TR3), and te act tat Σ is an equivalence o cateories sows tat, or any obect A, a distinuised trianle (,, ) induces a lon exact sequence upon application o te unctor C (A, ). Here we do not need te converse o (T2) because we are ree to replace A by Σ 1 A. In turn, by te ive lemma and te oneda lemma, tis implies te ollowin addendum to te previous lemma. Z k Σ Lemma 1.4. I i and in (TR3) are isomorpisms, ten so is k. Lemma 1.5 (T2 ). I (,, Σ) is distinuised, ten so is (,, ). Σi

3 THE AIOMS FOR TRIANGULATED CATEGORIES 3 Proo. Coose a distinuised trianle Z Σ. By (T2), te trianles ( Σ, Σ, Σ ) and ( Σ, Σ, Σ) are distinuised. By Lemmas 1.3 and 1.4, tey are isomorpic. By desuspension, (,, ) is isomorpic to (,, ). By (T1), it is distinuised. Similarly, we can derive te converse version, (T3 ) say, o Verdier s axiom (T3). Lemma 1.6 (T3 ). In te diaram o (T3), i and are iven suc tat (,, ) is distinuised, ten tere are maps and suc tat te diaram commutes and (,, ) is distinuised. Proo. Desuspend a braid o distinuised trianles enerated by = Σ. Lemma 1.7 (Te 3 3 lemma). Assume tat = i and te two top rows and two let columns are distinuised in te ollowin diaram. i i i Σ Σ Σ Σ Z k Σ Σi Z Σ k Σi Z Σ ΣZ k Σ Σ 2 Σi Ten tere is an obect Z and tere are dotted arrow maps,,, k, k, k suc tat te diaram is commutative except or its bottom rit square, wic commutes up to te sin 1, and all our rows and columns are distinuised. Proo. Te bottom row is isomorpic to te trianle ( Σ, Σ, Σ) and is tus distinuised by (T2); similarly te rit column is distinuised. Applyin (T1), we construct a distinuised trianle p V q Σ. Applyin (T3), we obtain braids o distinuised trianles enerated by and i. Tese ive distinuised trianles Z s V t Σ ΣZ suc tat s V t Z Σi Σ p = s, t p =, q s =, t = Σ q p = s i, t p =, q s = i, t = Σi q. Deine k = t s : Z Z. Ten k = and k = Σi, wic already completes te promised proo o Lemma 1.3. Deine = t s and apply (T1) to construct a distinuised trianle Z Σ.

4 4 J. P. MA Applyin (T3), we obtain a braid o distinuised trianles enerated by = t s. Here we start wit te distinuised trianles (s, t, Σi ) and (t, Σ, Σs), were te second is obtained by use o (T2). Tis ives a distinuised trianle Z k Z k V Σk ΣZ suc tat te squares let o and above te bottom rit square commute and t = k t and Σs k = Σs. Te commutativity (and anti-commutativity o te bottom rit square) o te diaram ollow immediately. It also ollows immediately tat (,, ) and (k, k, Σk) are distinuised. Lemma 1.5 implies tat (k, k, k ) is distinuised. 2. Weak pusouts and weak pullbacks In any cateory, weak limits and weak colimits satisy te existence but not necessarily te uniqueness in te deinin universal properties. Tey need not be unique and need not exist. Wen constructed in particularly sensible ways, tey are called omotopy limits and colimits and are oten unique up to non-canonical isomorpism. As we recall ere, tere are suc omotopy pusouts and pullbacks in trianulated cateories. Homotopy colimits and omotopy limits o sequences o maps in trianulated cateories are studied in [3, 13], but a complete teory o omotopy limits and colimits in trianulated cateories is not yet available. Te material in tis section is meant to clariy ideas and to describe a strentened orm o Verdier s axiom tat is important in te applications. Deinition 2.1. A omotopy pusout o maps : and : Z is a distinuised trianle (, ) Z (,k) W i Σ. A omotopy pullback o maps : W and k : Z W is a distinuised trianle Σ 1 W Σ 1i (,) Z (, k) W. Te sin is conventional and ensures tat in te isomorpism o extended trianles Σ 1 W Σ 1i (, ) Z (,k) W i Σ (id, id) Σ 1 W Σ 1i (,) Z (, k) W i Σ, te top row displays a omotopy pusout i and only i te bottom row displays a omotopy pullback. At tis point we introduce a eneralization o te distinuised trianles. Deinition 2.2. A trianle (,, ) is exact i it induces lon exact sequences upon application o te unctors C (, W ) and C (W, ) or every obect W o C. Te ollowin is a standard result in te teory o trianulated cateories [15]. Lemma 2.3. Every distinuised trianle is exact.

5 THE AIOMS FOR TRIANGULATED CATEGORIES 5 I (,, ) is distinuised, ten (,, ) is exact but enerally not distinuised. Tese exact trianles (,, ) ive a second trianulation o C, wic we call te neative o te oriinal trianulation. Problem 2.4. Te relationsip between distinuised and exact trianles as not been adequately explored in te literature. An additive cateory wit a iven Σ can admit several trianulations [5]. To see tis, deine a lobal automorpism o C to be a collection o automorpisms α : or all C wic commute wit all morpisms, α = α or all :, and satisy α Σ = Σ(α ). Te collection o trianles (,, ) suc tat (α, α, α) is distinuised in te oriinal trianulation ives a new trianulation on C. Te neative trianulation is an example. Dierent automorpisms can ive te same trianulation, but tere are trianulated cateories or wic tis construction ives ininitely many dierent trianulations. It is an open question weter or not (C, Σ) can admit two dierent trianulations tat are not obtained rom eac oter by a lobal automorpism. Te act tat te trianles in Deinition 2.1 ive rise to weak pusouts and weak pullbacks depends only on te act tat tey are exact, not on te assumption tat tey are distinuised. Tis motivates te ollowin deinition. Deinition 2.5. For exact trianles o te orm displayed in Deinition 2.1, we say tat te ollowin commutative diaram, wic displays bot a weak pusout and a weak pullback, is a puspull square. Lemma 2.6. Te central squares in any braid o distinuised trianles enerated by = are puspull squares. More precisely, wit te notations o (T3), te ollowin trianles are exact. Z W k (,) U Z (, ) V Σ V (, ) W Σ (, Σ) Proo. Altou rater lenty, tis is an elementary diaram case. Remark 2.7. We would like to conclude tat te trianles displayed in te lemma are distinuised and not ust exact. Examples in [12] imply tat tis is not true or all coices o and. Te braid in (T3) ives rise to a braid o distinuised trianles tat is coenerated by or, equivalently, enerated by Σ 1 ( ). Here Σ 1 ( ) = 0 since = Σ. Tis implies tat te central term in te braid splits as U Z. Application o (T3) ives a distinuised trianle Σ α U Z β V Σ. inspectin te relevant braid, we see tat α = (, ) and β = (, ). However, we cannot always replace and by and and still ave a distinuised trianle. Tis leaves open te possibility tat te trianles displayed in Lemma 2.6 are distinuised or some coices o and. It was stated witout proo in [2, ]

6 6 J. P. MA tat and can be so cosen in te main examples, and we sall explain wy tat is true in 3. It was suested in [2, ] tat tis conclusion sould be incorporated in Verdier s axiom i te conclusion were needed in applications. Tis course was taken in [10], and we believe it to be a sensible one. However, rater tan try to cane establised terminoloy, we oer te ollowin modiied deinition. Deinition 2.8. A trianulation o C is stron i te maps and asserted to exist in (T3) can be so cosen tat te two exact trianles displayed in Lemma 2.6 are distinuised. Remark 2.9. Neeman as iven an alternative deinition o a trianulated cateory tat is closely related to our notion o a stron trianulated cateory; compare [12, 1.8] and [13, 1.4]. It is based on te existence o particularly ood coices o te map k in (TR3). 3. How to prove Verdier s axiom We ere recall te standard procedure or provin Verdier s axiom (T3). Te exposition we ive is eneral, tinkin in terms o model cateories, but we also discuss te elementary down to eart version. Te reader unamiliar wit model cateory teory will be iven pointers to more direct proos in te literature. We assume tat our iven cateory C is te derived cateory or omotopy cateory obtained rom some Quillen model cateory B. One can ive eneral ormal proos o our axioms tat apply to te omotopy cateories associated to simplicial, topoloical, or omoloical model cateories tat are enriced over based simplicial sets, based spaces, or cain complexes, respectively. We sall be inormal, but we sall ive aruments in orms tat sould make it apparent tat tey apply equally well to any o tese contexts. An essential point is to be careul about te passae rom aruments in te point-set level model cateory B, wic is complete and cocomplete, to conclusions in its omotopy cateory C, wic enerally does not ave limits and colimits. We assume tat B is tensored and cotensored over te cateory in wic it is enriced. We ten ave canonical cylinders, cones, and suspensions, toeter wit teir Eckmann-Hilton duals. Te duals o cylinders are usually called pat obects in te model teoretic literature (altou in based contexts tat term mit more sensibly be reserved or te duals o cones). Wen we speak o omotopies, we are tinkin in terms o te canonical cylinder I or pat obect I, and we need not concern ourselves wit let versus rit omotopies in view o te adunction B( I, ) B(, I ). Hovey [6] ives an exposition o muc o te relevant backround material on simplicial model cateories. Discussions o topoloical model cateories appear in [4] and [9]. Homoloical model cateories appear implicitly in [6] and [8, III 1]. O course, we assume tat te unctor Σ on B induces a sel-equivalence o C. Te distinuised trianles in C are te trianles tat are isomorpic in C to a canonical distinuised trianle o te orm (3.1) i() C p() Σ in B. Here C = C, were C is te cone on, and i() and p() are te evident canonical maps. Ten (T1) is clear and (T2) is a standard arument wit

7 THE AIOMS FOR TRIANGULATED CATEGORIES 7 coiber sequences. One uses ormal comparison aruments (as in [15, II.1.3.2]) to reduce te veriication o (T3) in C to consideration o canonical coiber sequences in B. In B, one writes down te ollowin version o te braid in (T3). (3.2) i() Z C i() p() C Σ p() i() Σ C Σ Σi() ΣC p() Here =, and are evident canonical induced maps, = Σi() p(), and te diaram commutes in B. One proves (T3) by writin down explicit inverse omotopy equivalences ξ : C C and ν : C C suc tat = ν i() and = p() ξ. Details o te alebraic arument are in [15, pp ] and [16, p. 376], and te analoous topoloical arument is an illuminatin exercise. Tere is a standard and useul reormulation o te oriinal trianulation. Assumin, as can be arraned by coibrant approximation, tat is a coibration between coibrant obects, te quotient / is coibrant. Let M be te mappin cylinder o. Passae to pusouts rom te evident commutative diaram M ives a quotient map q() : C /. By [6, 5.2.6], we ave te ollowin standard result. It is central to our way o tinkin about trianulated cateories. Lemma 3.3. Let : be a coibration between coibrant obects. Ten te quotient map q() : C / is a weak equivalence. Now deine δ() : / Σ to be te map in C represented by te ormal connectin map q() (3.4) / C p() Σ in B. Observe tat (3.4) ives a unctor rom coibrations in B to diarams in B. Te composite q() i() : / is te evident quotient map, wic we

8 8 J. P. MA denote by (). Tereore, wen we pass to C, our canonical distinuised trianle (3.1) is isomorpic to te trianle represented by te diaram (3.5) () / δ() Σ in B, and our trianulation consists o all trianles in C tat are isomorpic to one o tis alternative canonical orm. Tis reormulation as distinct advantaes. Returnin to Verdier s axiom, we can replace te iven maps,, and tus = by coibrations between coibrant obects, and ten te quotient obects /, Z/ and Z/ are coibrant. Te point o Verdier s axiom now reduces to ust te observation tat Z/ is canonically isomorpic in B to (Z/)/(/). Usin our new canonical coibrations (3.5) startin rom,,, and te coibration : / Z/, we obtain te ollowin braid. (3.6) Z Z/ δ Z/ Σ δ Σ / Σ δ Σ Σ/ δ Expandin te arrows δ as in (3.4), we ind tat tis braid in C is represented by an actual commutative diaram in B, but o course wit some wron way arrows. Wit tis proo o Verdier s axiom, tere is no need to introduce te explicit omotopies ξ and ν o our irst proo. Modulo equivalences, te two central braids in (3.6) are as ollows. Here and later, we enerally write C(, ) instead o C or a iven coibration :. / Z Z/ C(Z, ) Σ C(Z, ) Tese are bot pusouts in wic te orizontal arrows are coibrations and all obects are coibrant. By te ollowin lemma, tis implies tat, in C, tese two squares ive puspull diarams tat arise rom distinuised trianles. We conclude tat C is stronly trianulated in te sense o Deinition 2.8. Lemma 3.7. Suppose iven a pusout diaram in B, Σ Z k W,

9 THE AIOMS FOR TRIANGULATED CATEGORIES 9 in wic and tereore k are coibrations and all obects are coibrant. Ten tere is a distinuised trianle (,) Z (,k) W Σ. in C. Tus te oriinal square ives rise to a puspull square in C. Proo. Standard topoloical aruments work model teoretically to ive a weak pusout (double mappin cylinder) M(, ) in B wic its into a canonical trianle (,k ) M(, ) Σ δ Σ Σ as in (3.5). It is easy to ceck tat δ = (, ) in C and tat tere is a weak equivalence M(, ) W under Z in B. Te conclusion ollows. Reerences [1] J.F. Adams. Stable omotopy and eneralized omoloy. Te University o Cicao Press [2] A.A. Beilinson, J. Bernstein, and P. Deline. Faisceaux pervers. Astérisque 100(1982), [3] M. Bökstedt and A. Neeman. Homotopy limits in trianulated cateories. Compositio Mat. 86(1993), [4] A. Elmendor, I. Kriz, M. A. Mandell, and J. P. May. Rins, modules, and alebras in stable omotopy teory. Amer. Mat. Soc. Matematical Surveys and Monoraps Vol [5] A. Heller. Bull. Amer. Mat. Soc. 74 (1968). [6] M. Hovey. Model cateories. Matematical Surveys and Monoraps Vol American Matematical Society. [7] M. Hovey, J. H. Palmieri, and N. P. Strickland. Axiomatic stable omotopy teory. Memoirs Amer. Mat. Soc. No [8] I. Kriz and J.P. May. Operads, alebras, modules, and motives. Astérisque. No [9] M. A. Mandell, J. P. May, S. Scwede, and B. Sipley. Model cateories o diaram spectra. Proc. London Mat. Soc. To appear. [10] H. Marolis. Spectra and te Steenrod alebra. Nort-Holland [11] J.P.May. Te additivity o traces in trianulated cateories. Advances in Matematics 163(2001), [12] A. Neeman. Some new axioms or trianulated cateories. J. Alebra 139(1991), [13] A. Neeman. Trianulated cateories. Preprint, 465 paes. [14] D. G. Quillen. Homotopical alebra. Spriner Lecture Notes in Matematics Volume [15] J.L. Verdier. Catéories dérivées, in Spriner Lecture Notes in Matematics Vol 569, [16] An introduction to omoloical alebra. Cambride University Press