MONOIDAL DERIVATORS MORITZ GROTH

Transcription

1 MONOIDAL DERIVATORS MORITZ GROTH Abstract. The aim of this paper is to develop some foundational aspects of the theory of monoidal derivators and modules over them. The passages from categories and model categories to derivators both respect monoidal objects and hence give rise to natural examples. As an illustration of these concepts, we discuss some derivators related to chain complexes and symmetric spectra. Contents 0. Introduction 1 1. The cartesian closedness of the 2-category of derivators and related notions The cartesian monoidal 2-category of (pre)derivators Bimorphisms and homotopy (co)limit preserving bimorphisms Homotopy coends and the Fubini-type theorem Adjunctions of two variables The closedness of the cartesian monoidal 2-category of derivators Monoidal derivators The 2-category of monoidal (pre)derivators Monoidal model categories induce monoidal derivators Weakly monoidal Quillen adjunctions Weighted homotopy (co)limits Modules over monoidal derivators The 2-Grothendieck fibration of modules over monoidal categories Actions by monoidal derivators Universal actions by endomorphism prederivators 37 Appendix A. The 2-categorical Grothendieck construction 37 Appendix B. Monoidal and closed monoidal 2-categories 39 B.1. The 2-categories of pseudomonoid objects and modules in a monoidal 2-category 39 B.2. Closed monoidal 2-categories 42 References Introduction The theory of derivators is an approach to abstract homotopy theory which sits between the theory of model categories and the more classical homotopy or derived categories. For many purposes the usual passage from a model category to its homotopy category looses too much information. Date: July 20,

2 2 MORITZ GROTH The homotopy category admits hardly any categorical (co)limits nor can we develop a good theory of homotopy (co)limits with the homotopy category only. The theory of derivators corrects this defect and captures the structure needed for a powerful calculus of homotopy (co)limits. But at the same time this theory simpler than say the theory of model categories in the sense that a derivator is a purely categorical notion. For more motivational comments on the theory we refer to [Gro11]. Derivators were introduced independently by Grothendieck [Gro], Heller [Hel88], and Franke [Fra96] and recently studied further by, among others, Maltsiniotis [Mal01], Cisinski, and the author [Gro11]. Many more references on the theory can be found on the webpage [Gro]. The aim of this paper 1 is to systematically develop some foundational aspects of the theory of monoidal derivators (cf. also to [Cis08]) and their actions. The study of monoidal derivators will be continued in a joint paper with Ponto and Shulman [GPS12]. Moreover, this paper provides the necessary foundation for its sequel [Gro12b] on enriched derivators. As we saw in [Gro11], two important classes of derivators are given by derivators associated to (combinatorial) model categories and derivators represented by bicomplete categories. Both classes of examples can be refined to give corresponding statements about situations where the input is suitably monoidal. We formalize the notion of a monoidal (pre)derivator and make these statements precise. It is well-known that homotopy categories of (combinatorial) monoidal model categories (in the sense of Hovey [Hov99], as opposed to the slightly different notion of [SS00]) can be canonically endowed with monoidal structures and similarly for suitably monoidal Quillen adjunctions. These statements are truncations of more structured results as we will see below. We will show that the derivator associated to a combinatorial monoidal model category can be canonically endowed with a monoidal structure. This easily generalizes to model categories which are suitably modules over a monoidal model category. In Section 1 we develop some general theory about the 2-category of (pre)derivators which is essential to the monoidal picture. Prederivators and derivators are respectively organized in a cartesian monoidal 2-category. We introduce the important related notion of a bimorphism and show that the bimorphism functor is corepresented by the cartesian product. With the concept of bimorphisms which preserve homotopy colimits separately in each variable at our disposal we can talk about adjunctions of two variables in the context of derivators. This latter notion is analyzed in some detail since it plays a key role both in later sections and in future work ([GPS12],[Gro12b]). We close this section with the observation that derivators actually form a cartesian closed 2-category (see also [Cis08]). In Section 2 we introduce the 2-categories of monoidal (pre)derivators. A monoidal prederivator is a pseudomonoid object in the cartesian 2-category of prederivators. For a monoidal derivator we demand the monoidal pairing to preserve homotopy colimits separately in each variable. We show that derivators associated to combinatorial monoidal model categories are canonically monoidal. More generally, a Brown functor between model categories (cf. Definition 2.6) induces a morphism of associated derivators and we mention some relevant examples. Weakly monoidal Quillen equivalences are shown to induce strongly monoidal equivalences of derivators and this will be illustrated by an example from stable homotopy theory. In the last subsection, we sketch how to associate a bicategory of profunctors (bimodules, correspondences) to a monoidal derivator. This bicategory has some pleasant formal properties and encodes important structure as will be studied in detail in [GPS12]. In particular, weighted homotopy (co)limits are subsumed by this structure and they will be taken up again in [Gro12b]. 1 This research was partially supported by the Deutsche Forschungsgemeinschaft within the graduate program Homotopy and Cohomology (GRK 1150).

3 MONOIDAL DERIVATORS 3 In Section 3 we consider actions by monoidal (pre)derivators. We begin by considering the 2- category of categories endowed with an action as a special case of the 2-categorical Grothendieck construction. Prederivators with an action by a monoidal prederivator turn out to be 2-functors taking values in this 2-category. Again there are the two classes of examples coming from suitable actions in the context of categories or model categories and we mention a few specific ones in the differential-graded and in the spectral context. As a consequence of the general theory as developed in Appendix B we obtain the following result: The endomorphism prederivator End(D) is canonically monoidal and comes with an action on D. Moreover, this action is the terminal example of a monoidal prederivator acting on D. This result will be used in [Gro12a]) to put into perspective the existence of certain linear structures on suitable additive derivators. There are two appendices. In Appendix A we briefly recall the Grothendieck construction and extend it to the context of a 2-category valued 2-functor. In Appendix B we consider (closed) monoidal 2-categories and establish a general result about them (Theorem B.11). In the special case of the closed monoidal 2-category of prederivators this result gives us the universal property of the monoidal action by endomorphism prederivators. Before we begin with the proper content of this paper let us make a comment on set-theoretical issues. In what follows we will frequently consider the category of categories and similar gadgets. Strictly speaking there are some size issues which were to be considered here but these problems could be circumvented by a use of Grothendiecks language of universes. Since we do not wish to add an additional technical layer to the exposition by keeping track of the different universes we decided to ignore these issues. Acknowledgments. It is a pleasure to thank André Joyal, Stefan Schwede, and Michael Shulman for fruitful discussions and their ongoing interest in the subject. 1. The cartesian closedness of the 2-category of derivators and related notions 1.1. The cartesian monoidal 2-category of (pre)derivators. In this subsection we mainly recall some basics about (pre)derivators and establish some notation. For more motivational comments, examples, and more details of the theory we refer to [Gro11]. The basic 2-categorical notions as employed here are discussed in [Bor94a, ML98, KS05] but nothing deep from that theory is used. A prederivator is a 2-functor D: Cat op CAT where Cat denotes the 2-category of small categories and CAT denotes the 2-category of (not necessarily small) categories. Spelling out this definition, we thus have for every small category J an associated category D(J), for a functor u: J K an induced functor D(u) u : D(K) D(J), and for a natural transformation α: u v between two such functors a natural transformation D(α) α : u v as indicated in the following diagram: J u α K, D(K) v u α D(J) These assignments are compatible with compositions and units in a strict sense, i.e., we have equalities of the respective expressions. If one considers 2-functors defined on suitable 2-subcategories Dia Cat (for example finite categories, finite and finite-dimensional categories or posets) [Gro11, Section 1] one is lead to the notion of a (pre)derivator of type Dia. For simplicity, we will stick to the case of all small categories in this paper but the reader is invited to replace (pre)derivator by (pre)derivator of type Dia throughout the paper. Prederivators assemble in a 2-category PDer with pseudo-natural transformations as morphisms and modifications as 2-cells. In more detail, a morphism F : D D consists of a functor v

4 4 MORITZ GROTH F J : D(J) D (J) for each small category J together with isomorphisms γu F : u F K F J u for each functor u: J K. These isomorphisms are subject to certain coherence conditions. To avoid awkward notation we will frequently suppress some of the indices in the constituents of such morphisms. Moreover, we will not distinguish notationally between the natural isomorphisms γ and their inverses. We also have lax and strong variants. Given a prederivator D and a functor u: J K the induced functor u : D(K) D(J) is called a restriction of diagram functor or precomposition functor. As a special case of these we have the evaluation functors: If we let e denote the terminal category then an object k K can be identified with the unique functor k : e K sending the unique object of e to k. Let us write f k : X k Y k for the image of a morphism f : X Y in D(K) under k : D(K) D(e). The prederivator admits homotopy left Kan extensions along u: J K if the induced functor u : D(K) D(J) has a left adjoint u! HoLan u : D(J) D(K). Similarly we speak of homotopy right Kan extensions which will be denoted by u or HoRan u. This specializes to homotopy (co)limits in the absolute case, i.e., if we consider functors p J : J e with target the terminal category. In this case we will use the notation Hocolim J and Holim J respectively. An important property of Kan extensions in bicomplete categories is that they can be calculated pointwise by (co)limits. In order to formulate this as an axiom for derivators let us consider a functor u: J K and an object k K. Associated to the slice categories J k/ and J /k we have the following two natural transformations: p Jk/ J k/ e pr k J K u In the case of J k/ the component of the natural transformation at (j, f : k u(j)) is f and similarly in the other case. Assuming D to be a prederivator admitting the necessary homotopy Kan extensions, the calculus of mates [KS05] gives rise to canonical 2-cells[Gro11, Subsection 1.2]: p J/k J /k Hocolim J/k pr k u! and k u Holim Jk/ pr Asking these maps to be isomorphisms is a convenient way to axiomatize the formulas for pointwise Kan extensions. Definition 1.1. A prederivator D is called a derivator if it satisfies the following axioms: (Der1) D sends coproducts to products. In particular, D( ) is trivial. (Der2) A morphism f : X Y in D(J) is an isomorphism if and only if f j : X j Y j is an isomorphism in D(e) for every object j J. (Der3) For every functor u: J K, there are homotopy left and right Kan extensions along u: (u!, u ): D(J) D(K) and (u, u ): D(K) D(J). (Der4) For every functor u: J K and every k K, the morphisms Hocolim J/k pr (X) α! α u! (X) k and u (X) k HolimJk/ pr (X) are isomorphisms for all X D(J). The full sub-2-category of PDer spanned by the derivators is denoted by Der. The category of morphisms between two (pre)derivators D and D is denoted by Hom(D, D ) while we write Hom strict (D, D ) for the category of strict morphisms. e pr k J K u

5 MONOIDAL DERIVATORS 5 Example 1.2. The Yoneda embedding y : CAT PDer sends a category C to the represented prederivator y(c): Cat op CAT: J C J. Here, C J denotes the category of functors from J to C. The prederivator y(c) is a derivator if and only if C is bicomplete. The 2-categorical Yoneda lemma implies that for an arbitrary prederivator D we have a natural isomorphism of categories Hom strict PDer(y(J), D) D(J). Similarly, the bicategorical Yoneda lemma gives us natural equivalences of categories Hom PDer (y(j), D) D(J). For simplicity, we will sometimes drop the embedding y from notation and again just write C for the prederivator represented by a category C. Example 1.3. Let M be a model category and let us denote the weak equivalences by W. For a small category J let us denote by W J the morphisms in M J which are levelwise weak equivalences. Cisinski has shown in [Cis03] that the assignment D M : Cat op CAT: J M J [W J 1 ] makes sense without change of universe and defines a derivator. combinatorial model category see [Gro11]. For a proof in the case of a In every 2-category we have the notion of adjoint 1-morphisms, equivalences, and Kan extensions (see Sections 1 and 2 of [Str72]). Since we will later introduce adjunctions of two variables let us recall the first notion in the 2-category Der of derivators (cf. [Gro11, Subsection 2.2]). A morphism L: D D of derivators is a left adjoint if and only if it is levelwise a left adjoint functor L K : D(K) D (K) and it preserves homotopy left Kan extensions. For convenience, let us be more precise about the second condition and let us consider a functor u: J K. Given a morphism L: D D of derivators we obtain a canonical natural transformation γ L u! : u! L L u! (cf. [Gro11, Subsection 1.2]): D (J) u D (K) L L D(J) u D(K) D (K) u! D (J) u D (K) L L D(J) u D(K) u! D(J) By definition, γ L u! : u! L L u! is given by the pasting of the above right diagram in which the two additional natural transformations are adjunction morphisms. We say that L preserves homotopy left Kan extensions if the natural transformation γ L u! : u! L L u! is an isomorphism for all functors u: J K. Now, given such a left adjoint morphism of derivators L: D D we choose levelwise right adjoint functors R K : D (K) D(K). These can be uniquely assembled into a morphism of derivators R: D D such that the adjunctions at the different levels are compatible. By this we mean that

6 6 MORITZ GROTH for a functor u: J K we obtain the following commutative diagram: hom D (K)(L K X, Y ) u hom D (J)(u L K X, u Y ) hom D(K) (X, R K Y ) u hom D(J) (u X, u R K Y ) γ L hom D (J)(L J u X, u Y ) hom D(J) (u X, R J u Y ) To give an important class of adjunctions of derivators let us recall from [Gro11] that there is the shift action by Cat op on PDer. In fact, for every prederivator D and every small category J we have the prederivator D J D(J ): Cat op J Cat op D CAT. This gives us an induced 2-functor ( ) ( ) : PDer Cat op PDer which turns PDer into a right Cat op module. Using this notation, the pointwise formulas for Kan extensions in a derivator imply the following result (see [Gro11, Section 2]). Lemma 1.4. Let D be a derivator and let u: J K be a functor. Then we obtain adjunctions of derivators: (u!, u ): D J D K and (u, u ): D K D J Let us define the ( internal ) product of two prederivators. Thus, let D and D be prederivators, then their product D D PDer is defined to be the composition of 2-functors Cat op Cat op Cat op D D CAT CAT CAT where denotes the diagonal. The product of morphisms of prederivators and natural transformations is defined similarly and this gives us the 2-product in the 2-category PDer of prederivators. This monoidal structure can also be obtained by Day convolution applied to the cartesian structure on both Cat and CAT. Recall from [Gro11, Section 3 and 4] that we also have the notions of a pointed derivator and a stable derivator. Lemma 1.5. Let D and D be derivators. Then D D is again a derivator which is pointed or stable if both D and D are. Proof. All axioms follow immediately from the fact that they are satisfied by D and D. To establish (Der4) for example it suffices to observe that the canonical morphisms under consideration can be chosen to be the products of the respective canonical morphisms of D and D which are hence isomorphisms. Thus the 2-categories PDer and Der are cartesian monoidal 2-categories. The unit e of the monoidal structure is the prederivator with constant value the terminal category e (consisting of one object and its identity morphism only). To simplify notation we will suppress the canonical associativity isomorphisms and hence also brackets from notation. In Section 2, we will introduce monoidal (pre)derivators as certain pseudomonoid objects in the respective 2-categories. However, γ R

7 MONOIDAL DERIVATORS 7 it turns out to be convenient to also use the exterior version of the monoidal structure. This version is based on bimorphisms which will be introduced in the next subsection Bimorphisms and homotopy (co)limit preserving bimorphisms. Since the product of two prederivators is the 2-categorical product we understand morphisms into them. But also maps out of a product of two prederivators are easy to describe: up to an equivalence of categories these are just the bimorphisms as we will define them now. Definition 1.6. Let D, E, and F be prederivators. A bimorphism B from (D, E) to F, denoted B : (D, E) F, consists of a family of functors B J1,J 2 : D(J 1 ) E(J 2 ) F(J 1 J 2 ), J 1, J 2 Cat, and for each pair of functors (u 1, u 2 ): (J 1, J 2 ) (K 1, K 2 ) a natural isomorphism γu B 1,u 2 as indicated in: D(K 1 ) E(K 2 ) u 1 u 2 D(J 1 ) E(J 2 ) B B F(K 1 K 2 ) (u 1 u 2) F(J 1 J 2 ) These data have to satisfy the following coherence conditions. Given a pair of composable pairs of functors (u 1, u 2 ): (J 1, J 2 ) (K 1, K 2 ) and (v 1, v 2 ): (K 1, K 2 ) (L 1, L 2 ) and a pair of natural transformations (α 1, α 2 ): (u 1, u 2 ) (u 1, u 2) we have γ idj1,id J2 id BJ1,J 2 and the commutativity of the following two diagrams: (u 1 u 2 ) (v 1 v 2 ) B γ (u 1 u 2 ) B(v1 v2) γ γ (u 1 u 2 ) B (u 1 u 2) B B(u 1 u 2)(v 1 v 2) B(u 1 u 2) B(u 1 u Now, given two parallel bimorphisms B, B : (D, E) F, a natural transformation τ : B B of bimorphisms consists of a family of natural transformations τ J1,J 2 : B J1,J 2 B J 1,J 2. These have to be compatible in the sense that given a pair of functors (u 1, u 2 ): (J 1, J 2 ) (K 1, K 2 ) the following diagram commutes: (u 1 u 2 ) B τ (u 1 u 2 ) B γ B(u 1 u 2) τ γ B (u 1 u 2) Let us quickly mention that three of the above coherence properties can be expressed by equalities between certain pasting diagrams. This observation combined with the nice behavior of the calculus of mates with respect to pasting (cf. [Gro11, Lemma 1.14]) proves to be useful in the discussion of adjunctions of two variables. Given three prederivators D, E, and F we obtain a category of bimorphisms from (D, E) to F which we denote by BiHom((D, E), F). In fact, given three such prederivators we can consider the exterior product D E of D and E and the 2-functor F ( ) which are respectively defined by (D E)(J 1, J 2 ) D(J 1 ) E(J 2 ) and (F ( ))(J 1, J 2 ) F(J 1 J 2 ). γ γ 2 )

8 8 MORITZ GROTH Then, we have an equality of categories BiHom((D, E), F) PsNat(D E, F ( )) where PsNat(, ) denotes the category of pseudo-natural transformations and modifications (cf. [Bor94a, Definition 7.5.2, Definition 7.5.3]). This observation shows that BiHom((, ), ) is functorial in all three arguments. Let us now show that BiHom((, ), ) is corepresentable by the product. For prederivators D and E, the universal bimorphism (D, E) D E has components induced by the projections: D(J 1 ) E(J 2 ) pr 1 pr 2 D(J 1 J 2 ) E(J 1 J 2 ) This bimorphism gives the right adjoint in the following proposition. Proposition 1.7. For prederivators D, E, and F we have natural isomorphisms between categories of strict (bi)morphisms and natural equivalences between categories of (bi)morphisms: BiHom strict ((D, E), F) Hom strict (D E, F) and BiHom((D, E), F) Hom(D E, F) Proof. We begin with the strict case and sketch a definition of functors in both directions which will be inverse to each other. Given a strict bimorphism B : (D, E) F we obtain a strict morphism l(b): D E F with components: l(b) J : D(J) E(J) B J,J F(J J) J F(J) Conversely, let us consider a strict morphism F : D E F. We can then construct an associated bimorphism r(f ): (D, E) F with components: r(f ) J1,J 2 : D(J 1 ) E(J 2 ) pr 1 pr 2 D(J 1 J 2 ) E(J 1 J 2 ) F J1 J 2 F(J 1 J 2 ) Thus, in the construction of the two functors we make use of the adjunction morphisms belonging to the adjunction (, ): Cat Cat Cat. In fact, the adjunction unit η : id is given by the diagonals while the adjunction counit ɛ: id is given by the projections. Let us note that if we take this adjunction and pass to Cat op and Cat op Cat op respectively then the direction of η and ɛ are inverted. Using (D E) D E, we can thus rewrite the functor l as the following composition: 2-Nat(D E, F ) 2-Nat((D E), F ) η 2-Nat(D E, F) If we depict this construction graphically (and add artificially an identity 2-cell), it reminds us of the calculus of mates and looks like: CAT CAT CAT F D E Cat op Cat op Cat op Cat op There is a similar reasoning for the functor r : Hom(D E, F) BiHom((D, E), F). In fact, r is given by the composition: 2-Nat((D E), F) 2-Nat((D E), F ) ɛ 2-Nat(D E, F )

9 MONOIDAL DERIVATORS 9 Again, this can be depicted diagrammatically as: CAT D E CAT Cat op Cat op CAT F Cat op Cat op Cat op Let us next show that the composition l r is the identity. For this purpose let us consider the diagram on the left which depicts the value of l r at a morphism F : D E F : CAT D E Cat op Cat op F CAT Cat op CAT F Cat op Cat op Cat op Cat op CAT D E Cat op Cat op Cat op Cat op Cat op More precisely, this value is by definition the pasting of that diagram. The triangular identity for the adjunction (, ) implies that this is just F as intended. A similar reasoning applies to the diagram on the right concluding the proof of the first statement. Note that there is a subtlety in the above pasting diagrams. To make this precise let us consider the following commutative diagram describing that pasting and which again shows that l r id. Using the labels of the arrows in that diagram, we have (l r)(f ) η (F ) (ɛ ) : (D E) ɛ (D E) η (D E) F F η F F The square is commutative in this case since F is a strict morphism as opposed to a more general morphism of prederivators. In the case where F happens to be a morphism of prederivators this square would only commute up to an invertible 2-cell given by the structure maps γ F belonging to F. This is the reason why we only have an equivalence between the respective categories of (bi)morphisms and not an isomorphism in that case. The details are left to the reader. In the context of derivators, we want to introduce bimorphisms which preserve homotopy colimits separately in its arguments. For this purpose, let B : (D, E) F be a bimorphism of derivators and let us consider functors u 1 : J 1 K 1 and u 2 : J 2 K 2. The calculus of mates applied to γ B u 1,id gives us a natural transformation γ B u 1,id! : (u 1 id)! B B (u 1! id)

10 10 MORITZ GROTH as given by the following pasting: (u 1 id)! F(K 1 J 2 ) B F(J 1 J 2 ) D(J 1 ) E(J 2 ) (u 1 id) F(K 1 J 2 ) B u 1 id D(K 1 ) E(J 2 ) u 1! id D(J 1 ) E(J 2 ) A similar construction gives the natural transformation γ B id,u 2! : (id u 2 )! B B (id u 2! ). Definition 1.8. Let D, E, and F be derivators. A bimorphism B : (D, E) F preserves homotopy left Kan extensions in the first variable (the second variable) if the natural transformations γu B ( 1,id! : (u 1 id)! B B (u 1! id) γ B id,u2! : (id u 2 )! B B (id u 2! ) ) are isomorphisms for all functors u 1 : J 1 K 1 (u 2 : J 2 K 2 ). For simplicity we will also say that a morphism of derivators is cocontinuous in one variable or in both variables separately. This notion will be important in the context of adjunctions of two variables between derivators (cf. Subsection 1.4) Homotopy coends and the Fubini-type theorem. Later, in the context of profunctors, we will need homotopy coends (with parameters). So, let us give their construction and also establish the basic result given by a Fubini-type theorem. As an intermediate step, let us recall the notion of the twisted arrow category associated to a category. Let K Cat and let us consider the associated functor hom K (, ): K op K Set. As a special case of a set-valued functor, hom K (, ) has an associated category of elements which is a discrete Grothendieck opfibration over K op K. The twisted arrow category of K which will be denoted by tw(k) is the opposite of this category of elements. Thus, an object in this category is just a morphism f : k 0 k 1 while a morphism f f is a commutative diagram as in: k 0 k 0 f k 1 f k 1 This category comes equipped with a functor (t, s): tw(k) K op K which sends an object f : k 0 k 1 to (k 1, k 0 ). By construction, this functor is a discrete Grothendieck fibration. Definition 1.9. Let D be a derivator and let J, K, and L be small categories. The homotopy coend functor (with parameters) K : D(J K op K L) D(J L) is defined by: K : D(J K op K L) (t,s) D(J tw(k) L) pr! D(J L) In the case of a represented derivator this reduces to the usual notion of coends (with parameters). Given a morphism of derivators F : D D and an object X D(J K op K L) we can use the calculus of mates in order to obtain a canonical map: K K F (X) F ( X)

11 MONOIDAL DERIVATORS 11 Let us say that F preserves homotopy coends if this map happens to be an isomorphism for all K and all X. We know from [Gro11, Proposition 2.4] that a morphism which preserves homotopy colimits already preserves homotopy left Kan extensions. This implies immediately the first statement of the next lemma. Lemma A homotopy colimit preserving morphism between derivators also preserves homotopy coends. In particular, homotopy coends are calculated pointwise. Similarly, a bimorphism which preserves homotopy colimits separately in each variable also preserves homotopy coends separately in each variable. The fact that homotopy coends are calculated pointwise can be suggestively written as follows. Given a derivator D, X D(J K op K L), and objects j J and l L the canonical map K K X(j,,, l) ( X)(j, l) is an isomorphism in D(e). In the context of derivators, the Fubini-type theorem about iterated homotopy coends (with parameters) takes the following form. Lemma Let D be a derivator, t: J K op K L op L M J (K L) op (K L) M the canonical isomorphism, and X D(J (K L) op (K L) M). Then there are natural isomorphisms: K L K L L K t X X t X Proof. This follows immediately from the observation that there is a canonical isomorphism between tw(k L) and tw(k) tw(l) which is compatible with the source and target maps in the sense that the following diagram commutes: tw(k) op tw(l) op tw(k L) op K op K L op L t (K L) op (K L) 1.4. Adjunctions of two variables. Our next aim is to introduce the notion of an adjunction of two variables between (pre)derivators. This will, in particular, allow us to talk about closed monoidal derivators and closed modules later. We begin by recalling this notion from classical category theory. Let D, E, and F be categories and let us agree that we call a bifunctor : D E F a left adjoint of two variables if there are functors Hom l : D op F E and Hom r : E op F D and natural isomorphisms as in: hom F (X Y, Z) home (Y, Hom l (X, Z)) homd (X, Hom r (Y, Z)) Interchangeably, we also denote these adjoints by X\Z Hom l (X, Z) and Z/Y Hom r (Y, Z) respectively. In the context of prederivators there is the following lemma about adjunctions of two variables. Lemma Let D, E, and F be prederivators and let : (D, E) F be a bimorphism which is levelwise part of an adjunction of two variables. Then there is a unique way to assemble any family

12 12 MORITZ GROTH of chosen adjoints Hom l (, ): D(J 1 ) op F(J 1 J 2 ) E(J 2 ) into a lax natural transformation Hom l such that the following squares commute for all functors u 1 : J 1 K 1, u 2 : J 2 K 2, and objects X D(K 1 ), Y E(K 2 ), Z F(K 1 K 2 ): hom(x Y, Z) hom(y, Hom l (X, Z)) hom((u 1 id) (X Y ), (u 1 id) Z) γ Hom l u 1,id γ u 1,id hom(u 1X Y, (u 1 id) Z) hom(x Y, Z) hom(y, Hom l (u 1X, (u 1 id) Z)) hom(y, Hom l (X, Z)) hom((id u 2 ) (X Y ), (id u 2 ) Z) γ id,u 2 hom(x u 2Y, (id u 2 ) Z) hom(u 2Y, u 2 Hom l (X, Z)) γ Hom l id,u 2 hom(u 2Y, Hom l (X, (id u 2 ) Z)) Proof. The uniqueness of the natural transformations γ Hom l u : Hom 1,id l(, ) Hom l (u 1, (u 1 id) ) and γ Hom l id,u 2 : u 2 Hom(, ) Hom l (, (id u 2 ) ) follows by taking Y Hom l (X, Z) in either of the diagrams and tracing the adjunction counit through the diagrams. If we spell out what we obtain that way then we see that γ Hom l u 1,id is defined as the pasting u 1 X\ (u 1 id) E(K 2 ) F(J 1 K 2 ) F(K 1 K 2 ) u 1 X X E(K 2 ) E(K 2 ) X\ F(K 1 K 2 ) where X lives in D(K 1 ). Similarly, γ Hom l id,u 2 is obtained by pasting the following diagram X\ (id u 2) E(J 2 ) F(K 1 J 2 ) F(K 1 K 2 ) X X E(J 2 ) u 2 E(K 2 ) X\ F(K 1 K 2 ) in which X is an object of D(K 1 ). The compatibility of the calculus of mates with pasting (cf. [Gro11, Lemma 1.14]) and the triangular identities for adjunctions imply that the constructed natural transformations satisfy the coherence conditions of a lax natural transformation. Again, the laxness in the statement refers to the fact that the natural transformations which were constructed are, in general, not invertible. We will come back to this in the context of derivators. And, of course, there is a similar result for arbitrary levelwise chosen adjoints Hom r (, ).

13 MONOIDAL DERIVATORS 13 In this lemma, we were very precise and split the construction of the structure morphisms in the two cases γ Hom l u and 1,id γhom l id,u 2. The point is that these two natural transformations play significantly different roles in the case of adjunctions of two variables between derivators as we want to discuss next. Motivated by the notion of a left adjoint morphism between derivators we give the following definition. Definition A bimorphism : (D, E) F between derivators is a left adjoint of two variables if : D(J 1 ) E(J 2 ) F(J 1 J 2 ) is part of an adjunction of two variables for all J 1, J 2 Cat and if preserves homotopy left Kan extensions separately in each variable. A bimorphism : (D, E) F is levelwise part of an adjunction of two variables if and only if this is the case for the morphism : D E F obtained by Proposition 1.7. In fact, this is an immediate consequence of the following easy but very convenient lemma. Lemma Let C 1, C 2, and D be categories and let : C 1 C 2 D be part of an adjunction of two variables. If L: D D, L 1 : C 1 C 1, and L 2 : C 2 C 2 are left adjoint functors then the composition L (L 1 L 2 ): C 1 C 2 C 1 C 2 D D is also part of an adjunction of two variables. In the context of the above definition, Lemma 1.12 guarantees that associated to the bimorphism we can construct two lax transformations Hom l (, ) and Hom r (, ). Let us again focus on the case of Hom l (, ) but similar remarks apply to Hom r (, ). Using the explicit construction of γ Hom l id,u 2 in the proof of that lemma and also [Gro11, Lemma 1.14] we see that the following natural transformations are conjugate: γ Hom l id,u 2 : u 2 Hom l (, ) Hom l (, (id u 2 ) ) and γid,u 2! : (id u 2 )! (id u 2! ) In particular, our assumption that preserves homotopy left Kan extensions in the second variable is equivalent to the fact that the structure morphisms γ Hom l id,u 2 of Hom l (, ) are isomorphisms for all functors u 2 : J 2 K 2. Now, using a similar reasoning and again [Gro11, Lemma 1.14] we can deduce that the natural transformation γ Hom l u 1,id is conjugate to the following pasting (u 1 id)! F(K 1 K 2 ) F(J 1 K 2 ) E(K 2 ) (u 1 id) F(K 1 K 2 ) u 1 X X E(K 2 ) E(K 2 ) In particular, γ Hom l u is an isomorphism if and only if the above map (u 1,id 1 id)! (u 1X Y ) X Y is an isomorphism. But, in general, there is no reason for this map being an isomorphism. Thus, the structure maps γ Hom l u 1,id are, in general, not invertible (we will see an example for this phenomenon at the end of this subsection). Let us collect these observations in the following lemma. Lemma Let : (D, E) F be a bimorphism between derivators which is levelwise a left adjoint of two variables and let Hom l and Hom r be the lax transformation guaranteed by Lemma Then preserves homotopy left Kan extensions in the second variable if and only if the structure maps γ Hom l id,u 2 are invertible for all u 2 : J 2 K 2. Similarly, preserves homotopy left Kan extensions in the first variable if and only if the structure morphisms γ Homr u are invertible for all u 1,id 1 : J 1 K 1.

14 14 MORITZ GROTH We now turn to examples of adjunctions of two variables for derivators. Here, we will cover the examples of represented derivators. The case of Quillen adjunctions of two variables will be taken up in the next section. Let : C D E be a functor of two variables. We can extend to a (strict) bimorphism : (C, D) E of the associated represented prederivators. In fact, for a pair of categories (J 1, J 2 ) let us define J1,J 2 : C J1 D J2 E J1 J2 by sending a pair (X, Y ) to: X Y : J 1 J 2 X Y C D E Let us call this bimorphism the bimorphism represented by. Proposition Let C, D be complete categories, E a category, and : C D E a left adjoint of two variables. The represented bimorphism : (C, D) E is then levelwise divisible on both sides. In particular, adjunctions of two variables between bicomplete categories induce adjunctions of two variables between represented derivators. Proof. Let us content ourselves by showing that the represented bimorphism : (D, E) F is levelwise divisible on the left. Thus, we give the construction of Hom l (, ) and the natural isomorphism expressing one half of the fact that we have an adjunction of two variables. So, let us consider a pair of categories (J 1, J 2 ) and let us construct a right adjoint Hom l (, ): (C J1 ) op E J1 J2 D J2. Using (C J1 ) op (C op ) J op 1, as an intermediate step we can associate a pair (X, Z) to the functor Hom l (, ) (X Z): J op 1 J 1 J 2 C op E D. Here, Hom l : C op E D is a functor expressing the fact that is divisible on the left. Forming the end over the category J 1 we can define Hom l (X, Z): J 2 D by: Hom l (X, Z)( ) Hom l (X(j 1 ), Z(j 1, )) j 1 Let us check that this gives us the desired adjunction. For this purpose let us consider a functor Y D J2. Using the fact that natural transformations are obtained by a further end construction we can make the following calculation: hom E J 1 J 2 (X Y, Z) hom E (X(j 1 ) Y (j 2 ), Z(j 1, j 2 )) (j 1,j 2) ( hom D Y (j2 ), Hom l (X(j 1 ), Z(j 1, j 2 )) ) (j 1,j 2) ( hom D Y (j2 ), Hom l (X(j 1 ), Z(j 1, j 2 )) ) j 2 j 1 ( hom D Y (j2 ), Hom l (X, Z)(j 2 ) ) j 2 ( hom D J 2 Y, Homl (X, Z) ) The third isomorphism follows from the Fubini-type theorem for ends and the fact that corepresented functors are end preserving, the second one is the adjunction isomorphism at the level of categories, while the first and the last one are given by the fact that natural transformations can be expressed as ends. This concludes the proof that the bimorphism is levelwise divisible on both sides.

15 MONOIDAL DERIVATORS 15 The statement in the context of bicomplete categories can be proved in two ways. One way is to check that the bimorphism : (C, D) E is cocontinuous in both variables. By Lemma 1.15 we could equivalently show that the canonical map u 2 Hom l (X, Z) Hom l (X, (id u 2 ) Z) is an isomorphism. But this is true since ends with parameters are calculated pointwise. We use this example and the details of the proof to illustrate that the structure maps belonging to Hom l (, ) in Lemma 1.15 are not necessarily isomorphisms, i.e., that we only obtain lax natural transformations as opposed to pseudo-natural transformations. So, let us consider a functor u 1 : J 1 K 1, two diagrams X : K 1 C and Z : K 1 J 2 E and let us have a look at the diagram: ( k 1 Hom l X(k1 ), Z(k 1, ) ) j 1 Hom l ( X(u1 (j 1 )), Z(u 1 (j 1 ), ) ) pr j 1,j 1 pr u1 (j 1 ),u 1 (j 1 ) ( Hom l X(u1 (j 1)), Z(u 1 (j 1 ), ) ) The upper left object is Hom l (X, Z) and the lower left one is Hom l (u 1X, (u 1 id) Z). The solid morphisms belong to the universal wedges of the respective end constructions. By the universal property of the lower wedge there is a unique dashed arrow as indicated which is compatible with all projection morphisms and this dashed arrow gives us To give a specific example such that γ Hom l u 1,id γ Hom l u 1,id : Hom l(, ) Hom l (u 1( ), (u 1 id) ( )). is not invertible let us consider the following situation. Let C D E be Set, the category of sets, and let us take the adjunction of two variables given by the cartesian closedness of Set. Moreover, let J 1 J 2 e be the terminal category and let u 1 k 1 : e K 1 classify an object k 1 K 1. Then, X and Z are just functors K 1 Set and the natural transformation γ Hom l u 1,id evaluated at X and Z is the map nat(x, Z) hom Set(X(k 1 ), Z(k 1 )) which evaluates a natural transformation at k 1. Clearly, this map is not an isomorphism in general The closedness of the cartesian monoidal 2-category of derivators. Recall from classical category theory that given a category D there is the monoidal category of endofunctors of D. Moreover, this is the universal example of a monoidal category acting from the left on D. In this paper we will establish corresponding results for prederivators (or more generally for objects in an arbitrary closed monoidal 2-category). Here, let us first show that the 2-category PDer is cartesian closed in a bicategorical sense. Thus, given three prederivators D, D, and D we want to construct a prederivator HOM(D, D ) of morphisms and a natural equivalence of categories Hom(D D, D ) Hom(D, HOM(D, D )). In more formal terms, we are looking for a biadjunction (see [Gra74], [Fio06, Chapter 9] or Appendix B.2). Note that we have Hom(, ) PsNat(, ) in our situation. For a category J let us again denote the represented prederivator by y(j). If we now assume that we were given such a construction of an internal hom HOM(, ) then for an arbitrary category J we would deduce the following chain of natural equivalences of categories: HOM(D, D )(J) PsNat(y(J), HOM(D, D )) PsNat(y(J) D, D ) PsNat(D, HOM(y(J), D ))

16 16 MORITZ GROTH The equivalences are given by the bicategorical Yoneda lemma and the assumed closedness property. Hence one possible description of the internal hom would be given by the second line. Alternatively, taking the last line, we have reduced the problem of describing the internal hom to giving an identification of HOM(y(J), D ) for a category J and a prederivator D. By similar arguments and for a category K we obtain natural equivalences of categories as follows: HOM(y(J), D )(K) PsNat(y(K), HOM(y(J), D )) PsNat(y(K) y(j), D ) PsNat(y(J K), D ) D (J K) D J (K) Putting these chains of natural equivalences together we would obtain as an upshot the following equivalences motivating the next definition: HOM(D, D )(J) Hom(y(J) D, D ) Hom(D, D J ) Definition The prederivator hom HOM is the 2-functor HOM: PDer op PDer PDer which is adjoint to PDer op PDer Cat op id ( ) ( ) PDer op PDer Hom CAT. Given two prederivators D and D the prederivator HOM(D, D ) is called the prederivator of morphisms from D to D. Moreover, for a single prederivator D we set END(D) HOM(D, D) PDer and call this the prederivator of endomorphisms of D. For derivators D and D we define HOM(D, D ) and END(D) using the underlying prederivators. More explicitly, for two prederivators D, D, and a small category J we have thus HOM(D, D )(J) Hom(D, D J ) CAT. Our next aim is to show that the bifunctor HOM defines an internal hom in the bicategorical sense (cf. Appendix B.2) for the cartesian monoidal 2-category PDer. As a preparation for that result let us construct pseudo-natural transformations which will be used in order to define the adjunction. Lemma For D, D PDer there are canonical morphisms η : D HOM(D, D D ) and ɛ: HOM(D, D ) D D. Moreover, η resp. ɛ is pseudo-natural in D resp. D. Proof. Let us begin with the construction of η : D HOM(D, D D ). For a category K we thus have to construct a functor η K : D(K) Hom(D, (D D ) K ). For an arbitrary category J, let us define the component η K ( ) J : D(K) Fun(D (J), D(K J) D (K J)) to be adjoint to the functor (pr 1, pr 2): D(K) D (J) D(K J) D (K J), i.e., we set η K (X) J (Y ) (pr 1(X), pr 2(Y )) for X D(K) and Y D (J). For a functor u: J 1 J 2 we can use the diagram K pr 1 K J 1 id u K K J 2 pr 1 pr 2 J 1 u pr 2 J 2

17 MONOIDAL DERIVATORS 17 to convince ourselves that the maps {η K ( ) J } J assemble to define a strict morphism of prederivators η K ( ): D (D D ) K. A similar reasoning shows that also the {η K } K together define a morphism of prederivators η : D HOM(D, D D ). Let us show now that η is pseudo-natural in D. For this purpose, let F : D 1 D 2 be a morphism of prederivators, let K be a category and let us consider the following non-commutative diagram: D 1 (K) η K Hom(D, (D 1 D ) K ) F K (F id) K D 2 (K) η K Hom(D, (D 2 D ) K ) For X D 1 (K) and Y D (J) we can use the natural isomorphisms belonging to the morphism F to deduce the following one: ( (F id)k η ) K (X)J (Y ) (F id) K (pr 1(X), pr 2(Y )) (F K J pr 1(X), pr 2(Y )) (pr 1 F K (X), pr 2(Y )) (η K F K )(X) J (Y ) One checks that these isomorphisms can be used to obtain a pseudo-natural transformation η as intended. Let us now construct a morphism ɛ: HOM(D, D ) D D. Thus, for a category K we have to define a functor ɛ K : Hom(D, D K ) D(K) D (K). This is defined to be the following composition Hom(D, D K ) D(K) ɛ K D (K) pr id Fun(D(K), D K (K)) D(K) ev K D K (K) D (K K) i.e., for F Hom(D, D K) and X D(K) we set ɛ K (F, X) K F K(X). To see that these ɛ K assemble to define a morphism of prederivators let us consider a functor u: J K and let us check that there is a canonical natural isomorphism γ ɛ u as in: Hom(D, D K) D(K) ɛ K D (K) D u D(u) D(u) Hom(D, D J) D(J) ɛ J D (J) But evaluated at F Hom(D, D K) and X D(K) we can use the natural isomorphisms belonging to F to obtain D(u) ɛ K (F, X) u KF K (X) J(u id) (id u) F K (X) J(u id) F J (u X) J(D u F ) J(u X) ɛ J (D u D(u))(F, X).

18 18 MORITZ GROTH Thus, slightly sloppy we have (γu) ɛ F,X (γu F ) X. Again one checks that these isomorphisms assemble together to define a morphism of prederivators ɛ: HOM(D, D ) D D. In order to show that ɛ is pseudo-natural let us consider a morphism of prederivators G: D D and let us construct a natural isomorphism as in: HOM(D, D ) D G id ɛ D HOM(D, D ) D ɛ D Using the natural isomorphisms belonging to G and a similar calculation as above we obtain for F Hom(D, D K) and X D(K) the following isomorphism: (ɛ (G id)) K (F, X) KG K K F K (X) G K KF K (X) G (G ɛ) K (F, X) These isomorphisms give us the desired natural isomorphisms turning ɛ into a pseudo-natural transformation which concludes the proof. With this preparation we can now give the following desired result. Proposition The prederivator of morphisms defines an internal hom in the cartesian monoidal 2-category PDer, i.e., for prederivators D, D, and D we have pseudo-natural equivalences of categories: Hom PDer (D D, D ) Hom PDer (D, HOM(D, D )) Proof. We use the pseudo-natural transformations of the last lemma to define functors l and r by l η HOM(D, ) and r ɛ ( D ) as depicted in: Hom(D D, D ) ɛ Hom(D D, HOM(D, D ) D ) HOM(D, ) D Hom(HOM(D, D D ), HOM(D, D )) η Hom(D, HOM(D, D )) Let us check that these are inverse equivalences of categories and let us begin by showing that we have a natural isomorphism r l id. For this purpose, let us consider a morphism F : D D D. We claim that we have the following diagram which commutes up to a natural isomorphism: D D η 1 HOM(D, D D ) D F id HOM(D, D ) D id ɛ D D By the lemma we only have to check that the triangle commutes. But using the explicit formulas of the last proof we can calculate for X D(K) and Y D (K) the following: (ɛ (η 1))(X, Y ) ɛ(η(x), Y ) Kη K (X) K (Y ) K(pr 1 X, pr 2 Y ) (X, Y ) Since the longer boundary path from D D to D calculates r l(f ) we conclude r l id. F ɛ D

19 MONOIDAL DERIVATORS 19 Let us show that we also have l r id. Thus, let us consider a morphism G: D HOM(D, D ) and let us show that the following diagram commutes up to natural isomorphisms: HOM(D, D D ) HOM(D,G 1) HOM(D, HOM(D, D ) D ) η η HOM(D,ɛ) HOM(D, D ) D G HOM(D, D ) By the last lemma it remains to show that the triangle commutes up to a natural isomorphism. But for F Hom(D, D K) and Y D J we can again use the formulas of the last proof to make the following calculation: ((ɛ ) K η K )(F ) J (Y ) ɛ K J (pr 1 F, pr 2 Y ) K J(pr 1) K J F K J pr 2Y K J(pr 1 1 1) (1 pr 2 ) F J (Y ) F J (Y ). Here, we used the natural isomorphism belonging to F and the commutativity of the following diagram in which the composition of the bottom row is just the identity: id D K(J) pr 2 D K(K J) (pr 1 ) (K J) D K J(K J) D (K J) (1 pr2 ) D (K K J) (pr 1 1 1) D (K J K J) D (K J) K J It follows that the triangle in the previous diagram also commutes up to natural isomorphism. Again, the longer path passing through the boundary from D to HOM(D, D ) is l r(g) and we can thus deduce that we have a natural isomorphism l r id. This concludes the proof of the proposition. From classical category theory we know that a functor category C J is (co)complete as soon as this is the case for the target category C. The corresponding result for derivators also holds true as we will show now. Proposition If D is a prederivator and D a derivator then the prederivator HOM(D, D ) is a derivator which is pointed if D is. In particular, the 2-categories of derivators and pointed derivators are cartesian closed. Proof. The axiom (Der1) is immediate. For axiom (Der2), let us consider a map φ: F G in HOM(D, D )(K). Then φ is an isomorphism if and only if φ J : F J G J is an isomorphism in nat(d(j), D (K J)) for all categories J. The fact that isomorphisms in D are detected pointwise shows that this is equivalent to all (φ J ) k (φ k ) J being isomorphisms. Thus, φ is an isomorphism if and only if all φ k are isomorphisms. For axiom (Der3), let us consider a functor u: J K. By Lemma 1.4 u induces an adjunction (u!, u ): D J D K of derivators. Then, since Hom(D, ) preserves adjunctions, we obtain the intended adjunction (u!, u ): HOM(D, D )(J) HOM(D, D )(K). One proceeds similarly for homotopy right Kan extensions. For the base change axiom, let u: J K be a functor and k K an object. Then, we have to show that the base change morphism in the

20 20 MORITZ GROTH square on the right-hand-side induced by the natural transformation on the left-hand-side is an isomorphism: J /k J Hom(D, D J /k ) Hom(D, D J) e K Hom(D, D e) Hom(D, D K) Evaluation of this base change morphism is just given by postcomposition with the base change morphism belonging to D. But this one is an isomorphism because D is a derivator by assumption. Thus, this together with a dual reasoning for homotopy right Kan extensions implies (Der4) for HOM(D, D ). Since homotopy Kan extensions are calculated pointwise it follows that HOM(D, D ) is pointed resp. additive if this is the case for D. Remark In the case of a stable derivator D there is the following comment concerning the internal derivator hom HOM(D, D ). By the last proposition we know that this gives us a pointed (even additive) derivator. Moreover, since homotopy Kan extensions are calculated pointwise it follows that this derivator has the additional property that the classes of cartesian and cocartesian squares coincide and hence that the suspension functor is invertible. But it is not known yet whether the internal hom HOM(D, D ) is again strong, i.e., if the partial underlying diagram functors associated to the ordinal [1] (0 1) HOM(D, D )(J [1]) HOM(D, D )(J) [1] are full and essentially surjective. Thus, we cannot deduce that the 2-category Der ex of stable derivators is closed monoidal with respect to the cartesian structure. This is a certain drawback of the notion of a stable derivator. In fact, the notion of a stable derivator can be thought of as a minimal notion which guarantees that one can construct the canonical triangulated structures on all of its values and the induced functors. 2. Monoidal derivators 2.1. The 2-category of monoidal (pre)derivators. Emphasizing similarity to the fact that a monoidal category ([EK66] or [ML98]) is just a pseudomonoid object (called a pseudo-monoid in [DS97]) in the cartesian 2-category CAT, we could just say that a monoidal prederivator is a pseudomonoid object in the cartesian 2-category PDer. Although we want to make this slightly more precise we will be sketchy here (but see also Appendix B). Definition 2.1. Let E be a prederivator. A monoidal structure on E is a 5-tuple (, S, a, l, r) consisting of two morphisms of prederivators : E E E and S: e E and natural isomorphisms l, a, and r expressing associativity and unitality of. This structure has to satisfy the usual coherence conditions. A monoidal prederivator is a prederivator endowed with a monoidal structure. There are similar notions of symmetric or braided monoidal prederivators [ML98, Bor94b]. We will often denote a monoidal prederivator simply by (E,, S) or even by E. This definition gives us the internal variant of a monoidal prederivator. Using the concept of bimorphisms and the obvious generalizations to more arguments (which could be called trimorphisms etc.) we can equivalently consider monoidal prederivators in their external variant. Recall from Proposition 1.7