Higher symmetries in abstract stable homotopy theories
aa r X i v : . [ m a t h . A T ] A p r HIGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPYTHEORIES
MORITZ GROTH AND MORITZ RAHN
Abstract.
This survey offers an overview of an on-going project on uniformsymmetries in abstract stable homotopy theories. This project has calcula-tional, foundational, and representation-theoretic aspects, and key featuresof this emerging field on abstract representation theory include the follow-ing. First, generalizing the classical focus on representations over fields, itis concerned with the study of representations over rings, differential-gradedalgebras, ring spectra, and in more general abstract stable homotopy theo-ries. Second, restricting attention to specific shapes, it offers an explanationof the axioms of triangulated categories, higher triangulations, and monoidaltriangulations. This has led to fairly general results concerning additivity oftraces. Third, along similar lines of thought it suggests the development ofabstract cubical homotopy theory as an additional calculational toolkit. Aninteresting symmetry in this case is given by a global form of Serre duality.Fourth, abstract tilting equivalences give rise to non-trivial elements in spec-tral Picard groupoids and hence contribute to their calculation. And, finally,it stimulates a deeper digression of the notion of stability itself, leading tovarious characterizations and relative versions of stability.
Contents
1. Introduction 22. A representation-theoretic perspective on triangulated categories 82.1. Functorial cones 92.2. Derived limits and derived Kan extensions 142.3. A -quivers and Barratt–Puppe sequences 202.4. A -quivers and refined octahedral diagrams 243. A crash course on derivators 313.1. Derivators 323.2. Stable derivators 383.3. Exponentials of derivators 453.4. Morphisms of derivators 484. Higher symmetries 544.1. Strong stable equivalences 544.2. Abstract representation theory of A n -quivers 584.3. Reflection functors 684.4. Digression: monoidal derivators 734.5. Universal tilting modules 83References 94 Date : April 2, 2019. Introduction
A good part of modern mathematics is centered around the study of symme-tries. Symmetries arise in various parts of mathematics, and in many specificsituations good control over the available symmetries is helpful in (if not crucialto) applications. Here we are interested in global symmetries which are common toall abstract stable homotopy theories. In one form or another (see further below),abstract stable homotopy theories are in the background of many areas of puremathematics. This certainly includes algebra and representation theory, homotopytheory and algebraic topology as well as algebraic geometry. The overall goal of thisproject is a detailed study of uniform symmetries which arise in these fields. Thecredo is that some of these symmetries provide a convenient calculational toolkitthat is common to all these situations. Other symmetries are more interesting froman abstract representation-theoretic perspective.In order to fill this first paragraph with more life, for the time being we focuson triangulated categories as one of the classical approaches to an “abstract stablehomotopy theory”. Triangulated categories originated in algebraic geometry in thestudy of derived categories D ( X ) of schemes [Ver67, Ver96] and in homotopy theoryin the study of the stable homotopy category of spectra SHC [Boa64, Pup67, Vog70].In both of these situations, the classical triangulations on D ( X ) and SHC encodeaspects of the calculus of cones or, equivalently, derived cokernels – as it is visibleto the respective category alone. And symmetries show up very prominently inthe definition of a triangulated category [Nee01, HJR10]. First, a triangulatedcategory T is by definition endowed with a suspension functor Σ : T ∼ −→ T whichis an equivalence of categories. Second, the rotation axiom allows us to rotatedistinguished triangles forward and backwards. Triangulated categories also encodeinformation about pairs of composable morphisms, but here the classical axiomsmiss some of the existing symmetries.Correspondingly, our first goal is to give a different explanation of the axioms ofa triangulated category. In order to achieve this in a fairly elementary way, in § R . Themain goal in that section is to give a somewhat unorthodox construction of theclassical Verdier triangulation on the derived category D ( R ). In fact, we take arepresentation-theoretic perspective and study representations of the A n -quivers ~A = 1 , ~A = (1 → , and ~A = (1 → → A n -quivers. For instance, a representation(1.1) x f → y g → z of ~A in chain complexes can be equivalently encoded by a diagram of chain com-plexes as in Figure 1. In that diagram all squares are derived pushouts (or, equiv-alently, derived pullbacks) and the diagram vanishes on the boundary stripes.A suitable restriction of Figure 1 gives rise to the octahedral diagram of the rep-resentation (1.1). For instance, the key distinguished triangle from the octahedral Always begin an introduction by drawing a big picture.
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 3 ... ... ... ... ...0 / / ˜ z (cid:15) (cid:15) / / ˜ v / / (cid:15) (cid:15) ˜ w (cid:15) (cid:15) / / (cid:15) (cid:15) / / x f / / (cid:15) (cid:15) y g / / (cid:15) (cid:15) z (cid:15) (cid:15) / / (cid:15) (cid:15) / / u / / (cid:15) (cid:15) v / / (cid:15) (cid:15) x ′ (cid:15) (cid:15) / / (cid:15) (cid:15) / / ... w / / ... y ′ / / ... u ′ / / ... 0 ... Figure 1.
A symmetric presentation of representations of ~A .axiom (“the cone of cones is a cone”) is induced by the square 1 which exhibits thecone C ( g ) as the cone of C ( f ) → C ( gf ) (see Remark 2.52 for more details). Butthe diagrams in Figure 1 have the advantage that there are two obvious symmetryoperations. For instance, there is the symmetry which shifts such diagrams alongthe diagonal. And this symmetry results in a relation between the octahedraldiagram of (1.1) and the one of the induced representation C ( f ) → C ( gf ) → Σ x. Also representations of ~A and ~A can be encoded by similar symmetric diagrams(see Figure 3 and Figure 7). And in that case the symmetries induce the familiarrotation symmetry and suspension at the level of D ( R ).Following this line of thought, we are led to a construction of the classical Verdiertriangulation on D ( R ). The way we present this construction suggests that thesearguments do not rely in an essential way on the specifics of chain complexes in R -modules. Instead, only formal aspects of the calculus of chain complexes were in-voked, namely the existence of a well-behaved calculus of derived limits and colimitswith the following additional key properties.(i) There is a zero object.(ii) A square is a derived pushout square if and only if it is a derived pullbacksquare.These two properties are the typical defining features of an “abstract stable ho-motopy theory”. There is an entire zoo of models all of which make such a no-tion precise. This zoo includes stable cofibration categories [Sch13, Len17], stablemodel categories [Hov99], stable ∞ -categories [Lur09, Lur14], and stable derivators[Hel97, Fra96, Mal01a] (see Disclaimer 3.1 for more references). For many of ourpurposes the precise choice of the model is not too relevant. In fact, many of theproofs are quite formal (and that is one of the points of the project) so that they MORITZ GROTH AND MORITZ RAHN are essentially model independent. Let us hasten to emphasize that we believe indiversity of technology.These abstract stable homotopy theories are enhancements of triangulated cat-egories. In fact, in the above cases one can use the defining exactness propertiesof stability in order to construct canonical triangulations. More interesting thanthis result itself are the techniques leading to its proof. These are developed in § ∞ -categories [Lur14,Prop. 1.1.3.1] but not by triangulated categories) is the following. Given a stablederivator D and a small category A there is the stable derivator D A of represen-tations of shape A with values in D (exponentials exist in stable derivators). Andthis opens the door for the study of symmetries of stable homotopy theories of rep-resentations of more complicated shapes than the A n -quivers for n = 1 , , § A we associate to it the 2-functor(1.2) ( − ) A : D ER St , ex → D ER : D D A which sends a stable derivator D to the stable derivator of A -shaped represen-tations in D . By specializing to specific D , this yields the homotopy theory ofrepresentations over fields, over rings, in quasi-coherent modules over schemes, ofdifferential-graded or of spectral representations (see Examples 4.2 for details). Wethink of ( − ) A as a gadget that neatly encodes the abstract representation theoryof the shape A . Correspondingly, given two small categories A and B (which couldbe the same), a strong stable equivalence Φ : A s ∼ B is a family of equivalencesΦ D : D A ≃ D B , D a stable derivator,which is pseudo-natural with respect to exact morphisms in D . As a slogan, such astrong stable equivalence shows that A and B have the same abstract representationtheory (see § A s ∼ B reveals a potentially interestingsymmetry in abstract stable homotopy theories. However, we also believe in thefollowing slogan. “Some shapes are more important than others.”In fact, some shapes are more foundational than others, and strong stable equiva-lences between such shapes buy us more. We illustrate this by the following threeshapes which are closely related to triangulated categories, higher triangulated cat-egories and the Waldhausen S • -construction, and monoidal triangulated categoriesand Goodwillie calculus.(i) The abstract representation theories of the Dynkin quivers ~A , ~A , and ~A areintimately related with the axioms of triangulated categories. This relation isprovided through the calculus of derived (co)kernels of morphisms and pairsof composable morphisms [Fra96, Mal01a, Gro13, Gro16]. Similarly, longerchains of composable morphisms are the same as abstract representations ofDynkin quivers ~A n = (1 → → . . . → n ) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 5
And their abstract representation theory is directly connected with higheroctahedral diagrams and ∞ -triangulated categories [BBD82, Mal05a, Bal11,GˇS16a]. Interesting symmetries are made visible by diagrams as in Wal-hausen’s S • -construction in algebraic K -theory [Wal85, GˇS16a]. Higher-dimensional versions of these diagrams have recently been studied in [Pog17,Bec18, DJW18].(ii) Similarly, the abstract representation theory of the commutative square (cid:3) = ~A × ~A is closely related to refined axioms for monoidal, triangulated or tensor trian-gulated categories. Of course, representations of the square arise in monoidalcontexts as pointwise products of pairs of morphisms. A prominent operationin this context is the pushout product operation (Example 4.62). Shadowsof this operation together with various compatibilities with rotations of thetwo morphisms, symmetry, and dualization have been axiomatized by May[May01]. The square is strongly stably equivalent to the trivalent sourceand this offers a more representation-theoretic perspective on May’s axioms[KN02, GˇS14].This calculus leads to applications in the study of dualizability phenom-ena. More specifically, part of Grothendieck’s original motivation to lookfor enhancements of triangulated categories was the following observation:traces of endomorphisms do not satisfy additivity at the level of triangu-lated categories (see Remark 4.72 for a precise formulation). In [GPS14a]a result concerning additivity of traces is established for stable, monoidalderivators (following May’s proof [May01] for model categories). And, again,possibly more interesting than the additivity result are the techniques devel-oped in that paper. (For further developments of this additivity result see[PS16, PS14, GAdS14, JY18].)(iii) Finally, also the abstract representation theory of the n -cube (cid:3) n = ~A × . . . × ~A and some of its subposets is fairly foundational. It leads to a detailed under-standing of the calculus of iterated partial cofiber constructions, total cofibers,and of an interpolation between cocartesian and strongly cocartesian n -cubes[BG18]. Such shapes show up naturally in monoidal contexts and also inGoodwillie calculus [Goo92, MV15]. Moreover, there is a global form of anabstract Serre duality on representations of these posets [BG18, § MORITZ GROTH AND MORITZ RAHN of tilting theory have interesting counterparts in the context of abstract stablehomotopy theories.First, this is the case for the classical BGP reflection functors. Let us recall thatBernˇste˘ın, Gel ′ fand, and Ponomarev [BGP73] introduced reflection functors andused them to give an elegant proof of Gabriel’s classification result of connectedhereditary representation-finite algebras over an algebraically closed field [Gab72].For every quiver Q and every source or sink q ∈ Q , the reflected quiver Q ′ = σ q Q of Q is obtained by reversing the orientations of all edges adjacent to q . Associatedwith these reflections at the level of quivers are reflection functorsMod( kQ ) → Mod( kQ ′ )between the module categories of the corresponding path algebras. While thesereflection functors fail to be equivalences, Happel [Hap87] showed that for acyclicquivers derived reflection functors are exact equivalences D ( kQ ) ∆ ≃ D ( kQ ′ ) . Later Ladkani [Lad07a] established a similar equivalence D ( A Q ) ≃ D ( A Q ′ ) forarbitrary abelian categories. It was shown in [GˇS16b, GˇS15] that Happel’s theoremcan be strengthened further by showing that reflection functors yield a strong stableequivalence Q s ∼ Q ′ . And this also holds for not necessarily finite or acyclic quivers. By purely combi-natorial arguments one concludes from this that if T is a finite oriented tree and if T ′ is an arbitrary reorientation of T , then there is a strong stable equivalence T s ∼ T ′ . Using the more flexible framework of stable ∞ -categories further generalizationswere recently constructed by Dyckerhoff, Jasso, and Walde [DJW19]. In loc. cit. certain procedures to glue stable ∞ -categories are developed, and this gluing worksneither for triangulated categories nor for stable derivators. These techniques arethen applied to offer a fairly general construction of reflection functors for stable ∞ -categories. Their result specializes both to the reflection functors in [GˇS16b, GˇS15]and also to additional examples considered by Ladkani [Lad07a, Lad07c, Lad07b].A second aspect from tilting theory that generalizes nicely is the calculus oftilting complexes. Recall that these are chain complexes of bimodules such thatthe corresponding derived tensor and hom functors are derived equivalences. Toobtain a counterpart of this for abstract stable homotopy theories, one invokes theuniversality of the homotopy theory of spectra: it is the free stable homotopy theorygenerated by the sphere spectrum. In the framework of derivators references relatedto this result include [Hel88, Hel97, Fra96, Cis08, Tab08, CT11, CT12] (but see also[Dug01b, Len10] and [Lur09, Lur14, GGN15] for closely related results in modelcategories and ∞ -categories, respectively). As a consequence of this universalityand Brown representability, every stable derivator is enriched over spectra. Andthis allows us to define tensor and hom functors associated to spectral bimodules(aka. weighted colimits and weighted limits [Kel05b]).It turns out that the strong stable equivalences mentioned above are inducedby universal tilting modules . These are certain spectral bimodules and the termi-nology is justified by the following two facts. First, these bimodules realize strong IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 7 stable equivalences in arbitrary stable derivators and, second, they are spectral re-finements of the classical tilting complexes. In fact, the classical tilting complexesare recovered from the universal tilting modules by smashing with the Eilenberg–MacLane spectrum. One convenient feature of these spectral bimodules is thatthey can be calculated quite explicitly. For instance, there are explicit spectralbimodules that govern the calculus of homotopy finite limits and colimits. We il-lustrate this in § equivalences and they are conse-quently invertible spectral bimodules. Considered from this perspective, the con-struction of strong stable equivalences contributes to the calculation of spectralPicard groupoids. To the best of the knowledge of the author, the only shapefor which the spectral Picard group is known is the trivial shape A = ∗ . In thatcase there is an isomorphism Z ∼ = Pic S p ( ∗ ) and a generator is given by S = Σ S ([HMS94] or [Str92, Thm. 2.2]). One of the current goals of this project is to obtaina calculation of spectral Picard groups at least for some shapes, but for the timebeing only first steps have been achieved (Remark 4.109).We refrain from giving a more detailed description of the content. Instead werefer the reader to the respective introductions of the individual sections § § § Philosophy of this writing.
It was our main goal to write an account which isreadable to representation theorists, to triangulated category theorists, and also to(abstract) homotopy theorists. This explains the high level of detail which is offeredat various places, and we want to apologize for this to these different communities.Still our feeling is that this is a reasonable compromise. As additional guidingprinciples, we tried to stress the main ideas in the project, we refer to the originalliterature for most of the proofs, we advertise a few loose ends which are to bepursued further, and we include many references to indicate the various connectionsto other parts of mathematics.The level of sophistication increases from section to section. In § § § Introduction of new coauthor.
It is very tempting to make various jokessuch as “the first author thanks the second author for having joined the projectwhile the second author in turn thanks the first author for fruitful discussions”.But to cut this short, the author recently got married and changed his last namefrom Groth to Rahn. All future publications will appear under this new last name.
Acknowledgments.
It is a great pleasure to begin by thanking my Ph.D.supervisor Stefan Schwede for having suggested to me to think about abstracthomotopy theory in the first place. My understanding of the subject was sharpenedthrough various more or less directly related cooperations, hence special thanksgo to my coauthors Falk Beckert [BG18], David Gepner and Thomas Nikolaus
MORITZ GROTH AND MORITZ RAHN [GGN15], Kate Ponto and Mike Shulman [GPS14b, GPS14a], again Mike Shulman[GS17] and, last but not least (watch the alphabetic order!), Jan ˇSˇtov´ıˇcek [GˇS14,GˇS16b, GˇS16a, GˇS15]. During the last years many colleagues and friends gave me(willingly or not) the opportunity to discuss this project and/or to have a good time.It is a great pleasure to thank Dimitri Ara, Peter Arndt, Denis-Charles Cisinski, IvoDell’Ambrogio, David Gepner, Drew Heard, Gustavo Jasso, Andr´e Joyal, PhilippJung, Bernhard Keller, Steffen K¨onig, Henning Krause, Rosie Laking, GeorgesMaltsiniotis, Thomas Nikolaus, Justin No¨el, Eric Peterson, Mona Rahn, GeorgeRaptis, Ulrich Schlickewei, Timo Sch¨urg, Stefan Schwede, Mike Shulman, JohanSteen, Greg Stevenson, and Jan ˇSˇtov´ıˇcek.The author also thanks the organizers of the International Conference on Repre-sentation theory of Algebras in Prague in August 2018 (ICRA2018). On occasionof that conference the author offered a series of three talks aiming for an overviewof this project. The sections of this account are in obvious bijection with the talksgiven at ICRA2018, but here we intend to draw a slightly more complete picture.2.
A representation-theoretic perspective on triangulatedcategories
Triangulated categories were introduced in the 1960’s, originally motivated bysituations in algebraic geometry [Ver67, Ver96] and topology [Boa64, Pup67, Vog70].Since their invention triangulated categories have become a valuable tool in manyareas of pure mathematics and this ubiquity is one of the features of triangulatedcategories. In algebraic geometry triangulated categories arise as derived categoriesof schemes ([Ver67, Ver96] or [Huy06]), in representation theory they come up asderived categories of algebras (see [Hap88] or [AHHK07]), in modular representationtheory as stable module categories [BCR97], and in homotopy theory as homotopycategories of spectra or related (stable model) categories ([Vog70] or [Hov99]). Nicesurveys on this ubiquity of triangulated categories can be found in [HJR10], in[Bal10], and from a different perspective in [SS03].In this first section we want to contribute to the understanding of the axioms oftriangulated categories. The main idea we intend to transport is that the axiomsof triangulated categories are closely related to (abstract) representation theoryof A -quivers (!), A -quivers, and A -quivers. To keep things simple and veryexplicit, in most of this section we specialize to chain complexes of modules over aring R . Assuming only elementary techniques from homological algebra, we revisitthe classical Verdier triangulation on the derived category D ( R ) of a ring and offera somewhat unorthodox construction of it. In § § § § ~A -representations. In § ~A and encodethese in terms of refined ocatahedral diagrams. We discuss at some length in which IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 9 sense these diagrams are more symmetric versions of the diagrams in the classicaloctahedral axiom.2.1.
Functorial cones.
To set the stage, let us begin by establishing some basicnotation.
Notation 2.1.
Throughout this section, A denotes an abelian category. From § A = Mod( R ), the category of (not necessarily finitelygenerated) left modules over a (not necessarily commutative) ring R .We recall that abelian categories essentially axiomatize the usual calculus of finitedirect sums of modules and kernels and cokernels of homomorphisms of modules.In classical homological algebra, the focus is on additive functors between abeliancategories and on an investigation of their behavior with respect to short exactsequences. In particular, various techniques are developed in order to approximatea given functor by exact functors in a universal way. To make such a statementmore specific, we collect the following lemma. Notation 2.2.
For every abelian category A , we denote by Ch( A ) the category ofunbounded chain complexes in A . Lemma 2.3.
Let A and B be abelian categories and let F : A → B be an additivefunctor. The following are equivalent.(i) The functor F : A → B is exact.(ii) The induced functor F : Ch( A ) → Ch( B ) preserves quasi-isomorphisms.Proof. The passage to homology objects is obtained by the formation of subquo-tients, and the proof is hence straightforward. (cid:3)
This suggests that the above-mentioned ‘exact approximations’ of more generaladditive functors are defined at the level of derived categories.
Construction . For every abelian category A , we denote by W A the class of quasi-isomorphisms in Ch( A ). Up to set-theoretic considerations, there is the localizationfunctor(2.5) Ch( A ) γ / / D ( A ) = Ch( A )[( W A ) − ] , which universally inverts the quasi-isomorphisms W A [GZ67]. In the case of amodule category A = Mod( R ) we use the standard notation Ch( R ) → D ( R ).It is common to think of the derived category D ( A ) as a rather refined invariantof the abelian category A . A good deal of information about homological invariantsis encoded by derived categories, and many such invariants are invariant underderived equivalences (see, for instance, [Kel07b] and the many references therein).Here we want to take a different, but related perspective and focus on formalproperties of derived categories. The reason that we drew the morphism in (2.5) insuch a long way is that we want to stress that the categories Ch( A ) and D ( A ) livein rather different worlds. The category Ch( A ) is an abelian category and henceenjoys rather nice properties. In particular, it has all finite limits and finite colim-its. In contrast to this, the category D ( A ) is rather ill-behaved (the 1-categoricallocalization procedure is an act of violence, and it destroys these nice properties).There are two relevant ways to make this more precise. (i) First, derived categories D ( A ) tend to not have many limits or colimits. Infact, the only typical (co)limits which exist are finite biproducts and in somecases also infinite (co)products.(ii) More importantly, the calculus of derived (co)limits is not visible to the cat-egory D ( A ) alone .There are various ways to deal with these observations (see Disclaimer 3.1 andreferences there), and here we begin by following Verdier and Grothendieck. Thefollowing classical theorem offers one way to react upon the above issues, and theremaining goal of this first section is to explain that result from the perspective ofsymmetries. Theorem 2.6 (Verdier [Ver67, Ver96]) . The derived category D ( A ) “is” a trian-gulated category. In the formulation of this theorem, we were rather picky and put the verb intoquotation marks. This is in order to stress that a triangulation amounts to the spec-ification of additional structure on D ( A ) (in contrast to asking for mere properties of D ( A )). In particular, this means that there are(i) an auto-equivalence Σ : D ( A ) ∼ −→ D ( A ) and(ii) a class of distinguished triangles x f → y → z → Σ x in D ( A ),and these are subject to certain axioms [Nee01, HJR10]. Given such a distinguishedtriangle, the object z is referred to as “the” cone or cofiber of f . This time, thequotation marks are meant to allude to the following well-known fact. Approachedthis way, cones are only unique up to non-canonical isomorphism and only weaklyfunctorial.The following proposition describes an alternative approach to the cone con-struction. It exhibits the cone as the total left derived cokernel functor, therebyguaranteeing that the cone is a canonical and functorial construction. Proposition 2.7.
Let A be an abelian category, let [1] be the poset (0 < , let A [1] be the category of morphisms in A , and let cok: A [1] → A be the cokernel functor.(i) The cokernel functor is right exact.(ii) The cokernel functor is exact on monomorphisms.(iii) The cokernel functor has a total left derived functor given by the cone con-struction, C ∼ = L cok: D ( A [1] ) → D ( A ) . Proof.
A short exact sequence in A [1] is simply a morphism of short exact sequencesin A , 0 / / x ′ / / f ′ (cid:15) (cid:15) x / / f (cid:15) (cid:15) x ′′ / / f ′′ (cid:15) (cid:15) / / y ′ / / y / / y ′′ / / . (i) This is an immediate consequence of the snake lemma, which yields a 6-termexact sequence0 → ker( f ′ ) → ker( f ) → ker( f ′′ ) → cok( f ′ ) → cok( f ) → cok( f ′′ ) → . (ii) In the case of monomorphisms, the above sequence reduces to a short exactsequence, so also this statement is immediate from the snake lemma. IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 11 (iii) While the third statement does not follow as directly from the snake lemma,the snake lemma suggests what we are supposed to do. Namely, we shouldapproximate an arbitrary chain map up to quasi-isomorphism by a monomor-phic one and apply the cokernel to it instead. The details of this are as follows.Given a chain map f : x → y , we note that cok( f ) is the pushout in thequadrilateral on the right in(2.8) x f / / i (cid:15) (cid:15) y cof ( f ) (cid:23) (cid:23) ✵✵✵✵✵✵✵ x f / / (cid:15) (cid:15) y (cid:25) (cid:25) ✹✹✹✹✹✹✹✹ Cx ' ' ❖❖❖❖❖❖❖ ( ( PPPPPPP Cf / / cok( f ) . The intended monomorphic approximation up to quasi-isomorphism is dis-played in the remaining quadrilateral. Therein, x Cx denotes the clas-sical functor which sends a chain complex to its cone (see, for instance,[GM03, Wei94]). The relevant key facts about this construction are that,first, the cone Cx is an acyclic complex and, second, it comes with a naturalmonomorphic chain map i : x → Cx . Now, the cone Cf of the chain map f is defined by the pushout square on the left in (2.8). Note that, since theinduced map x → Cx ⊕ y is a monomorphism, by the first two parts the result-ing functor C : Ch( A [1] ) → Ch( A ) : f Cf preserves quasi-isomorphisms.Consequently, it descends to a cone functor at the level of derived categories C : D ( A [1] ) → D ( A ) . In order to exhibit this as the total left derived functor of the cokernel, itonly remains to construct a corresponding natural transformation. Since thecone Cx is acyclic, the unique chain map Cx → x and id y induces a levelwise quasi-isomorphism( Cx ← x → y ) → (0 ← x → y ) , i.e., the intended monomorphic resolution. At the level of pushouts this yieldsa natural chain map Cf → cok( f ) (the dotted arrow in (2.8)), which in turngives rise to the intended natural transformation inCh( A [1] ) C / / γ (cid:15) (cid:15) Ch( A ) γ (cid:15) (cid:15) D ( A [1] ) C / / D ( A ) . ✡✡✡✡ A I ε Standard techniques imply that (
C, ε ) is a model for L cok . (For more details and a more systematic explanation we refer to [Gro18, § (cid:3) We want to briefly discuss this result.
Remark . For every abelian category A the category of morphisms A [1] is againabelian. It is important to distinguish the two categories D ( A [1] ) and D ( A ) [1] , the derived category of the morphism category and the morphism category of thederived category . A relation between these categories is provided by a forgetfulfunctor dia [1] : D ( A [1] ) → D ( A ) [1] which is constructed as follows. An object X ∈ D ( A [1] ) is a chain complex ofmorphisms which we can hence rewrite as a functor X : [1] → Ch( A ). Written thisway, we can compose X with the localization functor γ : Ch( A ) → D ( A ), and wemake the definition dia [1] ( X ) = γ ◦ X : [1] → D ( A ) . The crucial observation now is that this functor discards relevant information. Inparticular, the functorial cone from Proposition 2.7 does not factor through thisunderlying diagram functor, D ( A [1] ) C / / dia [1] (cid:15) (cid:15) D ( A ) D ( A ) [1] . ∄ C : : ✉✉✉✉✉✉✉✉✉ (This fails already for vector spaces over a field as is detailed in [Gro18, § derived category of the morphism category but not on the morphism categoryof the derived category . In order to develop more intuition for these underlyingdiagram functors, we collect the following straightforward result [BG18, Cor. 3.9]. Example . Let R be a semi-simple ring, let B be a category with finitely manyobjects, and let RB be the R -linear category algebra. There is a similar underlyingdiagram functor dia B : D (Mod( R ) B ) → D ( R ) B : X γ ◦ X, which is equivalent to the Z -graded homology functor H ∗ : D ( RB ) → Mod( RB ) Z .This example illustrates very nicely that underlying diagram functors discardrelevant information. It turns out that not only these underlying diagram functorstend to fail to be equivalences, but also that the categories D ( A B ) and D ( A ) B are not equivalent (see [Gro18, §
5] for an explicit example). Moreover, there is nosimple recipe which allows us to reconstruct derived categories of diagram categories(categories of the form D ( A B )) from the derived category D ( A ). If we want tostudy derived (co)limits at the level of derived categories only (in contrast to theapproaches in Disclaimer 3.1), then we have to keep track of the categories D ( A B )themselves.Before we state a general theorem about the existence of derived (co)limits(Theorem 2.19), we give an elementary proof in the following central example. We IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 13 denote by p the full subposet of the cube looking like:(2.11) (0 , / / (cid:15) (cid:15) (0 , , A p is the category of spans in A and thecolimit functor(2.12) colim p : A p → A simply forms pushouts. Proposition 2.13.
Let A be an abelian category. The pushout functor (2.12) hasa total left derived functor L colim p : D ( A p ) → D ( A ) . Proof.
The strategy of the proof is very similar to the one in Proposition 2.7, andwe are more sketchy. As a left adjoint the pushout functor is right exact, and itturns out to be exact on those spans x / / (cid:15) (cid:15) yz, such that the induced map x → y ⊕ z is a monomorphism. Functorial resolutionsadapted to the pushout are hence given by any of the spans x / / (cid:15) (cid:15) y x / / (cid:15) (cid:15) Cx ⊕ y x / / (cid:15) (cid:15) Cx ⊕ yCx ⊕ z, z, Cx ⊕ z, together with the obvious maps collapsing the cones. Choosing for instance theresolution of the left, the derived pushout functor L colim p : D ( A p ) → D ( A )is induced by applying colim p to these resolution. The canonical natural transfor-mation Ch( A p ) colim p / / γ (cid:15) (cid:15) Ch( A ) γ (cid:15) (cid:15) D ( A p ) L colim p / / D ( A ) ✠✠✠✠ @ H ε again comes from the chain maps which collapse the cones. (For a more systematicapproach see [Gro18, § (cid:3) Derived limits and derived Kan extensions.
In this short subsection wecollect key formal properties of the calculus of derived limits in categories of mod-ules. More generally, restriction functors between derived diagram categories haveadjoints on both sides. These derived Kan extension functors can be calculated interms of derived (co)limits. It was the philosophy of Grothendieck [Gro83, pp. 196-200] that the “entire triangulated information” of a derived category is encoded bythis system of derived diagram categories and derived limit functors.
Notation 2.14.
For every small category B , we denote the derived category of thediagram category Mod( R ) B by D R ( B ) = D (Mod( R ) B ) . As a special case D R ( ∗ ) is simply the derived category D ( R ). The diagonal functor∆ B : Mod( R ) → Mod( R ) B is exact and hence induces a similar diagonal functor∆ B : D R ( ∗ ) → D R ( B ) . Recall that (co)limits are simply adjoints to diagonal functors [ML98, § IV.2].Generalizing Proposition 2.13 there is the following derived version of this.
Theorem 2.15.
For every small category B there are adjunctions ( L colim B , ∆ B ) : D R ( B ) ⇄ D R ( ∗ ) and (∆ B , R lim B ) : D R ( ∗ ) ⇄ D R ( B ) . Proof.
This is part of the statement of Theorem 2.19. (cid:3)
In order to build towards a further generalization, we recall the following defini-tion. Given a functor u : A → B between small categories, there is the restriction or precomposition functor (2.16) u ∗ : Mod( R ) B → Mod( R ) A : X X ◦ u. In particular, for an object b ∈ B , we obtain the evaluation functor b ∗ : Mod( R ) B → Mod( R ) . In this case we simplify notation by writing f b : X b → Y b for the value of f : X → Y under the evaluation functor. Kan extensions are some kind of relative versions of (co)limits. To explain whatwe mean by this it suffices to identify diagonal functors as restriction functors. Inmore detail, for every small category B let π B : B → ∗ be the unique functor tothe terminal category. Note that under the isomorphism Mod( R ) ∗ ∼ = Mod( R ) thefunctors ∆ B and ( π B ) ∗ are identified. Thus, to put this differently, the (co)limitfunctors are adjoints to restriction along these particular functors. It turns outthat the existence of limits and colimits in Mod( R ) also guarantees the existence ofadjoints to arbitrary restriction functors (2.16). Given a functor u : A → B betweensmall categories, there are adjunctions( u ! , u ∗ ) : Mod( R ) A ⇄ Mod( R ) B and ( u ∗ , u ∗ ) : Mod( R ) B ⇄ Mod( R ) A . The functor u ! is the left Kan extension functor along u , and u ∗ is referred toas the right Kan extension along u . For the basic theory of these Kan extensionfunctors see [ML98, § X] or [Bor94b, § § IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 15
The restriction functors (2.16) are again exact. Hence, for trivial reasons weobtain derived restriction functors u ∗ : D R ( B ) → D R ( A ) . As a relative version of Theorem 2.15, also the Kan extension adjunctions havederived versions.
Theorem 2.17.
For every functor u : A → B between small categories there areadjunctions ( u ! , u ∗ ) : D R ( A ) ⇄ D R ( B ) and ( u ∗ , u ∗ ) : D R ( B ) ⇄ D R ( A ) . Proof.
This is part of the statement of Theorem 2.19. (cid:3)
For every functor u : A → B we hence have a derived left Kan extension functorsand a derived right Kan extension functor u ! : D R ( A ) → D R ( B ) and u ∗ : D R ( A ) → D R ( B ) . To be of any use, it is important to be able to calculate these derived Kan extensionfunctors. More specifically, given X ∈ D R ( A ) and b ∈ B we would like to be ableto express the values u ! ( X ) b , u ∗ ( X ) b ∈ D R ( ∗ ) ∼ = D ( R )of the derived Kan extensions u ! ( X ) , u ∗ ( X ) ∈ D R ( B ) in terms of X , u , and b only.Before dealing with this derived situation, let us recall the corresponding fact formodule categories. Construction . Let u : A → B be a functor between small categories and let X : A → Mod( R ) be a diagram of modules. In order to construct the left Kanextension u ! ( X ) : B → Mod( R ) we recall the definition of slice categories . Forevery object b ∈ B the slice category ( u/b ) of objects u -over b has the followingdescription. Objects in ( u/b ) are pairs ( a, f : ua → b ) consisting of an object a ∈ A and a morphism f : ua → b in B . A morphism g : ( a, f ) → ( a ′ , f ′ ) in ( u/b ) is amorphism g : a → a ′ in A such that f ′ ◦ u ( g ) = f . Note that there is forgetfulfunctor p : ( u/b ) → A : ( a, f ) a , and to construct u ! ( X ) b it suffices to form acolimit over this slice category. More precisely, for every b ∈ B we define u ! ( X ) b = colim ( u/b ) X ◦ p ∈ Mod( R ) . One checks that this defines a diagram u ! ( X ) : B → Mod( R ), the left Kan extensionof X along u .Dually, for every b ∈ B there is the slice category ( b/u ) of objects u -under b . Andalso this category comes with a forgetful functor q : ( b/u ) → A : ( a, f : b → ua ) a .It turns out that also the right Kan extension u ∗ ( X ) can be constructed pointwiseby the formula u ∗ ( X ) b = lim ( b/u ) X ◦ q ∈ Mod( R ) . This concludes the recap of the pointwise formulas for Kan extensions of dia-grams of modules (which, of course, generalizes to arbitrary complete and cocom-plete categories). The following theorem shows that similar formulas extend tothe derived Kan extensions in Theorem 2.17. For convenience, we also summarizesome additional key properties of the calculus of derived limits, derived colimits,and derived Kan extensions over a ring.
Theorem 2.19 (Grothendieck) . Let R be a ring. The formation of derived cate-gories of diagram categories defines a -functor D R : B D R ( B ) = D (Mod( R ) B ) from small categories to not necessarily small categories. The -functor D R enjoysthe following properties.(Der1) The canonical inclusion functors B j → ` i ∈ I B i , j ∈ I, induce an equivalenceof categories D R ( a i ∈ I B i ) ∼ −→ Y i ∈ I D R ( B i ) . (Der2) A morphism f : X → Y in D R ( B ) is an isomorphism if and only if themorphisms f b : X b → Y b , b ∈ B, are isomorphisms in D R ( ∗ ) = D ( R ) . (Der3) For every functor u : A → B , the restriction functor u ∗ : D R ( B ) → D R ( A ) has a left adjoint u ! and a right adjoint u ∗ , ( u ! , u ∗ ) : D R ( A ) ⇄ D R ( B ) , ( u ∗ , u ∗ ) : D R ( B ) ⇄ D R ( A ) . (Der4) For every functor u : A → B , the functors u ! , u ∗ : D R ( A ) → D R ( B ) can becalculated pointwise. More specifically, for X ∈ D R ( A ) and b ∈ B there arecanonical isomorphisms L colim ( u/b ) p ∗ ( X ) ∼ −→ u ! ( X ) b and u ∗ ( X ) b ∼ −→ R lim ( b/u ) q ∗ ( X ) . Proof.
While an elementary proof building on classical homological algebra onlydoes not seem to exist in the literature, we think that it would be nice to expandon the techniques behind Proposition 2.7 and Proposition 2.13 in order to obtainsuch a proof. Currently, this result is a consequence of more general theorems. Thecategory Mod( R ) is Grothendieck abelian and Ch( R ) can hence be endowed witha combinatorial Quillen model structure (see [Bek00, Prop. 3.13], [Hov01, § §§ § (cid:3) Remark . We want to stress that, by definition, D R sends a small category B to the derived category D (Mod( R ) B ). In this assignment the derived category isconsidered as a plain category only and hence not as a triangulated category. Infact, the main goal of § § D ( R ) (Theorem 2.6) using certain nice properties of this2-functor D R only. Those arguments will indicate the above-mentioned intimaterelation between the triangulation and symmetries.Formally, in the remainder of this section we focus on modules over a ring.We want to stress, however, that variants of Theorem 2.19 arise in many othersituations as well. Moreover, many of the arguments which are sketched furtherbelow apply almost verbatim to those more general situations (see § Proposition 2.21.
Let R be a ring and let u : A → B a fully faithful functor. TheKan extension functors u ! , u ∗ : D R ( A ) → D R ( B ) are fully faithful.Proof. This is a special case of Proposition 3.21. (cid:3)
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 17
This result is of central importance in many constructions, and to explain itsrelevance we collect the following reformulation (based on standard results on ad-junctions): For every fully faithful u : A → B and every X ∈ D R ( A ) the adjunctionunits and counits η : X ∼ −→ u ∗ u ! X and ε : u ∗ u ∗ X ∼ −→ X are invertible. Thus, this result allows us to extend diagrams defined on smallercategories to larger ones without affecting the diagram on the smaller category, andin this sense Kan extensions are proper “extensions”.As a first example of the derived Kan extension functors from Theorem 2.19,we reformulate the proof of Proposition 2.7 in terms of these functors. This needssome preparation.
Notation 2.22.
We denote by (cid:3) the category [1] × [1],(0 , / / (cid:15) (cid:15) (0 , (cid:15) (cid:15) (1 , / / (1 , . For every ring R , a square in D R is an object X ∈ D R ( (cid:3) ). Such a square is hencea commutative square of chain complexes x / / (cid:15) (cid:15) y (cid:15) (cid:15) z / / w considered as an object in the derived category D (Mod( R ) (cid:3) ) . Associated to everysquare, there are the following canonical maps.(i) Ignoring the chain complex w for a moment, there is the corresponding left de-rived pushout ( L colim p )( X | p ) (Proposition 2.13). This chain complex comeswith a canonical comparison map(2.23) ( L colim p )( X | p ) → w. (ii) Dually, ignoring the chain complex x , there is the right derived pullback R lim y ( X | y ), and this chain complex comes with a canonical comparison map(2.24) x → ( R lim y )( X | y ) . Definition 2.25.
Let R be a ring and let X ∈ D R ( (cid:3) ).(i) The square X is cocartesian if (2.23) is an isomorphism in D R ( ∗ ) = D ( R ).(ii) The square X is cartesian if (2.24) is an isomorphism in D R ( ∗ ) = D ( R ).The following two properties of D R are crucial to all later constructions. Proposition 2.26 (Stability of D R ) . Let R be a ring.(i) The category D R ( ∗ ) = D ( R ) has a zero object.(ii) A square X ∈ D R ( (cid:3) ) is cocartesian if and only if it is cartesian. We referto these squares as bicartesian squares . Proof.
The first statement is obvious since D ( R ) is an additive category. We invitethe reader to come up with a direct proof of the second statement. That resultalso follows more indirectly from the fact that Σ : D ( R ) → D ( R ) is an equivalencecombined with Theorem 2.19 and Theorem 3.29. (cid:3) Whenever we want to stress that a given square is bicartesian, we decorate it bya square in the middle such as in x / / (cid:15) (cid:15) (cid:3) y (cid:15) (cid:15) z / / w. Remark . On a first view, it seems to be a bit strange, that in Definition 2.25 weintroduce the notions of cocartesian and cartesian squares in D R for every ring R ,in order to then observe in Proposition 2.26 that these two classes coincide. Thepoint is, of course, that these notions make sense in many other situations as well(see § C with a zero object, the classes of pushout and pullbacksquares agree if and only if C is equivalent to the terminal category (which isequivalent to C being a contractible groupoid). In this precise sense the conjunctionof the two properties in Proposition 2.26 are invisible to ordinary category theoryand they require more refined techniques (see also Remark 3.31).With these basic notions in place, we can now revisit Proposition 2.7. Definition 2.28.
Let R be a ring. A cofiber square in D R is a cocartesian square X ∈ D R ( (cid:3) ) which vanishes at the lower left corner,(2.29) x f / / (cid:15) (cid:15) (cid:3) y (cid:15) (cid:15) / / z. We denote by D R ( (cid:3) ) cof ⊆ D R ( (cid:3) ) the full subcategory spanned by the cofibersquares.We want to stress that the lower left corner is a zero object in D ( R ), so itreally could be any acyclic chain complex. The intuition is that a cofiber squareis essentially determined by the upper horizontal morphism ( f : x → y ). Thismorphism is obtained formally from the cofiber square by restriction along(2.30) k : [1] → (cid:3) : i (0 , i ) . Proposition 2.31.
For every ring R restriction along (2.30) induces an equivalenceof categories k ∗ : D R ( (cid:3) ) cof ∼ −→ D R ([1]) . Proof.
We sketch the proof and begin by noting that the functor (2.30) factors as k = j ◦ i : [1] → p → (cid:3) . Here, p is again suggestive notation for the poset (2.11) which corepresents spans.By Theorem 2.19 there are derived Kan extension adjunctions( i ∗ , i ∗ ) : D R ( p ) ⇄ D R ([1]) and ( j ! , j ∗ ) : D R ( p ) ⇄ D R ( (cid:3) ) . And in this case the construction of i ∗ and j ! can be given very explicitly. IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 19 (i) Before passing to derived categories, it is straightforward to check that thefunctor i ∗ which sends a chain map ( x → y ) to the span (0 ← x → y ) is rightadjoint to i ∗ (alternatively, this also follows from the pointwise formulas inConstruction 2.18). Since both functors are exact, this immediately yieldsthe first adjunction.(ii) In the case of the second adjunction a direct argument can be sketched asfollows. While pushouts fail to be exact on all spans, they are exact on spanssuch that at least one of the maps is a monomorphism. Hence, as in the proofof Proposition 2.13, we can resolve an arbitrary span by adding inclusions intocones on at least one of the edges, and the corresponding pushout squaresare models for the left derived pushout square functor j ! : D R ( p ) → D R ( (cid:3) ) . With these adjunctions at our disposal we can now consider the composition j ! ◦ i ∗ : D R ([1]) → D R ( p ) → D R ( (cid:3) ) , and this yields the desired equivalence. In more detail, the functors i ∗ and j ! areboth fully faithful (Proposition 2.21), and one checks that the essential image oftheir composition consists precisely of the cofiber squares. (cid:3) The above sketch has shown that the left quadrilateral in (2.8) can be constructedin terms of derived Kan extension functors. This justifies the terminology cofibersquare since for every such square (2.29) there is a canonical isomorphism between z and the cone or cofiber of f , Cf ∼ −→ z. There is the following important special case of cofiber squares.
Definition 2.32.
Let R be a ring. A suspension square in D R is a cocartesiansquare which vanishes at the lower left and the upper right corner,(2.33) x / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / x ′ . Remark . For every ring R we denote by D R ( (cid:3) ) Σ ⊆ D R ( (cid:3) ) the full subcate-gory spanned by the suspension squares. Similar to the case of Proposition 2.31,evaluation at (0 , ∈ (cid:3) induces an equivalence of categories(0 , ∗ : D R ( (cid:3) ) Σ ∼ −→ D R ( ∗ ) = D ( R ) . We invite the reader to adapt the arguments of the previous case in order to sketcha proof of this result. An inverse of this equivalence is given by forming the cofibersquare associated to ( x → x → Cx , which is clearly isomorphic to theusual suspension Σ x of the chain complex x . The terminology suspension squares is hence justified by the fact that every such square (2.33) induces a canonicalisomorphism Σ x ∼ −→ x ′ . A -quivers and Barratt–Puppe sequences. The point of the construc-tions at the end of the previous subsection was the following. A morphism in D R can be equivalently encoded by a cofiber square (Proposition 2.31) and, similarly,an object in D R is as good as a suspension square (Remark 2.34). In this subsectionwe want to expand a bit on this observation for the case of morphisms. Representa-tions of ~A in D R turn out to be equivalent to Barratt–Puppe sequences . Moreover,the symmetry of these Barratt–Puppe sequence corresponds to the rotation at thelevel of triangulated categories.As a warmup we consider cofiber sequences which are essentially obtained byforming two cofiber squares. To carry this out in detail, we introduce the followingnotation.
Notation 2.35.
For every natural number n , we denote by [ n ] the finite linearorder (0 < < . . . < n ). The product (cid:25) = [1] × [2] hence agrees with the shape oftwo adjacent commuting squares(0 , / / (cid:15) (cid:15) (0 , / / (cid:15) (cid:15) (0 , (cid:15) (cid:15) (1 , / / (1 , / / (1 , . In representation theoretic parlance, the poset [ n ] is the linearly oriented A n +1 -quiver ~A n +1 = (1 → . . . → n + 1) . Note that the labeling conventions only match up to a shift by one. There is aunique isomorphism between the two shapes which will always be used implicitly,hopefully not arising in a confusion concerning the various labels.
Definition 2.36.
Let R be a ring. A cofiber sequence in D R is an object X ∈ D R ( (cid:25) ) such that the square on the left and the square on the right arecocartesian and such that X vanishes at (1 ,
0) and (0 , x f / / (cid:15) (cid:15) (cid:3) y g (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / z / / x ′ . We denote by D R ( (cid:25) ) cof ⊆ D R ( (cid:3) ) the full subcategory spanned by the cofibersquares.Morally, a cofiber sequence is as good as a morphism only. In fact, given acofiber sequence (2.37), the morphism z → x ′ can be reconstructed from g : y → z by Proposition 2.31. Similarly, by that same result also the morphism g : y → z is determined by f : x → y . This makes the following result plausible in which wedenote the inclusion of the upper left morphism by(2.38) k : [1] → (cid:25) : i (0 , i ) . Proposition 2.39.
For every ring R restriction along (2.38) induces an equivalenceof categories k ∗ : D R ( (cid:25) ) cof ∼ −→ D R ([1]) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 21
Proof.
The strategy is very similar to the one in the earlier cases. The functor k in(2.38) factors as a composition j ◦ i : [1] → B → [1] × [2]of fully faithful inclusions where B is given by(0 , / / (cid:15) (cid:15) (0 , / / (0 , , . At the level of chain complexes it is straightforward to check that the restrictionfunctor i ∗ has a right adjoint i ∗ which simply adds two zero chain complexes and theunique chain maps. Alternatively, this follows immediately from Construction 2.18since the corresponding slice categories are empty. Thus, both i ∗ and i ∗ are exact,and the induced functor i ∗ : D R ([1]) → D R ( B ) is hence under control. It is fullyfaithful (Proposition 2.21) and the essential image consists of all X ∈ D R ( B ) whichvanish at (1 ,
0) and (0 , j ! to j ∗ is given by the functor which adds two pushout squares. The correspondingleft derived functor j ! hence adds two derived pushout squares, and one checks thatthe essential image of this fully faithful functor can be characterized by this. As anupshot, both functors in j ! ◦ i ∗ : D R ([1]) → D R ( B ) → D R ( (cid:25) )are fully faithful and the essential image of their composition consists precisely ofthe cofiber sequences. (cid:3) In order to analyze cofiber sequences in more detail, we recall the followingpasting and cancellation property of bicartesian squares in D R . There are threeobvious inclusions of the square (cid:3) in (cid:25) . Let us denote by ι : (cid:3) → (cid:25) , ι : (cid:3) → (cid:25) , and ι : (cid:3) → (cid:25) the inclusion of the left square, the right square, and the composed square, respec-tively. Proposition 2.40.
Let R be a ring and let X ∈ D R ( (cid:25) ) . If two of the squares ι ∗ ( X ) , ι ∗ ( X ) , and ι ∗ ( X ) are bicartesian, then so is the third square.Proof. This is a special case of [Gro13, Prop. 4.6]. (cid:3)
Construction . Let R be a ring and let X ∈ D R ( (cid:25) ) cof be a cofiber sequencelooking like (2.37). By Proposition 2.40 the composed square ι ∗ ( X ) is again bi-cartesian and it takes the form x / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / x ′ . Thus, ι ∗ ( X ) is a suspension square (Definition 2.32), and by Remark 2.34 we ob-tain a canonical isomorphism Σ x ∼ −→ x ′ . It follows that associated to every cofibersequence there is a triangle x f → y g → z h → Σ x. ... ... ... ...( − , − / / ( − , − (cid:15) (cid:15) / / ( − , / / (cid:15) (cid:15) ( − , (cid:15) (cid:15) (0 , − / / (0 , / / (cid:15) (cid:15) (0 , / / (cid:15) (cid:15) (0 , (cid:15) (cid:15) (1 , / / (1 , / / (cid:15) (cid:15) (1 , / / (cid:15) (cid:15) (1 , (cid:15) (cid:15) (2 , / / ... (2 , / / ... (2 , / / ... (2 , Figure 2.
The full subposet M ⊆ Z × Z This is an ordinary diagram in D R ( ∗ ) = D ( R ), and the morphism h is obtainedfrom the unlabeled morphism z → x ′ in (2.37) thanks to the isomorphism Σ x ∼ −→ x ′ .This already suggests that cofiber sequences are closely related to distinguishedtriangles, and we will come back to this at the end of § R , at thelevel of derived categories of diagram categories there are equivalences of categories D R ( (cid:25) ) cof ∼ −→ D R ( (cid:3) ) cof ∼ −→ D R ([1]) . These equivalences are induced by suitable restriction functors, and inverse equiv-alences are obtained by extending a morphism to a cofiber square or by iteratingthis construction twice. We now simply iterate this construction countably manytimes in both directions. The combinatorial details are as follows.
Notation 2.42.
Let M ⊆ Z × Z be the full subposet given by all ( i, j ) ∈ Z × Z such that i − ≤ j ≤ i + 2.A part of this poset is displayed in Figure 2. The reason that we consider thisposet is that, for every ring R , this shape allows us to simultaneously encode amorphism ( f : x → y ) ∈ D R ([1]), all its iterated derived cokernels, and also all itsiterated derived kernels. This is achieved by forming Barratt–Puppe sequences inthe following sense.
Definition 2.43.
Let R be a ring. A diagram X ∈ D R ( M ) is a Barratt–Puppesequence in D R if the diagram X vanishes on the two boundary stripes and allsquares in X are bicartesian. We denote by D R ( M ) ex ⊆ D R ( M ) the full subcat-egory spanned by all Barratt–Puppe sequences.The terminology is motivated by similar constructions with pointed topologicalspaces. In that case this notion yields the classical Barratt–Puppe sequences as in[Pup58], which provide the space-level origin of many long exact sequences in alge-braic topology. The following result makes precise that Barratt–Puppe sequencesare determined by their restrictions along the inclusion(2.44) i : [1] → M : k (0 , k ) . Theorem 2.45.
For every ring R restriction along (2.44) induces an equivalenceof categories i ∗ : D R ( M ) ex ∼ −→ D R ([1]) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 23
Proof.
We only sketch the arguments and refer the reader to [GˇS16a, §
4] for a de-tailed proof of a more general result. The idea is to construct an inverse equivalence F [1] : D R ([1]) ∼ −→ D R ( M ) ex as a composition of suitable derived Kan extensions. To this end we observe thatthe inclusion (2.44) factors as i : [1] i → K i → K i → K i → M , where the K j are the following full subposets of M and the i j are the obviousinclusions.(i) K is obtained from the image of i by adding those objects on the boundarystripe which sit under the image of i , i.e., the objects ( n, n + 2) , n ≥ , andthe objects ( n, n − , n > . (ii) K contains besides the objects in K also all remaining objects under theimage of i , i.e., the objects ( n, n ) and ( n, n + 1) for n > . (iii) K is obtained from K by also adding the remaining objects on the boundarystripes, i.e., the objects ( n, n + 2) , n < , and the objects ( n, n − , n ≤ . We know from Proposition 2.21 that the following four functors D R ([1]) ( i ) ∗ → D R ( K ) ( i ) ! → D R ( K ) ( i ) ! → D R ( K ) ( i ) ∗ → D R ( M )are fully faithful. Invoking arguments similar to the previous cases in combinationwith homotopy (co)finality arguments, one checks that(i) ( i ) ∗ : D R ([1]) → D R ( K ) simply adds zero objects at the new components,(ii) ( i ) ! : D R ( K ) → D R ( K ) inductively adds bicartesian squares,(iii) ( i ) ! : D R ( K ) → D R ( K ) again adds zero objects at the new components,(iv) and that ( i ) ∗ : D R ( K ) → D R ( M ) constructs new bicartesian squares.Thus, the composition of these four functors indeed extends ( f : x → y ) ∈ D R ([1])to a Barratt–Puppe sequence. Moreover, in each of these four steps the abovedescribes precisely the corresponding essential image, and we hence conclude that F [1] = ( i ) ∗ ◦ ( i ) ! ◦ ( i ) ! ◦ ( i ) ∗ is the desired inverse equivalence of i ∗ : D R ( M ) ex → D R ([1]). (cid:3) Let us discuss these Barratt–Puppe sequences to some extent.
Remark . Let R be a ring and let X ∈ D R ( M ) ex be a Barratt–Puppe se-quence looking like Figure 3. In that Barratt–Puppe sequence any two adjacentsquares determine a cofiber sequence and hence give rise to a triangle in D R ( ∗ )(Construction 2.41). In particular, the cofiber sequence determined by the squares1 , f : x → y . Similarly, the squares 2 , g : y → z , and this is a rotatedtriangle of the previous one. As an upshot, the Barratt–Puppe sequence in Figure 3encodes all iterated rotations of triangles associated to f : x → y .The defining exactness properties of Barratt–Puppe sequences (vanishing onboundary stripes and all squares are bicartesian) are invariant under the obvi-ous symmetries of the shape M . Hence, as alluded to in the introduction, therotation at the level of distinguished triangles is a shadow of certain symmetrieson representations of ~A = (1 → ∼ = [1] with values in chain complexes. As a ... ... ... ...0 / / ˜ y (cid:15) (cid:15) / / ˜ z / / (cid:15) (cid:15) (cid:15) (cid:15) / / x f / / (cid:15) (cid:15) y / / (cid:15) (cid:15) (cid:15) (cid:15) / / z / / (cid:15) (cid:15) x ′ / / (cid:15) (cid:15) (cid:15) (cid:15) / / ... y ′ / / ... z ′ / / ... 0 ... Figure 3.
Iterated derived (co)kernels of ( f : x → y ) ∈ D R ([1])variant of this, let us consider the square 1 in Figure 3 as a “triangular fundamen-tal domain” of the Barratt–Puppe sequence. The flip symmetry of the shape M identifies the squares 1 and 4 . At the level of diagrams this flip symmetry corre-sponds to the suspension, and Figure 3 encodes the fundamental domain and all itsiterated (de)suspensions. Of course, the suspension is simply the cube of the cofiber([GPS14b, Lem. 5.13]), and in that generality this can be interpreted as an exam-ple of the abstract fractionally Calabi–Yau property of ~A n -quivers ([Kel08],[GˇS16a,Cor. 5.20]). Remark . The strategy behind the constructions discussed so far is fairly typicaland it relies on two ingredients. First, for every fully faithful functor u : A → B thederived Kan extension functors u ! , u ∗ : D R ( A ) → D R ( B ) are again fully faithful.In particular, they induce equivalences onto their essential images. Second, forsuitable classes of fully faithful functors, the functors u ! , u ∗ simply add zero chaincomplexes at the new objects. We refer the reader to [Gro13] for a more systematicdiscussion of this.2.4. A -quivers and refined octahedral diagrams. In this subsection, we con-clude the unorthodox proof of Theorem 2.6 and there are only two main steps left.First, we repeat to some extent the discussion in § ~A instead. Second, for suitable shapes the underlying diagram functorsfrom Example 2.10 are full and essentially surjective, and this can be invoked toestablish the existence of the Verdier triangulation.The octahedral axiom of triangulated categories asks for a version of the thirdNoether isomorphism for cones in triangulated categories. Hence, it is essentially astatement about the situation of two composable morphisms, which is to say of arepresentation of the linearly oriented A -quiver ~A ∼ = [2] = (0 < < . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 25 ... ... ... ... ...( − , − / / ( − , − (cid:15) (cid:15) / / ( − , / / (cid:15) (cid:15) ( − , (cid:15) (cid:15) / / ( − , (cid:15) (cid:15) (0 , − / / (0 , / / (cid:15) (cid:15) (0 , / / (cid:15) (cid:15) (0 , (cid:15) (cid:15) / / (0 , (cid:15) (cid:15) (1 , / / (1 , / / (cid:15) (cid:15) (1 , / / (cid:15) (cid:15) (1 , (cid:15) (cid:15) / / (1 , (cid:15) (cid:15) (2 , / / ... (2 , / / ... (2 , / / ... (2 , / / ... (2 , Figure 4.
The full subposet M ⊆ Z × Z The following is a variant for ~A of the above discussion of Barratt–Puppe se-quences. Notation 2.48.
Let M ⊆ Z × Z be the full subposet given by all ( i, j ) ∈ Z × Z such that i − ≤ j ≤ i + 3.A part of this poset is displayed in Figure 4, and this poset is the shape of refinedoctahedral diagrams in the following precise sense. Definition 2.49.
Let R be a ring. A diagram X ∈ D R ( M ) is a refined octa-hedral diagram in D R if the diagram X vanishes on the two boundary stripesand all squares in X are bicartesian. We denote by D R ( M ) ex ⊆ D R ( M ) the fullsubcategory spanned by all refined octahedral diagrams.Similar to the case of Barratt–Puppe sequences, a refined octahedral diagramis determined by restriction along various embeddings of ~A ∼ = [2] to M . To bespecific, we consider the standard embedding(2.50) i : [2] → M : k (0 , k ) . Theorem 2.51.
For every ring R restriction along (2.50) induces an equivalenceof categories i ∗ : D R ( M ) ex ∼ −→ D R ([2]) . Proof.
It is straightforward to adapt the strategy of the proof of Theorem 2.45 inorder to also cover this situation. For a detailed proof we refer to [GˇS16a, § F [2] : D R ([2]) ∼ −→ D R ( M ) ex . (cid:3) The connection to the octahedral axiom is as follows, and this justifyies theabove terminology.
Remark . Let R be a ring and let X ∈ D R ( M ) ex be a refined octahedraldiagram looking like Figure 5. This diagram is determined by the restriction i ∗ ( X ) ∈ D R ([2]), and this restriction in turn gives rise to the three morphisms f : x → y, g : y → z, and g ◦ f : x → z. Given three such morphisms in D R ( ∗ ), let us recall that the octahedral axiom relatestriangles of f , g , and g ◦ f by an additional triangle incorporating the various cones.These four triangles can be deduced from Figure 5 as follows. ... ... ... ... ...0 / / ˜ z (cid:15) (cid:15) / / ˜ v / / (cid:15) (cid:15) ˜ w (cid:15) (cid:15) / / (cid:15) (cid:15) / / x f / / (cid:15) (cid:15) y g / / (cid:15) (cid:15) z (cid:15) (cid:15) / / (cid:15) (cid:15) / / u / / (cid:15) (cid:15) v / / (cid:15) (cid:15) x ′ (cid:15) (cid:15) / / (cid:15) (cid:15) / / ... w / / ... y ′ / / ... u ′ / / ... 0 ... Figure 5.
The refined octahedral diagram of ( x f → y g → z ) ∈ D R ([2])(i) The square 1 is a cofiber square, and there is hence a canonical isomor-phism Cf ∼ −→ u . Similarly, if we consider the square 1 and the horizontalcomposition 2 + 3 , then we obtain by Proposition 2.40 a cofiber sequence x f / / (cid:15) (cid:15) (cid:3) y g (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / u / / x ′ . This cofiber sequence of f induces by Construction 2.41 the first of the fourtriangles.(ii) Similarly, the vertical composition 2 + 4 is by Proposition 2.40 a cofibersquare and we obtain the identification Cg ∼ −→ w . Jointly with the verticalcomposition 3 + 5 this yields the desired cofiber sequence of g .(iii) As of the cofiber sequence of g ◦ f , we note that the horizontal composition1 + 2 is a cofiber square for g ◦ f , leading to C ( g ◦ f ) ∼ −→ v . In combinationwith the square 3 we obtain the cofiber sequence of g ◦ f .(iv) Finally, the square 4 and the horizontal composition 5 + 6 define a cofibersequence u / / (cid:15) (cid:15) (cid:3) v (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / w / / u ′ . Invoking the above identifications of u, v, and w as cones, this cofiber sequenceinduces the fourth triangle Cf → C ( g ◦ f ) → Cg → Σ Cf.
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 27 ... ... ...( − , − / / ( − , − (cid:15) (cid:15) / / ( − , (cid:15) (cid:15) (0 , − / / (0 , / / (cid:15) (cid:15) (0 , (cid:15) (cid:15) (1 , / / (1 , / / (cid:15) (cid:15) (1 , (cid:15) (cid:15) (2 , / / ... (2 , / / ... (2 , Figure 6.
The full subposet M ⊆ Z × Z One checks directly that all the necessary compatibilities of the octahedral axiomare satisfied (see [Gro13, § ~A ∼ = ∗ . Remark . Let R be a ring. We denote by M ⊆ Z × Z the full subposet givenby all ( i, j ) ∈ Z × Z such that i − ≤ j ≤ i + 1. A part of this poset is displayedin Figure 6. Moreover, let us write D R ( M ) ex ⊆ D R ( M )for the full subcategory spanned by all diagrams which vanish on the boundarystripes and which make all squares bicartesian. Evaluation at (0 , ∈ M inducesan equivalence of categories(2.54) (0 , ∗ : D R ( M ) ex ∼ −→ D ( R ) , as one checks by copying the proof of Theorem 2.45 (see [GˇS16a, § X ∈ D R ( M ) ex is displayed in Figure 7, and therein all squares aresuspension squares (Remark 2.34).The shape M has two obvious generating symmetries, namely translation alongthe diagonal direction and the swap symmetry which keeps the diagonal fixed whileit interchanges the objects ( n, n + 1) and ( n + 1 , n ) for all n . These symmetriesof M induce symmetries on D R ( M ) ex since the defining exactness properties areclearly invariant under these symmetries. Interpreted at the level of D ( R ) (bya conjugation with (2.54)), we obtain induced self-equivalences on D ( R ). Thefirst symmetry gives rise to the suspension equivalence Σ : D ( R ) ∼ −→ D ( R ), andthe second symmetry can be identified with − id : D ( R ) ∼ −→ D ( R ). One can makeprecise in which sense this latter symmetry is responsible for the sign in the rotationaxiom and we will come back to this in § D R ([1]) = D (Mod( R ) [1] ) and D R ([2]) = D (Mod( R ) [2] ) . ... ... ...0 / / ˜ x (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / x / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / x ′ / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / ... x ′′ / / ... 0 ... Figure 7.
A diagram X ∈ D R ( M ) ex In contrast to this, in the classical Verdier triangulation we consider morphisms orpairs of composable morphisms in D ( R ), which is to say objects in the categories D ( R ) [1] and D ( R ) [2] . In order to connect these two perspectives, the following result proves crucial.
Proposition 2.55.
For every ring R and every n ≥ the underlying diagramfunctor dia [ n ] : D R ([ n ])) = D (Mod( R ) [ n ] ) → D ( R ) [ n ] is essentially surjective and full.Proof. We invite the reader to check this directly. Alternatively, see for instance[Cis10, Prop. 2.15] or [RB06, Thm. 10.3.3] for much more general statements. (cid:3)
Remark . Let R be a ring.(i) The functor dia [0] is, of course, an equivalence of categories, but already inthe case of n = 1 the functor dia [1] is not faithful. For instance, in the case ofa field R = k , one can invoke Auslander–Reiten theory to make precise thatthe extension group Ext k ~A (( k → , (0 → k )) ∼ = k is precisely the reason forthis failure (see also [Gro18, § F the underlyingdiagram functors dia F is essentially surjective and full ([Cis10, Prop. 2.15] or[RB06, Thm. 10.3.3]). (Let us recall that a free category is the path categoryof an oriented graph.)(iii) In contrast to this, already in the case of a field R = k and the non-freecategory (cid:3) , the underlying diagram functor dia (cid:3) does not preserve isomor-phism types (see [BG18, Example 3.17]) and is hence not full and essentiallysurjective.We now sketch the slightly unorthodox proof of Theorem 2.6. IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 29
Proof. (of Theorem 2.6)
We carry out the sketch proof in the case of Mod( R ),but the arguments also apply to arbitrary abelian categories (see Remark 2.59).Moreover, we focus on those aspects which allow for nice “representation theoreticexplanations”. Let us take for granted that the derived category D ( R ) is additive(see also Remark 3.34). The suspension functor Σ : D ( R ) → D ( R ) is simply theshift functor (Remark 2.34), and it clearly is an equivalence of categories.We now define the class of distinguished triangles in D ( R ). To this end, forevery ( f : x → y ) ∈ D R ([1]) we can consider by Proposition 2.39 the correspondingcofiber sequence X ∈ D R ([1] × [2]) cof as in (2.37). As detailed in Construction 2.41we obtain an associated triangle(2.57) x f → y → z → Σ x. This is defined to be the standard triangle of f ∈ D R ([1]), and a triangle in D ( R )is distinguished when it is isomorphic to a standard triangle. It is worthwhile tosummarize the construction by the following diagram D R ([1]) ∼ F [1] / / dia [1] (cid:15) (cid:15) tria & & ▲▲▲▲▲▲▲▲▲▲ D R ( M ) ex (cid:15) (cid:15) D ( R ) [1] D ( R ) [3] . The horizontal functor is the equivalence constructed in the proof of Theorem 2.45.The unlabeled vertical functor takes a Barratt–Puppe sequence, restricts it to theappropriate cofiber sequence and then passes to the underlying distinguished tri-angle. Note that the vertical functors and hence also the diagonal functor amountto a loss of information since in all three cases underlying diagram functors areinvolved.In order to show that every morphism f : x → y in D ( R ) extends to a distin-guished triangle we invoke that dia [1] is essentially surjective (Proposition 2.55).Thus, we can find Y ∈ D R ([1]) and an isomorphism α : dia [1] ( Y ) ∼ −→ f , and a com-bination of the standard triangle tria( Y ) of Y and the isomorphism α yields theintended distinguished triangle. Similarly, the weak functoriality of distinguishedtriangles is a consequence of dia [1] being essentially surjective and full. Up to thesign issue, the rotation axiom is essentially a consequence of the symmetries ofBarratt–Puppe sequences (see also Remark 2.46). The sign issue will be taken upagain § B bethe shape of the diagrams showing up in the octahedral axiom. For every X = ( x → y → z ) ∈ D R ([2])we can consider the corresponding refined octahedral diagram F [2] ( X ) ∈ D R ( M ) ex (Theorem 2.51). The above construction of standard triangles and Remark 2.52imply that from F [2] ( X ) we can construct the intended octahedral diagramocta( X ) : B → D ( R ) . Similarly to the previous case, the situation can be summarized by the diagram(2.58) D R ([2]) ∼ F [2] / / dia [2] (cid:15) (cid:15) octa & & ▲▲▲▲▲▲▲▲▲▲ D R ( M ) ex (cid:15) (cid:15) D ( R ) [2] D ( R ) B . Here, F [2] is the equivalence from Theorem 2.51 and the remaining functors amountto a loss of information. Since also the functor dia [2] is essentially surjective(Proposition 2.55), this establishes the octahedral axiom. (cid:3) Let us collect some of the benefits which we obtain from this alternative con-struction of the classical Verdier triangulation on D ( R ). Remark . Let R be a ring, let ( x f → y g → z ) ∈ D R ([2]), and let F [2] ( X ) be theassociated refined actahedral diagram.(i) The shape M (Figure 4) has two obvious generating symmetries (translationand flip symmetry). By definition refined octahedral diagrams are preservedby restriction along these symmetries. This implies, for instance, that X alsoencodes the refined octahedral diagram associated to the canonical represen-tation of ~A looking like( Cf → C ( g ◦ f ) → Σ x ) . And this leads to a relation between the respective octahedral diagrams.(ii) The refined octahedral diagram F [2] ( X ) encodes additional distinguished tri-angles teaching us something about the cone C ( g ◦ f ) of the composition. Infact, let us recall that every bicartesian square x / / (cid:15) (cid:15) (cid:3) y (cid:15) (cid:15) z / / w in D R gives rise to a cofiber square x / / (cid:15) (cid:15) (cid:3) y ⊕ z (cid:15) (cid:15) / / w, and hence to a distinguished triangle x → y ⊕ z → w → Σ x. See, for instance, [GPS14b] for a construction of these
Mayer–Vietoris se-quences . Let us specialize this to bicartesian squares occurring in F [2] ( X )(Figure 5). In this situation, the square 2 gives rise to an additional distin-guished triangle y → z ⊕ Cf → C ( g ◦ f ) → Σ y. Similarly, the Mayer–Vietoris sequence associated to the square 5 looks like C ( g ◦ f ) → Σ x ⊕ Cg → Σ y → Σ C ( g ◦ f ) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 31 (iii) The proof of the octahedral axiom (as summarized by (2.58)) used only thatdia [2] is essentially surjective. Since this functor is also full, we concludethat octahedral diagrams depend weakly functorially on pairs of composablemorphisms in derived categories.(iv) For simplicity, we presented the proof in the case of Mod( R ) only. How-ever, there are variants of the main ingredients (such as Theorem 2.45 andTheorem 2.51) for arbitrary abelian categories A . In fact, by (co)finalityarguments the constructions only rely on the existence of finite limits andcolimits, and these exist in all abelian categories. Also the above remarksgeneralize to this more general situation.This sketch proof exhibits some of the axioms of triangulated categories as cer-tain shadows of symmetries of Barratt–Puppe sequences and of refined octahedraldiagrams. Put differently, this strategy makes precise the connection of these ax-ioms to abstract representations of the quivers ~A and ~A . The quivers ~A n for n ≥ § A crash course on derivators
In this section we give a short introduction to derivators, which offer one ofthe many approaches to higher category theory, abstract homotopy theory, or ho-motopical algebra. The precise choice of this model is not too relevant for ourlater purposes (see Disclaimer 3.1), and here we essentially only present what isnecessary for our later discussion of strong stable equivalences of quivers (or smallcategories). In fact, the main goal of this section is to provide enough backgroundin order to turn the following pseudo-definition into an actual definition.
Pseudo-definition 3.0.
Two quivers Q and Q ′ are strongly stably equivalent if for every stable homotopy theory D there is an equivalence between the homo-topy theory D Q of representations of shape Q and the homotopy theory D Q ′ ofrepresentations of shape Q ′ , Φ D : D Q ∼ −→ D Q ′ , which is pseudo-natural with respect to exact morphisms F : D → E of stablehomotopy theories.This goal pretty much dictates what we are supposed to do. In particular, as afirst step we have to make precise what we mean by an “abstract homotopy theory”.Here we choose to work with derivators but see the following disclaimer. Disclaimer 3.1.
By now there are various approaches to formalize an “abstracthomotopy theory”. All of these notions enhance certain defects of the more classicalhomotopy categories (which arise as 1-categorical localizations).(i) One of the most classical approaches is given by Quillen model categories[Qui67, Hov99, Hir03]. The homotopy theory is in this case encoded by a classof weak equivalences accompanied by classes of fibrations and cofibrationswhich are subject to certain axioms.(ii) There is the zoo of different models for a theory of ( ∞ , ∞ , categorical localization” of a category with weak equivalences. A very promi-nent model is given by ∞ -categories, and this model was developed to animpressive extent most notably in [Joy08] and [Lur09, Lur14]. With all thesetechniques at hand, this is currently the most flexible notion. Among the al-ternative approaches to such a theory are simplicial categories [Ber07], Segalcategories [HS01], and complete Segal spaces [Rez01]. Additional referencesand more details can for example be found in [Ber10, Sim12, Cam13, Gro10].Moreover, a model-independent approach to higher category theory can befound in [RV] and the sequels.(iii) Derivators [Gro, Hel88] axiomatize key properties of the calculus of homotopylimits and homotopy Kan extensions. In this language, these are character-ized by ordinary universal properties making them accessible to elementarytechniques.The notion of a derivator is a compromise: a derivator encodes more informationthan a mere homotopy category but somewhat less information than a full homo-topy theory. Some of the defects of classical homotopy categories and triangulatedcategories are addressed successfully by the theory of derivators – and this is donein a reasonably elementary way by means of (at most 2-) categorical techniquesonly. For instance, derivators allow for a construction of exponentials and for aformal study of stability, thereby offering a first framework for this project.At the same time there are obvious limitations to the theory of derivators (suchas the absence of gluing of derivators or the incoherence of morphisms and naturaltransformations), and for various purposes it is more convenient to use the moreflexible theory of ∞ -categories. Going even beyond this, for instance for the calculusof universal tilting modules (as in § theory of ( ∞ , ∞ -categories will enter the picture more prominently (see already [DJW18, DJW19]).The structure of this section is dictated by the above-mentioned main goal. In § § § § Derivators.
In this section we discuss the philosophy of derivators which offerone of the many approaches to abstract homotopy theory. Derivators were intro-duced independently by Grothendieck [Gro] and Heller [Hel88], and closely relatednotions were also considered by Franke [Fra96] and Keller [Kel91] (see Remark 3.14for some additional references). The main focus of derivators is on diagram cate-gories and the related calculus of (derived or homotopy) limits, colimits, and Kanextensions.
Notation 3.2.
We denote by C at the 2-category of small categories, functors, andnatural transformation, and similarly, by C AT the 2-category of not necessarilysmall categories. IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 33
In this section we use very basic terminology related to 2-categories. The dis-cussions in [ML98, § XII] and [Bor94a, §
7] suffice for our purposes (but see also[KS74, Lac10]).
Definition 3.3. A prederivator is a 2-functor D : C at op → C AT .Thus, a prederivator D associates to every small category A ∈ C at a category D ( A ), the category of coherent diagrams of shape A in D . For the trivialshape ∗ , we refer to D ( ∗ ) as the underlying category of D . Moreover, everyfunctor u : A → B induces a restriction functor or precomposition functor u ∗ : D ( B ) → D ( A ). Similarly, every natural transformation α : u → v of functors u, v : A → B induces a natural transformation α ∗ : u ∗ → v ∗ , but in these notes weon purpose decide to not be too explicit about these 2-dimensional aspects. Warning . The above is only terminology. Given a prederivator D and A ∈ C at ,an object X ∈ D ( A ) is just an abstract object, and not a diagram in any precisesense. However, as we will see, every such X gives rise to a usual diagram A → D ( ∗ ),and it is crucial to distinguish between these two. Example . Let C be a category and A ∈ C at . We denote by C A the category offunctors X : A → C and natural transformation between them. The formation ofthese diagram categories defines a 2-functor y C : C at op → C AT : A
7→ C A , the prederivator represented by C . The underlying category is C itself.More interesting examples arise from this one by localization . The followingexample collects two key examples of a construction which applies to arbitraryrelative categories (categories with a class of weak equivalences), and this relies onthe fact that localizations are 2-localizations. Examples . (i) Let A be an abelian category. The prederivator of A is the2-functor D A : C at op → C AT : B D ( A B ) . The underlying category of D A is the derived category D ( A ).(ii) Let M be a Quillen model category with class of weak equivalences W . Forevery B ∈ C at we denote by W B those natural transformations α : X → Y between diagrams X, Y : B → M such that the components α b : X b → Y b are weak equivalences for all b ∈ B , i.e., W B is the class of levelwise weakequivalences. The homotopy prederivator of M is the 2-functor H o M : C at op → C AT : B Ho( M B ) = M B [( W B ) − ] . The underlying category of H o M is the homotopy category Ho( M ). Construction . Let D be a prederivator, B ∈ C at , and b ∈ B . We can considerthe functor b : ∗ → B which sends the unique object to b ∈ B , and the correspondingrestriction functor b ∗ : D ( B ) → D ( ∗ )is referred to as an evaluation functor (at b ). For every f : X → Y in D ( B ) wewrite f b : X b → Y b for its image under the evaluation functor. As an exercise in2-functoriality, we invite the reader to check that every X ∈ D ( B ) gives rise to anunderlying diagram dia B ( X ) : B → D ( ∗ ) : b X b , and that this defines an underlying diagram functor dia B : D ( B ) → D ( ∗ ) B . Warning . Let D be a derivator and B ∈ C at . In general, the underlying diagramfunctor dia B fails to be an equivalence. In fact, in many cases the categories D ( B )and D ( ∗ ) B are not equivalent. We refer to the category D ( ∗ ) B as the category of incoherent diagrams of shape B in D .However, the underlying diagram functors are useful in order to visualize abstractcoherent diagrams, i.e., we often draw dia B ( X ) and say that X ∈ D ( B ) looks likedia B ( X ). To reissue this warning, it is important to distinguish between these two.In specific situations the underlying diagram functors take the following form. Example . Let B ∈ C at .(i) In the case of a represented prederivator y C , the underlying diagram dia B isthe isomorphism of categoriesdia B : C B ∼ −→ ( C ∗ ) B . (ii) For every abelian category A and the corresponding prederivator D A , theunderlying diagram functorsdia B : D ( A B ) → D ( A ) B were already considered in § M , the corresponding underlyingdiagram functor of H o M takes the formdia B : Ho( M B ) → Ho( M ) B . In general, homotopy categories of diagram categories and diagram categoriesof homotopy categories are not equivalent, and these underlying diagramfunctors hence again fail to be equivalences.The point is that coherent diagrams (such as objects in D ( A B ) and H o ( M B ))carry more information than incoherent ones. This information is crucial when onewants to calculate their derived or homotopy (co)limits. In fact, as we have seenin §
2, morphisms in derived categories do not suffice to canonically determine theirderived (co)kernels, and similarly for derived pushouts. The key properties listedin Theorem 2.19 are now turned into an abstract definition, thereby capturing aformal calculus of abstract limits and colimits. Axiom (Der4) will be made moreprecise in the discussion that follows the definition.
Definition 3.10.
A prederivator D : C at op → C AT is a derivator if it enjoys thefollowing properties.(Der1) The canonical inclusion functors B j → ` i ∈ I B i , j ∈ I, induce an equivalenceof categories D ( a i ∈ I B i ) ∼ −→ Y i ∈ I D ( B i ) . (Der2) A morphism f : X → Y in D ( B ) is an isomorphism if and only if the mor-phisms f b : X b → Y b , b ∈ B, are isomorphisms in D ( ∗ ) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 35 (Der3) For every functor u : A → B , the restriction functor u ∗ : D ( B ) → D ( A ) hasa left adjoint u ! and a right adjoint u ∗ ,( u ! , u ∗ ) : D ( A ) ⇄ D ( B ) , ( u ∗ , u ∗ ) : D ( B ) ⇄ D ( A ) . (Der4) For every functor u : A → B , the functors u ! , u ∗ : D ( A ) → D ( B ) can becalculated pointwise. Remark . Some discussion of this definition is in order.(i) Axiom (Der1) makes precise that that coherent diagrams of shape given by adisjoint union are determined by the canonical restrictions to the respectivesummands. Axiom (Der2) is motivated by the following two examples.(a) A natural transformation α : X → Y in a diagram category C B is an iso-morphism if and only if all components α b : X b → Y b are isomorphisms.(b) For every abelian category A and B ∈ C at there is an obvious iso-morphism Ch( A B ) ∼ −→ Ch( A ) B . Under this isomorphism the quasi-isomorphisms W A B in A B correspond precisely to the levelwise quasi-isomorphisms W B A .(ii) Axioms (Der3) and (Der4) jointly encode, first, a “ homotopical completenessand cocompleteness property”, thereby guaranteeing that “homotopical ver-sions” of limits, colimits, and Kan extensions exist, and, second, formulaswhich allow us to calculate the (homotopy) Kan extensions. Let us expanda bit on this.As a special case, for every A ∈ C at , there is the unique functor π A : A → ∗ .Correspondingly, by (Der3) the functor π ∗ A : D ( ∗ ) → D ( A ) has a left adjoint( π A ) ! and a right adjoint ( π A ) ∗ , which we respectively also denote bycolim A = ( π A ) ! : D ( A ) → D ( ∗ ) and lim A = ( π A ) ∗ : D ( A ) → D ( ∗ ) . These abstract colimit and limit functors generalize ordinary categorical(co)limits, derived (co)limits, and homotopy (co)limits (see Examples 3.13).More generally, the adjoints u ! and u ∗ are referred to as left and right Kanextensions , respectively. To be able to work with these adjoints, it is crucialthat they can be calculated as in the classical case (Construction 2.18). Toformulate this abstractly let us assume that D is a prederivator which satisfies(Der3). For every functor u : A → B , for every b ∈ B , and every coherentdiagram X ∈ D ( A ) there are canonical maps(3.12) colim ( u/b ) p ∗ ( X ) → u ! ( X ) b and u ∗ ( X ) b → lim ( b/u ) q ∗ ( X ) . (Here, p and q are the projection functors associated to slice categories as inConstruction 2.18.) Axiom (Der4) asks that these canonical maps are isomor-phisms, and in this precise sense Kan extensions in derivators are pointwise.(iii) The construction of the canonical maps in (3.12) is an instance of the calculusof canonical mates [KS74]. While working with derivators one often runs intothe situation that outputs of certain universal constructions “obviously areisomorphic”. In many cases, the formalism of mates and related notion of ho-motopy exact squares allows one to actually conclude this by, first, providingcanonical maps between such gadgets and, second, guaranteeing that thesemaps are isomorphisms [Gro13, Mal12]. For a more detailed discussion of theformalism behind it we refer to [Gro18, § § square is homotopy exact (see [Gro83, Gro] and [Cis04, Cis06, Mal05b] formany more advanced examples).(iv) All axioms of a derivator ask for certain properties of the underlying pre-derivators, which is the only actual structure . In contrast to this, in otherapproaches such as triangulated categories some non-canonical structure isput on an underlying category. Examples . Let us take up again the above examples of prederivators.(i) Let C be an ordinary category. The category C is complete and cocompleteif and only if y C : B
7→ C B is a derivator, the derivator represented by C .In this case the abstract (co)limits are the usual ones from ordinary categorytheory, and the more general adjoints u ! , u ∗ are the classical Kan extensions.For an introduction to these Kan extensions we refer to [Gro18, §
6] and fora detailed proof that y C indeed is a derivator to [Gro18, § A , D A : B D ( A B ) is a derivator,the derivator of (chain complexes in) A . In this case abstract (co)limitsare derived (co)limits, and similarly for Kan extensions. The fact that D A isa derivator follows from the next example.As special cases, there are hence the derivator D k of a field k and thederivator D R of a ring R (see Theorem 2.19). In contrast to the derivedcategory D ( k ) of a field, the derivator D k of a field already is quite interestingas it encodes, for instance, derived categories of path algebras, incidencealgebras, and group algebras. As an additional class of examples, associatedto every scheme X there is the derivator D X of (chain complexes of) quasi-coherent O X -modules. In fact, by [EE05, §
3] the category of quasi-coherent O X -modules is Grothendieck abelian for arbitrary schemes X .(iii) For every Quillen model category M , the prederivator H o M : B Ho( M B )is a derivator, the homotopy derivator of M . In full generality this was es-tablished by Cisinski as [Cis03, Thm. 6.11], and related to this see [CS02]. Foran even more general version we also refer to [Cis10, Thm. 2.21, Cor. 2.24, andCor. 2.28]. For combinatorial Quillen model categories an alternative proofcan be found in [Gro13, § A , the category Ch( A ) admitsa combinatorial model structure with quasi-isomorphisms as weak equiv-alences (see [Bek00, Prop. 3.13], [Hov01, § §§ D A of A can be described as homotopy derivator H o Ch( A ) .(b) As an universal example there is the derivator of spaces S = H o Top , which arises homotopy derivator of the classical Serre model structure[Qui67]. For alternative approaches to S we also refer to [Tho80, Gro83,Mal05b, Cis06]. In the case of S abstract limits are the classical ho-motopy limits of topological spaces. Systematic classical accounts arein [BK72, BV73] and special instances can, for example, be found in[Mat76]. Similarly, the derivator of pointed spaces is S ∗ = H o Top ∗ . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 37 (c) An important role in this project is played by the derivator of spectra.This derivator is defined as S p = H o Sp , the homotopy derivator with respect to the classical model structureon sequential spectra [BF78]. The underlying category of S p is the stable homotopy category SHC [Boa64, Pup67, Vog70]. Alternative (andmonoidal) approaches to this derivator are based on [HSS00, EKMM97,MMSS01].There is also a variant of homotopy derivators of complete and cocomplete ∞ -categories [RV17b, Len17]. Morally, the same should be true for all other ap-proaches to a theory of complete and cocomplete ( ∞ , ∞ , Remark . A few more comments related to Definition 3.10 are in order.(i) There is some flexibility with respect to the choice of allowable shapes inthe definition of a derivator. In this paper we always allow for arbitrarysmall shapes, but in some situations more restrictive assumptions are useful.For instance, if one uses bounded derived categories of diagram categories ofsuitably finite shapes only, then by a result of Keller [Kel07a] for every exactcategory in the sense of Quillen [Qui73, §
2] there is a corresponding derivator.(ii) Following the convention of Anderson [And79], Heller [Hel88], and Franke[Fra96], the axioms in Definition 3.10 encode aspects of the calculus of di-agrams (covariant functors) in a fixed abstract homotopy theory. Conse-quently, the domain of definition is C at op . Grothendieck [Gro], Cisinski[Cis03, Cis08], Maltsiniotis [Mal01b, Mal07] consider presheaves (contravari-ant functors) instead, and consequently their domain of definition is C at coop ,which is obtained from C at by changing the orientation of functors and nat-ural transformations. The resulting theories are equivalent.(iii) Up to the fact that we give preference to diagrams and not presheaves, theprecise form of Definition 3.10 is due to Grothendieck [Gro, pp. 43-46]. Sim-ilar axioms were also proposed by Anderson [And79, § Example . Let D be a derivator. The opposite derivator D op of D is givenby D op : C at op → C AT : B D ( B op ) op . As in ordinary category theory, this example is important because of the resulting duality principle . In many later statements we allow ourselves to focus on resultson colimits and left Kan extensions. An application to opposite derivators yieldsthe corresponding dual statement. The formation of opposites of derivators iscompatible with the formation of opposites of complete and cocomplete categories,abelian categories and model categories.
Stable derivators.
Having introduced derivators as a compromise model forabstract homotopy theories in § § Definition 3.16.
A derivator D is pointed if the underlying category D ( ∗ ) has azero object. Remark . Let D be a derivator.(i) As a consequence of (Der1), the underlying category D ( ∗ ) has an initial objectand a final object, and these objects are hence asked to be isomorphic. Asusual, any such object is denoted by 0 ∈ D ( ∗ ).(ii) It follows from (Der3) that for every B ∈ C at also D ( B ) has a zero object, andthat all restriction and Kan extension functors preserve these zero objects. Examples . Let us take up again Examples 3.13.(i) Let C be a complete and cocomplete category. The underlying category ofthe represented derivator y C is C , and y C is hence pointed if and only if C ispointed (has a zero object).(ii) Let A be a Grothendieck abelian category. The underlying category of thederivator D A is the derived category D ( A ). As an additive category, D ( A )clearly has a zero object, and D A is hence pointed.(iii) Similarly, for a pointed Quillen model category M , the homotopy derivator H o M is pointed. Among the specific examples S , S ∗ , and S p only the lasttwo are pointed. Definition 3.19.
Let D be a derivator and let X ∈ D ( (cid:3) ).(i) The square X is cocartesian if it lies in the essential image of the left Kanextension functor ( i p ) ! : D ( p ) → D ( (cid:3) ) . (ii) The square X is cartesian if it lies in the essential image of the right Kanextension functor ( i y ) ∗ : D ( y ) → D ( (cid:3) ) . Remark . There are the following remarks related to Definition 3.19.(i) Let D be a derivator and let X ∈ D ( (cid:3) ) be a square in D . It turns out that X is cocartesian if and only if a certain canonical mapcolim p ( i p ) ∗ X → X , is an isomorphism. The formalism behind these canonical maps (based onthe calculus of canonical mates) will not be made explicit in this paper (butsee [KS74] or [Gro13]). In this paper, we will occasionally refer to certainmaps in derivators as “canonical”, and in such situations they always ariseby means of this calculus.(ii) Let us take up again our standard examples (Examples 3.13). A square ina represented derivator is cocartesian if and only if it is a pushout square.For every Grothendieck abelian category A , a square in D A is cocartesian ifand only if is a derived pushout square. In particular, in the derivator D R of a ring we recover Definition 2.25, which played a key role in §
2. Finally, a
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 39 square in the homotopy derivator of a Quillen model category is cocartesianif and only if it is a homotopy pushout square.
Proposition 3.21.
Let D be a derivator and u : A → B a fully faithful functor.The Kan extension functors u ! , u ∗ : D ( A ) → D ( B ) are fully faithful.Proof. For a proof based on the calculus of canonical mates and the correspondingformalism of homotopy exact squares we refer to [Gro13, Prop. 1.20]. (cid:3)
This property applied to D R (Proposition 2.21) was central to many construc-tions in §
2. In order to develop some intuition for derivators we generalize some ofthese constructions to pointed derivators.
Construction . Let D be a pointed derivator.(i) We begin with a derivator version of Proposition 2.31. Let f ∈ D ([1]) be amorphism looking like ( f : x → y ), which is to say that dia [1] ( f ) : [1] → D ( ∗ )takes the form ( f : x → y ) (see Warning 3.8). In order to extend f to acofiber square, we again consider the fully faithful inclusion functors i p ◦ i : [1] → p → (cid:3) and their corresponding Kan extension functors( i p ) ! ◦ i ∗ : D ([1]) → D ( p ) → D ( (cid:3) ) . The image ( i p ) ! ◦ i ∗ ( f ) ∈ D ( (cid:3) ) can be restricted along the inclusion k : [1] → (cid:3) pointing at the vertical morphism on the right. We define the cone functor cof = k ∗ ◦ ( i p ) ! ◦ i ∗ : D ([1]) → D ([1]) . The object obtained from cof ( f ) by an evaluation at 1 ∈ [1] is also referredto as the cone of f , but the corresponding construction is distinguished no-tationally by C = 1 ∗ ◦ cof : D ([1]) → D ( ∗ ) . With this notation, the cofiber square ( i p ) ! ◦ i ∗ ( f ) associated to f looks like(3.23) x f / / (cid:15) (cid:15) y cof ( f ) (cid:15) (cid:15) / / C ( f ) . ❴✤ We invite the reader to dualize these construction in order to define fiberfunctors fib : D ([1]) → D ([1]) and F : D ([1]) → D ( ∗ ) . (ii) As a minor variant of the previous case, we now construct suspension andloop functors in pointed derivators, thereby generalizing Remark 2.34. Thereare at least two ways of defining the suspension. First, for x ∈ D ( ∗ ) we wouldlike to make the definition Σ x = C ( x → . To explain the right hand side in terms of Kan extensions, it suffices to con-sider the functor 0 : ∗ → [1] pointing at zero and the corresponding right
Kan extension functor 0 ∗ : D ( ∗ ) → D ([1]). In fact, 0 ∗ extends x to a mor-phism with 0 as target as follows from (Der4). Correspondingly, we definethe suspension functor asΣ = C ◦ ∗ : D ( ∗ ) → D ( ∗ ) . Second, a canonically isomorphic way of defining the suspension is by meansof Σ = (1 , ∗ ◦ ( i p ) ! ◦ (0 , ∗ : D ( ∗ ) → D ( p ) → D ( (cid:3) ) → D ( ∗ ) . In both cases the suspension Σ x sits as lower right corner in the corresponding suspension square (3.24) x / / (cid:15) (cid:15) (cid:15) (cid:15) / / Σ x. ❴✤ It is straightforward to dualize these constructions in order to obtain a loopfunctor
Ω : D ( ∗ ) → D ( ∗ ) . Examples . Let us specialize these constructions in our examples of pointedderivators (Examples 3.18).(i) Let C be a pointed, complete and cocomplete category and y C its representedderivator. The cofiber squares (3.23) and the suspension squares (3.24) in y C are ordinary pushout squares. Hence, (3.23) implies that cof and C in y C are the usual cokernel functors (once considered with the structure map andonce without). As a special case of this, the suspension squares (3.24) implythat Σ x is isomorphic to a zero object, and Σ is hence naturally isomorphicto the constant functor on the zero object,(3.26) Σ ∼ = 0 : C → C . (ii) Let A be a Grothendieck abelian category and D A its derivator. In thiscase the cone functor C : D ( A [1] ) → D ( A ) reproduces the functorial coneconstruction from Proposition 2.7 (see also the proof of Proposition 2.13).As noted in §
2, the functor cof : D ( A [1] ) → D ( A [1] )is in the background of the rotation of distinguished triangles (as a shadowof the symmetries of Barratt–Puppe sequences in Figure 3). The suspen-sion functor in D A specializes to the usual shift functor Σ : D ( A ) → D ( A )(Remark 2.34).(iii) As a final example, we consider the homotopy derivator S ∗ of pointed spaceswhich also provides the motivation for some of the terminology employedhere. Given a morphism ( f : X → Y ) ∈ Ho(Top [1] ∗ ) of pointed topologicalspaces, the cone C ( f ) is the usual mapping cone construction. Moreover, theabstract suspension specializes to the reduced suspension functorΣ : Ho(Top ∗ ) → Ho(Top ∗ ) , and Ω : Ho(Top ∗ ) → Ho(Top ∗ ) IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 41 is the usual loop space functor (see also Remark 3.32).
Proposition 3.27.
Let D be a pointed derivator.(i) There is an adjunction ( cof , fib ) : D ([1]) ⇄ D ([1]) . (ii) There is an adjunction (Σ , Ω) : D ( ∗ ) ⇄ D ( ∗ ) . Proof.
For a proof we refer to [Gro13, § (cid:3) In triangulated categories we are used to be able to rotate distinguished trian-gles without a loss of information and similarly that the suspension functor is anequivalence. With the few exceptions of exotic triangulated categories [MSS07], formost triangulated categories arising in nature these features are consequences ofproperties of stable homotopy theories in the background.
Definition 3.28.
A pointed derivator D is stable if every square in D is co-cartesian if and only if it is cartesian, and these squares are then referred to as bicartesian .In order to find examples of stable derivators, it is useful to have simpler char-acterizations of stability. Theorem 3.29.
The following are equivalent for a pointed derivator D .(i) The derivator D is stable.(ii) The adjunction ( cof , fib ) : D ([1]) ⇄ D ([1] is an equivalence.(iii) The adjunction (Σ , Ω) : D ( ∗ ) ⇄ D ( ∗ ) is an equivalence.Proof. If D is stable, then cofiber squares (3.23) and fibre squares are essentiallythe same, and similarly for suspension squares (3.24) and loop squares. From thisit is fairly straightforward to deduce that (i) implies (ii) and (iii). The converseimplications are more involved and we refer the reader to [GPS14b, Thm. 7.1]. (cid:3) Examples . With this preparation we can revisit our examples of pointed deriva-tors (Examples 3.18).(i) Let C be a pointed, complete and cocomplete category and y C its representedderivator. In this case the abstract suspension functor is naturally isomorphicto the constant functor on the zero object (3.26). Hence, by Theorem 3.29 weconclude that y C is stable if and only if Σ ∼ = 0 is an equivalence of categoriesif and only if C is equivalent to the terminal category ∗ .(ii) Let A be a Grothendieck abelian category and D A its derivator. In D A theabstract suspension functor agrees with the shift functor Σ : D ( A ) → D ( A ),which clearly is an equivalence of categories. Hence, derivators of the form D A are stable.(iii) A stable model category is a pointed model category such that the ab-stract suspension functor is an equivalence in its homotopy derivator ([Hov99,Def. 7.1.1]). By Theorem 3.29 we deduce that homotopy derivators of stablemodel categories are stable (as it also follows from [Hov99, Rmk. 7.1.12]).As a special case, the derivator S p of spectra is stable, and there are manyadditional interesting examples (see Examples 4.2 or [SS03]).Additional classes of examples arise as homotopy derivators of exact categories[Gil11, ˇSˇto14], stable cofibration categories [Sch13, Len17], or stable ∞ -categories[Lur09, Lur14, Len17]. Remark . We want to stress the observation that a represented derivator y C is stable if and only if C ≃ ∗ . This essentially says that stability is invisible toordinary category theory . In order to capture the phenomenon of stability as itarises in interesting examples in homological algebra, stable homotopy theory orhomotopical algebra we have to use more refined techniques. Among these aretriangulated categories, stable derivators, stable model categories, and stable ∞ -categories (see again Disclaimer 3.1).We conclude this subsection by sketching the relation to triangulated categories.Many arguments are completely parallel to the discussion in §
2, and here we hencefocus on the remaining arguments. We begin by sketching that stability implies ad-ditivity. In order to motivate the general approach, we revisit the classical situationin topology.
Remark . Let (
X, x ) be a pointed topological space. The loop space Ω X of X at x is the space of paths [0 , → X which send both boundary points 0 , x . The loop space is the homotopy pullback (in the usual Serre model structure)of the cospan on the left in ∗ x (cid:15) (cid:15) Ω X / / (cid:15) (cid:15) ❴✤ P X ev (cid:15) (cid:15) ∗ x / / X, ∗ x / / X. In fact, if one wants to calculate this homotopy pullback, then it suffices to replaceone of the maps x : ∗ → X by a weakly homotopy equivalent Serre fibrationand then to calculate the categorical pullback. The standard example of such anapproximation is the path space P X of paths [0 , → X starting at x endowedwith the evaluation map ev : P X → X at the end point 1. This space is weaklycontractible since all such paths are homotopic to the constant path at x , and thepullback of the cospan on the right gives the above description of the loop space.The concatenation of loops can be used to show that the loop space Ω X is a groupobject in the homotopy category Ho(Top ∗ ).In order to motivate the general approach in pointed derivators, we explain apurely categorical way of modelling the inversion of loops in topology. And for thispurpose it is convenient to use a different model for the above homotopy pullback,which is obtained by replacing both maps by the above Serre fibration and thencalculating the pullback(3.33) Ω X ❴✤ p / / p (cid:15) (cid:15) P X ev (cid:15) (cid:15) P X ev / / X. Up to an obvious homeomorphism, this model of the loop space has as pointspaths [ − , → X which send − , x , and in this model the inversion of loops ι : Ω X → Ω X is given by a reparametrization via the reflection at 0 ∈ [ − , P X in the pullback square (3.33). More formally, the outer commutative square
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 43 in the diagram Ω X p ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ p (cid:25) (cid:25) ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ " " ❋❋❋❋ Ω X p / / p (cid:15) (cid:15) ❴✤ P X ev (cid:15) (cid:15) P X ev / / X induces by the universal property of the pullback square a canonical dashed mor-phism making both triangles commute. And this morphism of course is the abovereparametrization map ι : Ω X → Ω X .This can be abstracted to pointed derivators as follows. Remark . Let D be a pointed derivator. For every x ∈ D ( ∗ ), the loop ob-ject Ω x can be naturally turned into a group object. The vague idea is to modelthe concatenation of loops from topology purely categorically. The details are abit more involved (see [Gro13, § A ∞ -multiplications [Seg74]. (In fact, using fibrational techniques the argumentsin [Gro13, § − ) − : Ω x → Ω x comes fromthe swap symmetry of loop squares (the duals of (3.24)) which interchanges (1 , , Eckmann–Hilton argument [EH62, Thm. 4.17] impliesthat two-fold loop objects Ω x are abelian group objects. Corollary 3.35.
The underlying category of a stable derivator is additive.Proof.
In a stable derivator D , the suspension Σ : D ( ∗ ) → D ( ∗ ) is an equivalence(Theorem 3.29), and for every object x ∈ D ( ∗ ) there is a natural isomorphism x ∼ = Ω Σ x . By Remark 3.34 this implies that every object in D ( ∗ ) is an abeliangroup object, showing that D ( ∗ ) is additive. (cid:3) With this preparation the construction of triangulations on stable derivators(Theorem 3.40) is fairly parallel to §
2. As a variant of the underlying diagramfunctor (Construction 3.7), for every prederivator D there are partial underlyingdiagram functors dia A,B : D ( A × B ) → D ( A ) B , A, B ∈ C at , making a diagram incoherent in the B -direction only. More formally, for a coherentdiagram X ∈ D ( A × B ) and b ∈ B we setdia A,B ( X ) b = (id A × b ) ∗ ( X ) ∈ D ( A ) , the restriction along id A × b : A ∼ = A × ∗ → A × B . Definition 3.36.
A derivator D is strong if the partial underlying diagram functordia A,F : D ( A × F ) → D ( A ) F is essentially surjective and full for all free categories F ∈ C at and all A ∈ C at . Construction . Let D be a stable derivator. By Construction 3.22 the under-lying category D ( ∗ ) can be endowed with a suspension functor(3.38) Σ : D ( ∗ ) → D ( ∗ ) which is an equivalence by Theorem 3.29. For every morphism ( f : x → y ) ∈ D ([1]),completely parallel to the sketch proof of Theorem 2.6 (see the discussion around(2.57)) one constructs the corresponding standard triangle of f . A triangle(3.39) x f → y g → z h → Σ x in D ( ∗ ) is distinguished if it is isomorphic to a standard triangle.The following theorem has some history, and we refer to work of Franke [Fra96],Maltsiniotis [Mal01a], and the author [Gro13]. Theorem 3.40.
For every strong and stable derivator D , the underlying category D ( ∗ ) admits a triangulation given by (3.38) and (3.39) .Proof. With this preparation, the proof is essentially the same as the proof ofTheorem 2.6. We expand a bit on the rotation axiom in order to explain wherethe sign comes from, and refer the reader to loc. cit. for the remaining aspects.For every morphism ( f : x → y ) ∈ D ([1]) there is an associated Barratt–Puppesequence (Figure 3). Let us redraw the relevant part of it and add decorations tothe various zero objects in order to distinguish them notationally. So let B be thefollowing poset and let Y ∈ D ( B ) look like(3.41) x f / / (cid:15) (cid:15) (cid:3) y / / g (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / z h / / (cid:15) (cid:15) (cid:3) x ′ f ′ (cid:15) (cid:15) / / y ′ . In order to identify the underlying morphism of f ′ as Σ f , we make the identifications ϕ : Σ x ∼ −→ x ′ and ψ : Σ y ∼ −→ y ′ , and these lead to the commutative square(3.42) Σ x ϕ ∼ / / Σ f (cid:15) (cid:15) x ′ f ′ (cid:15) (cid:15) Σ y ψ ∼ / / y ′ . Strictly speaking these identification ϕ and ψ respectively are the canonical mapsassociated to the suspension squares x / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) y / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / x ′ , / / y ′ . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 45
Similarly, in order to identify the final object in the distinguished triangles of f and g as suitable suspensions, we restrict (3.41) in order to obtain the cofiber sequences x f / / (cid:15) (cid:15) (cid:3) y / / g (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) y g / / (cid:15) (cid:15) (cid:3) z / / h (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / z h / / x ′ / / x ′ f ′ / / y ′ , paste the respective bicartesian squares in order to obtain the suspension squares x / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) y / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / x ′ , / / y ′ , and deduce from these the resulting identifications ϕ : Σ x ∼ −→ x ′ and ψ : Σ y ∼ −→ y ′ . Now, the suspension squares giving rise to the identifications ϕ and ϕ are literallythe same and we deduce ϕ = ϕ . In contrast to this, the suspension squaresinducing ψ and ψ differ by a restriction along the symmetry exchanging (1 , , ψ agrees with the negative of ψ .Thus, in order to write the third morphism in the distinguished triangle associatedto g as a morphism Σ x → Σ y , invoking (3.42) we are led toΣ x ϕ ∼ / / − Σ f (cid:15) (cid:15) x ′ f ′ (cid:15) (cid:15) Σ y ψ ∼ / / y ′ . Consequently, the distinguished triangle associated to g takes the form y g → z h → Σ x − Σ f → Σ y, as asked for by the rotation axiom of triangulated categories. (cid:3) Exponentials of derivators.
In this short subsection we discuss the con-struction of derivators of representations or exponentials in more categorical termi-nology. This is a derivatorish version of diagram categories and these exponentialsgovern the calculus of parametrized limits and parametrized Kan extensions. Theformation of represented derivators, of derivators of abelian categories and of ho-motopy derivators are compatible with the passage to exponentials.Let D be a derivator and let A ∈ C at . Taking the philosophy of derivatorsserious, we should not be happy with a category D ( A ) of coherent diagrams ofshape A in D , but we should rather ask for a derivator D A of such diagrams. Theheuristics are as follows: diagrams of shape B in D A are diagrams of shape A indiagrams of shape B in D which by the exponential law should be the same asdiagrams of shape A × B in D . This suggests the following construction. Construction . Let D be a prederivator and let A ∈ C at . We denote by D A the 2-functor D A : C at op → C AT : B D ( A × B ) . This is the prederivator of diagrams of shape A in D . The underlying category is D A ( ∗ ) = D ( A × ∗ ) ∼ = D ( A ).The following example illustrates the relation to representation theory. In thatexample we already invoke the notion of a morphism of (pre-)derivators which weintroduce formally in § Example . Let k be a field, let Q be a quiver with finitely many vertices only,and let kQ be the k -linear path algebra. Identifying the quiver Q with the freecategory Q generated by it, we want to describe the prederivator D Qk : C at op → C AT : B D k ( Q × B )differently. For this purpose, given an abelian category A , let us write Ho(Ch( A ))for the localization of Ch( A ) at the class of quasi-isomorphisms, so that we have D ( A ) = Ho(Ch( A )). With this notation, by means of the exponential law and theequivalence Mod( k ) Q ≃ Mod( kQ ) there is the following chain of equivalences ofcategories D Qk ( B ) = D k ( Q × B )= Ho(Ch( k ) Q × B ) ∼ = Ho (cid:0) (Ch( k ) Q ) B (cid:1) ≃ Ho (cid:0) Ch( kQ ) B (cid:1) = D kQ ( B ) . This holds for every B ∈ C at in a compatible way, and we hence conclude thatthere is an equivalence of derivators D Qk ≃ D kQ . Thus the formation of shifted derivators models the passage to the path algebra,and there are of course variants for incidence algebras, group algebras, and moregeneral category algebras.In the above example, the prederivator D Qk of Q -shaped diagrams in D k is againa derivator. Let us recall from classical category theory that functor categories C A inherit “exactness properties” from C by means of pointwise constructions. Thefollowing is a derivatorish version of this. Theorem 3.45.
Let D be a derivator and let A ∈ C at . The prederivator D A is aderivator, the derivator of A -shaped diagrams in D . Moreover, if D is pointed,stable, or strong, then D A is pointed, stable, or strong, respectively.Proof. Most of the axioms of a derivator are straightforward. In order to show that D A also satisfies (Der4), some basics on the calculus of homotopy exact squares areneeded [Gro13, Thm. 1.25]. The fact that D A is strong if this is the case for D isimmediate from Definition 3.36. Moreover, by Remark 3.17 the property of beingpointed is inherited as well, and for the remaining case of stable derivators we referto [Gro13, Prop. 4.3]. (cid:3) We also refer to these derivators D A as exponentials or shifted derivators . Remark . Let us expand a bit on the calculus of Kan extensions in shiftedderivators. Let D be a derivator, let A ∈ C at , and let u : B → B ′ be a functor. The IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 47 proof of Theorem 3.45 shows that left Kan extension along u in D A is given by leftKan extension along id A × u in D , u D A ! = (id A × u ) D ! : D ( A × B ) → D ( A × B ′ ) . Moreover, Kan extensions and restrictions in unrelated variables commute ([Gro13,Prop. 2.5]). This specializes to the fact that for every a ∈ A the following square D ( A × B ) (id × u ) ! / / ( a × id) ∗ (cid:15) (cid:15) ∼ = D ( A × B ′ ) ( a × id) ∗ (cid:15) (cid:15) D ( B ) u ! / / D ( B ′ )commutes up to a canonical isomorphism. In particular, for every X ∈ D ( A × B )there is a canonical isomorphismcolim D B ( a ∗ X ) ∼ −→ a ∗ colim D A B X in D ( ∗ ). In this precise sense (co)limits and Kan extensions in D A treat the A -variable as parameters. Remark . This closure under the formation of exponentials is one of the tech-nical advantages of derivators over triangulated categories. Recall that, in general,for a triangulated category T and a small category A there is no natural trian-gulation on the functor category T A (which, for instance, is compatible with allevaluations). And even if there would be such a triangulation, in most examples,the category T A is not the one we would like to study (for instance, for a field k and a quiver Q , we are interested in D ( kQ ) which is different from D ( k ) Q , thecategory of Z -graded representations of Q ).Similarly to derivators, ∞ -categories also enjoy this closure property [Lur09,Prop. 1.2.7.3], while for arbitrary Quillen model category the situation is somewhatmore subtle (related to this see [BK72, Hel88, Jar87] for early special cases, and[Hir03, Thm. 11.6.1] or [Lur09, § A2.8] for published general results).
Examples . Let B be a small category.(i) For every complete and cocomplete category C the derivators y B C and y C B are equivalent. In these derivators the abstract calculus of limits and Kanextensions agrees with the classical parametrized versions of limits and Kanextensions ([ML98, § V.3]).(ii) For every Grothendieck abelian category A there is an equivalence of deriva-tors D B A ≃ D A B . (iii) For every model category M there is an equivalene of derivators H o M B ≃ H o B M . There is a similar variant for ∞ -categories. Remark . Let D be a derivator and A ∈ C at .(i) The underlying category of D A is D ( A ), which is to say that every stageof a derivator is the underlying category of a derivator. In particular, thisoften allows us to focus on the underlying category when we want to estab-lish certain properties for arbitrary stages. For instance, the class of strong and stable derivators is closed under exponentials (Theorem 3.45), and weconclude by Theorem 3.40 that all stages of such derivators admit canoni-cal triangulations. These triangulations are compatible with each other as itfollows from the discussion of morphisms in § D A is the calculus of parametrized limits and Kan extensions in D . Implicitly, this parametrized calculus alreadyappeared in proof of the rotation axiom. In fact, the vertical coherent mor-phism f ′ in (3.41) really is the value of the parametrized suspension functorΣ : D ([1]) → D ([1]) at f ∈ D ([1]). In fact, by [GPS14b, Lem. 5.13] there is anatural isomorphism(3.50) Σ ∼ = cof : D ([1]) → D ([1]) . Morphisms of derivators.
In this subsection we briefly discuss morphisms ofderivators, the interaction of morphisms with limits and the relation to exact func-tors of triangulated categories. We also introduce natural transformations betweenmorphisms of derivators and the related notions of adjunctions and equivalences ofderivators. To begin with, there is the following definition.
Definition 3.51. A morphism of derivators is a pseudo-natural transformation.Let us unravel this definition. For derivators D and E , a morphism F : D → E consists of(i) functors F A : D ( A ) → E ( A ) for A ∈ C at and(ii) natural isomorphisms γ u : u ∗ F B ∼ −→ F A u ∗ for u : A → B, D ( A ) F A / / E ( A ) ✻✻✻✻ W _ ∼ = D ( B ) u ∗ O O F B / / E ( B ) , u ∗ O O satisfiying certain coherence axioms. For instance, for every pair of composablefunctors u : A → B and v : B → C the natural isomorphism γ vu agrees with thefollowing pasting of γ u and γ v , D ( A ) F A / / E ( A ) ✻✻✻✻ W _ ∼ = D ( B ) u ∗ O O F B / / E ( B ) , ✻✻✻✻ W _ ∼ = u ∗ O O D ( C ) v ∗ O O F C / / E ( C ) . v ∗ O O And there is the axiom γ id A = id and one additional axiom concerning the inter-action of γ u and γ v with natural transformations α : u → v . A morphism is strict if all components γ u are identities, which is to say that the functors F A commutewith restrictions on the nose (and not only up specified, coherent isomorphisms). Examples . Let us take up again some of Examples 3.13.
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 49 (i) Every functor F : C → D between complete and cocomplete categories inducesa strict morphism of represented derivators F = y F : y C → y D . In fact, for every A ∈ C at , we define F A as the postcomposition functorinduced by F , F A : C A → D A : X F ◦ X, and it is straightforward to verify the axioms of a strict morphism.(ii) Every left Quillen functor F : M → N of model categories induces a totalleft derived morphism of homotopy derivators LF : H o M → H o N ;see [KM08] and [Cis10, § LF ) A is given by( LF ) A = L ( F A ) : Ho( M A ) → Ho( N A ) . The passage to total left derived functors is pseudo-natural [Hov99, § γ u . There is, of course,a similar result for right Quillen functors, and these results yield extrinsicconstructions of many interesting examples of morphisms of derivators. Thisincludes derived tensor products or derived hom functors, and we refer thereader to the literature for plenty of examples.(iii) A relevant intrinsic example is given by restriction morphisms. In fact, forevery derivator D and every u : A → B there is an induced restrictionmorphism u ∗ : D B → D A . For every C ∈ C at the corresponding component is defined as u ∗ C = ( u × id C ) ∗ : D ( B × C ) → D ( A × C ) . These functors define a strict morphism u ∗ as it follows immediately from2-functoriality of D . (There are also Kan extension morphisms as we will seein a bit.) Definition 3.53.
A morphism of derivators F : D → E preserves colimits of shape A if for every X ∈ D ( A ) the canonical mapcolim A F X ∼ −→ F colim A X is an isomorphism in E ( ∗ ).These canonical maps are further instances of the canonical mates mentioned inRemark 3.20. In all detail, these are morphisms of the formcolim E A F A ( X ) ∼ −→ F ∗ colim D A ( X ) , where F ∗ : D ( ∗ ) → E ( ∗ ) is the underlying functor of F . In represented derivatorsthis notion reduces to the usual notion of preservation of colimits, while in homotopyderivators of abelian categories or model categories this captures the preservationof derived colimits or homotopy colimits, respectively. Of course, there are refinednotions such as the preservation of the A -shaped colimit of a fixed diagram only. Examples . The following classes of morphisms are important. (i) A morphism of pointed derivators is pointed if and only if it preserves zeroobject (in the usual sense).(ii) A morphism of derivators is right exact if it preserves initial objects andpushouts, i.e., colimits of shape p . There is the dual notion of a left exactmorphism and the combined notion of an exact morphism .(iii) Finally, a morphism is cocontinuous if it preserves all colimits and contin-uous if it preserves all limits. Lemma 3.55.
A morphism of stable derivators is left exact if and only if it is rightexact if and only if it is exact.Proof.
The key is the following elementary observation: a morphism of derivatorspreserves colimits of shape A if and only if it is preserves colimiting cocones of shape A [Gro16, Prop. 3.9]. As a special case this implies that a morphism of derivatorspreserves pushouts if and only if it is preserves cocartesian squares, and the claimsfollow by definition of stability. (cid:3) Exact morphisms are the good notion of morphisms of stable derivators. Toprovide some first evidence for this claim, we include the following discussion. Recallthat in ordinary category theory all colimits can be constructed from coproductsand coequalizers [ML98, § V.2]. More relevant to our current discussion is a variantof this for finite colimits. To begin with, let us recall that a category is finite if ithas finitely many objects and morphisms only.
Finite colimits can be constructedfrom finite coproducts and coequalizers or, alternatively, from initial objects andpushouts. Similarly, a functor preserves finite colimits if and only if it is right exact,i.e., it preserves initial objects and pushouts.A variant of these results also holds for derivators, but in this case we have toinvoke a more restrictive notion of finiteness. Suggestively, this is due to the factthat in derivator land colimits also include derived colimits and homotopy colimits.
Definition 3.56.
A category A is strictly homotopy finite if A is finite, if everyendomorphism f : x → x in A is equal to id x , and if isomorphic objects in A areequal ( A is skeletal). A category is homotopy finite if it is equivalent to a strictlyhomotopy finite category.The following result is due to Ponto–Shulman. There is also a variant for theconstruction of colimits, but here we only formulate the result for morphisms. Theorem 3.57.
A morphism of derivators is right exact if and only if it preserveshomotopy finite colimits.Proof.
This is [PS16, Thm. 7.1]. (cid:3)
Corollary 3.58.
A morphism of stable derivators is exact if and only if it preserveshomotopy finite limits and homotopy finite colimits.Proof.
This is immediate from Theorem 3.57 and Lemma 3.55. (cid:3)
Remark . Generalizing Definition 3.53, a morphism of derivators F : D → E preserves left Kan extensions along u : A → B in C at if for every X ∈ D ( A ) thecanonical morphism u ! F ( X ) ∼ −→ F u ! ( X )is an isomorphism in E ( B ). Since Kan extensions are pointwise, one checks that amorphism is cocontinuous if and only if it preserves all left Kan extensions [Gro13, IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 51
Prop. 2.3]. As a variant, a morphism of stable derivators is exact if and only if itpreserves all sufficiently finite left and right Kan extensions (see [Gro16, Thm. 9.14]for more details).In order to offer a second justification that exact morphisms are the good mor-phisms for stable derivators we include the following discussion. Since our mainfocus is on stable derivators, in the following construction we immediately special-ize to this context.
Construction . Let G : D → E be a pointed morphism of stable derivators.(i) For every x ∈ D ( ∗ ) there is the defining suspension square X ∈ D ( (cid:3) ) on theleft in x / / (cid:15) (cid:15) (cid:15) (cid:15) Gx / / (cid:15) (cid:15) (cid:15) (cid:15) / / Σ x, ❴✤ / / G Σ x. Since G preserves zero objects, the image GX ∈ E ( (cid:3) ) looks like the abovediagram on the right (here, we invoke that morphisms of derivators commutewith evaluation functors as special cases of restriction functors). In general, GX will not be cocartesian. Hence, a comparison of this square against thesuspension square of Gx yields a canonical comparison map ψ : Σ Gx → G Σ x. This comparison map is an isomorphism if and only if GX is cocartesian,which is certainly the case if G is not only pointed but even exact. In thatcase, we can consider its inverse(3.61) ϕ = ψ − : G Σ x ∼ −→ Σ Gx which allows us to ‘pull out the suspension’.(ii) As a variant of the previous case, let ( f : x → y ) ∈ D ([1]) be a morphism andlet X ∈ D ( (cid:3) ) be the cofiber square associated to it, x f / / (cid:15) (cid:15) y (cid:15) (cid:15) Gx Gf / / (cid:15) (cid:15) Gy (cid:15) (cid:15) / / Cf, ❴✤ / / GCf.
The image square GX ∈ E ( (cid:3) ) again vanishes at the lower left corner, but,in general, it fails to be cocartesian. This time this leads to a canonicalcomparison map ψ : CGf → GCx, which is invertible if G is exact.The details of these constructions are carried out in [Gro16, § Proposition 3.62.
Let F : D → E be an exact morphism of strong stable deriva-tors. The natural isomorphism ϕ : F Σ ∼ −→ Σ F defined by (3.61) turns the underlyingfunctor F : D ( ∗ ) → E ( ∗ ) into an exact functor. Proof.
This is [Gro16, Thm. 10.6]. (cid:3)
Remark . The direct proof of Proposition 3.62 is a lengthy direct calculation.There is a more systematic approach based on the notion of exact formulas instable derivators [BG18, § cof ∼ = Σ : D ([1]) → D ([1]) from (3.50), and Proposition 3.62 followsfrom the compatibility of exact morphisms with this formula. However, as detailedin loc. cit. there are many additional such exact formulas (see for instance [BG18,Ex. 13.14]).Having discussed the notion of exact morphisms, in order to fulfill our duties inthis section it only remains to talk about equivalences of derivators. This notionis, of course, defined internally in the 2-category of derivators, and correspondinglywe begin by defining natural transformations. Definition 3.64.
A natural transformation of morphisms of derivators is a modi-fication.Again, for basics on this notion, we refer to [Gro13, §
2] and [Gro16, Lem. 3.11],but for convenience we unravel this definition. Let
F, G : D → E be morphisms ofderivators. A natural transformation α : F → G consists of natural transformations α A : F A → G A : D ( A ) → E ( A ) , A ∈ C at . These components α A are supposed to satisfy the obvious compatibility with thepseudo-naturality constraints of F and G . Remark . There is a 2-category D ER of derivators, morphisms of derivators,and natural transformations. We denote by D ER St , ex the sub-2-category given bystable derivators, exact morphisms, and all natural transformations. (The basiccalculus of the interaction of morphisms with colimits implies, for instance, thatexact morphisms are closed under composition.) This 2-category plays a key rolein abstract representation theory, as we discuss in § Definition 3.66. An adjunction of derivators is an adjunction internally to D ER .An equivalence of derivators is an equivalence internally to D ER .Thus, an adjunction consists of morphisms L : D → E and R : E → D togetherwith natural transformations η : id → RL and ε : LR → id which are subject to thetriangular identities. We will denote adjunctions by( L, R ) : D ⇄ E . There is the following result which often allows us to exhibit a given morphism aspart of an adjunction.
Proposition 3.67.
A morphism F : D → E of derivators is a left adjoint if andonly if all components F A : D ( A ) → E ( A ) , A ∈ C at , are left adjoints and the mor-phism F is cocontinuous.Proof. This is [Gro13, Prop. 2.9], but we want to sketch one direction of the proof.Given a morphism F : D → E such that all components F A are left adjoints, webegin by choosing right adjoints G A and corresponding levelwise adjunctions( F A , G A ) : D ( A ) ⇄ E ( A ) IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 53 for all A ∈ C at independently. It turns out that the cocontinuity of F allowsus to define pseudo-naturality constraints u ∗ G B ∼ −→ G A u ∗ , thereby obtaining theintended adjunction ( F, G ) : D ⇄ E . (cid:3) In order to provide some additional feeling for the notion we mention the follow-ing compatibility of the various adjunctions ( F A , G A ) constituting an adjunction( F, G ) : D ⇄ E of derivators. For every u : A → B in C at , X ∈ D ( B ), and Y ∈ E ( B ) the diagramhom E ( B ) ( F X, Y ) ∼ / / u ∗ (cid:15) (cid:15) hom D ( B ) ( X, GY ) u ∗ (cid:15) (cid:15) hom E ( A ) ( u ∗ F X, u ∗ Y ) ∼ (cid:15) (cid:15) hom D ( A ) ( u ∗ X, u ∗ GY ) ∼ (cid:15) (cid:15) hom E ( A ) ( F u ∗ X, u ∗ Y ) ∼ / / hom D ( A ) ( u ∗ X, Gu ∗ Y )commutes. Examples . In order to obtain key examples of adjunctions we revisit the situ-ations considered in Examples 3.52.(i) A functor F : C → D between complete and cocomplete categories is a leftadjoint if and only if the morphism y F : y C → y D is a left adjoint. Thisfollows from the obvious fact that the passage to represented derivators is2-functorial.(ii) For every Quillen adjunction ( F, G ) : M ⇄ N there is an induced adjunction( LF, RG ) : H o M ⇄ H o N . In particular, if the model categories M and N are stable, then LF and RG are exact morphisms (by Proposition 3.67 and Corollary 3.58), and this leadsto a rich supply of exact morphisms of stable derivators.(iii) Let D be a derivator, let u : A → B in C at , and let u ∗ : D B → D A be therestriction morphism. Clearly, all components ( u × id C ) ∗ , C ∈ C at , of therestriction morphism have adjoints on both sides, and the morphism is alsocontinuous and cocontinuous. In fact, as a further application of the calculusof canonical mates, Kan extensions and restrictions in unrelated variablescommute up to canonical isomorphisms [Gro13, Prop. 2.5]. As an upshot, byProposition 3.67 there are adjunctions of derivators( u ! , u ∗ ) : D A ⇄ D B and ( u ∗ , u ∗ ) : D B ⇄ D A . The components of these
Kan extension morphisms are the Kan extensionfunctors ( u × id C ) ! and ( u × id C ) ∗ of D . Corollary 3.69.
Let D be a strong, stable derivator and let u : A → B be in C at .The functors u ∗ : D ( B ) → D ( A ) , u ! : D ( A ) → D ( B ) , and u ∗ : D ( A ) → D ( B ) are exact functors with respect to the canonical triangulations. Proof.
As adjoint morphisms, u ! and u ∗ are cocontinuous, while u ∗ and u ∗ arecontinuous (Proposition 3.67). Since stability is inherited by the shifted deriva-tors D A and D B (Theorem 3.45), all three morphisms u ∗ , u ! , and u ∗ are exact(Lemma 3.55), and the claim hence follows from Proposition 3.62. (cid:3) There is the following more refined version of this result.
Remark . Let T riaCAT be the 2-category of triangulated categories, exactfunctors, and exact natural transformations. For every strong and stable derivator D , the formation of canonical triangulations and canonical exact structures yieldsa lift of D against the forgetful functor T riaCAT → C AT ,(3.71) T riaCAT (cid:15) (cid:15) C at op D rrrrr D / / C AT ;see [Gro16, Thm. 10.14]. We want to use this result to stress once more the dis-tinction between properties and structures. As we discussed, the property of beingstrong and stable implies the existence of canonical triangulations (Theorem 3.40).Similarly, the property of preserving certain basic (co)limits, implies the existence ofcanonical exact structures (Proposition 3.62). In particular, equivalences of deriva-tors always are exact.Along these lines, there is the following remark. Let us recall that an exactnatural transformation α : F → G of exact functors between triangulated cate-gories is a natural transformation α which commutes with the natural isomorphisms F Σ ∼ −→ Σ F and G Σ ∼ −→ Σ G . Since in derivator land these isomorphisms arise canon-ically, there is no counterpart for the notion of an exact natural transformation forderivators. In fact, every natural transformation between exact morphisms is com-patible with the canonical morphisms (3.61) (see [Gro16, Cor. 10.12]). This resultis used implicitly in the construction of the lifts in (3.71).4. Higher symmetries
In this section we give an overview over some main results of this project onhigher symmetries. In § § A . In § § § Strong stable equivalences.
Motivated by the compatibility of the forma-tion of derivators of abelian categories and exponentials (Example 3.44), in thissubsection we define strong stable equivalences as a variant of the classical derivedequivalences of quivers.
Construction . Let A be a small category and let D be a stable derivator.We again denote by D A the 2-functor constructed in Construction 3.43, whichby Theorem 3.45 is a derivator, the derivator of representations of shape A with IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 55 values in D . The motivation for this terminology stems from Example 3.44. It isstraightforward to check that the formation D D A extends to a 2-functor( − ) A : D ER St , ex → D ER : D D A , where D ER St , ex ⊆ D ER again denotes the 2-category of stable derivators, exactmorphisms, and arbitrary transformations (Remark 3.65).The slogan is that this 2-functor encodes the abstract representation theory ofthe small category A . To fill this slogan with more life, we collect the followingexamples of stable derivators and specialize the shape A to the case of (the path-category of) a quiver Q . Examples . As part of the structure encoded by the 2-functor ( − ) Q of abstractrepresentations of a quiver Q there are the following stable derivators of morespecific representations.(i) For every ordinary, not necessarily commutative ring R there is the stablederivator D R of the ring (Examples 3.13). In fact, this derivator arises forinstance from the projective model structure [Hov99, § R ) of unbounded chain complexes over R . From this we obtain by shiftingthe derivator D QR of representations of Q in D R , and there is an equivalenceof stable derivators D QR ≃ D RQ . One way to see this is by observing that Mod( RQ ) ≃ Mod( R ) Q induces aQuillen equivalence at the level of unbounded chain complexes.(ii) This generalizes immediately to arbitrary Grothendieck abelian categories A .In fact, the injective model structure on the category Ch( A ) (see e.g. [Hov01]or [Lur14, Chapter 1]) induces a stable derivator D A . For example, for everyquasi-compact, quasi-separated scheme X there is the stable derivator D X of unbounded chain complexes of quasi-coherent O X -modules [Hov01]. Moregenerally, associated to every such A there is the derivators D Q A of represen-tations of Q with values in D A .(iii) Still sticking to the framework of classical homological algebra, this can begeneralized further to exact categories in the sense of Quillen [Qui73, § E there is by [Kel07a], [Gil11] or [ˇSˇto14] the bounded derivator D E of E enhancing the bounded derived category. Correspondingly, for finitequivers Q there is the derivator D Q E of representations.(iv) As additional interesting variants, given a differential-graded algebra A overan arbitrary ground ring we can consider differential-graded representations of Q over A which is to say functors from Q to dg-modules over A . In orderto import this to derivators, we recall that the category of dg-modules over A admits suitable stable model structures (see for instance [Hin97, SS00,Fre09]), and consequently we obtain the stable derivator D A of dg-modulesover A . For our quiver Q there is an equivalence D QA ≃ D AQ , where AQ is adifferential-graded version of the usual path-algebra.(v) Another algebraic context giving rise to stable derivators is stable moduletheory and representation theory of groups . For every quasi-Frobenius ringor, more generally, Iwanaga–Gorenstein ring [EJ00, § R ) of modules can be endowed with the Gorenstein projective and the Gorenstein injective model structure [Hov02, Theorem 8.6]. TheseQuillen equivalent model structures are stable (see for instance [Bec14, Corol-lary 1.1.16]), and hence induce up to equivalence the same stable derivator D Gor R . Correspondingly, associated to Q there is the stable derivator ( D Gor R ) Q of representations. A special case occurs when R is the group algebra kG of afinite group G over a field k , and this important special case was for instancestudied in [BCR97, Ric97, BIK11].(vi) Going beyond the algebraic context, we can pass to spectra in the sense oftopology (see for example [HSS00, EKMM97, MMSS01]). For concreteness,let us stick to one of these monoidal models and assume that E is a symmetricring spectrum. Then the category of E -module spectra can be endowed with astable model structure [HSS00], and we obtain the associated stable derivator D E of E -module spectra. Correspondingly, we obtain the derivator D QE of spectral representations of Q over E .(vii) Finally, as mentioned in § ∞ -categories [Lur14], or stable cofibra-tion categories [Sch10]. Among others many examples of interest arise inequivariant stable homotopy theory [MM02, LMSM86], motivic stable homo-topy theory [Voe98, MV99, Jar00] or parametrized stable homotopy theory[MS06, ABG +
09, ABG10]. For many more examples of stable model cate-gories arising in various areas of algebra, geometry, and topology see [SS03].Before defining strong stable equivalences we recall the following classical defi-nition.
Definition 4.3.
Two quivers Q and Q ′ are derived equivalent over a field k if the path-algebras kQ and kQ ′ are derived equivalent, i.e., if there is an exactequivalence of derived categories D ( kQ ) ∆ ≃ D ( kQ ′ ) . Such derived equivalences are usually obtained by means of tilting theory (seethe handbook [AHHK07] and the many references therein) and have been studiedsystematically (also for more general finite dimensional algebras over a field).
Remark . In Definition 4.3 we were very careful and stressed that the existenceof derived equivalences potentially depends on the choice of the field or ring ofcoefficients. It turns out that certain derived equivalences are “more combinatorialin nature” and they even extend to representations with values in arbitrary abeliancategories (hence they are universal derived equivalences in the sense of Ladkani[Lad07a, Lad08]).Following the line of though of Ladkani’s universal derived equivalences one stepfurther, we are led to the following definition [GˇS14, Def. 5.1].
Definition 4.5.
Two small categories A and A ′ are strongly stably equivalent ,in notation A s ∼ A ′ , if there is a pseudo-natural equivalenceΦ : ( − ) A ≃ ( − ) A ′ : D ER St , ex → D ER . We call such a pseudo-natural equivalence a strong stable equivalence . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 57
In more down-to-earth terms this means the following. Given two small cate-gories A and A ′ , a strong stable equivalence Φ : A s ∼ A ′ consists of(i) for every stable derivator D an equivalence of derivatorsΦ D : D A ≃ D A ′ (ii) and for every exact morphism of stable derivators F : D → E a naturalisomorphism γ F : F ◦ Φ D → Φ E ◦ F, D A Φ D ≃ / / F (cid:15) (cid:15) ☎☎☎☎ ~ (cid:6) ∼ = D A ′ F (cid:15) (cid:15) E A ≃ Φ E / / E A ′ . And this datum is supposed to satisfy some obvious coherence axioms. Let uscomment a bit on this definition.
Remark . The notion of strong stable equivalences is in various respects morerestrictive than the notion of a derived equivalence or a universal derived equiva-lence.(i) Strongly stably equivalent small categories have equivalent homotopy theoriesof representations in Grothendieck abelian categories, of differential-gradedrepresentations, of spectral representations and of more general abstract rep-resentations (by choosing specific examples in Examples 4.2).(ii) The components Φ D of a strong stable equivalence are equivalences of ho-motopy theories and not merely of homotopy categories (together with theclassical triangulation). This means that also the higher-order homotopytheoretic information is supposed to be preserved.(iii) The various equivalences Φ D are suitably compatible with exact morphisms.For instance, Quillen adjunctions between stable model categories induce ex-act morphisms of homotopy derivators and, similarly, exact functors betweenstable ∞ -categories induce exact morphisms of homotopy derivators. Forstrongly stably equivalent quivers or categories this implies that the equiva-lences commute with various kinds of restriction of scalar functors, inductionand coinduction functors as well as (Bousfield) localizations and colocaliza-tions.To put it as a slogan, strongly stably equivalent have the same abstract stablerepresentation theory.We also want to point the following. Definition 4.5 is formulated in the languageof derivators. However, for every stable equivalence A s ∼ B we can conclude thefollowing (related to this see [Ren09], [Dug01a], and [Lur09]).(i) For every combinatorial model category M the diagram categories M A and M B are Quillen equivalent.(ii) For every presentable ∞ -category C the diagram categories C A and C B areequivalent.In §§ Lemma 4.7.
Let
A, A ′ , B, B ′ , and A i , A ′ i , i ∈ I, be small categories. (i) The relation of ‘being strongly stably equivalent’ s ∼ defines an equivalencerelation.(ii) Equivalent categories are strongly stably equivalent.(iii) If A s ∼ A ′ and B s ∼ B ′ , then A × B s ∼ A ′ × B ′ .(iv) If A i s ∼ A ′ i for i ∈ I , then F A i s ∼ F A ′ i .Proof. The straightforward proof is left as an exercise. (cid:3)
On the other hand, classical results from representation theory provide us witha non-trivial necessary condition for quivers to be strongly stably equivalent.
Proposition 4.8.
If two finite quivers without oriented cycles are strongly stablyequivalent, then the underlying non-oriented graphs are isomorphic.Proof.
This is [GˇS14, Prop. 5.3]. (cid:3)
Abstract representation theory of A n -quivers. In this subsection we il-lustrate the notion of a strong stable equivalence by a few examples related toDynkin quivers of type A . We construct reflection functors and briefly study therelated Coxeter and Serre functors in this case. This subsection is largely based onthe paper [GˇS16a] which is joint with Jan ˇSˇtov´ıˇcek.Let us begin by a few toy examples which make the connection to stability veryobvious. The first examples are discussed in quite some detail, but later we allowourselves to be a bit more concise. Example . The source ( • ← • → • ) and the sink ( • → • ← • ) of valence twoare strongly stably equivalent. In fact, let D be a stable derivator and let X be anabstract representation of the source of valence two with values in D as displayedon the left in x / / (cid:15) (cid:15) y x / / (cid:15) (cid:15) y (cid:15) (cid:15) y (cid:15) (cid:15) z, z / / w, z / / w. The idea is that the strong stable equivalence is obtained by first forming thecocartesian square in the middle and then restricting it to the sink of valence twoas displayed on the very right, thereby obtaining Φ( X ). To formalize this idea, werecall the following two facts.(i) Every functor u : A → B between small categories induces by Examples 3.68Kan extension and restriction morphisms of derivators u ! : D A → D B , u ∗ : D B → D A , and u ∗ : D A → D B . (ii) Moreover, Kan extensions along fully faithful functors (Proposition 3.21) arefully faithful and hence induce equivalences on their images.We are interested in the special case of the fully faithful inclusions i p : p → (cid:3) and i y : y → (cid:3) . The left Kan extension morphisms ( i p ) ! : D p → D (cid:3) sends an abstractrepresentation of the source to the corresponding cocartesian square. Denotingby D (cid:3) , cocart the full subderivator of D (cid:3) consisting of the cocartesian squares, weobtain the equivalence of derivators on the left in( i p ) ! : D p ∼ −→ D (cid:3) , cocart , ( i y ) ∗ : D y ∼ −→ D (cid:3) , cart . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 59
Similarly, the formation of cartesian squares yields the equivalence on the right, andin both cases inverse equivalences are given by the corresponding restriction mor-phisms. Now, by definition of stability (Definition 3.28), the derivators D (cid:3) , cocart and D (cid:3) , cart agree, and we obtain the desired equivalence Φ D : D p ∼ −→ D y as acomposition of equivalences of derivatorsΦ D = ( i y ) ∗ ◦ ( i p ) ! : D p ∼ −→ D (cid:3) , cocart = D (cid:3) , cart ∼ −→ D y . Since only restrictions and sufficiently finite Kan extensions are involved in this con-struction, it is straightforward to verify (invoking Remark 3.59) that these equiv-alences are pseudo-natural with respect to exact morphisms. Consequently, weobtain the desired strong stable equivalenceΦ : p = ( • ← • → • ) s ∼ y = ( • → • ← • ) . The following example is similar, but it involves an additional homotopy finalityargument.
Example . The source of valence two p = ( • ← • → • ) and the linearly orientedquiver [2] = ( • → • → • ) are strongly stably equivalent. In fact, let D be a stablederivator and let X be an abstract representation of the source of valence two withvalues in D which is displayed on the left in: x g / / f (cid:15) (cid:15) y F f fib ( f ) / / (cid:15) (cid:15) (cid:3) x f (cid:15) (cid:15) g / / y F f fib ( f ) / / x g / / yz, / / z, The idea is to simply replace the morphism f : x → z by its fiber fib f : F f → x ,thereby obtaining the above representation Φ( X ) on the right. In more detail,starting from either side, by means of fully faithful Kan extension morphisms wecan pass to a representation as displayed in the middle. In fact, starting with ourrepresentation X = ( z ← x → y ) we first add a zero object and then the cartesiansquare as in the diagram x g / / f (cid:15) (cid:15) y x f (cid:15) (cid:15) g / / y F f fib ( f ) / / (cid:15) (cid:15) (cid:3) x (cid:15) (cid:15) g / / yz, / / z, / / z. To re-express this in terms of Kan extensions, let C ⊆ [1] × [2] be the full subposetobtained by removing the lower right corner (1 , B ⊆ C be the resultof also removing the upper left corner (0 , i : p → B and i : B → C with corresponding Kan extension morphisms( i ) ∗ : D p → D B and ( i ) ! : D B → D C . By Proposition 3.21 both Kan extension morphisms are fully faithful and theyinduce equivalences onto their respective images. The morphism ( i ) ! preciselyamounts to adding a zero object (as a consequence of (Der4)) while ( i ) ∗ adds acartesian square (and this step invokes a simple homotopy finality argument). Asan upshot we obtain an equivalence of derivators( i ) ∗ ◦ ( i ) ! : D p ∼ −→ D C, ex where D C, ex ⊆ D C is the full subderivator spanned by all diagrams which vanishon the lower left corner and which make the square cartesian (the square is henceessentially a fiber square).If we instead begin with a representation of the linearly oriented A -quiver, wesimply add the cofiber square to the first of the two morphisms. Thus, in suggestivenotation we carry out the constructions: w f / / x g / / y, w f / / (cid:15) (cid:15) x g / / y, w f / / (cid:15) (cid:15) (cid:3) x cof ( f ) (cid:15) (cid:15) g / / y / / Cf Denoting by B ⊆ C the full subposet obtained by removing (1 , j : [2] = ( • → • → • ) → B and j : B → C . Arguments similarto the previous case imply that we obtain an equivalence of derivators( j ) ! ◦ ( j ) ∗ : D [2] ∼ −→ D C, ex where D C, ex ⊆ D C is the full subderivator spanned by all diagrams which vanishon the lower left corner and which make the square cocartesian (the square is hencea cofiber square). Now, since D is a stable derivator, the two subderivators D C, ex and D C, ex agree, and we obtain the desired equivalenceΦ D = ( j ) ∗ ◦ ( j ) ∗ ◦ ( i ) ∗ ◦ ( i ) ! : D p ∼ −→ D [2] . Since only restrictions and sufficiently finite Kan extensions are involved, theseequivalences are pseudo-natural with respect to exact morphisms (Remark 3.59),and we obtain the desired strong stable equivalenceΦ : p = ( • ← • → • ) s ∼ [2] = ( • → • → • ) . Corollary 4.11.
All A -quivers are strongly stably equivalent.Proof. We have to show that the quivers Q = (1 → → Q = (1 ← → Q = (1 → ← Q = (1 ← ←
3) are strongly stably equivalent. There arestrong stable equivalences Q ∼ Q (Example 4.10) and Q ∼ Q (Example 4.9),and since Q and Q are equivalent we also deduce Q ∼ Q (Lemma 4.7). Since s ∼ is an equivalence relation (Lemma 4.7), we are done. (cid:3) There is a version of Corollary 4.11 for longer A n -quivers as well, and a proof ofthis essentially follows the above pattern. Given two differently oriented A n -quivers Q and Q and a stable derivator D , we construct a certain poset P = P Q ,Q together with suitable combinations of fully faithful Kan extensions morphisms D Q → D P and D Q → D P . The stability of D will then imply that in both cases the essential image consist ofprecisely the same representations of P (which are determined by certain exactnessconditions such as the vanishing on certain objects or the fact that certain squaresare bicartesian ). In fact, in Example 4.9 the poset P = (cid:3) was enough, while inExample 4.10 we considered the subposet P = C ⊆ [1] × [2] obtained by removingthe final vertex (1 , A n -version of Corollary 4.11 we referthe reader to [GˇS14, § § IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 61 canonical triangulations in § n thereis poset which conveniently encodes all A n -quivers with arbitrary orientations, andthis poset will be described in detail in the following construction. Construction . We recall that every quiver Q has associated to it a repetitivequiver b Q with the following description: vertices in it are pairs ( k, q ) with k ∈ Z and q ∈ Q while associated to every edge α : q → q in Q there are the two edges α : ( k, q ) → ( k, q ) and α ∗ : ( k, q ) → ( k + 1 , q ) in b Q . Here we are only interestedin the special case of the linearly oriented A n -quiver ~A n = (1 < . . . < n ). In thespecial case of n = 3 the repetitive quiver of ~A takes the form: ●●●●● ( − , β ∗ % % ❏❏❏ (0 , β ∗ ❍❍❍ (1 , β ∗ ❍❍❍ (2 , ! ! ❉❉❉❉ · · · ( − , β rrrr α ∗ & & ▲▲▲▲ (0 , β ; ; ✈✈✈ α ∗ ❍❍❍ (1 , β ; ; ✈✈✈ α ∗ ❍❍❍ (2 , β ; ; ✈✈✈ α ∗ ❍❍❍ · · · ; ; ✇✇✇✇✇ (0 , α ttt (1 , α ; ; ✈✈✈ (2 , α ; ; ✈✈✈ (3 , = = ③③③③ Let M ~A n be the category which is obtained from the repetitive quiver of ~A n bymaking all squares commutative. By abuse of terminology, we call this poset M ~A n the mesh category .We want to show that, for stable derivators, representations of ~A n can be equiv-alently encoded by suitable representations of mesh categories. It is worth to com-pare the following to the discussion of the rotation axiom in § § Construction . For every n ≥ M n = M [ n +1] for [ n + 1] = (0 < . . . < n + 1) . Given a stable derivator D , we denote by D M n , ex ⊆ D M n the full subderivatorspanned by all representations which(i) vanish on the boundary stripes (i.e., at ( k, , ( k, n + 1) for all k ∈ Z )(ii) and which make all squares bicartesian.The fact that this is a derivator is a consequence of Theorem 4.15 since derivatorsare closed under equivalences of prederivators. In order to relate D M n , ex to D ~A n ,we note that there is the fully faithful functor(4.14) i : ~A n → M n : l (0 , l )The following is a derivatorish version of Theorem 2.45 and Theorem 2.51. Infact, the proofs of these two theorems were modeled after the proof of the followingresult. Theorem 4.15.
For every stable derivator D and n ≥ restriction along i (4.14) induces an equivalence of derivators i ∗ : D M n , ex ∼ −→ D ~A n . This equivalence is pseudo-natural with respect to exact morphisms, and the inclu-sion D M n , ex → D M n is exact. Proof.
We only sketch the proof and refer the reader to [GˇS14, Thm. 4.6] for details.For every stable derivator D we want to construct an inverse equivalence of i ∗ . Thus,given a representation X of ~A n with values in D we want to obtain a coherentdiagram of shape M n satisfying the defining exactness conditions of D M n , ex . Tothis end, similarly to the sketch proofs of Theorem 2.45 and Theorem 2.51, we notethat the inclusion (4.14) factors as a composition of inclusions of full subcategories i : ~A n i → K i → K i → K i → M n where(i) K contains all objects from ~A n and the objects ( k, n + 1) for k ≥ k, k > K is obtained from K by adding the objects ( k, l ) , k >
0, and(iii) K contains all objects from K and the objects ( k, n + 1) for k < k, k ≤ i thus adds the remaining objects in the negative k -direction. ByProposition 3.21, associated to these fully faithful functors there are fully faithfulKan extension functors D ~A n ( i ) ∗ / / D K ( i ) ! / / D K ( i ) ! / / D K ( i ) ∗ / / D M n . These Kan extension morphisms can be analyzed in turn, and the conclusion isthat ( i ) ∗ adds zero objects on the boundary stripes in the positive direction, ( i ) ! adds bicartesian squares in the positive direction, ( i ) ! adds zero objects in on theboundary stripes in the negative direction, and finally ( i ) ∗ fills up by bicartesiansquares in the negative direction. Thus the essential image agrees with D M n , ex ,and this concludes the construction of an equivalence F ~A n : D ~A n ∼ −→ D M n , ex whichis inverse to i ∗ . Again, since we only used restrictions and sufficiently finite Kanextension morphisms, both i ∗ and its inverse F ~A n are pseudo-natural with respectto exact morphisms (Remark 3.59). (cid:3) Variants of this theorem for different orientations of A n -quivers lead to the fol-lowing result. Theorem 4.16.
Let n ≥ be fixed. All A n -quivers are strongly stably equivalent.Proof. We sketch very roughly the main idea of the proof, and for this purposewe consider an arbitrarily oriented A n -quiver Q . For every such Q there are ad-missible embeddings i Q : Q → M n and the corresponding restriction morphisms( i Q ) ∗ : D M n → D Q again restrict to equivalences ( i Q ) ∗ : D M n , ex ∼ −→ D Q . In thiscase it is more tricky to write down in closed form the inverse equivalence F Q (whichdepends on Q and the choice of the admissible embedding i Q ).To illustrate this step, we content ourselves by one example, but we invite thereader to come up with additional examples. In the case of n = 3 and the sourceof valence two Q = (1 ← → i Q : Q → M given by2 (0 , , (1 , , and 2 (0 , . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 63
The corresponding equivalence F Q sends a representation X looking like x g / / f (cid:15) (cid:15) yz to a coherent diagram as in Figure 8. Therein, the boundary stripes are populatedby zero objects and all squares are bicartesian. To put it differently, the diagram F Q ( X ) is a refined octahedral diagram of ( F f → x → y ) (as in § / / • (cid:3) (cid:15) (cid:15) / / • (cid:3) / / (cid:15) (cid:15) • (cid:3) (cid:15) (cid:15) / / (cid:15) (cid:15) / / • / / (cid:15) (cid:15) (cid:3) x g / / f (cid:15) (cid:15) (cid:3) y (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / z / / (cid:15) (cid:15) (cid:3) • / / (cid:15) (cid:15) (cid:3) • (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / ... • / / ... • / / ... • / / ... 0 ... Figure 8.
The equivalence F Q for Q = ( • ← • → • )Now, given an additional A n -quiver Q ′ we choose an admissible embedding i Q ′ : Q ′ → M n , and obtain the corresponding inverse equivalence F Q ′ of ( i Q ′ ) ∗ .In order to conclude the proof it suffices to make the definitionΦ Q ′ ,Q = ( i Q ′ ) ∗ ◦ F Q : D Q ∼ −→ D M n , ex ∼ −→ D Q ′ . As a composition of pseudo-natural equivalences, this defines a strong stable equiv-alence Φ Q ′ ,Q : Q s ∼ Q ′ . (cid:3) The strong stable equivalences constructed in Theorem 4.16 arise as finite com-positions from certain basic building blocks. These building blocks admit a fairlyelementary description as we discuss next, and they turn out to be special cases ofthe more general reflection morphisms from § Construction . Let n ≥
0, let D be a stable derivator, and let M n be themesh category. Choosing two A n -quivers Q, Q ′ and admissible embeddings i Q , i Q ′ of the quivers to M n , the resulting strong stable equivalence Φ Q ′ ,Q : D Q ∼ −→ D Q ′ measures the difference between the restriction morphisms ( i Q ) ∗ and ( i Q ′ ) ∗ . Theabove-mentioned basic building blocks arise when the two embeddings “differ byone square only”. Instead of making this precise by explicit combinatorial formulas,we illustrate this by an example related to Figure 8. As in the proof of Theorem 4.16, let Q = ( • ← • → • ) and let i Q : Q → M bethe embedding indicated by Figure 8. If Q ′ = ( • → • ← • ) is the sink of valencetwo, then there is an obvious embedding of i Q ′ : Q ′ → M which has two objects incommon with i Q but differs from it by one square. In this case, the resulting strongstable equivalence Φ Q ′ ,Q agrees with the one from Example 4.9. Alternatively, wecan consider the linearly oriented quiver ~A = ( • → • → • ) with the standardembedding i : ~A → M as in (4.14). Also these embeddings differ by one squareonly and we reproduce the strong stable equivalence from Example 4.10. Finally,there is the additional vertical embedding j : ~A → M and in this case the strongstable equivalence Q s ∼ ~A forms the fiber of the morphism labeled by g in Figure 8.All these three cases are special cases of reflection functors. Given an A n -quiver Q and a source a ∈ Q , let σ a Q be the A n -quiver obtained by reorienting all edgesadjacent to a such that the vertex a ∈ σ a Q now is a sink. By Theorem 4.16the quivers Q and Q ′ = σ a Q are strongly stably equivalent, and a preferred suchequivalence is constructed as follows. Let i Q : Q → M n be an admissible embeddingand let i Q ′ = i σ a Q : Q ′ = σ a Q → M n be the induced admissible embedding whichdiffers from i Q by one square only. The strong stable equivalence s − a = Φ Q ′ ,Q = ( i Q ′ ) ∗ ◦ F Q : D Q ∼ −→ D Q ′ and its inverse s + a = Φ Q,Q ′ = ( i Q ) ∗ ◦ F Q ′ : D Q ′ ∼ −→ D Q are reflection functors . It is easy to see that these morphisms do not dependon the choice of the embeddings i Q and i Q ′ as long as they differ by precisely onesquare.The strong stable equivalences from Theorem 4.16 can be described as compo-sitions of the above reflection morphisms, and by the following remark the order ofthe reflections does not matter. Remark . Let Q be an A n -quiver and let a , a ∈ Q be two distinct sinks.In that case the vertex a is also a sink in the reflected quiver σ a Q and we canhence iterate the reflection thereby obtaining σ a σ a Q . A straightforward argumentshows that at the level of the quivers the order of the reflections is irrelevant anddefinition σ { a ,a } Q = σ a σ a Q = σ a σ a Q hence makes sense. Playing a bit with the corresponding embeddings, we note thatfor the corresponding reflection functors there are natural isomorphisms s + a s + a ∼ = s + a s + a : D Q ∼ −→ D σ { a ,a } Q . Of course there are similar results for sinks instead of sources.
Definition 4.19. An admissible sequence of sinks in a finite quiver Q is a totalordering ( a , . . . , a n ) of all vertices of Q such that a is a sink in Q and a i is a sinkin σ a i − . . . σ a Q for all 2 ≤ i ≤ n . Admissible sequences of sources are defineddually.It turns out that every finite, acyclic quiver admits an admissible sequence ofsources and sinks [BGP73, Lemma 1.2(1)]. Moreover, one can show that the quiver σ a n . . . σ a Q agrees with the original quiver Q and similarly in the case of sources.In what follows we only need these results for Dynkin quivers of type A , in which IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 65 case the arguments are more straightforward. While admissible sequences alwaysexist, they are by no means unique as simple examples show. However, at the levelof iterated reflection functors there is the following result.
Construction . Let n ≥ Q be an A n -quiver. For every stable derivator D and every admissible sequence of sinks ( a , . . . , a n ) in Q there is the iteratedreflection functor Φ +( a ,...,a n ) = s + a n ◦ . . . ◦ s + a : D Q → D Q . A combination of Remark 4.18 with some combinatorial arguments imply that upto natural isomorphism this iterated reflection functor is independent of ( a , . . . , a n )(see the proof of [BGP73, Lemma 1.2(3)]). Of course, the same observation appliesto iterated reflections Φ − ( b ,...,b n ) at admissible sequences of sources.By Construction 4.20 the following functors are well-defined up to natural iso-morphisms. Definition 4.21.
Let D be a stable derivator and let Q be an A n -quiver.(i) The Coxeter functor Φ + = Φ + Q : D Q → D Q is Φ + = Φ +( a ,...,a n ) for someadmissible sequence of sinks ( a , . . . , a n ) in Q .(ii) The Coxeter functor Φ − = Φ − Q : D Q → D Q is Φ − = Φ − ( b ,...,b n ) for someadmissible sequence of sources ( b , . . . , b n ) in Q .We illustrate these reflections and Coxeter functors by an example. Example . The linearly oriented A -quiver ~A = (1 → →
3) has a uniqueadmissible sequence of sinks (3 , , D the Coxeter functor is given byΦ + = s +1 ◦ s +2 ◦ s +3 : D ~A ∼ −→ D ~A . Given a representation X = ( x f → y g → z ) in D , to calculate Φ + ( X ) we firstdetermine s +3 ( X ) by considering the diagram on the left in F g (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) F ( gf ) / / (cid:15) (cid:15) (cid:3) F g (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) x f / / y g / / z, x f / / y g / / z. In order to describe s +2 s +3 ( X ) we next reflect ( x → y ← F g ) at the vertex decoratedby y . This is achieved by extending the above diagram by one more bicartesiansquare. Since the pasting of the squares is again bicartesian, the upper left corner isindeed populated by F ( gf ). As a final step, it remains to reflect the representation( x ← F ( gf ) → F g ) at the vertex decorated by x . For this purpose we extend thediagram by one more fiber square in order to obtainΩ z / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:3) F ( gf ) / / (cid:15) (cid:15) (cid:3) F g (cid:15) (cid:15) / / (cid:3) (cid:15) (cid:15) / / x f / / y g / / z. Again, the outer most square is bicartesian and hence a loop square, and the upperleft corner consequently agrees with Ω z . As an upshot we conclude that the Coxeterfunctor Φ + is given by(4.23) Φ + ( x f → y g → z ) = (cid:0) Ω z → F ( gf ) → F g (cid:1) . Similarly, the quiver ~A = (1 →
2) has (2 ,
1) as unique admissible sequence ofsinks. We leave it to the reader to check that in this simpler case the Coxeterfunctor Φ + is given by(4.24) Φ + ∼ = fib : D ~A ∼ −→ D ~A . Example . Let D be a stable derivator and let Q = (1 ← →
3) be the sourceof valence two. This quiver has (1 , ,
2) and (3 , ,
2) as admissible sequences ofsinks. In order to calculate the Coxeter functor Φ + : D Q ∼ −→ D Q we consider anarbitrary representation X = ( z ← x → y ) and extend it to the diagram w / / (cid:15) (cid:15) F g / / (cid:15) (cid:15) (cid:15) (cid:15) F f / / (cid:15) (cid:15) x g / / f (cid:15) (cid:15) y (cid:15) (cid:15) / / z / / y ∪ x z consisting of bicartesian cubes. Ignoring the lower right square for the moment, eachof the remaining three squares will be used to define a reflection at a sink and thelabels match the corresponding vertices. This illustrates again the commutativityof reflections at different sinks, since independently of the order we have s +1 s +3 ( X ) ∼ = s +3 s +1 ( X ) ∼ = ( F f → x ← F g ) . To determine Φ + ( X ) it suffices to understand the upper left corner. For this purposewe also drew the remaining auxiliary cocartesian square 0 , since it allows us toconclude that the total pasting of these squares is a loop square. Consequently,there is a canonical isomorphism w ∼ = Ω( y ∪ x z ) and we obtain(4.26) Φ + ( z f ← x g → y ) ∼ = ( F f ← Ω( y ∪ x z ) → F g ) . Motivated by the classical situation (see [RVdB02, § I.2]) we make the followingdefinition.
Definition 4.27.
Let D be a stable derivator and let Q be an A n -quiver. The Serre functor of D Q is S = ΣΦ + ∼ = Φ + Σ : D Q → D Q . In order to put this definition into context we recall the notion of a Serre func-tor. Serre functors have been formalized in various contexts and references in-clude [BK89, BO01] in the framework of k -linear categories, [Che11] in the contextof k -linear triangulated categories, and [LM08] in the realm of (pretriangulated) A ∞ -categories (see also [BK90, KS09, BLM08]). Here we are mainly interestedin the case of sufficiently well-behaved triangulated categories (see [RVdB02, § I.1]and [BK89]).
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 67
Definition 4.28.
Let k be a field and let T be a k -linear triangulated category withfinite-dimensional mapping vector spaces. An exact auto-equivalence S : T → T isa
Serre functor if it is endowed with an isomorphismhom T ( x, y ) ∼ = −→ (cid:0) hom T ( y, Sx ) (cid:1) ∗ which is natural in x and y .Here, ( − ) ∗ of course denotes the vector space duals. As a consequence of theYoneda lemma we see that Serre functors are essentially unique if they exist, so theexistence of a Serre functor is really a property of the triangulated category. Letus recall the following two typical examples of Serre functors. Examples . (i) The example which is of key interest to here is given bybounded derived categories of finitely generated modules over finite dimen-sional algebras of finite global dimension ([Hap87, § derived Nakayama functors [Hap87, § § k be an algebraically closed field, let X be a smooth, projective varietyof dimension n over k , and let D b ( X ) be the bounded derived category ofcoherent sheaves on X . The derived tensor product with the shifted canonicalline bundle S X = ω X [ n ] ⊗ − : D b ( X ) ∼ −→ D b ( X )is a Serre equivalence on D b ( X ) [Huy06, Thm. 3.12]. Relevant classical ref-erences include [BK89, BO01, Ser55] and also [Muk81, Huy06].Let us now return to the situation in abstract representation theory, and let usrevisit Example 4.22 and Example 4.25. Examples . There are the following examples of Serre functors.(i) The Serre functor of ~A is S ∼ = cof : D ~A → D ~A as it easily follows from(4.24) and (3.50).(ii) In the case of the linearly oriented A -quiver ~A it follows from (4.23) thatthe Serre functor is given by S ( x f → y g → z ) = (cid:0) z → C ( gf ) → Cg (cid:1) . (iii) Similarly, in the case of Q = (1 ← →
3) we can recycle (4.26) in order toconclude that the Serre functor has the description S ( z f ← x g → y ) ∼ = ( Cf ← y ∪ x z → Cg ) . Remark . The justification for the terminology Serre functors in abstract rep-resentation theory is as follows. Specialized to the derivator D k of a field, theseSerre functors induce on underlying categories the classical Serre functors fromrepresentation theory. For an additional example of Serre functors in abstract rep-resentation we refer to [BG18]. Jointly with Falk Beckert, we intend to study theseabstract Serre functors more systematically elsewhere.For more details on the abstract representation theory of Dynkin quivers of type A we refer to [GˇS16a]. In particular, therein is also a discussion of an abstractfractional Calabi–Yau property [Kel05a, Kel08, KS12, vR12] which plays a keyrole in the context of spectral Picard groups. These Picard groups will be briefly discussed in Remark 4.109. We conclude this subsection by a remark on highertriangulations . Remark . Despite the fact that triangulated categories are extremely usefulin plenty of situations, from the early days on people were pointing out defectsof the theory (see for instance the introduction to the early [Hel68]). Besidesthe enhancements in the sense of stable model categories, stable ∞ -categories ordifferential-graded categories, there are the attempts to fix some of the deficienciesof triangulated categories by also keeping track of higher triangles . The idea ofconsidering higher triangles already appears in [BBD82, Remark 1.1.14], while amore recent definition of strong triangulations is in [Mal05a] (see also the closelyrelated notion in [Bal11] as well as the thesis of Beckert [Bec18]).We sketch the rough idea and consider an additive category A together with anauto-equivalence Σ : A ∼ −→ A . An n -angle in ( A , Σ) is essentially an ordinary dia-gram X : M n → A which vanishes on the boundary stripes and such that the trian-gular fundamental domains match up to suitable powers of the suspension Σ. Thus,2-angles correspond to the usual triangles in the sense of Puppe and Verdier, while3-angles are closely related to octahedral diagrams; we refer to [Mal05a] for moredetails. A strong triangulation or ∞ -triangulation on ( A , Σ) is given by classes ofdistinguished n -angles for n ≥ D admit canonical strong triangulations. Infact, analogously to Remark 3.70, there is a lift of D to the 2-category of strongtriangulated categories (as follows from the details in [GˇS16a, §
13] and [Gro16]).Moreover, the coherent versions of these distinguished n -triangles (thus objects in D M n , ex ) in combination with the Mayer–Vietoris triangles from [GPS14b] give riseto many additional triangles in the underlying category D ( ∗ ).4.3. Reflection functors.
In this subsection we discuss additional examples ofstrong stable equivalences given by reflection functors. Happel [Hap87, § Proposition 4.33.
If the finite acyclic quivers Q and Q ′ are strongly stably equiv-alent, then the underlying unoriented graphs are isomorphic.Proof. This is [GˇS14, Prop. 5.3] which in turn relies essentially on [Hap87]. (cid:3)
Thus, in the context of finite acyclic quivers all that can happen by strong stableequivalences are reorientations of some of the edges. These can not be carriedout arbitrary as the case of oriented cycles shows (see Corollary 4.46). However, avalid reorientation is the classical reflection of (acyclic) quivers at sources and sinks(see, for instance, [ASS06, § VII.5], or [Kra08] for more detail). Let us recall thisconstruction.
Construction . Let Q be a quiver and let q ∈ Q . The vertex q is a source ifthere are no edges in Q with q as target, and there is dual notion of a sink of aquiver. Given a source q in a quiver, the reflected quiver σ q Q is obtained from IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 69 Q by turning the source into a sink. Thus, both quivers have the same vertices andtheir is a bijection between the edges. The only difference is that the orientationsof all edges in Q that start at the source q ∈ Q have been reversed in σ q Q so thatthey now end at q ∈ σ q Q .Special cases of such reflections already occurred in § ~A -quivers, Example 4.9 and Example 4.10 provide examples of strong stableequivalences (1 ← → s ∼ σ (1 ← →
3) = (1 → ← ← → s ∼ σ (1 ← →
3) = (1 → → . These reflections at the level of quivers are accompanied by the following reflectionfunctors.
Construction . Let Q be a finite quiver, let q ∈ Q be a source, and let k bea field. For every representation M : Q → Mod( k ) we obtain an induced represen-tation s − q M : σ q Q → Mod( k ) of the reflected quiver as follows. For all vertices q ′ = q ∈ σ q Q we set ( s − q M ) q ′ = M q ′ . To describe the remaining component, let q , . . . , q n denote the finitely many vertices of Q such that there are edges q → q i for i = 1 , . . . , n . Then we define( s − q M ) q = cok( M q → M q ⊕ . . . ⊕ M q n ) , and the canonical homomorphisms yield the required maps ( s − q M ) q i → ( s − q M ) q .This construction is clearly functorial in M , and we hence obtain the reflectionfunctor s − q : Mod( kQ ) → Mod( kσ q Q ) associated to Q and the source q . Dually,if q is a sink, then there is a reflection functor s + q : Mod( kQ ) → Mod( kσ q Q ).At the level of abelian categories of modules these reflection functors fail to beequivalences (see, for example, [ASS06, Thm. VII.5.3]), but there is the followingpositive derived version of it. Theorem 4.36.
Let k be a field, let Q be a finite acyclic quiver, let q ∈ Q be asource, and let Q ′ = σ q Q be the reflected quiver. The reflection functors induce anexact equivalence of bounded derived categories ( Ls − q , Rs + q ) : D b ( kQ ) ∆ ≃ D b ( kQ ′ ) . Proof.
This is due to Happel and can be found in [Hap87, § (cid:3) It can be shown that these reflection functors extend to abstract stable homo-topy theories, and by now there are two approaches to such a generalization. Let usrecall from Construction 4.34 and Construction 4.35, that these constructions relyon some “local modifications at the level of shapes and representations”. Corre-spondingly, the idea of gluing shapes and representations is central to this business.Depending on the formal framework, there are different ways of dealing with it.(i) Similar to the case of triangulated categories, gluing is not available in the2-category D ER St , ex of stable derivators. But in that framework (and thisproject began in this framework of derivators), one can partially sail aroundthis defect by more involved combinatorial arguments. This approach wastaken in [GˇS14, GˇS16b, GˇS15], where representation-theoretic techniqueswere developed (one-point extensions and homotopical epimorphisms). (ii) In contrast to this, the more flexible language of stable ∞ -categories allows,among many other things, for gluing. At the price of quoting results from themore involved machinery [Lur09, Lur14], a very elegant construction of re-flection functors in stable ∞ -categories was worked out by Dyckerhoff, Jasso,and Walde in [DJW19].Because of its importance, we single out the special case of a source and a sinkof finite valence. We begin by recalling some notation from [GˇS16b] and [BG18]. Notation 4.37.
For n ≥ (cid:3) n the n -cube which is to say the n -foldproduct (cid:3) n = [1] × . . . × [1] . The n -cube (cid:3) n can be realized as the power set of { , . . . , n } with the orderinggiven by inclusion. Correspondingly, the n -cube admits the following filtration byfull subcategories. For 0 ≤ l ≤ n we denote by (cid:3) n ≤ l the full subposet spanned bythe subsets of cardinality at most l . The subposet (cid:3) nk ≤ l is defined similarly, andthere are fully faithful inclusion functors ι k,l : (cid:3) nk ≤ l → (cid:3) n . As special cases, weobtain the inclusion of the source and the sink of valence n , ι , : (cid:3) n ≤ → (cid:3) n and ι n − ≤ n : (cid:3) nn − ≤ n → (cid:3) n . Similarly, there are the inclusions of the two possible punctured n -cubes ι ,n − : (cid:3) n ≤ n − → (cid:3) n and ι ≤ n : (cid:3) n ≤ n → (cid:3) n . Definition 4.38.
Let D be a derivator, let n ≥
1, and let X ∈ D ( (cid:3) n ). The n -cube X is cocartesian if it lies in the essential image of ( ι ,n − ) ! : D ( (cid:3) n ≤ n − ) → D ( (cid:3) n ).The n -cube X is strongly cocartesian if it lies in the essential image of the functor( ι , ) ! : D ( (cid:3) n ≤ ) → D ( (cid:3) n ).Of course, there are dual notions of (strongly) cartesian n -cubes . For squaresthere is obviously no difference between the two notions in Definition 4.38. In higherdimension every strongly cocartesian n -cube is cocartesian since ι , : (cid:3) n ≤ → (cid:3) n factors through (cid:3) n ≤ n − . But there is no converse to this and, in fact, there is thefollowing more precise result. Theorem 4.39.
Let D be a derivator, let n ≥ , and let X ∈ D ( (cid:3) n ) . The n -cube X is strongly cocartesian if and only if every subsquare of it is cocartesian.Proof. This is an entirely combinatorial problem and the details can be found in[GˇS14, Thm. 8.4]. (cid:3)
Corollary 4.40.
Let D be a stable derivator and let n ≥ . An n -cube in D isstrongly cocartesian if and only if it is strongly cartesian.Proof. This follows immediately from Theorem 4.39 and Definition 3.28. (cid:3)
We refer to this common class of n -cubes as the class of strongly bicartesian n -cubes. Corollary 4.41.
For every fixed n ≥ the source and the sink of of valence n arestrongly stably equivalent, (cid:3) n ≤ ∼ (cid:3) nn − ≤ n . Proof.
It is straightforward to adapt the proof of Example 4.9 to our current situ-ation (invoking Corollary 4.40 instead of Definition 3.28). (cid:3)
IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 71 ∅ / / (cid:3) n =0 / / (cid:3) n ≤ / / . . . / / (cid:3) n ≤ n − / / (cid:3) n ∅ / / O O (cid:3) n =1 / / O O . . . / / O O (cid:3) n ≤ n − / / O O (cid:3) n ≤ n O O . . . O O / / . . . O O / / . . . O O / / . . . O O . . . O O / / (cid:3) n = n − / / O O (cid:3) nn − ≤ n O O ∅ / / O O (cid:3) n = n O O ∅ O O Figure 9.
The cardinality filtration of the n -cube (cid:3) n Remark . The way we have presented Corollary 4.41, it is an immediate con-sequence of the calculus of n -cubes in stable derivators. This cubical calculus wasdeveloped more systematically in [BG18], and here we want to mention some of thekey results. Using suggestive notation (compare to Notation 4.37), the cardinalityfiltration of the n -cube ∅ / / (cid:3) n =0 / / (cid:3) n ≤ / / . . . / / (cid:3) n ≤ n − / / (cid:3) n leads to an interpolation between cocartesian and strongly cocartesian n -cubes. Thedifferences between these individual filtration layers are given by the full subposets (cid:3) nk ≤ l ⊆ (cid:3) n , ≤ k ≤ l ≤ n, which we refer to as chunks of n -cubes . The chunksand the inclusion functors between them are neatly organized by Figure 9. In loc. cit. the authors study in detail the calculus of iterated partial cofiber construc-tions of n -cubes and their interaction with functors in Figure 9. As a generalizationof Corollary 4.41 there is a strong stable equivalence (cid:3) nk ≤ l s ∼ (cid:3) nn − l ≤ n − k ;see [BG18, Thm. 10.15]. Moreover, for every fixed n these strong stable equivalencesfor the various 0 ≤ k ≤ l ≤ n are suitably compatible with each other [BG18,Thm. 12.15].An additional interesting symmetry comes from a further incarnation of Serreequivalences. For every n ≥ ≤ k ≤ l ≤ n there is a Serre equiva-lence S = S k,l : (cid:3) nk ≤ l s ∼ (cid:3) nk ≤ l . Again, also these Serre equivalences turn out to be compatible with the morphismsbetween the chunks. More specifically, in this situation, for every inclusion functor i : (cid:3) nk ′ ≤ l ′ → (cid:3) nk ≤ l in Figure 9 there is a canonical isomorphism S k,l ◦ i ! ∼ = i ∗ ◦ S k ′ ,l ′ ; see [BG18, Thm. 12.6]. To put this into plain English, in this case left and rightKan extensions match up to a conjugation by Serre equivalences. As a special case([BG18, Cor. 12.12]), for every stable derivator D there are canonical isomorphismscolim (cid:3) nk ≤ l ∼ = lim (cid:3) nk ≤ l ◦ S k,l : D (cid:3) nk ≤ l → D . These compatibilities of local Serre equivalences with Kan extensions are thestarting point of an on-going project on global Serre dualities . Jointly with FalkBeckert we intend to come back to this more systematically elsewhere.Let us now return to the case of acyclic quivers. The approach to reflectionfunctors for acyclic quivers in stable derivators taken in [GˇS14, GˇS16b, GˇS15] relieslargely on the special case in Corollary 4.41. More precisely, the classical construc-tion of reflection functors uses finite biproduct in vector spaces (Construction 4.35).In stable derivators finite biproducts and all the related inclusion and projectionmaps can be organized in suitable cubical diagrams. For instance, in the case oftwo objects x, y ∈ D ( ∗ ) in the underlying category of a stable derivator D , there isthe associated diagram 0 / / (cid:15) (cid:15) (cid:3) x / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) y / / (cid:15) (cid:15) (cid:3) x ⊕ y / / (cid:15) (cid:15) (cid:3) y (cid:15) (cid:15) / / x / / , in which all squares are bicartesian. Given a finite acyclic quiver and a source,such cubes can be glued to the quiver in order to then mimic the construcion ofreflection functors in Construction 4.35. One of the main results of [GˇS15] is thefollowing. Theorem 4.43.
Let Q be a finite acyclic quiver, let q ∈ Q be a source, and let Q ′ = σ q Q be the reflected quiver. The reflection functors define a strong stableequivalence ( s − q , s + q ) : Q s ∼ Q ′ . Proof.
We refer the reader to [GˇS15] for a detailed construction of these reflectionfunctors. The theorem appears as [GˇS15, Thm. 10.3] as a specializiation of a moregeneral result ([GˇS15, Thm. 9.11]). (cid:3)
Corollary 4.44.
Let T be a finite oriented tree and let T ′ be an arbitrary reorien-tation of T . There is a strong stable equivalence T s ∼ T ′ . Proof.
It is a purely combinatorial argument that a reorientation of a finite treecan be obtained by a finite number of reflections at sources and sinks ([BGP73,Thm. 1.2(1)]). The claim hence follows from Theorem 4.43. (cid:3)
In fact, there is also a converse to this result as follows from a specialization tothe derivator D k of a field k . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 73
Remark . Theorem 4.43 is obtained by specialization of the following moregeneral result. Let A be a small category and let q , . . . , q n ∈ A be finitely many,not necessarily pairwisely different objects. The two categories which are obtainedfrom A by freely gluing a source or a sink of valence n to the given objects arestrongly stable equivalent [GˇS15, Thm. 9.11]. In particular, this shows that avariant of Theorem 4.43 also works for quivers which fail to be finite or acyclic.In contrast to the case of trees, for acyclic quivers the underlying graphs tonot determine the strong stable equivalence type. To illustrate this, we revisit thefollowing classical example of orientations Q of an unoriented n -cycle n ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ · · · n − n − . ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ Such quivers are called
Euclidean or extended Dynkin quivers of type e A n − [Rin84, SS07]. For every orientation Q of the n -cycle we write c ( Q ) = { p, q } , if p arrows in Q are oriented clockwise and q are oriented counterclockwise. Corollary 4.46. If Q and Q ′ are n -cycles with c ( Q ) = c ( Q ′ ) , then there is a strongstable equivalence Q s ∼ Q ′ . Proof.
This is [GˇS15, Prop. 10.5]. (cid:3)
Again, the case of the derivator of a field can be used to show that the converseimplication also holds, and c ( Q ) hence determines the strong stable equivalencetype of an n -cycle Q . Remark . Ladkani studied additional interesting examples of reflection functorsfor representations of posets with values in arbitrary abelian categories. For theseuniversal derived equivalences we refer to [Lad07a, Lad07c, Lad07b]. Dyckerhoff,Jasso, and Walde [DJW19] obtain a general construction of reflection functors instable ∞ -categories which also generalizes the examples of Ladkani to the homo-topical setting.4.4. Digression: monoidal derivators.
In this subsection we include a digres-sion on monoidal derivators, the calculus of tensor products of functors, and en-riched derivators. We briefly discuss the universality of the derivator of spectra andthe resulting canonical enrichment of stable derivators over spectra [Hel88, Hel97,Cis08, Tab08, CT11, CT12]. These techniques are central to the construction ofuniversal tilting modules in § § VII].The starting point of this theory is the following elementary observation.
Remark . The 2-category D ER of derivators admits (2-)products. In moredetail, given two derivators D and D the pointwise product D × D : C at op → C AT : A D ( A ) × D ( A )is again a derivator. The derivator D × D enjoys the universal property of aproduct. This simple observation allows us to speak about pseudo-monoid objects andtheir pseudo-actions in the world of derivators. For instance, a first attempt todefine a monoidal derivator would be as a pseudo-monoid object in D ER , i.e., aderivator V which is endowed with coherently associative and unital multiplications ⊗ : V × V → V . Such a pseudo-monoid structure is equivalent to a lift of V to the2-category of monoidal categories, strong monoidal functors and monoidal transfor-mations. However, since we are mostly interested in monoidal structures which arecompatible with colimits, we want to build this into the basic notion. Consequently,we begin by making this compatibility precise and, with later applications in mind,we immediately consider arbitrary morphisms of two variables. Remark . Let D , D , and D be derivators. A morphism of two variables is a morphism of derivators ⊗ : D × D → D . By definition of the product D × D such a morphism has components(4.50) ⊗ A : D ( A ) × D ( A ) → D ( A ) , A ∈ C at . These components are related by pseudo-naturality isomorphisms. In particular,given diagrams X ∈ D ( A ) and Y ∈ D ( A ), there are canonical isomorphisms( X ⊗ A Y ) a ∼ = X a ⊗ Y a , a ∈ A. We refer to this description of a morphism of two variables as the internal version .Given two categories A and B , we can restrict diagrams along the projections A × B → A and A × B → B and then apply ⊗ A × B . This way we obtain functors(4.51) ⊗ : D ( A ) × D ( B ) → D ( A × B ) , A, B ∈ C at , which again are endowed with suitable coherence isomorphisms. For instance, fordiagrams X ∈ D ( A ) and Y ∈ D ( B ), these specialize to canonical isomorphisms( X ⊗ Y ) ( a,b ) ∼ = X a ⊗ Y b , a ∈ A, b ∈ B. One checks that ⊗ : D × D → D can be recovered from this datum by alsoinvoking restrictions along diagonal functors ∆ : A → A × A . In fact, these twodescriptions are equivalent [GPS14a, Theorem 3.11], and we refer to the secondone as the external version of the morphism of two variables. Note that wedistinguish these two versions notationally.One point of this external reformulation is that it allows for the following simpledefinition of cocontinuity. Definition 4.52.
Let D , D and D be derivators and let ⊗ : D × D → D bea morphism of two variables. The morphism ⊗ preserves left Kan extensionsalong u : A → A in the first variable if the canonical morphisms(4.53) ( u × ! ( X ⊗ Y ) ∼ −→ u ! ( X ) ⊗ Y, X ∈ D ( A ) , Y ∈ D ( B ) , are isomorphisms.This notion has various relevant variants. In particular, we say that ⊗ is cocon-tinuous in the first variable if (4.53) are invertible for all u : A → A . Similarlyto [Gro13, Prop. 2.3], one only has to verify that colimits are preserved. With thispreparation we now make the following definition. Definition 4.54. A monoidal derivator is a pseudo-monoid object ( V , ⊗ , S ) in D ER such that the monoidal structure ⊗ : V × V → V preserves left Kan extensionsin both variables separately. IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 75
Just to stress this, our convention is that a monoidal derivator is obtained bycategorifying the notion of a monoid and imposing a cocontinuity condition. Thereare obvious variants of braided or symmetric monoidal derivators (see also[ML98, § XI] for the classical case).
Examples . There are the following expected examples of monoidal derivators.(i) Let V be a complete and cocomplete category together with a monoidal struc-ture ⊗ : V × V → V which preserves colimits in both variables separately(this is for example the case when V is closed monoidal). Then the repre-sented derivator y V inherits a monoidal structure. In the external versionthis monoidal structure sends diagrams X : A → V and Y : B → V to( X ⊗ Y ) ( a,b ) = X a ⊗ Y b , a ∈ A, b ∈ B. (ii) The homotopy derivator of a monoidal model category is a monoidal deriva-tor. For a more general statements related to Quillen adjunctions of twovariables see [GPS14a, Example 3.23].(iii) Let us again consider a commutative ring R . The category Ch( R ) of un-bounded chain complexes over R endowed with the projective model struc-ture [Hov99, § D R of the ring. With respect to theusual tensor product ⊗ R of chain complexes, Ch( R ) is a stable, symmetricmonoidal model category. We conclude that the derivator D R together withthe derived tensor product is stable and symmetric monoidal. This applies,in particular, to the derivator D k of a field k .(iv) There are various Quillen equivalent stable, symmetric monoidal closed modelcategories of spectra such as the ones in [HSS00, EKMM97, MMSS01]. Takingany of these as a model for the derivator of spectra S p , we conclude that S p endowed with the derived smash product is a stable, symmetric monoidalderivator. Similarly, also the derivators S and S ∗ are symmetric monoidal.(One can show that there is an intrinsic approach to these monoidal structuresbased on universal properties of the derivators under consideration.)Besides the internal and the external versions of morphisms of two variables,there also is the canceling version which we discuss next. These specialize to acategorification of the usual tensor products of bimodules over rings and play a keyrole in § coends , so we begin by extending coends to derivators. Recall that in classicalcategory theory there are various equivalent ways of defining coends [ML98, § IX].In homotopical situations one has to be a bit careful which one to choose, sincenot all of them lead to the good notion (see [BG18, Rmk. 13.21]). The followingapproach works perfectly well (see [GPS14a, §
5] and [GPS14a, Appendix A] formore details).
Construction . Let A be a small category. The twisted arrow category tw( A ) of A has as objects all morphisms f : a → b in A . A morphism f → f in tw( A ) is a commutative diagram a f / / b (cid:15) (cid:15) a f / / O O b , and the “twist” is the reorientation of the first coordinate. To put this in plainenglish, a morphism f → f is a 2-sided factorization of f through f . We notethat tw( A ) is simply the category of elements of hom A : A op × A → Set.The category tw( A ) comes with a functor( s, t ) : tw( A ) → A op × A given by the source and target functors. As in the case of ordinary category theory,in order to define the coend construction we need essentially the opposite of thisfunctor, namely the composition( t op , s op ) : tw( A ) op ( s,t ) op → ( A op × A ) op ∼ = A op × A. Definition 4.57.
Let D be a derivator, let A ∈ C at , and let X ∈ D ( A op × A ). The coend R A X ∈ D ( ∗ ) of X is given by Z A X = colim tw( A ) op ( t op , s op ) ∗ X. As a composition of functors, the coend functor R A : D ( A op × A ) → D ( ∗ ) is Z A : D ( A op × A ) ( t op ,s op ) ∗ → D (tw( A ) op ) colim → D ( ∗ ) . Examples . In the case of a represented derivator, this definition of coendsreduces to a formula in [ML98]. Hence, the notion reproduces the classical one in a(complete and) cocomplete category. There is also a different classical descriptionof coends as certain coequalizers. This, however, does not extend that directly tohomotopical frameworks. In fact, in that case coequalizers have to be replaced bygeometric realizations of simplicial bar constructions and for an extension of thisreformulation to derivators we refer to [GPS14a, Appendix A].The motivation for us to discuss coends here is the following construction.
Construction . Let D , D , and D be derivators and let ⊗ : D × D → D be a morphism of two variables. Based on the coend, there is the following thirdversion of such a morphism. Given a small category A and diagrams X ∈ D ( A op )and Y ∈ D ( A ), the external product X ⊗ Y lives in D ( A op × A ), and qualifies asan input for the coend. The canceling version of ⊗ is X ⊗ [ A ] Y = Z A X ⊗ Y, X ∈ D ( A op ) , Y ∈ D ( A ) . Thus, as a functor ⊗ [ A ] is the composition(4.60) ⊗ [ A ] : D ( A op ) × D ( A ) ⊗ → D ( A op × A ) R A → D ( ∗ ) , IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 77 and we refer to it as the (canceling) tensor product of functors . Note that wedistinguish this canceling version notationally from the internal version (4.50) andthe external one (4.51). For the canceling one the subscript [ A ] is added in orderto indicate which “category is canceled”.In order to develop some first intuition for this calculus, we unravel the notionof a twisted arrow category for posets. Lemma 4.61.
Let P be a small poset. The functor ( s, t ) : tw( P ) → P op × P induces an isomorphism between tw( P ) and the up-set generated by the diagonal ∆ P ⊆ P op × P .Proof. The functor ( s, t ) is injective on objects and fully faithful, so it induces anisomorphism onto its image. By reflexivity the diagonal ∆ P = { ( p, p ) | p ∈ P } liesin the image. Let f : p → q be an object in tw( P ), which is to say that p, q ∈ P and p ≤ q . Then in P op × P we have ( p, p ) , ( q, q ) ≤ ( p, q ), and we deduce that the imageof ( s, t ) lies in the up-set generated by the diagonal. Conversely, let p, q , q ∈ P be such that ( p, p ) ≤ ( q , q ) in P op × P . Then q ≤ p and p ≤ q in P , and hence q → q defines an object in tw( P ) which under ( s, t ) is mapped to ( q , q ). (cid:3) In the calculation of coends we essentially use the opposite functor of ( s, t ). Thusin order to calculate tensor products of functors over posets we first form the exter-nal product, then restrict the diagram to the down-set generated by the diagonaland finally calculate the colimit. We illustrate this by the following prominent ex-ample of Construction 4.59. Therein and in what follows, our drawing conventionfor diagrams of two variable is that the first coordinate is drawn horizontally andthe second one vertically.
Example . Let D , D , and D be derivators and let ⊗ : D × D → D be amorphism of two variables. Given coherent morphisms X = ( f : x → y ) ∈ D ([1] op )and X ′ = ( f ′ : x ′ → y ′ ) ∈ D ([1]), the external product X ⊗ X ′ ∈ D ([1] op × [1])takes by pseudo-naturality the form y ⊗ x ′ id ⊗ f ′ (cid:15) (cid:15) x ⊗ x ′ f ⊗ id o o id ⊗ f ′ (cid:15) (cid:15) y ⊗ y ′ x ⊗ y ′ . f ⊗ id o o The restriction of X ⊗ X ′ to the down-set generated by the diagonal is the span( y ⊗ x ′ ← x ⊗ x ′ → x ⊗ y ′ ), and the tensor product of functors X ⊗ [[1]] X ′ ∈ D ( ∗ )hence sits in a defining cocartesian square y ⊗ x ′ (cid:15) (cid:15) x ⊗ x ′ f ⊗ id o o id ⊗ f ′ (cid:15) (cid:15) X ⊗ [[1]] X ′ x ⊗ y ′ . o o In this special case the universal object X ⊗ [[1]] X ′ = y ⊗ x ′ ∪ x ⊗ x ′ x ⊗ y ′ is often referred to as the pushout product of the two morphisms. In § Remark . Let D , D , and D be derivators and let ⊗ : D × D → D bea morphism of two variables. The internal (4.50), external (4.51), and cancelingversions (4.60) of ⊗ can be suitably combined. For instance there is the functor D ( A × B op ) × D ( A × B × C ) → D ( A × C )which treats the A -variable internally and the C -variable externally, while the B -variable is canceled by means of a coend. Taking the philosophy of derivatorsserious, we should not be happy with such functors but instead ask for correspondingmorphisms of derivators. In the first two cases this leads to parametrized internaland parametrized external tensor products [GPS14a, § ⊗ A : D A × D A → D A and ⊗ : D A × D B → D A × B . As a preparation for the remaining case, one notes that for every derivator D thereare parametrized coends Z A : D A op × D A → D . These are derivatorish versions of the usual coends with parameters [ML98, § IX.7].With this in place, the parametrized canceling tensor products ⊗ [ A ] : D A op × D A → D can now be defined by the same formula as in Construction 4.59.For every monoidal derivator V , the monoidal structure ⊗ : V × V → V isassociative and unital up to coherence isomorphisms. This is the case by definitionfor the internal version and it is straightforward to also check this for the externalversion. We now turn towards a similar result for the canceling tensor product. Construction . Let V be a monoidal derivator and let A, B ∈ C at . We also callobjects in V ( A × B op ) bimodules in V or ( A, B ) -bimodules in case we want tobe more specific. Given a third small category C , we can define a compositionfunctor (over B ) as the parametrized canceling tensor product(4.65) ⊗ [ B ] : V ( A × B op ) × V ( B × C op ) → V ( A × C op )which treats A and C op externally. For every small category B there is also apreferred ( B, B )-bimodule I B (which somehow corresponds to the regular bimoduleover a ring in algebra). To build towards these bimodules, let us recall that, as partof the monoidal structure, V is endowed with a monoidal unit S . This is a pseudo-functor S : y ∗ = ∗ → V defined on the terminal derivator. More concretely, S amounts to a pseudo-functorial choice of objects S A ∈ V ( A ) , A ∈ C at , all of whichare monoidal units with respect to the internal tensor products ⊗ A . The pseudo-functoriality constraint for the projection π A : A → ∗ gives a preferred isomorphism S A ∼ = π ∗ A ( S ∗ ), which is to say that S A is constant on the monoidal unit S ∗ ofthe underlying category V ( ∗ ). With this preparation we can define the Yonedabimodule or identity profunctor of B as the bimodule(4.66) I B = ( t, s ) ! S tw( B ) ∈ V ( B × B op ) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 79
Here ( t, s ) : tw( B ) → B × B op sends ( f : a → b ) to ( b, a ) (see Construction 4.56).By an application of the pointwise formulas for left Kan extensions one checks thatfor a, b ∈ B there is a canonical isomorphism( I B ) ( b,a ) ∼ = a hom B ( a,b ) S ∗ . In plain English, the value at ( b, a ) is the coproduct of copies of the unit S ∗ parametrized by the morphism set hom B ( a, b ), and this justifies the terminology Yoneda bimodule .The composition of bimodules in monoidal derivators is associative and unitalin the following sense. For basic terminology on bicategories we refer the reader to[B´en67], [Bor94a, § §§ XII.6-7].
Theorem 4.67.
For every monoidal derivator V there is a bicategory P rof ( V ) with the following description. Its objects are small categories, the category of ho-momorphisms from A to B is the category of bimodules V ( A × B op ) , the compositionfunctors are given by (4.65) and the identity 1-cells are the Yoneda bimodules (4.66) .We refer to P rof ( V ) as the bicategory of bimodules in V .Proof. The proof of the associativity of composition functors is a fairly direct con-sequence of a derivatorish version of the Fubini lemma on iterated coends [GPS14a,Lem. 5.3]. It turns out that the proof of the unitality is more involved [GPS14a,Appendix B]. For details we refer the reader to [GPS14a, Theorem 5.9]. (cid:3)
In [GPS14a] we refer to P rof ( V ) as the bicategory of profunctors (and thisexplains the notation), while here we prefer the terminology of bimodules. Forlater reference we include the following example of Yoneda bimodules. Example . Let V be a pointed, monoidal derivator and let P be a poset. TheYoneda bimodule I P ∈ V ( P × P op ) restricted to the up-set generated by the diagonal∆ P is constant with value the monoidal unit S ∗ ∈ V ( ∗ ) and it vanishes on thecomplement [GˇS16a, Lem. 7.4].By now we have a reasonably solid understanding of the basic formalism of tensorproducts of functors in monoidal derivators (more examples will be discussed in § § Definition 4.69.
Let D , D , and D be derivators. A morphism of two variables ⊗ : D × D → D is a two-variable left adjoint if(i) the external components ⊗ : D ( A ) × D ( B ) → D ( A × B ) are two-variableleft adjoints for all A, B ∈ C at and(ii) the morphism ⊗ : D × D → D is cocontinuous in both variables separately.Thus, for every A, B ∈ C at we ask for the existence of functors ⊲ [ B ] : D ( B ) op × D ( A × B ) → D ( A ) and ⊳ [ A ] : D ( A × B ) × D ( A ) op → D ( B )and for natural isomorphismshom D ( A × B ) ( X ⊗ Y, Z ) ∼ = hom D ( A ) ( X, Y ⊲ [ B ] Z ) ∼ = hom D ( B ) ( Y, Z ⊳ [ A ] X ) . The notational convention follows [GPS14a] and it has the feature that it preservesthe cyclic structure of the arguments
X, Y, and Z . The notation of the functors ⊲ [ B ] and ⊳ [ A ] again indicated which “category is canceled”. One can show thatthere are morphisms of derivators ⊳ : D op2 × D → D and ⊲ : D × D op1 → D and that the functors ⊳ [ B ] and ⊲ [ A ] arise from these by certain ends (see [GPS14a, §§ Definition 4.70. A closed monoidal derivator is a monoidal derivator suchthat the monoidal structure is a two-variable left adjoint.There is the obvious variant of symmetric closed monoidal derivators. Examples . (i) Derivators represented by closed monoidal categories are alsoclosed monoidal. More generally, two-variable adjunctions between completeand cocomplete categories induce two-variable adjunctions of representedderivators [GPS14a, Ex. 8.9].(ii) Homotopy derivators of cofibrantly-generated monoidal model categories areclosed monoidal [GPS14a, Thm. 9.11]. Two-variable Quillen left adjoint func-tors between combinatorial model categories induces two-variable adjunctionsbetween homotopy derivators [GPS14a, Ex. 8.11].(iii) The derivator D R of a commutative ring R , the derivator S of spaces, thederivator S ∗ of pointed spaces, and the derivator S p of spectra are symmetricclosed monoidal.(iv) Let ⊗ : D × D → D be a two-variable left adjoint morphism of deriva-tors. Then also the internal, external, and canceling versions of ⊗ fromRemark 4.63 are two-variable left adjoints [GPS14a, § V the bicategory P rof ( V ) of bimodules isclosed. Remark . Many interesting examples of triangulated categories arising in na-ture come with an additional monoidal structure (see for instance [Bal10, § duality phenomena (see [DP80]as well as [BG99] and the references there). Let T be monoidal, triangulatedcategory and let x ∈ T be a dualizable object. Every endomorphism ϕ : x → x hasa trace tr( ϕ ) : S → S which is an endomorphism of the monoidal unit. Morally,one would expect that traces of dualizable objects are additive with respect todistinguished triangles. More precisely, given an endomorphism of a distinguishedtriangle x / / ϕ x (cid:15) (cid:15) y / / ϕ y (cid:15) (cid:15) z / / ϕ z (cid:15) (cid:15) Σ x Σ ϕ x (cid:15) (cid:15) x / / y / / z / / Σ x IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 81 such that x, y, and z are dualizable, the formulatr( ϕ x ) + tr( ϕ z ) = tr( ϕ y )should hold. However, as a further reminiscence of the non-functoriality of cones,such a formula does not hold at the level of triangulated categories [Fer06]. And,in fact, this failure was part of the original motivation of Grothendieck to come upwith the notion of a derivator in the first place.May [May01] proposed very carefully chosen compatibility axioms for monoidal,triangulated categories which can be used to establish additivity of Euler charac-teristics (traces of identities), and a representation theoretic perspective on theseappears in work of Keller–Neeman [KN02]. In [GPS14a] the basic formalism ofmonoidal, stable derivators is developed, and canonical monoidal triangulations inthe sense of May are constructed for such derivators. As an additional applicationof these techniques, it is shown that traces of coherent morphisms are additive withrespect to cofiber sequences [GPS14a].The credo suggested by this is that all the compatibility is already encoded ina stable, monoidal derivator (or stable, monoidal ∞ -category). Hence, instead ofimposing more and more complicated axioms on monoidal, triangulated categories,one should try to prove lemmas about, say, stable, monoidal derivators and thecorresponding calculus of bimodules. This was illustrated successfully in the pa-pers [PS16, PS14, GAdS14] where the above additivity formula was extended tohomotopy finite colimits of dualizable objects (see also the closely related [JY18]). Definition 4.73.
Let V be a monoidal derivator. A V -module is a cocontinuouspseudo-module over V . A V -enriched derivator is a V -module such that theaction is a two-variable left adjoint.Thus, a V -module is a derivator D together with a morphism of two variables(4.74) ⊗ : V × D → D , the action or module structure , such that(i) the action is associative and unital up to coherence isomorphisms and(ii) the action is cocontinuous in both variables separately.If V and D are stable derivators, then these are derivatorish versions of actionsof triangulated categories (as in [Ste13]). For a V -enriched derivator the action(4.74) is supposed to be a two-variable left adjoint (Definition 4.69). Thus, for a V -enriched derivator D besides the action morphisms we also have internal homs or internal mapping objects ⊲ : D op × D → V and cotensors ⊳ : D × V op → D . Examples . (i) If V is complete and cocomplete closed symmetric monoidalcategory and C is a complete and cocomplete V -enriched category in theclassical sense [Kel05b], then y C is a y V -enriched derivator (Examples 4.71).(ii) If V is symmetric monoidal model category and M is a V -model category inthe sense of [Hov99], then H o M is a H o V -enriched derivator.(iii) Any closed monoidal derivator is enriched over itself. This gives rise to many interesting examples of enriched derivators. We in-vite the reader to come up with additional closure properties of enriched deriva-tor such as the passage to shifted derivators or the behavior under monoidal ad-junctions. We conclude this subsection by a discussion of the universality of theclassical homotopy theories of spaces, pointed spaces, and spectra, and the result-ing canonical enrichments of stable derivators over spectra. These are deep theo-rems and they are crucial to our applications in § Remark . The homotopy derivator of spaces S is freely generated under colimitsby the singleton ∆ [Cis08, Theorem 3.24]. In more detail, given derivators D , E we denote by Hom ! ( D , E ) the category of cocontinuous morphisms D → E and allnatural transformations. In particular, given a cocontinuous morphism F : S → D ,we can evaluate the underlying functor S ( ∗ ) → D ( ∗ ) at ∆ . The universality ofspaces is made precise by the statement that the evaluation induces an equivalenceof categories Hom ! ( S , D ) ∼ → D ( ∗ ) : F F (∆ ) . This universal property explains the ubiquity of spaces in abstract homotopy theory.A formally correct proof of this result is highly non-trivial, and we refer the readerto [Hel88] and [Cis08]. But to provide some evidence for this result let us recall thefollowing.(i) Every topological space has a CW-approximation.(ii) CW-complexes are constructed from n -cells under coproducts, pushouts andcountable colimits.Since the n -cells are all weakly contractible, this means that every space can bebuilt from ∆ using homotopy colimits only. These heuristics make the universalityat least quite plausible.There are variants of this result for pointed spaces and spectra. The derivator ofpointed spaces S ∗ is the free pointed derivator generated by the 0-sphere S . Andthe derivator S p is the free stable derivator generated by the sphere spectrum S [CT11, Theorem A.11]. Thus, for every stable derivator D the evaluationHom ! ( S p , D ) ∼ → D ( ∗ ) : F F ( S ) . is an equivalence of categories. Theorem 4.77.
Every stable derivator is enriched over the derivator S p of spectra.Proof. The proof is essentially a consequence of the above-mentioned universalproperty of the derivator of spectra, and we refer the reader to [CT11, Appen-dix A.3] for details. It turns out that the action ⊗ : S p × D → D which belongs to this enrichment is characterized by two properties. First, theaction preserves colimits in both variables separately, and, second, the sphere spec-trum acts trivially (which is to say S ⊗ − ∼ = id D as it is the case for all actions). (cid:3) In § IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 83
Remark . Let D be a stable derivator, let A ∈ C at , and let X, Y ∈ D ( A ). Werefer to X ⊲ [ A ] Y ∈ S p ( ∗ ) as the mapping spectrum of X, Y . This is a higher-structured version of the categorical morphisms hom D ( A ) ( X, Y ). In fact, these canbe recovered as π ( X ⊲ [ A ] Y ) ∼ = hom S p ( ∗ ) ( S , X ⊲ [ A ] Y ) ∼ = hom D ( A ) ( S ⊗ X, Y ) ∼ = hom D ( A ) ( X, Y ) . Similarly, the extension groups can be described as homotopy groups of the mappingspectrum. The formalism of two-variable adjunctions allows us to calculate themapping spectrum X ⊲ [ A ] Y as an end which is useful in many situations.4.5. Universal tilting modules.
In this section we discuss universal tilting mod-ules: certain explicitly constructed spectral bimodules realizing our strong stableequivalences. These universal tilting modules are spectral refinements of the classi-cal tilting complexes and they are non-trivial invertible elements in the bicategoryof spectral bimodules. Considered this way, abstract representation contributes tothe calculation of spectral Picard groupoids.Thus, analogously to the situation in tilting theory, here the focus shifts fromthe equivalences themselves to the representing bimodules.We begin this section by defining the class of morphisms which are associatedto spectral bimodules.
Definition 4.79.
Let
A, B ∈ C at and M ∈ S p ( B × A op ). For every stable derivator D , the weighted colimit with weight M is the morphism M ⊗ [ A ] − : D A → D B . There is the dual notion of a weighted limit (using cotensors and ends insteadof tensors end coends). In this subsection we will always focus on the case ofweighted colimits.
Remark . Let D be a stable derivator. We say that a morphism F : D A → D B is a weighted colimit if there is a spectral bimodule M ∈ S p ( B × A op ) and a naturalisomorphism F ∼ = M ⊗ [ A ] − : D A → D B . If we do not merely have a morphism F : D A → D B for a fixed stable derivator,but a pseudo-natural transformation F : ( − ) A → ( − ) B : D ER St , ex → D ER between the corresponding abstract representation theories, then the weight isuniquely determined by F . In fact, it suffices to consider the universal stablederivator D = S p of spectra. The natural isomorphism F S p ∼ = M ⊗ [ A ] − specializesto a natural isomorphism of functors S p A ( A op ) = S p ( A × A op ) → S p B ( A op ) = S p ( B × A op ) . Plugging in the Yoneda bimodule I A , we can invoke the unitality constraint fromTheorem 4.67 in order to obtain F ( I A ) ∼ = M ⊗ [ A ] I A ∼ = M. This remark shows that in favorable cases the weight is obtained if one evaluatesthe morphism on a suitable Yoneda bimodule. This idea is also central to thefollowing two results. The first of these two results also justifies the terminology ofweighted colimits . Theorem 4.81.
Let D be a stable derivator and let u : A → B be a functor.(i) The restriction morphism u ∗ : D B → D A is a weighted colimit.(ii) The left Kan extension morphism u ! : D A → D B is a weighted colimit. Inparticular, the colimit morphism colim A : D A → D is a weighted colimit.Proof. The stable derivator D is by Theorem 4.77 a S p -enriched derivator, and letus denote the corresponding adjunction of two variables by ⊗ : S p × D → D . Moreover, for arbitrary categories
C, D, E it follows from Remark 4.63 that alsothe external-canceling version of this action ⊗ [ D ] : S p C × D op × D D × E op → D C × E op is a left adjoint of two variables. In particular, the morphism ⊗ [ D ] is compatiblewith restrictions and left Kan extensions in both variables separately. To concludethe proof it suffices to specialize this to the two situations under consideration.In the first case, let X ∈ D ( B ) and let I B ∈ S p ( B × B op ) be the Yonedabimodule. By left unitality and pseudo-naturality we obtain u ∗ ( X ) ∼ = u ∗ ( I B ⊗ [ B ] ⊗ X ) ∼ = (cid:0) ( u × id B op ) ∗ I B (cid:1) ⊗ [ B ] X, and this defines the intended natural isomorphism(4.82) u ∗ ∼ = (cid:0) ( u × id B op ) ∗ I B (cid:1) ⊗ [ B ] − : D B → D A , exhibiting the restriction morphism as a weighted colimit.Similarly, in the second case, let X ∈ D ( A ) and let I A ∈ S p ( A × A op ) be theYoneda bimodule. Using the fact that the morphism − ⊗ [ A ] X preserves colimitsand hence left Kan extensions, we obtain the isomorphisms u ! ( X ) ∼ = u ! ( I A ⊗ [ A ] X ) ∼ = (cid:0) ( u × id A op ) ! I A (cid:1) ⊗ [ A ] X. This yields the intended natural isomorphism(4.83) u ! ∼ = (cid:0) ( u × id A op ) ! I A (cid:1) ⊗ [ A ] − : D A → D B exhibiting left Kan extensions as weighted colimits. Of course, the case of colimitsis obtained by specializing to u = π A . (cid:3) This first result generalizes to more general enriched derivators in not necessarilystable situations. In contrast to this, Theorem 4.85 relies crucially on stability.
Remark . Let us recall that a left exact morphism of derivators is defined as amorphism which preserves terminal objects and cartesian squares (Examples 3.54).By Theorem 3.57 such morphisms preserve homotopy finite limits. Since right Kanextensions in derivators can be calculated pointwise in terms of limits (by (Der4)),it follows that right exact morphisms also preserve right homotopy finite right Kanextension (see [Gro16, Thm. 9.14]). This includes those right Kan extensions forwhich the corresponding slice categories are homotopy finite, and the followingresult hence covers a large class of examples.
Theorem 4.85.
Let D be a stable derivator and let u : A → B be a functor.(i) If u is a sieve, then the right Kan extension morphism u ∗ : D A → D B is aweighted colimit. IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 85 (ii) If u : A → B is right homotopy finite, then the right Kan extension morphism u ∗ : D A → D B is a weighted colimit. In particular, for every homotopy finitecategory A the limit morphism lim A : D A → D is a weighted colimit.Proof. As in the proof of Theorem 4.81 for small categories
C, D, E the action ofspectra yields a left adjoint of two variables ⊗ [ D ] : S p C × D op × D D × E op → D C × E op between stable derivators. In particular, for a fixed object X ∈ D ( A ) the morphism − ⊗ [ A ] X : S p A op → D is exact and hence preserves right Kan extensions along sieves and, more generally,right homotopy finite right Kan extensions ([Gro16, Thm. 9.17]). Thus, in bothsituations we obtain isomorphisms u ∗ ( X ) ∼ = u ∗ ( I A ⊗ [ A ]) X ) ∼ = (cid:0) ( u × id A op ) ∗ I A (cid:1) ⊗ [ A ] X. Letting the diagram X vary we obtain the natural isomorphism(4.86) u ∗ ∼ = (cid:0) ( u × id A op ) ∗ I A (cid:1) ⊗ [ A ] − : D A → D B showing that these particular right Kan extensions are weighted colimits. (cid:3)
Remark . We want to include a short discussion of these two theorems.(i) The proofs of Theorem 4.81 and Theorem 4.85 are constructive in that theyoffer formulas which allow us to calculate the weights for restrictions andsuitable Kan extensions. In fact, as detailed by the formulas (4.82), (4.83),and (4.86), in all cases we start with a suitable Yoneda bimodule. One ofthe variables is bound by the canceling tensor product (or the correspondingcoend), and we simply apply the corresponding operation to the remainingfree variable. This allows us to explicitly calculate the representing weightsas we illustrate a bit further below.(ii) Restrictions and all sufficiently finite left and right Kan between stable deriva-tors are weighted colimits, and there are dual statements for weighted limits .The corresponding representing spectral bimodules admit a fairly rich calcu-lus, and we refer the reader to [GPS14a, GˇS16a, GS17, Shu08, PS16] for moredetails.(iii) The motivation to break up the above examples of weighted colimits into twoclasses is twofold. First, Theorem 4.81 extends to more general V -modulesand V -enriched derivator, while only in Theorem 4.85 we rely on additionalexactness properties. In fact, the result about right Kan extensions alongsieves is also valid in modules over pointed monoidal derivators. Second, theresults in Theorem 4.85 indicate that in stable land the distinction betweenconstructions on the left and constructions on the right is blurred to someextent (sufficiently finite limits are weighted colimits). It turns out that thisis a defining feature of stability and this perspective offers an interestinginvitation to a formal study of abstract stability (see [GS17] for first stepsalong these lines). Example . For every diagram of spectra M ∈ S p ( A ) and every stable derivatorthe action M ⊗ − : D → D A is a weighted colimit. As a special case, for the suspension of the sphere spectrumΣ S ∈ S p ( ∗ ) and X ∈ D ( B ) we have(Σ S ) ⊗ X ∼ = Σ( S ⊗ X ) ∼ = Σ X. Consequently, the suspension morphism Σ : D → D is a weighted colimit withweight Σ S , Σ S ⊗ − ∼ = Σ : D → D . There is a similar description of Σ n , n ∈ Z as weighted colimit with weight Σ n S .In the following examples we study special cases of weighted colimits of mor-phisms. In those cases we start from the Yoneda bimodule I [1] and follow the aboverecipe to construct weights (see Remark 4.87). To also get more used to the calculusof coends from § Example . The Yoneda module I [1] ∈ S p ([1] × [1] op ) looks like S id / / SS / / O O S . id O O Example . For every stable derivator the evaluation morphism 0 ∗ : D [1] → D is a weighted colimit (Theorem 4.81). The weight P is obtained from I [1] by eval-uation at 0 in the covariant variable and is hence given by P = (0 × id [1] op ) ∗ I [1] = ( S ← ∈ S p ([1] op ) . To double-check this result, for X = ( f : x → y ) ∈ D ([1]) we calculate the cancelingtensor product P ⊗ [[1]] X = ( S ← ⊗ [[1]] ( x → y ). By Example 4.62 this is simplythe pushout-product of these two morphisms whose calculation is displayed in thefollowing diagram S ⊗ x (cid:15) (cid:15) ⊗ x o o ∼ = (cid:15) (cid:15) x ∼ = (cid:15) (cid:15) ✤✤✤ (cid:3) o o ∼ = (cid:15) (cid:15) S ⊗ y ⊗ y, o o x . o o ❴ ❴ ❴ Since isomorphisms are stable under cobase change [Gro13, Prop. 3.12], the de-sired pushout is simply x . This calculation hence confirms that there is a naturalisomorphism 0 ∗ ∼ = P ⊗ [[1]] − = ( S ← ⊗ [[1]] − : D [1] → D . Similarly, the weight P for the evaluation morphisms 1 ∗ : D [1] → D is given by P = (1 × id [1] op ) ∗ I [1] = ( S ← S ) ∈ S p ([1] op ) . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 87
Analogously to the previous case, in order to calculate P ⊗ [[1]] X we contemplatethe diagrams S ⊗ x (cid:15) (cid:15) S ⊗ x ∼ = o o (cid:15) (cid:15) x (cid:15) (cid:15) ✤✤✤ (cid:3) x ∼ = o o f (cid:15) (cid:15) S ⊗ y S ⊗ y, ∼ = o o y y. ∼ = o o ❴ ❴ ❴ Again, this calculation confirms the above claim and we obtain a natural isomor-phism 1 ∗ ∼ = P ⊗ [[1]] − = ( S ← S ) ⊗ [[1]] − : D [1] → D . This example generalizes as follows.
Remark . Let D be a stable derivator, A ∈ C at , and a ∈ A . The evaluationmorphism a ∗ : D A → D is a weighted colimit with weight P a ∈ S p ( A op ) , the freediagram generated at a by S ∈ S p ( ∗ ) . More formally, let a : ∗ → A op be the functorclassifying the object a , then P a is given by P a ∼ = a ! ( S ) ∈ S p ( A op ) . This conclusion also holds in arbitrary derivators, and for this it suffices to considerthe corresponding weight in spaces instead of in spectra.
Example . Let D be a stable derivator and let C : D [1] → D be the conemorphism. We recall from Construction 3.22 that C is a finite composition ofsuitably finite Kan extensions and evaluation morphisms. By Theorem 4.81 andTheorem 4.85 the cone morphism is a weighted colimit. Moreover, the weight isobtained from the Yoneda bimodule I [1] ∈ S p ([1] × [1] op ) by an application of thecone with parameters in [1] op . The isomorphisms C (0 → S ) ∼ = S and C ( S ∼ −→ S ) ∼ = 0imply that the weight M of C is isomorphic to (0 ← S ) ∈ S p ([1] op ). We againverify this description. Given a morphism X = ( f : x → y ) ∈ D ([1]) the followingcalculation 0 ⊗ x (cid:15) (cid:15) S ⊗ x o o (cid:15) (cid:15) (cid:15) (cid:15) ✤✤✤ (cid:3) x o o f (cid:15) (cid:15) ⊗ y S ⊗ y, o o Cf y, o o ❴ ❴ ❴ whose details are left to the reader double-checks the claim (compare again toConstruction 3.22). Thus, we obtain the intended natural isomorphism C ∼ = (0 ← S ) ⊗ [[1]] − : D [1] → D . Remark . This description of cones as weighted colimits generalizes to pointedderivators. In fact, in that case the derivator S ∗ of pointed spaces is universalRemark 4.76, and every pointed derivator is a S ∗ -module. Of course, in contrastto this the cone is not an ordinary colimit but merely a weighted colimit. This iseasy to see already for represented derivators because in that case the cone agreeswith the usual cokernel (Examples 3.25). Example . For every stable derivator D there is the morphism G : D [1] → D [1] × [2] which sends a morphism to a (coherent) cofiber sequence. Let us recall from theproof of Theorem 3.40 that G is a finite composition of sufficiently finite homo-topy Kan extensions (see Proposition 2.39 for the classical case which serves as ablueprint for arbitrary stable derivators). Consequently, G is a weighted colimitwhose weight M lives in S p (([1] × [2]) × [1] op ). As one easily verifies by calculationssimilar to the previous cases, the representing weight M looks like Figure 10. Inthat figure the paper plane corresponds to the coordinates in [1] × [2] while thediagonal direction corresponds to the copy of [1] op . Evaluating at (0 , , (0 , , and(1 , ∈ [1] × [2], we obtain the coherent morphisms M , = ( S ← , M , = ( S ← S ) , and M , = (0 ← S ) ∈ S p ([1] op )which already occured as weights for 0 ∗ , ∗ , and C , respectively (see Example 4.90and Example 4.92). Let us only double-check the remaining non-trivial evaluation M , ∈ S p ([1] op ) which is given by(Σ S ← ∼ = (Σ S ) ⊗ ( S ← . S / / (cid:15) (cid:15) S (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) ^ ^ ❂❂❂❂❂❂ S (cid:15) (cid:15) ^ ^ ❃❃❃❃❃❃ / / (cid:15) (cid:15) ` ` ❆❆❆❆❆❆❆ / / / / Σ S ^ ^ ❂❂❂❂❂❂ / / S ^ ^ ❃❃❃❃❃❃ / / ` ` ❆❆❆❆❆❆ Figure 10.
The universal constructor for cofiber sequences.Invoking Example 4.88 and Example 4.90, the corresponding weighted colimitsends X = ( f : x → y ) ∈ D ([1]) to(Σ S ← ⊗ [[1]] X ∼ = (cid:0) (Σ S ) ⊗ ( S ← (cid:1) ⊗ [[1]] X ∼ = (Σ S ) ⊗ (cid:0) ( S ← ⊗ [[1]] X (cid:1) ∼ = (Σ S ) ⊗ x ∼ = Σ x, as it is supposed to be the case. Since M is a representing weight for G , we alsorefer to M as the universal constructor for cofiber sequences .If we restrict M along the functor i : [1] → [1] × [2] which classifies the morphism(0 , → (1 , N = i ∗ M ∈ S p ([1] × [1] op ) looking like S / / S / / O O S . O O IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 89
The corresponding weighted colimit is the morphism cof : D [1] → D [1] ,N ⊗ [[1]] − ∼ = cof : D [1] → D [1] , which at the level of canonical triangulations corresponds to the rotation of trian-gles.From these considerations we can also calculate the weight which describes thefiber morphism fib : D [1] → D [1] as a weighted colimit . In fact, by (3.50) there is anatural isomorphism cof ∼ = Σ : D [1] → D [1] , and in combination with Theorem 3.29this implies fib ∼ = Ω ◦ cof . The weight for cof can be calculated as N ⊗ [[1]] N . More efficiently, it can simplybe read off from M (see Figure 10) by restriction along the functor j : [1] → [1] × [2]classifying the morphism (1 , → (1 , fib is given by Ω( j ∗ M ) and looks like0 / / (cid:3) S Ω S / / O O . O O This example concludes the little detour which was included in order to developsome first feeling for the calculus of weighted colimits and canceling tensor products(for additional examples we refer to [GPS14a] or [GˇS16a]). We now turn to the casewhich is of particular interest to abstract representation theory namely the case ofweights such that the corresponding weighted colimits are equivalences. After allour goal is to describe strong stable equivalences by means of spectral bimodules.
Proposition 4.95.
Let
A, B ∈ C at and let M ∈ S p ( B × A op ) , N ∈ S p ( A op × B ) be spectral bimodules. The following are equivalent.(i) There are isomorphisms M ⊗ [ A ] N ∼ = I B and N ⊗ [ B ] M ∼ = I A .(ii) The weighted colimits M ⊗ [ A ] − : D A → D B and N ⊗ [ B ] − : D B → D A areinverse equivalences for all stable derivators D .(iii) The weighted colimits M ⊗ [ A ] − : S p A → S p B and N ⊗ [ B ] − : S p B → S p A are inverse equivalences.In this situation we say that M is an invertible spectral bimodule and that N is an inverse bimodule of M .Proof. Statement (ii) follows easily from (i) since weighted colimits associated toYoneda bimodules are naturally isomorphic to the corresponding identity mor-phisms. Clearly, (iii) is a consequence of (ii), and it remains to show that (iii)implies (i). This implication is immediate from the uniqueness of representingweights for weighted colimits (see Remark 4.80). (cid:3)
Remark . This proposition of course only makes explicit the notion of equiva-lences internal to the bicategory P rof ( S p ) of spectral bimodules. The point is thatthe strong stable equivalences discussed in this paper give rise to such representinginvertible spectral bimodules. In that context we also refer to these bimodules as universal tilting modules for the following two reasons.(i) The word “universal” alludes to the fact that invertible bimodules realizesimultaneous symmetries in all stable derivators. (ii) These spectral bimodules are spectral refinements of the more classical tiltingcomplexes. In more detail, for every commutative ring k extensions of scalarsalong Z → k defines a cocontinuous monoidal morphism k ⊗ − : D Z → D k .Moreover, let H Z be the integral Eilenberg–MacLane spectrum (realized asa symmetric ring spectrum) and let D H Z be the derivator of H Z -modulespectra. There is a zigzag of weakly monoidal Quillen equivalences relatingCh( Z ) and Mod( H Z ) [Shi07], and this induces a monoidal equivalence ofderivators D H Z ≃ D Z . As a final ingredient, inducing up along S → H Z yields a cocontinuous monoidal morphism H Z ∧ − : S p → D H Z , and we endup with a cocontinuous monoidal morphism of derivators(4.97) k ⊗ − : S p H Z ∧− → D H Z ≃ D Z k ⊗− → D k . In the case of a field k , an application of this morphism to invertible spec-tral bimodules yields tilting complexes as in tilting theory (see for example[APR79, BB80, HR82, Ric91, Kel94] or the survey articles in [AHHK07]).We illustrate this notion of invertible spectral bimodules by some examples. Example . For every stable derivator the morphisms Σ n : D → D for n ∈ Z areinvertible (Theorem 3.29) and weighted colimits (Example 4.88). By Proposition 4.95the corresponding weights Σ n S ∈ S p ( ∗ ) are invertible. Example . For every stable derivator D the iterated cofiber morphisms cof n : D [1] → D [1] are invertible for n ∈ Z (Theorem 3.29) and these are by Example 4.94 weightedcolimits. In more detail, the weights for cof and fib are respectively given by S / / / / (cid:3) SS / / O O S , O O Ω S / / O O , O O and these bimodules are inverse to each other (Proposition 4.95). Example . By Corollary 4.11 all A -quivers are strongly stably equivalent.The proof of this statement shows that these strong stable equivalences are weightedcolimits which by Proposition 4.95 give rise to invertible spectral bimodules. Toconsider a specific example, let Q = p = ( • ← • → • ) be the source of valencetwo and let ~A = ( • → • → • ) be the linearly oriented A -quiver. We explicitlydescribe the universal tilting modules associated to the strong stable equivalenceΦ : ~A ∼ Q and its inverse as constructed in Example 4.10.The universal tilting module T Q, ~A ∈ S p ( Q × ~A op3 ) is obtained from the Yonedabimodule I ~A by an application of the strong stable equivalence Φ : S p ~A → S p Q (Remark 4.87). The bimodule I ~A takes by Example 4.68 the form as shown on the IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 91 left in: S / / S / / S S / / o o SI ~A : 0 / / O O S / / O O S O O T Q, ~A : S O O S / / o o O O S O O / / O O / / O O S O O O O / / o o O O S O O The strong stable equivalence Φ simply forms the cofiber of the first morphism, andthis amounts in the three cases to S / / (cid:15) (cid:15) (cid:3) S (cid:15) (cid:15) / / S / / (cid:15) (cid:15) (cid:3) S (cid:15) (cid:15) / / S / / (cid:15) (cid:15) (cid:3) (cid:15) (cid:15) / / S / / , / / S , / / . Consequently, for every stable derivator D there is a natural isomorphismΦ D ∼ = T Q, ~A ⊗ [ ~A ] − : D ~A ∼ −→ D Q , where T Q, ~A takes the form as displayed in the above diagram on the right.Similarly, also the inverse strong stable equivalence Φ − : Q s ∼ ~A is given by auniversal tilting module T ~A ,Q ∈ S p ( ~A × Q op ). This invertible spectral bimoduleis obtained from I Q by an application of Φ − which amounts to forming fibers ofthe morphisms pointing from right to left. We leave it to the reader to double-checkthat I Q and T ~A ,Q look like: S (cid:15) (cid:15) / / o o (cid:15) (cid:15) (cid:15) (cid:15) Ω S / / (cid:15) (cid:15) (cid:15) (cid:15) / / (cid:15) (cid:15) I Q : S S / / o o S T ~A ,Q : 0 / / S / / S O O O O o o / / S O O O O / / / / O O S O O These universal tilting modules T Q, ~A and T ~A ,Q are inverse to each other.Additional interesting examples of universal tilting bimodules arise from theCoxeter and the Serre functors discussed in § Example . For every stable derivator D the Serre functor S : D ~A ∼ −→ D ~A isnaturally isomorphic to the cofiber morphism cof (Examples 4.30). The universaltilting bimodule is by Example 4.99 given by S / / S / / O O S . O O We invite the reader to directly calculate the universal tilting bimodules forthe Serre functors in the cases ~A and Q = ( • ← • → • ) (see Example 4.22 and Example 4.25). Here, instead we turn to the following different perspective on thesebimodules. Let us recall from our discussion of opposite derivators that the Yonedabimodule I A ∈ S p ( A × A op ) can also be considered as an object in S p ( A × A op ) op ∼ = S p op (( A × A op ) op ) ∼ = S p op ( A op × A ) . The point of this reformulation is that the sphere spectrum S defines a morphism − ⊲ S : S p op → S p . Definition 4.102.
For every small category A the canonical duality bimodule D A ∈ S p ( A × A op ) is the image of the Yoneda bimodule I A ∈ S p ( A × A op ) under S p op ( A op × A ) − ⊲ S → S p ( A op × A ) ∼ = → S p ( A × A op ) . Example . For every poset P the canonical duality bimodule D P admits thefollowing description. Let P × P op be endowed with the product order and let∆ P ⊆ P × P op be the diagonal. The bimodule D P ∈ S p ( P × P op ) restricted tothe down-set generated by ∆ P is constant with value the sphere spectrum S and itvanishes on the complement [GˇS16a, Lem. 7.4].We illustrate this class of examples in the following specific cases. As in previouscases, for the bimodules D P ∈ S p ( P × P op ) the P -coordinate is drawn horizontallyand the P op -coordinate vertically. Example . The canonical duality bimodule D ~A ∈ S p ( ~A × ~A op2 ) looks like S / / S / / O O S . O O Example . The canonical duality bimodule D ~A ∈ S p ( ~A × ~A op3 ) takes theform S / / / / S O O / / S O O / / O O S O O / / S / / O O S . O O Example . Let Q = ( • ← • → • ) be the source of valence two. The canonicalduality bimodule D Q ∈ S p ( Q × Q op ) is given by S (cid:15) (cid:15) S / / o o (cid:15) (cid:15) (cid:15) (cid:15) S / / o o O O S O O o o / / S O O Definition 4.107.
Let D be a stable derivator. The Nakayama functor associ-ated to A ∈ C at is D A ⊗ [ A ] − : D A → D A . IGHER SYMMETRIES IN ABSTRACT STABLE HOMOTOPY THEORIES 93
This definition is of course motivated by the corresponding situation in repre-sentation theory (see for instance [Hap88, § ~A theSerre functor and the Nakayama functor are naturally isomorphic (Example 4.101and Example 4.104). This holds in full generality. Theorem 4.108.
Let D be a stable derivator and let Q be a Dynkin quiver oftype A . The Serre functor and the Nakayama functor are naturally isomorphic S ∼ = D Q ⊗ [ Q ] − : D Q ∼ −→ D Q . Proof.
This theorem is in [GˇS16a]. The special case of linearly oriented Dynkinquivers is established by direct calculation as [GˇS16a, Thm. 7.7]. And the generalcase is deduced from it as an application of the general calculus of bimodules [GˇS16a,Thm. 10.9]. (cid:3)
Additional universal tilting bimodules arise from more general reflection functors(as in § Remark . A different way of thinking of abstract representation theory isthat it contributes to the calculation of spectral Picard groups (or groupoids). Forevery closed monoidal derivator V there is the bicategory P rof ( V ) of bimodules(Theorem 4.67). In particular, for every small category A we obtain the monoidalcategory P rof ( V )( A, A ) = V ( A × A op )of ( A, A )-bimodules with values in V . The monoidal structure is the cancelingtensor product ⊗ [ A ] and the monoidal unit is the Yoneda bimodule I A . Corre-spondingly, we define the Picard group
Pic V ( A ) of A relative to V as thePicard group of the monoidal category P rof ( V )( A, A ),Pic V ( A ) = Pic (cid:0) P rof ( V )( A, A ) (cid:1) . Elements are simply isomorphism classes of invertible bimodules and the isomor-phism class of I A is the neutral element. Particularly interesting cases arise forthe stable, closed monoidal derivators S p , D Z , and D k for arbitrary commutativerings k . Correspondingly, we obtain the spectral Picard group Pic S p ( A ), the integral Picard group Pic D Z ( A ), and also the k -linear Picard group Pic D k ( A ).In particular, in the case of a field k for special choices of A the group Pic D k ( A )agrees with the derived Picard group of Miyachi and Yekutieli [MY01]. For everystable, closed monoidal derivator V the suspension of the monoidal unit Σ S definesan element in Pic V ( ∗ ) (Example 4.88).To the best of the knowledge of the author, the only shape for which the spectralPicard group is known is A = ∗ . In that case, the Picard group Pic S p ( ∗ ) is thePicard group of the stable homotopy category which is isomorphic to Z with Σ S as generator ([HMS94] or [Str92, Thm. 2.2]). Every construction of a strong stableequivalence A s ∼ A and its universal tilting bimodule T A,A yields an element inPic S p ( A ), and the hope is that at least for some shapes A this leads to a calculationof the spectral Picard group.One nice feature of these Picard groups is their functoriality in V . In fact, everycocontinuous, monoidal morphism F : V → W of monoidal derivators induces amorphism of bicategories F : P rof ( V ) → P rof ( W ) (as follows from the fact thatcocontinuous morphisms preserve coends). In particular, for every A ∈ C at we obtain a monoidal functor P rof ( V )( A, A ) → P rof ( W )( A, A ) and hence a grouphomomorphism F : Pic V ( A ) → Pic W ( A ). A particularly interesting case is givenby the morphism (4.97). Hence, associated to every commutative ring k a grouphomomorphism k ⊗ − : Pic S p ( A ) → Pic D k ( A ) . This functoriality of Picard groups is calculationally useful. For instance,incombination with calculations of derived Picard groups by Miyachi and Yekutieli[MY01, Theorem 4.1] and an abstract fractionally Calabi–Yau property of Dynkinquivers of type A [GˇS16a, Cor. 5.20], this was used to show that for every suchquiver Q the homomorphism k ⊗ − : Pic S p ( Q ) → Pic D k ( Q )is a split epimorphism [GˇS16a, Thm. 12.6]. Conjecturally, there is no kernel, and,jointly with Jan ˇSˇtov´ıˇcek, we intend to come back to this and related calculationselsewhere. References [ABG +
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Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Mathematisches Institut, Ende-nicher Allee 60, 53115 Bonn, Germany, and Johannes Gutenberg-Universit¨at Mainz,Institut f¨ur Mathematik, Staudingerweg 9, 55128 Mainz, Germany
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