On the Tensor Product of Large Categories

Faculty of Sciences Mathematics

On the tensor product of large categories

Over het tensor product van grote categorieën

Thesis for the degree of Doctor of Science: Mathematics at the University of Antwerp to be defended by

Julia RAMOS GONZÁLEZ

Supervisors:

prof. dr. Wendy LOWEN

prof. dr. Boris SHOIKHET Antwerp, 2017 ii

COVERDESIGN: Original idea: Juan Ramos González Composition: Susana González Marín

Photography: Agustín Ramos Guerreira Contents

Acknowledgements vii

Introduction ix

Nederlandse samenvatting xv

1 Preliminary results1 1.1 Topos theory: On Grothendieck topoi...... 2 1.1.1 Linearized Grothendieck topologies...... 2 1.1.2 Linear sites and categories of sheaves...... 5 1.1.3 Gabriel-Popescu theorem...... 9 1.2 Locally presentable categories...... 11 1.3 Triangulated categories...... 13 1.3.1 Deﬁnition and axioms...... 14 1.3.2 Triangulated subcategories...... 16 1.3.3 Well-generated triangulated categories...... 18 1.3.4 Localization theory in well-generated triangulated categories 19 1.3.5 The inadequacy of triangulated categories...... 21 1.4 Dg categories...... 21 1.4.1 Basics on dg categories...... 22 1.4.2 Homotopy category of small dg categories...... 24

iii iv Contents

1.4.3 Pretriangulated dg categories...... 28 1.4.4 Quotient of dg categories...... 32

2 Tensor product of linear sites and Grothendieck categories 35 2.1 Tensor product of linear sites...... 39 2.1.1 The poset of linear sites...... 40 2.1.2 The poset of localizing subcategories...... 40 2.1.3 Equivalent approaches to localization...... 43 2.1.4 The tensor product topology...... 44 2.1.5 Tensor product of localizing subcategories...... 46 2.1.6 Tensor product of strict localizations...... 46 2.1.7 Relation between the three tensor products...... 47 2.1.8 Exact categories...... 48 2.2 Functoriality of the tensor product of linear sites...... 49 2.2.1 LC functors...... 50 2.2.2 Tensor product of functors...... 51 2.3 Tensor product of Grothendieck categories...... 56 2.3.1 Tensor product of Grothendieck categories...... 56

2.3.2 Tensor product of Z-algebras...... 58 2.3.3 Quasicoherent sheaves on projective schemes...... 61 2.4 Relation with other tensor products...... 62 2.4.1 Tensor product of locally presentable categories...... 63 2.4.2 Relation with Deligne’s tensor product...... 66 2.4.3 The α-Deligne tensor product...... 67 2.4.4 Relation with the tensor product of topoi...... 69

3 The tensor product of Grothendieck categories as a ﬁltered bicolimit 71 3.1 Generalities on the 2-ﬁltered bicolimit of categories...... 73 3.2 Locally presentable categories as bicolimits of small categories... 77 Contents v

3.3 The tensor product of Grothendieck categories as a ﬁltered bicolimit 78 3.4 The tensor product of Grothendieck categories: Functoriality, associativity and symmetry...... 83

4 Localization of sites with respect to LC morphisms 87 4.1 Bicategories of fractions...... 89 4.2 The 2-category of Grothendieck categories and the 2-category of sites 92 4.3 Bilocalization of the 2-category of sites with respect to LC morphisms 94 4.4 The 2-category of Grothendieck categories as a bilocalization of the 2-category of sites...... 105 4.5 Monoidal bilocalization...... 109

5 Tensor product of well-generated pretriangulated categories 111 5.1 Hom and tensor of dg categories...... 114 5.2 Quotient of cocomplete pretriangulated dg categories...... 120 5.3 Two variables setting...... 122 5.4 Well generated pretriangulated dg categories...... 123 5.5 Localization of well-generated dg categories...... 125 5.5.1 Localizing subcategories generated by a set...... 126 5.5.2 Strict localizations...... 127 5.5.3 Equivalent approaches to localization...... 128 5.6 Tensor products and quotients...... 129 5.7 Tensor product of well-generated dg categories...... 132 5.8 Localization theory of well-generated dg categories and tensor product...... 138 5.8.1 Tensor product of localizing subcategories...... 138 5.8.2 Tensor product of strict localizations...... 142 5.8.3 Relation between the tensor product in both approaches.. 143 5.9 Tensor product in terms of α-continuous derived categories..... 144 5.10 From small to large dg categories...... 145 vi Contents

5.10.1 Tensor product of homotopically α-cocomplete dg categories 146 5.10.2 Relation with the tensor product of well-generated dg categories...... 149

Bibliography 151 Acknowledgements

In ﬁrst place I would like to thank Wendy and Boris for giving me this opportunity. This thesis would not have been possible without you two. Thank you, Wendy, for your constant support, your guidance and generosity with time, for your comments, your careful corrections of this text and for transmitting your love for mathematics. Thank you, Boris, for your enthusiasm, for always providing a new perspective of the problems I have treated and for the nice discussions along these years. Thank you as well for your help while preparing the exercise sessions from Lie Groups, I learnt a lot from our discussions. I would also like to thank FWO for providing the founding of my scholarship. I would like express my gratitude to the members of the jury for their careful reading of the thesis, their feedback and their relevant and interesting questions. Bureaucracy is inevitably present in almost every job nowadays. I would like to show my gratitude to all the colleagues of the University of Antwerp who are in charge of the different types of administration related to a PhD. During these years I have found nothing but help and kind words from them, which has made dealing with paperwork a lot easier. I would like to thank my colleagues for the nice atmosphere at the department. I am very grateful to all colleagues with whom I had the pleasure to enjoy seminars, lunch and coffee breaks, card games and nice both mathematical and non-mathematical conversations. In particular, I would like to thank Pieter for many useful discussions, as well as for his active organization of learning and reading seminars. I would also like to thank Matthias for organizing the Gauge Theory seminar with me and for sharing his enthusiasm for the topic. My professors from the Mathematics Department at Universidad de Salamanca deserve a special mention. I am very grateful to Fernando, with whom I started my research career; to Esteban, from whom I learnt so many things during the work on

vii viii Acknowledgements my Master’s thesis; and to Ángel, without whom I would never had known about this opportunity to study my PhD in Belgium. In these sad times of cutting policies happening in many countries in Europe, and which I have unfortunately experienced in Spain, I would like to thank those who ﬁght for the high-quality public educative system every community deserves. This thesis would had not been possible without the help from many other people which, although not related with the research activity itself, have been essential for the ﬁnishing of this work. First of all, I would like to thank my parents for their constant support and help. Thank you for happily accepting my being far away so that I could do what I really enjoy doing. I would also like to thank my brother for his trust and his exceptional sense of humor. Thank you for your immense help. My tía Espe has been a constant help and support during these last four years, and my cousins Carmen and Pedro have been like a sister and a brother for me. Thank you all for your Skype calls and your visits, and thanks for sharing with me your joy of the little Carmen and Pablo. I know my grandfather Pedro and my tío Goyo, who both saw the beginning of this doctorate, would have been proud, and I know my tío Goyo, in the view of this thesis, would not have missed the chance to discuss and tease me about more or less arguable terminology that we mathematicians use. I would also like to thank my tías Marisol, Pili and specially Marola, who has always been so caring during these years, and my tío Manolo, who came together with her to visit. My mathematical part of the family also deserves my gratitude. Thanks Ana, Cristina and tío Ángel for your advice. I am also very grateful to my friends in Spain. Thank you for your visits to Antwerp, our meetings when I am back and your care from the distance. You have made the whole process a lot easier. In particular, I would like to thank Paula, Alba, Marina and Jorge for their immense support. I would also like to thank the friends I have had the fortune to make in Belgium: the climbers, the cinema-goers, the pintje-drinkers, the Dutch-learners, the board game-players and all those with whom I have enjoyed many great times. I would like to specially thank Balansstraat folks for making of their home my second home. Last but not least, I would like to express my gratitude to Dries. Thanks for your patience during the hard moments, for believing in me and keeping my feet on the ground, and for having made the last stressful period of the thesis a way easier. Introduction

Category theory is of fundamental importance in algebraic geometry. In particular, its role is manifest in noncommutative algebraic geometry, where one of the approaches to the theory consists of considering certain types of categories as models for noncommutative spaces. The aim of this thesis is to define and analyse a suitable notion of tensor product for certain large categories of importance in noncommutative algebraic geometry in order to provide a generalization of the product of schemes to the noncommutative setting. With the geometrical picture as our beacon throughout the whole text, the mathematical content of this thesis belongs, in many of its parts, to category theory. That is why we would like to start by briefly sketching the role played by category theory in noncommutative algebraic geometry, and by this, making more precise the first two sentences of this very introduction. Category theory is one of the most abstract branches of mathematics. It provides a conceptual language that allows to analyse and study different mathematical structures and their interrelations. Since the notions of category, functor and natural transformation were introduced by Eilenberg and Mac Lane in 1945 [25], category theory has had a great and fast developement both as a theory of its own and in its scope of applications. The versatility and generality of the language of categories make of it a very useful tool in order to approach problems in many different areas, such as algebraic geometry, topology, analysis, probability theory... and even be- yond mathematics, with applications in theoretical computer science, mathematical physics, linguistics... One of the main ideas at the core of category theory is that the main role is now played by arrows. Mac Lane, referring to category theory, remarks in [58, p. vii] the following:

Since a category consists of arrows, our subject could also be described as learning how to live without elements, using arrows instead.

ix x Introduction

In other words, in category theory one studies a certain mathematical object by its relations with others, i.e. by the arrows that map to and from it, instead of looking at the “elements that constitute the object”. As a consequence, the set-theoretical perspective gets difuminated in the picture. This has lead to foundational research, started by Lawvere, proposing category theory, instead of set theory, as foundation for mathematics. On the other hand, this idea has also played an especially important role in particular branches of mathematics, for instance in algebraic geometry. Classically, geometrical objects had been studied via the sets of points that “con- stituded” them. However, since the 1950’s and more particularly since the work of Grothendieck’s school, with the categorical philosophy at hand, algebraic geometers started to study and analyse the properties of geometrical objects by considering them as elements of a category of geometrical objects and observing their relations with other objects in this category via the arrows or morphisms. One of the key principles in this “categorical” perspective of algebraic geometry, is that the geometry of a scheme X can be completely recovered from its category of quasi-coherent sheaves Qch(X ), which is a Grothendieck abelian category. On the other hand, if one is also interested in higher cohomological information, then the key category to be understood is the derived category of the quasi-coherent sheaves D(Qch(X )), which is a well-generated triangulated category. Note that both Qch(X ) and D(Qch(X )) are large categories. This principle is at the heart to an approach to noncommutative algebraic geometry. We can provide a couple of motivational examples: Given an afﬁne scheme SpA, where A is a commutative ring, one would like to replace it with “something” that also works for noncommutative rings. Such a thing as the spectrum of a noncommutative ring is no longer at our disposal. However, if we consider the category of quasi-coherent sheaves, we have

Qch(A) Mod(A), ∼= and the category of (right) modules Mod(A) of a noncommutative ring A does make perfect sense.

Similarly, Serre’s theorem [71] shows that for a nice graded ring A, we have

Qch(Proj(A)) Qgr(A), ∼= where Qgr(A) is the quotient of the category of graded modules Gr(A) modulo the category of torsion modules Tors(A), both of which can be also deﬁned for noncommutative rings. Introduction xi

Following this philosophy, noncommutative algebraic geometry considers spaces to be modelled by categories that look like (derived) categories of quasi-coherent sheaves. The categories in the title of the thesis are thought of as models for noncommutative spaces. In particular, we will focus on models given by abelian categories such as in the work of Artin, Stafford, Van den Bergh and others; and models given by dg enhancements of triangulated categories, such as in the work of Bondal, Kapranov, Kontsevich, Toën and others. In classical algebraic geometry, a very natural operation that can be performed with schemes is taking their product. The goal of this thesis is the developement of a good notion of tensor product of noncommutative spaces, such that we recover the product of schemes from the classical setting. We deﬁne and construct a tensor product of both Grothendieck abelian categories and pretriangulated dg categories with well-generated homotopy category by making essential use of the Gabriel-Popescu type theorems that exist in both settings. From the Gabriel-Popescu theorem [65], we know that Grothendieck abelian categories are precisely the categories of sheaves on linear sites. On the other hand, thanks to the Gabriel-Popescu theorem for triangulated categories [66] we know that pretriangulated dg categories with well-generated homotopy category are precisely the localizations of dg derived categories of small dg categories with respect to localizing categories generated by a set. These representations are the starting point for the construction of both tensor products. The structure of the thesis is as follows. In Chapter1 we give an overview of the basic theory and properties of the main types of categories that will be used throughout the thesis:

Grothendieck abelian categories; • Locally presentable categories; • Triangulated categories; • Differential graded categories. • The material presented in this chapter is expository and well-known, but serves as a summary of the necessary results that are used in the rest of the text. The thesis can then be divided in two main parts, corresponding to the two different models for noncommutative spaces considered, Grothendieck abelian categories and pretriangulated dg categories. xii Introduction

Grothendieck abelian categories

This part is treated in Chapters2 to4. Chapter2 is devoted to the deﬁnition and construction of a suitable tensor product of Grothendieck categories in terms of their representations as categories of linear sheaves, provided by the generalized Gabriel-Popescu theorem [51]. The tensor product is deﬁned via the construction of an underlying tensor product of linear sites, of which we analyse its functoriality using a certain type of morphisms between sites, called LC morphisms, that induce equivalences between the sheaf categories. Then, we describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes X , Y , we have

Qch(X ) Qch(Y ) = Qch(X Y ), ∼ × which proves that our tensor product generalizes the product of projective schemes to the noncommutative setting. We additionally analyse the relation of the constructed tensor product with other well-known tensor products in the literature. More concretely, we show that our tensor product is a special case of the tensor product of locally presentable linear categories, and hence we can also relate it to Kelly’s tensor product of α-cocomplete small categories [42, 43]. We can additionally relate it to an α-version of Deligne’s tensor product of abelian categories from [21]. In Chapter3, we take the analysis of the relation between the tensor product of Grothendieck categories and the tensor product of α-cocomplete small categories one step further. We prove that the tensor product of Grothendieck categories can be represented as a ﬁltered bicolimit [23] of Kelly’s tensor product of the underlying α-cocomplete categories of α-presented objects. In particular, we explicitely describe the functoriality of the tensor product of Grothendieck categories by translating the functoriality of Kelly’s tensor product via this bicolimit construction. On the other hand, the analysis of the functoriality of the tensor product of linear sites in terms of LC morphsims carried out in Chapter2 naturally leads to consider the possibility of recovering the category of Grothendieck categories in terms of a localization of the category of linear sites with respect to LC morphisms. This is investigated in Chapter4 using the bicategories of fractions from [68]. Introduction xiii

Well-generated pretriangulated categories

This part is treated in Chapter5. Our objects of study are pretriangulated dg categories with well-generated homotopy category, which we refer to as well-generated pretriangulated categories. Among the geometrical examples of relevance, one finds the enhancements of derived categories of Grothendieck categories. Based on the homotopy theory of dg categories developed by Tabuada and Toën, we define a tensor product of well-generated pretriangulated categories in terms of a universal property. Then, by means of Porta’s Gabriel-Popescu type theorem for triangulated categories and Toën’s derived Morita theory, we show that the tensor product exists and we provide a construction. We then provide a construction of the tensor product of well-generated pretriangulated categories in terms of the two approaches to localization theory of (dg enhancements of) triangulated categories at our disposal, namely (enhanced) Bous- field and Verdier localization.

Considering the α-continuous derived category of a small dg category [66], one observes that homotopically α-cocomplete small dg categories are to well-generated pretriangulated categories what α-cocomplete small categories are to locally presentable categories. We proceed to define the tensor product of homotopically α-cocomplete small dg categories and prove its existence by giving a concrete construction. To finish the chapter, we provide the relation of this latter tensor product with the tensor product of well-generated pretriangulated categories we have constructed. We expect the tensor product of well-generated pretriangulated categories to be compatible with Lurie’s tensor product of presentable infinity categories [57]. Since infinity categories are outside the scope of this text, this will be addressed elsewhere. In contrast, the relation between the tensor product of Grothendieck categories from Chapter2 and the tensor product of well-generated pretriangulated categories from Chapter5 is quite subtle due to flatness issues and it is topic of current investi- gations. xiv Introduction Nederlandse samenvatting

Het product van schema’s is een essentiële operatie in de klassieke algebraïsche meetkunde. Het doel van deze thesis is een product van niet-commutatieve ruimten te definiëren dat het product van schema’s veralgemeent. Een mogelijke benadering van niet-commutatieve algebraïsche meetkunde bestaat uit het beschouwen van bepaalde types van categorieën als modellen voor niet- commutatieve ruimten. In deze tekst maken we gebruik van deze benadering. We focussen op twee verschillende modellen: abelse Grothendieck categorieën en well- generated dg categorieën (pre-getrianguleerde dg categorieën met well-generated homotopiecategorie). Ons doel is van een geschikt tensor product te definiëren, zowel van Grothendieck categorieën als van well-generated dg categorieën, waardoor het product van schema’s veralgemeend wordt. Gabriel-Popescu type stellingen zijn ons belangrijkste middel om deze twee tensor producten te definiëren. De structuur van de thesis is als volgt. In hoofdstuk 1 geven we een overzicht van de fundamentele theorie en eigen- schapen van de verschillende type’s van categorieën die een belangrijke rol spelen in de rest van de thesis: Grothendieck categorieën, lokaal-gepresenteerde categorieën, getrianguleerde categorieën en dg categorieën. Het materiaal in dit hoofdstuk is bekend en dient als een samenvatting van de nodige resultaten die in de rest van de tekst gebruikt worden. De rest van de thesis is verdeeld in twee onderdelen, overeenkomstig met de twee modellen die beschowd worden, namelijk Grothendieck categorieën en well- generated dg categorieën. In de abelse context stelt de klassieke Gabriel-Popescu stelling dat Grothendieck categorieën precies de categorieën van schoven over lineaire sites zijn. We definiëren in hoofdstuk 2 een tensor product van lineaire sites en bewijzen dat het een goed- gedefinieerd tensor product van Grothendieck categorieën induceert. Verder to-

xv xvi Nederlandse samenvatting nen we dat dit tensor product het product van projectieve schema’s veralgemeent. Bovendien vergelijken we het met andere bekende tensor producten uit de liter- atuur, waaronder het tensor product van lokaal-gepresenteerde categorieën en het Deligne tensor product van abelse categorieën. Ten slotte analyseren we de monoï- dale structuur geïnduceerd door dit tensor product in de categorie van Grothendieck categorieën via twee verschillende benaderingen: enerzijds via geﬁlterde bicolimi- eten (hoofdstuk 3) en anderzijds via breukenbicategorieën (hoofdstuk 4). In de dg context toont een verrijking van de getrianguleerde Gabriel-Popescu stelling van Porta dat well-generated dg categorieën precies de quotiënten van afgeleide dg categorieën zijn door lokaliserende deelcategorieën die door een verza- meling voortgebracht zijn (op het niveau van de homotopiecategorie). Deze rea- lisaties laten ons toe het tensor product via een universele eigenschap binnen de homotopiecategorie van well-generated dg categorieën te deﬁniëren en ook een constructie te geven. Dit wordt in hoofdstuk 5 gedaan. CHAPTER 1

Preliminary results

Our aim in this thesis is the deﬁnition, construction and study of the notion of tensor product for certain large categories in relation with geometry. Different types of categories come into play along the text. A brief account on the basic theory of each of them seemed therefore convenient for the sake of completeness and readability. The material in this chapter is not new. We refer in each of the sections to the main references used. Four different types of categories will play an important role in the development of the thesis. These are:

Grothendieck abelian categories • Locally presentable categories • Triangulated categories • Dg categories • In particular, ﬁxed k a commutative ground ring, we are interested in the k-linear versions of the categories above. In other words, we are interested in working in a categorical Mod(k)-enriched setting. Enrichment over Mod(k) is very well behaved because the forgetful functor Mod(k) Set is conservative (i.e. is isomorphism- reﬂecting), and hence the intuition from−→ classical category theory works usually without further complications in the Mod(k)-enriched setting. Nevertheless, it is convenient for the reader to consider all the notions we treat as k-linear, unless otherwise stated. The structure of this chapter is as follows. In section 1.1 we introduce the basic notions and results of linear topos theory, as a geometrical approach to Grothendieck

1 2 Chapter 1. Preliminary results categories. In section 1.2 we give a brief account on locally presentable categories. In section 1.3 we provide a summary of the basic theory of triangulated categories with the focus on well-generated ones and their localization theory. Finally, in section 1.4, we give an overview about dg categories and their homotopy theory.

1.1 Topos theory: On Grothendieck topoi

As explained for example in [51], Grothendieck abelian categories are precisely the topoi over linear sites. In this section we discuss the linear counterparts to the classical notions of Grothendieck topology, site and topos. For the classical setting, we refer the reader to the original source [76]. A nice account on topos theory can also be found in [9] or [35], among some others. The main references for the linear setting are [10] and [52].

1.1.1 Linearized Grothendieck topologies

The material from this section is based on [52, §2] where the classical notion of Grothendieck topology from [76] is treated in the k-linear setting. We also refer to [10, §1] for the notion of Grothendieck topology in more general enriched category theory.

Let a be a k-linear category. We will denote by Mod(a) the k-linear category op op Funk (a ,Mod(k)) of k-linear functors from a to Mod(k). Elements in Mod(a) are called a-modules, or just modules.

Deﬁnition 1.1. Let A a.A sieve on A is a subobject R of the representable module a( , A) on A in the category∈ Mod(a). − This means that R A is a k-submodule of a A , A for each A a and that for any ( 0) ( 0 ) 0 f R A and for any morphism g a A , A , one has that f g ∈a A , A belongs to ( 0) ( 00 0) ( 00 ) R ∈A . ∈ ◦ ∈ ( 00)

Definition 1.2. Given F = (fi : Ai A)i I a family of morphisms in a, the sieve → ∈ generated by F is the smallest sieve R on A such that fi R(Ai ) for all i I . We denote it by F f . One can esily see that, for every A a∈, F A consists∈ of the family = i i I 0 ( 0) of all〈 finite〉 〈 sums〉 ∈ of morphisms g : A A that factor∈ through〈 〉 some f of F . 0 i −→ Definition 1.3. A cover system on a consists of providing for each A a a family of sieves (A) on A. The sieves inR a cover system are called covering sieves∈ or simply R R covers (for ). We will say that a family (fi : Ai A)i I is a cover, or a covering family, ∈ if the sieveRfi i I it generates is a cover. −→ 〈 〉 ∈ Chapter 1. Preliminary results 3

Deﬁnition 1.4. Given R a sieve on A a and g : A A a morphism in a, the pullback 0 of R along g , denoted g 1R, is the sieve∈ on A obtained→ as the pullback − 0

g 1R a , A − ( 0) − g (1.1) ◦− R a( , A) − in Mod a . One can easily see that g 1R is given as follows: ( ) −

1 g − R(A00) = f : A00 A0 g f R(A00) (1.2) { → | ◦ ∈ } for all A a. 00 ∈ Deﬁnition 1.5. Let R be a sieve on A and for every morphism f : A f A in R(A) −→ consider a sieve R f on A f . We deﬁne the glueing of the sieves (R f )f R along the sieve R to be the sieve ∈

R (R f )f R = g f f R,g R f . (1.3) ◦ ∈ 〈 ◦ 〉 ∈ ∈ A cover system is closed under glueings if given R (A) and a collection of R ∈ R R f (A f ) for each f : A f A in R(A), the glueing R (R f )f R belongs to (A). ∈ R −→ ◦ ∈ R We have already introduced all the necessary notions in order to deﬁne a (k-linear) Grothendieck topology.

Deﬁnition 1.6. A cover system on a is localizing if it satisﬁes the following: T

(Id) Identity axiom: Given any object A a, the sieve IdA is a cover for , i.e. a( , A) (A) for all A a; ∈ 〈 〉 T − ∈ T ∈ (Pb) Pullback axiom: Given a covering sieve R A and g : A A a morphism ( ) 0 in a, the pullback sieve g 1R is also a covering⊆ T sieve for .→ − T If moreover also satisﬁes the following: T (Glue) Glueing axiom: Let R be a sieve on A. If there exists a sieve S on A such that for all morphism g : A A in S the pullback sieve g 1R A , then 0 − ( 0) R (A); → ∈ T ∈ T we say is a Grothendieck topology. T 4 Chapter 1. Preliminary results

Observe that the axioms of a Grothendieck topology above are a direct analogue in the k-linear setting of the axioms of a Grothendieck topology in the classical setting as defined in [76]. Remark 1.7. Notice that the glueing axiom as stated above coincides with the one in [10, Def 1.2. Axiom (T3)] or with the property of being closed under glueings described in [52, §2.2 Axiom (Glue’)], but is stronger than the glueing axiom described in [52, §2.2. Axiom (Glue)]. Hence, a priori, the notion of Grothendieck topology above and in [10] is stronger than the one in [52]. However, if (Id) and (Pb) hold for a cover system , it is then easy to see that both versions of the axiom become equivalent, and henceR so do both definitions of Grothendieck topology. Our choice for this version of the glueing axiom is based in our preference to keep the definition of Grothendieck topology in the linear setting as parallel as possible to the definition on the classical setting.

The following properties of a cover system in a are also interesting. T (Up) We say that a cover system is upclosed if for any R (A), we have that R S a( , A) implies that ST (A). ∈ R ⊆ ⊆ − ∈ R (Int) We say that is closed under intersections if given two sieves R,S (A), the pullbackT ∈ R R S S ∩ (1.4)

R a( , A) − in Mod(a) belongs to (A) as well. T In particular, one can show that Grothendieck topologies satisfy these last two properties as well.

Proposition 1.8. Let be a cover system on a. Then the following hold: T 1. If satisﬁes (Id) and (Glue), then it satisﬁes (Up). In particular, this implies thatT every Grothendieck topology is upclosed.

2. If is a Grothendieck topology, then satisfies (Int). T T Definition 1.9. Given a k-linear category a, and two Grothendieck topologies and on a, we say that is finer than , or is coarser than , if for all A a andT all 0 0 0 RT A , R A .T T T T ∈ 0( ) ( ) ∈ T ∈ T Chapter 1. Preliminary results 5

Given a localizing cover system on a, we will be interested in constructing the coarsest Grothendieck topology thatR contains it. This construction can be performed as a combination of a closure under glueings and property (Up) and it is analogous to the construction performed in the classical setting (see for example [9, §3.2]). Definition 1.10. Let be a cover system on a. The upclosure of is a cover system up on a defined as follows:R R R up (A) = R a( , A) S (A) with S R (1.5) R { ⊆ − | ∃ ∈ R ⊆ } Observe that up is the smallest upclosed cover system containing . R R Definition 1.11. Let be a cover system on a. We define the cover system + on a as follows: R R + A R R R A ,R A (1.6) ( ) = ( f )(f :A f A) R ( ) f ( f ) R { ◦ → ∈ | ∈ R ∈ R } for A a. ∈ Let be a cover system on a. Consider the following transfinite construction process:R

up Take 0 = ; • R R + up For every successor ordinal λ, define λ+1 = ( λ ) ; • R R S For every limit ordinal γ, define γ = λ<γ λ. • R R This transfinite sequence stabilizes and we denote by upglue the cover system on a obtained as the stable value. R

Proposition 1.12. Let be a cover system on a. Then upglue is the smallest upclosed cover system closed underR glueings that contains . R R Moreover, if is localizing, then upglue is a Grothendieck topology and it is the coarsest one containingR . R R

1.1.2 Linear sites and categories of sheaves

In this section we introduce the linear counterparts of Grothendieck site and topos as in [10] and [52]. We further introduce the linear notions of continuous and cocontinuous morphisms between sites and provide their main properties. The main source for this last part is [76, Exposé iii] and we present its linear counterpart. 6 Chapter 1. Preliminary results

Let a be a k-linear category endowed with a localizing system . As in the classical setting, we can deﬁne the linear versions of presheaves, separatedT presheaves and sheaves over (a, ). T Deﬁnition 1.13.

A presheaf F on (a, ) is simply an a-module, i.e. F is an object in Mod(a). • T A separated presheaf F on (a, ) is a presheaf such that the restriction functor • T F (A) = Mod(a)(a( , A), F ) Mod(a)(R, F ) (1.7) ∼ − −→ is a monomorphism for all A a and all R (A). ∈ ∈ T We denote by Sep(a, ) Mod(a) the full subcategory of separated linear presheaves. T ⊆

A sheaf F on (a, ) is a presheaf such that the restriction functor • T F (A) = Mod(a)(a( , A), F ) Mod(a)(R, F ) (1.8) ∼ − −→ is an isomorphism for all A a and all R (A). ∈ ∈ T We denote by Sh(a, ) Sep(a, ) Mod(a) the full subcategory of linear sheaves. T ⊆ T ⊆

Deﬁnition 1.14. A linear site is a pair (a, ) where a is a k-linear category and is a Grothendieck topology on a. T T

Remark 1.15. It is important to notice that we do not require our (linear) topologies to come induced by (linear) pretopologies (see [76, Exposé ii, §1]). In order to work in the full generality that we need, we are not assuming that our sites come equipped with pullbacks. This is for example essential for geometrical purposes: Z-algebras, which in general do not have pullbacks, are used extensively in noncommutative geometry and endowed with relevant topologies (see for example [20]).

Given a localizing system , recall that upglue is the coarsest topology containing . The following shows thatT when we haveT a category of sheaves we can always safely assumeT it is a category of sheaves on a linear site.

Theorem 1.16. Given a k-linear category endowed with a localizing system , we upglue have that Sh(a, ) = Sh(a, ). T T T Given a k-linear functor φ : a b between two k-linear categories a and b, we have the following induced functors−→ between their module categories: Chapter 1. Preliminary results 7

: Mod b Mod a : F F ; φ∗ ( ) ( ) φ • −→ 7−→ ◦ Its left adjoint, denoted by φ! : Mod(a) Mod(b); • −→ Its right adjoint, denoted by φ : Mod(a) Mod(b). • ∗ −→ Deﬁnition 1.17. Consider k-linear categories a and b endowed with localizing cover systems a, b respectively. A k-linear functor φ : a b is continuous, if any of the followingT equivalentT properties hold: −→

1. The functor : Mod b Mod a : F F preserves sheaves; φ∗ ( ) ( ) φ −→ 7−→ ◦ 2. There exists a functor φs : Sh(b, b) Sh(a, a) such that the diagram T −→ T

φ∗ Mod(a) Mod(b)

ia ib

Sh(a, a) Sh(b, b) T φs T

commutes, where ia : Sh(a, a) , Mod(a) and ia : Sh(b, b) , Mod(b) denote the respective natural embeddingsT −→ of the category of sheavesT −→ in the category of modules;

s 3. There exists a colimit preserving functor φ : Sh(a, a) Sh(b, b) such that the diagram T −→ T φ a b

Ya Yb

φ! Mod(a) Mod(b)

#a #b φs Sh(a, a) Sh(b, b) T T commutes, where Ya : a , Mod(a), Yb : b , Mod(b) are the corresponding −→ −→ Yoneda embeddings, and #a : Mod(a) Sh(a, a), #b : Mod(b) Sh(b, b) denote the respective sheaﬁﬁcation functors−→ (seeT Theorem 1.24 below).−→ T

s In addition, if any of the previous properties holds, we have necessarily that φ φs s a and φ = #b φ! ia. ∼ ◦ ◦ Deﬁnition 1.18. Consider k-linear categories a and b endowed with localizing cover systems a, b respectively. With the notations above, a k-linear functor φ : a b is cocontinuousT T , if any of the following equivalent properties hold: −→ 8 Chapter 1. Preliminary results

1. For all object A a and all covering sieve R b(φ(A)), there exists a covering ∈ ∈ T sieve S a(A) with φS R. ∈ T ⊆ 2. The functor φ : Mod(a, a) Mod(b, b) preserves sheaves. ∗ T −→ T In addition, if any of previous properties holds we have:

1. The functor # i is colimit preserving and exact and the diagram φe∗ = a φ∗ b ◦ ◦

φ∗ Mod(a) Mod(b)

#a #b

Sh(a, a) Sh(b, b) φ T e∗ T

is commutative up to canonical isomorphism.

2. There exists a functor φe : Sh(a, a) Sh(b, b) such that the diagram ∗ T −→ T φ Mod(a, a) ∗ Mod(b, b) T T ia ib

Sh(a, a) Sh(b, b) φe T ∗ T

commutes up to canonical isomorphism and is an adjoint pair. φe∗ φe a ∗ Remark 1.19. In [52] the term cover continuous is used for what we call here cocontinuous.

Deﬁnition 1.20. Given a k-linear category a, a Grothendieck topology on a is said to be subcanonical if all the representable presheaves are sheaves for T . The ﬁnest Grothendieck topology on a for which all representable presheaves areT sheaves is called the canonical topology on a.

Deﬁnition 1.21. A k-linear category C is called a linear Grothendieck topos if it there exists a site (a, a) such that T C = Sh(a, a) (1.9) ∼ T as k-linear categories. Chapter 1. Preliminary results 9

1.1.3 Gabriel-Popescu theorem

In this section, we state Gabriel-Popescu theorem [65] and the main result in [10], which combined show that linear topoi are precisely the Grothendieck categories. This statement is a linear counterpart of the classical Giraud Theorem that characterizes Grothendieck topoi in the classical setting. We further state the generalization of Gabriel-Popescu theorem from [51], which characterizes the the linear functors a C from a small linear category to a topos which induce an equivalence −→ C = Sh(a, a) Mod(a), where a is a certain topology on a. ∼ T ⊆ T A k-linear Grothendieck abelian category C is a cocomplete abelian k-linear category with a generator and exact ﬁltered colimits. Given a a small k-linear category, it is a well-known result that Mod(a) is a Grothendieck category. Recall that a fully- faithful functor i : A , B is a localization if it has a left adjoint that commutes with ﬁnite limits. −→

Theorem 1.22 (Gabriel-Popescu). Given a k-linear category C, the following are equivalent:

1. C is a Grothendieck category;

2. C is a localization of a category of presheaves.

Remark 1.23. Given a Grothendieck category C, in order to construct such a localization, one takes a small k-linear full subcategory a C with objects given by a set of generators of C. Then the fully faithful functor a , ⊆ C, gives rise to a localization C Mod(a) : C C( ,C ). −→ −→ 7−→ − The following theorem is part of the main result in [10]. It proves that linear Grothendieck topoi are precisely the localizations of presheaf categories of linear categories.

Theorem 1.24. Consider (a, a) a site. Then the functor i : Sh(a, a) , Mod(a) is a T T −→ localization functor and the exact left adjoint #: Mod(a) Sh(a, a) is called the sheaﬁﬁcation functor. −→ T

Conversely, given a localization i : C , Mod(a) for a k-linear category a with exact left adjoint #: Mod(a) C, there exists−→ a unique topology on a such that i induces an equivalence C Sh−→(a, ). The topology is such thatT R (A) if and only if ∼= #(R) a( , A) is an isomorphismT in C. T ∈ T ⊆ − Hence, from the combination of Theorem 1.22 and Theorem 1.24 above it follows that Grothendieck categories are precisely the linear topoi. 10 Chapter 1. Preliminary results

Theorem 1.25 (Linear Giraud theorem). Let C be a k-linear category. The following are equivalent:

1. C is a linear Grothendieck topos;

2. C is a Grothendieck category.

Observe, nevertheless, that the classical Gabriel Popescu theorem does not provide us with all the possible realizations of Grothendieck categories as localizations of categories of presheaves. Such result is provided by the generalization of Gabriel- Popescu theorem in [51]: it characterizes the linear functors u : a C such that C Mod(a) : C C(u( ),C ) gives a localization (observe that−→ in the Gabriel- Popescu−→ theorem only7−→ fully− faithful functors u where considered, see Remark 1.23 above). In order to state this characterization precisely, we ﬁrst introduce some necessary notions. Consider u : a C a k-linear functor from a small k-linear category a to a Grothendieck category−→ C.

Deﬁnition 1.26. We say that a family of morphisms (Ai A)i I in a is epimorphic ` −→ ∈ with respect to u if the induced morphism i I u(Ai ) u(A) is an epimorphism in C. ∈ −→

Consider such a functor u : a C. −→

(G) We say u satisfies (G) if the family of objects (u(A))A a generates C; ∈ (F) We say u satisfies (F) if for every morphism c : u A u A , there exists ( ) ( 0) −→ an epimorphic collection (fi : Ai A)i I in a and a family of morphisms g : A A such that c u f −→u g ∈; ( i i 0)i I ( i ) = ( i ) −→ ∈ ◦ (FF) We say u satisfies (FF) if for each morphism f : A A such that u f 0, 0 ( ) = −→ there exists an epimorphic family (Ai A)i I such that f fi = 0. −→ ∈ ◦ Theorem 1.27 (General Gabriel-Popescu theorem). Let u : a C be a k-linear functor from a small k-linear category a to a Grothendieck category−→ C. Then, the following are equivalent:

1. The induced functor C Mod(a) : C C( ,C ) is a localization; −→ 7−→ − 2. The functor u satisﬁes (G),(F) and (FF). Chapter 1. Preliminary results 11

Remark 1.28. Observe that if C Mod(a) : C C( ,C ) is a localization, applying Theorem 1.24 above, there exits−→ a unique Grothendieck7−→ − topology in a such that the essential image of C is equivalent to Sh a, . One can show thatT u 1 , where ( ) = − C T T T C is the topology on C of jointly epimorphic sieves, that is R C(C ) if and only if TL C C is an epimorphism. ∈ T (f :C f C ) R f → ∈ −→

1.2 Locally presentable categories

Locally presentable categories are present in many fields of mathematics. Among the important examples we find varieties and quasi-varieties of algebras, categories of presheaves, the category Set of sets, the category SSet of simplicial sets and many more. They also enjoy a very nice set of properties and features and they are closed under certain categorical constructions such as limits or comma-categories. In particular, of great relevance for our work is the fact that Grothendieck categories are locally presentable categories (see, for example, [9, Prop 3.4.16]). In this section we give a short account on locally presentable categories. We refer the reader to the classical [27] and the nice monograph [1] for the results provided below. In both references the notion of locally presentable categories is considered in a classical non-enriched setting. Further, we refer the reader to [43] for a treatment of the notion of locally presentable categories in enriched category theory. As already mentioned in the introduction of this chapter, we are interested in locally presentable categories in the k-linear setting. In the case of Mod(k)-enriched categories, the classical and the enriched notion of locally presentable category coincide. Nevertheless, we remind the reader that it is essential to understand all constructions in a k-linear sense. Definition 1.29. Let α be a regular cardinal. Given C a k-linear category, we say that an object C C is α-presentable if the k-linear functor ∈ C(C , ) : C Mod(k) (1.10) − −→ preserves α-filtered colimits. Remark 1.30. Observe that if an object C C is α-presentable for some regular cardinal α, then it is β-presentable for all∈ regular cardinals β α. This follows directly from the fact that β-filtered colimits are α-filtered for all≥ regular cardinal β α. ≥ Definition 1.31. Given a regular cardinal α, a k-linear category C is called locally α-presentable if it is cocomplete and has a set G of α-presentable objects such that every object C C is an α-filtered colimit of objects in G. ∈ 12 Chapter 1. Preliminary results

We say C is locally presentable if it is locally α-presentable for some regular cardinal α. Remark 1.32. Observe that if a k-linear category C is locally α-presentable, then it is locally β-presentable for every regular cardinal β α. This is direct consequence of Remark 1.30 above. ≥ Notation 1.33. Given C a locally presentable category, we denote by Cα the class of α-presentable objects of C. Proposition 1.34. Let C be a locally presentable category. Given any object C C, there exists a regular cardinal αC such that C is β-presentable for all β αC . In particular∈ S α ≥ we have a ﬁltration C = α C .

We have the following useful characterization of locally presentable categories:

Theorem 1.35. A k-linear category C is locally α-presentable if and only if it is cocomplete and is strongly generated by a set of α-presentable objects. Theorem 1.36. Let C be a locally α-presentable category. Then Cα is small and α- cocomplete (i.e. closed under α-small colimits) and given a set of strong generators G Cα, Cα is the closure of G in C under α-small colimits. ⊆ Deﬁnition 1.37. Consider a k-linear category c.

A free cocompletion of c is a full embedding i : c , C such that: • −→ 1. C is a cocomplete k-linear category; 2. Given any k-linear functor F : c B with B cocomplete can be extended to a cocontinuous (i.e. colimit−→ preserving) k-linear functor F : C B which is unique up to natural isomorphism. −→

Let α be a regular cardial. An α-cocompletion of c is a full embedding i : c , C • such that: −→

1. C is a cocomplete k-linear category, i preserves α-small colimits which exist in c; 2. Given any α-cocontinuous (i.e. α-small colimit preserving) k-linear functor F : c B with B cocomplete can be extended to a cocontinuous k-linear functor−→ F : C B which is unique up to natural isomorphism. −→

Given a regular cardinal α, we denote by Lexα(a) the full subcategory of Mod(a) consisting of α-continuous modules, i.e. those that send α-small colimits in aop to α-small limits in Mod(k). Chapter 1. Preliminary results 13

Proposition 1.38. Let a be a small k-linear category. Then:

The Yoneda embedding Y : a Mod(a) is a free cocompletion of a. • −→ For each regular cardinal α, the corestriction Y : a Lexα(a) of the Yoneda • embedding is an α-cocompletion of a. −→

We have in addition the following useful characterization of locally presentable categories.

Theorem 1.39 (Representation theorem). Let α be a regular cardinal. Given a k-linear category C, the following are equivalent:

1. C is locally α-presentable;

2. C is equivalent to the category Lexα(a) for some small category a;

3. C is equivalent to a reﬂective subcategory of Mod(a) (i.e. a full subcategory for which the inclusion functor has a left adjoint) which is closed under α-ﬁltered colimits for some small category a;

4. C is an α-cocompletion of a small category a.

The following proposition states an essential property of locally presentable categories.

Proposition 1.40. Let C be a locally α-presentable k-linear category. Then α-ﬁltered colimits commute with α-small limits in C.

1.3 Triangulated categories

Triangulated categories were introduced by Verdier in his thesis [84] in the setting of the study of derived categories of abelian categories. More precisely, given an abelian category, one can consider its category of complexes. By identifying homotopic morphisms of complexes one obtains the homotopy category. This category is not abelian any more, but it still has some structure derived from its abelian origin. This structure is what Verdier studied and axiomatized, and he deﬁned a triangulated category as any category having this kind of structure. By inverting quasi-isomorphisms in the homotopy category, the derived category of the abelian category is obtained, which is again a triangulated category. 14 Chapter 1. Preliminary results

Hence from Verdier’s thesis we obtain the first two examples of triangulated categories, the homotopy and derived categories of an abelian category, of great relevance for geometry. However, there are some other categories that carry a natural triangulated structure in other settings, for example the stable homotopy category in topology, the stable module category of a self-injective algebra in representation theory, the homotopy category of a stable infinity category and many others. Since the work of Verdier, triangulated categories have played a very important role in many fields. In this section we will give a basic introduction to the field, focusing on well-generated triangulated categories and their localization theory.

We refer the reader to [61] and [47], as the main references for the overview below.

1.3.1 Deﬁnition and axioms

Deﬁnition 1.41. A triangulated category T is an additive category together with an 1 auto-equivalence [1] : T T which is called shift functor , and a ﬁxed set of diagrams of the form → f g h X Y Z X [1] (1.11) which are called distinguished triangles, or exact triangles, and where morphisms between two distinguished triangles are given by morphisms u, v, w such that the diagram f g h X Y Z X [1]

u v w u[1] (1.12) f g X Y Z h X 1 0 0 0 0[ ] is commutative. Moreover, the following axioms hold:

TR1. Every diagram of the form of (1.11) isomorphic to a distinguished triangle • is a distinguished triangle.

IdX 0 0 The diagram X X 0 X [1] is a distinguished trian- • gle.

1We are giving the abstract deﬁnition of a triangulated category, but some of the terminology recalls us the setting where Verdier introduced them. That is the case of the "shift functor", as in derived categories the actual shift functor in complexes is the one playing this role. Chapter 1. Preliminary results 15

Any morphism f : X Y of objects in T can be embbeded in a distin- • →f g h guished triangle X Y Z X [1] where Z is called the 2 cone of the morphism f and usually denoted cone(f ).

f g h TR2. The diagram X Y Z X [1] is a distinguished triangle if g h f [1] and only if Y Z X [1] − Y [1] is a triangle.

TR3. Given X Y Z X 1 and X Y Z X 1 two distinguished [ ] 0 0 0 0[ ] triangles,→ and u→: X →X and v : Y →Y two→ morphisms→ such that the ﬁrst 0 0 square in → → X Y Z X [1]

u v w u[1] (1.13) X Y Z X 1 0 0 0 0[ ] is commutative, then there is a morphism w : Z Z , not necessarily unique, 0 completing the diagram such that u, v, w form a→ morphism of triangles.

TR4. (The octahedral axiom) The idea of the axiom is that given three triangles with some of their objects in common, there exists a triangle which ﬁts perfectly with the previous three providing a nice commutative diagram. As it is a rather technical axiom when stated, we will not go through it here. The exact for- mulation can be found in any reference dealing with triangulated categories. However it is worthwhile mentioning that there is a nice interpretation that allows us to see it as an analogue on triangulated categories for Noether’s Third Isomorphism Theorem.

Notation 1.42. We ﬁx the following notations:

n [n] = [1] ...( ) [1] for all n N; • ◦ ◦ ∈ [0] = IdT; • n 1 1 ...(n) 1 1 for all n . [ ] = [ ]− [ ]− N • − ◦ ◦ ∈ Remark 1.43. Verdier’s axioms have been deeply analysed and some variations of them can be found in certain references. For example in [61], the original octahedral axiom (or Verdier’s axiom) is replaced by an equivalent statement. Moreover, in [59,

2The name of cone is given after the cone complex, which plays the role of the cone in the setting of the homotopy category of an abelian category. 16 Chapter 1. Preliminary results

§1] it is proven that we do not need to require all of them, as TR3 and the inverse implication of TR2 can be deduced from the others. Despite the fact, they are usually written including these “redundant” parts. The main reason is to simplify the theory; if we did not consider them as part of the axioms, they would need to be stated as results immediately after, as they are required for working with triangulated categories from the beginning. Remark 1.44. Triangulated categories are additive categories and hence they live in an enriched setting, namely, they are Ab-enriched. In particular, k-linear categories are a “further” enrichement of Ab-categories, as every k-module has in particular a group structure with the sum. It follows that one can safely work with k-linear triangulated categories. From now on, we will provide the k-linear counterpart of the classical statements for triangulated categories, which is a direct translation from the classical theory. We may though sometimes omit the term k-linear in order to keep the language ﬂuent.

Given a triangulated category T, observe that Top is also triangulated with the natural distinguished triangles and the shift functor given by 1 1 1 . [ ]− = [ ] − Deﬁnition 1.45. A k-linear functor F : T T between two k-linear triangulated 0 categories T and T is called exact if it sends−→ distinguished triangles to distinguished 0 triangles.

Observe that a functor is exact if and only if it sends cones to cones and shifts to shifts.

Deﬁnition 1.46. Given a triangulated k-linear category T and an abelian k-linear category A, we say a functor H : T A is homological if it sends distinguished triangles to long exact sequences. We−→ say a functor H : Top A is cohomological if it is homological in Top. −→

Given a triangulated k-linear category T, the classical example of homological functor is the functor H : T Mod(k) : X T(X , ). Similarly, the classical op example of cohomological functor−→ is H : T 7−→Mod(k)−: X T( , X ). −→ 7−→ −

1.3.2 Triangulated subcategories

We introduce here the main notions concerning subcategories of (k-linear) triangulated categories. The deﬁnitions and results stated here can be found in any basic account on triangulated categories. We refer the reader for example to [61]. Chapter 1. Preliminary results 17

Deﬁnition 1.47. A k-linear subcategory H of a triangulated category T is called a triangulated subcategory if it is closed under isomorphisms, shifts (i.e. closed under taking [n] for all n Z) and such that for every triangle ∈ X Y Z X [1] −→ −→ −→ in T, with X , Y H, one has that Z H as well. ∈ ∈ Given H T a triangulated subcategory of a triangulated category T, observe that H is also a⊆ triangulated category with the triangulated structure induced by T and that the natural inclusion H , T is an exact functor. −→ Deﬁnition 1.48. A full k-linear triangulated subcategory of a triangulated category is called thick (or épaisse) if it is closed under taking direct summands of its objects.

Proposition 1.49. Every full triangulated subcategory of a triangulated category with countable coproducts is thick.

Deﬁnition 1.50. Let T a k-linear triangulated category with small coproducts. A k-linear triangulated subcategory H T is called localizing (resp. α-localizing) if it is closed under small coproducts (resp.⊆ under α-small coproducts).

Let T be a triangulated category with coproducts. Given a class of objects N in T, we denote by N (resp. by N α) the smallest localizing (resp. α-localizing) k-linear subcategory of〈 T〉containing〈 N〉 . We say an object X is left orthogonal to an object Y or Y is right orthogonal to X if T(X , Y ) = 0 and we denote it by X Y . For a full subcategory H T, we obtain the following k-linear subcategories of⊥T: ⊆

H X T H X H H ⊥ = • { ∈ | ⊥ ∀ ∈ } H X T X H H H ⊥ = • { ∈ | ⊥ ∀ ∈ } Remark 1.51. This notation for the right and left orthogonals is the most common in the literature, though it is not standard. For example, the notation in [61] is reversed (see [61, Def 9.1.10 & 9.1.11]).

Deﬁnition 1.52. Let T be a triangulated category with countable coproducts and let

j1 j2 X0 X1 X2 ... (1.14) −→ −→ −→ 18 Chapter 1. Preliminary results be a sequence of objects in T. The homotopy colimit of this sequence, denoted by hocolimXi , is the cone of the distinguished triangle

a∞ f a∞ a∞ Xi Xi X Xi [1] (1.15) i =0 −→ i =0 −→ −→ i =0 where the morphism f sends each element x X to x , j x ` X . Observe i ( i +1( )) i∞=0 i that hocolimXi is deﬁned up to non-canonical∈ isomorphism.− ∈ Remark 1.53. Notice that localizing subcategories are closed under taking homotopy colimits.

1.3.3 Well-generated triangulated categories

Well-generated triangulated categories were introduced by Neeman in [61] as a generalization to higher cardinals of the notion of compactly generated triangulated categories. Further, an elegant characterization of them is given in [48]. They enjoy many of the nice properties of the compactly generated categories, for example, Brown’s representability theorem holds for them. Additionally, they are closed under some nice constructions such as taking certain localizing subcategories or localizations. The main references used in this section are [61], [48] and [47]. Deﬁnition 1.54. Let T be a k-linear triangulated category with coproducts and α a regular cardinal. We say T is α-compactly generated if there exists a set G in T of α-good generators, i.e. a set of objects G in T closed under shifts such that:

G1. Let X be an object in T. If T(G , X ) = 0 for all G G, then X = 0; ∈ G2. Given a set of morphisms Xi Yi in T, if T(G , Xi ) T(G , Yi ) is surjective for all G G, then −→ −→ ∈ a a T(G , Xi ) T(G , Yi ) (1.16) i −→ i is surjective; ` G3. All the objects G G are α-small, i.e. given any morphism G i I Xi , it ∈ ` −→ ∈ factors through a subsum j J X j for some J I with J < α. ∈ ⊆ | | A triangulated category T with coproducts is called well-generated if there exists a regular cardinal α for which it is α-compactly generated.

Given T a triangulated category with coproducts. Given a class N of objects in T, we say it fulﬁlls G4 if the following holds: Chapter 1. Preliminary results 19

` G4. Given any family of objects (Xi )i I , any morphism N i I Xi with N N ` ∈ ` −→ ∈ ∈ factors through a morphism i I Ni i I Xi , with Ni N for all i I . ∈ −→ ∈ ∈ ∈ Given a regular cardinal α, there exists a maximal class Tα consisting of α-small objects for which G4 holds. We denote it by Tα. The objects in Tα are called α- compact. Proposition 1.55. If T is an α-compactly generated triangulated category, the class of objects Tα is essentially small.

The terminology “α-compactly generated” is a natural choice once one observes the following fact. Proposition 1.56. Let T be a well-generated triangulated category and let G be a set of α-good generators. Then we have that:

α α G = T . (1.17) 〈 〉 Moreover, the objects of the skeleton of Tα form a set of α-good generators of T. Proposition 1.57. Let T be a well-generated triangulated category. Then, a set G of objects closed under shifts in T generates T (i.e. G1 holds for G) if and only if G classically generates T (i.e. G = T). 〈 〉 One of the most important properties of well-generated triangulated categories is the following. Theorem 1.58 (Brown’s representability theorem). Let T be a well-generated k-linear triangulated category and H : Top A a functor from T to an abelian k-linear category A. Then H is representable−→ if and only if it is cohomological and sends coproducts to products.

1.3.4 Localization theory in well-generated triangulated categories

An excellent survey on the localization of triangulated categories can be found in [47], which is our main reference for this section. There are two ways of approaching localization in the realm of triangulated categories: Verdier localization and Bousﬁeld localization. Deﬁnition 1.59. Given a k-linear triangulated subcategory W of a k-linear triangulated category T, the Verdier quotient of T with respect to W is a k-linear triangulated category T/W together with an exact functor T T/W that annihilates the elements in W and is universal with this property. −→ 20 Chapter 1. Preliminary results

Definition 1.60. A Bousfield localization of a k-linear triangulated category T is given by a k-linear exact endofunctor L : T T together with a natural transformation −→ µ : IdT L such that L µ = µ L. ⇒ ◦ ◦ Observe that a Bousfield localization functor L : T T, or localization functor for short, gives rise to two full triangulated subcategories−→ of T, Ker(L) and Im(L). In fact, they are nicely related as follows:

T/Ker(L) Im(L), (1.18) ∼= and Ker(L) is a thick subcategory. We deduce that a localization functor is nothing but a composition of a Verdier localization functor T T/W with respect to a thick subcategory and a fully faithful right adjoint T/W −→T to it. −→ One may be interested in the triangulated subcategories W that appear as kernels of localization functors. The following provides a characterization.

Theorem 1.61. Given T a k-linear triangulated category and W T a thick subcategory. Then the following are equivalent: ⊆

1. There exists a localization functor L : T T with Ker(L) = W; −→ 2. The inclusion W , T has a right adjoint; −→ 3. Given any X T, there exists a distinguished triangle ∈

X 0 X X 00 X 0[1] −→ −→ −→ with X W and X W ; 0 00 ⊥ ∈ ∈ 4. The Verdier localization functor T T/W has a right adjoint; −→ 5. The composition W T T W is an equivalence; ⊥ , / −→ −→ 6. The inclusion W T has a left adjoint and W W. ⊥ , ⊥( ⊥) = −→ Both approaches to localization are nicely related when we deal with well-generated triangulated categories. We revise below the most relevant aspects of the theory for our purposes. As a result of Brown’s representability theorem, one has the following.

Theorem 1.62. Let T be a k-linear triangulated category with small coproducts and consider W a localizing subcategory. Then, if W is well-generated, there exists a localization functor L : T T with Ker(L) = W. −→ Chapter 1. Preliminary results 21

In addition, well-generated triangulated categories are closed under taking certain localizing subcategories and localizations:

Theorem 1.63. Let T be a k-linear well-generated triangulated category and W a localizing subcategory generated by a small set of objects. Fix α such that T is α⊆- compactly generated and W is generated by α-compact objects. Then both W and the Verdier quotient T/W are α-compactly generated.

A direct consequence of Theorem 1.62 and Theorem 1.63 is the following.

Corollary 1.64. Let T be a well-generated triangulated category and W T a localizing subcategory generated by a set. Then all the equivalent statements of⊆ Theorem 1.61 above hold true.

Hence, in well-generated triangulated categories, Verdier and Bousﬁeld localization are different approaches to the same localization phenomenon provided that the corresponding localizing subcategory is generated by a set.

1.3.5 The inadequacy of triangulated categories

Triangulated categories present certain problems, which sometimes makes the triangulated realm inadequate to work in. One of the most well-known difﬁculties that appears is the fact that the cone is not functorial, which makes triangulated structures not rigid enough for certain constructions. Another well-known problem is the fact that functor categories or tensor products of triangulated categories are no longer triangulated. If one works with derived categories (our main instance of triangulated categories), some other problems also appear, for example the impos- sibility to recover certain invariants that only depend on the derived category from the derived category itself, the non-local behaviour of the triangulated structure and some others. A very nice overview of the problems one faces when working with derived categories can be found in [78, §1]. The approach to solve these problems goes through considering different types of enhancements of triangulated categories (dg, A ,...) which rigidify the structure and for which the problems mentioned above no longer∞ exist.

1.4 Dg categories

As mentioned in the previous section, differential graded (dg for short) categories were introduced in order to overcome the difﬁculties that appear when we work 22 Chapter 1. Preliminary results with triangulated categories. We provide below an overview of some basic aspects of their theory. In §1.4.1 we review the basic concepts and results on dg categories. We provide, in §1.4.2, a brief summary of the basics on the homotopy theory of dg categories. In §1.4.3 we give a short account about pretriangulated dg categories and dg enhancements. Finally, in §1.4.4, we sketch the main properties of the dg quotient of dg categories. The references used will be provided in the corresponding sections.

1.4.1 Basics on dg categories

The material of this section is based on [39], [40] and [78]. Fix k a commutative ground ring. Deﬁnition 1.65. A differential graded category (or dg category for short) is a category A enriched over C(k), where C(k) denotes the category of cochain complexes of k-modules.

In other words, given two objects A, B A, the morphism space A(A, B) is a chain complex over k, and given any three∈ objects A, B,C A the composition ∈ A(B,C ) k A(A, B) A(A,C ) is a morphism of cochain complexes (i.e. a homoge- nous map⊗ of degree−→ 0 which preserves the differential) where k denotes the usual tensor product of complexes over k. ⊗ Remark 1.66. Observe that the notion is relative to the ground ring k, hence it would be more correct to talk about dg categories over k. Nevertheless, we always work over the ﬁxed commutative ground ring k, thus for convenience we will frequently omit mentioning k. Deﬁnition 1.67. Let A and B be two dg categories. A dg functor F : A B is a C(k)-enriched functor, i.e. the associated maps −→

A(A, A0) B(F (A), F (A0)) (1.19) −→ are morphisms of complexes.

Let A be a dg category. Its opposite dg category Aop is the dg category with the same op objects as A and morphisms given by A (X , Y ) = A(Y , X ) such that the composition of g Aop X , Y m with f Aop Y ,Z n is given by 1 n m g f . ( ) ( ) ( ) · ∈ ∈ − ◦ Given any k-algebra A, we deﬁne C(A) to be the dg category with objects the com- n plexes of right A-modules and for any two such complexes M and N , C(A)(M ,N ) is given by the graded morphisms of degree n and the the differential of C A M ,N is ( )( )• given by n d (f ) = dN f ( 1) f dM , (1.20) ◦ − − ◦ Chapter 1. Preliminary results 23

n where dN and dM are the differentials of N and M and f C(A)(M ,N ) . ∈ Deﬁnition 1.68. Let A be a dg category.

0 We denote by Z (A) the category with the same objects as A and morphisms • given by 0 0 Z (A)(A, B) = Z (A(A, B)), (1.21) 0 0 1 where Z (A(A, B)) = Ker(A(A, B) A(A, B) ). −→ 0 We denote by H (A) the category with the same objects as A and morphisms • given by 0 0 H (A)(A, B) = H (A(A, B)), (1.22) 0 th where H (A(A, B )) denotes the 0 cohomology of the complex A(A, B ). We call 0 H (A) the homotopy category of A. 0 Remark 1.69. Observe that Z (C(A)) = C(A) is the category of complexes of A-modules 0 and that H (C(A)) = H(A) is the homotopy category of complexes of A-modules. Deﬁnition 1.70. Consider a dg category A and two objects A, A A. We say that A 0 and A are homotopically equivalent if they are equivalent as objects∈ in the homotopy 0 0 category H (A). Deﬁnition 1.71. Let A,B be two dg categories. A dg functor F : A B is: −→ quasi-fully faithful if for any two objects A, A A, the morphism of complexes 0 • A A, A B F A , F A is a quasi-isomorphism;∈ ( 0) ( ( ) ( 0)) −→ 0 0 quasi-essentially surjective if the induced functor H (A) H (B) between • the underlying categories is essentially surjective; −→

a quasi-equivalence if it is quasi-fully faithful and quasi-essentially surjective. • Deﬁnition 1.72. Let A be a small dg category. A (right) A-dg module is a dg functor op F : A C(k). −→ We denote by dgMod(A) the dg category of A-dg modules, where objects are the n A-dg modules and f dgMod(A)(F,G ) is such that ∈ n f (A) C(k)(F (A),G (A)) ∈ for all A A and natural in A, and its differential is given by ∈ n d (f )(A) = dG (A) f (A) ( 1) f (A) dF (A). (1.23) ◦ − − ◦ 24 Chapter 1. Preliminary results

Deﬁnition 1.73. Let A be a small dg category.

0 We denote by C(A) = Z (dgMod(A)); • 0 We denote by H(A) = H (dgMod(A)). • Deﬁnition 1.74. A morphism in C(A) or in H(A) is called a quasi-isomorphism if it induces an isomorphism in cohomology.

Deﬁnition 1.75. We deﬁne the derived category of a dg category A and denote it by D(A) to be the localization of H(A) with respect to quasi-isomorphisms.

Deﬁnition 1.76. A morphism in dgMod(a) is called a quasi-isomorphism if it induces a quasi-isomorphism (in the sense of Deﬁnition 1.74 above) in C(A).

Deﬁnition 1.77. We deﬁne the derived dg category of a dg category A and denote it 3 by D(A) to the localization of dgMod(A) with respect to quasi-isomorphisms .

In particular, we have that:

0 H (D(A)) = D(A). (1.24)

1.4.2 Homotopy category of small dg categories

Given a ﬁxed universe U, we can consider the category U-dgcat of U-small dg categories over k with morphisms given by dg functors. For the rest of the chapter we will work with this ﬁxed universe U and hence will omit to mention it.

Deﬁnition 1.78. Let A, B be two small dg categories. We deﬁne the tensor product of dg categories A and B and denote it by A k B to the dg category with objects given by ⊗

Obj(A k B) = Obj(A) Obj(B) ⊗ × and morphisms given by

A k B((A, B),(A0, B 0)) = A(A, A0) k B(B, B 0). ⊗ ⊗

Observe that A k B is again a small dg category. ⊗ 3 Dg localization along a class of morphisms is described in [79, §8.2]. Chapter 1. Preliminary results 25

In addition, notice that dgcat(A,B) is also a small dg category, with objects the dg functors A B and morphisms given by the C(k)-enriched natural transformations. −→ These two operations endow the category of small dg categories with a monoidal structure, i.e. we have

dgcat(A k B,C) = dgcat(A,dgcat(B,C)). (1.25) ⊗ ∼ For many purposes, it is a very natural choice to consider dg categories up to quasi- equivalences. The natural procedure then is to formally invert quasi-equivalences in the category of small dg categories dgcat. As pointed out in [78], the localization of categories is normally not well-behaved and usually hard to describe. In many situations, we lose control on the morphisms of the localization. However, things get simpler when we are in presence of a model structure. In this section we sketch briefly what a model category is. We point the reader to [32] for a nice monograph on the topic. We proceed then to describe, based on [75] and [79], the model structure in the category dgcat of small dg categories and the corresponding homotopy category obtained. Finally we discuss the existence of a monoidal structure in the homotopy category of dgcat, for which we refer the reader to [79]. A model category is a complete and cocomplete category with a model structure. The concept was introduced by Quillen in [69] and is a fundamental part of the foundations of homotopy theory. A model structure on a category consists of three families of morphisms (weak-equivalences, fibrations and cofibrations) closed under composition, for which certain axioms hold. The added value of the structure is that when we localize with respect to the weak-equivalences, we still have complete control over the morphisms in the localization category: morphisms between two objects X and Y are just the morphisms in the original category, up to homotopy, between suitable replacements of X and Y , i.e. objects with certain properties which are weak-equivalent to X and Y . These objects with nice localization properties are:

1. Coﬁbrant objects: The objects such that the functor from the initial object to them is a coﬁbration;

2. Fibrant objects: The objects such that the morphism from them to the terminal object is a ﬁbration.

In a model category one can choose a functor Q, called cofibrant replacement, which assigns to every object a cofibrant object and such that we have a natural morphism Q(X ) X which is both a weak equivalence and a fibration. Analogously, we −→ 26 Chapter 1. Preliminary results can choose a functor R, called fibrant replacement which assigns to every object a fibrant object and such that we have a natural morphism X R(X ) which is both a weak equivalence and a cofibration. The category obtained−→ by localizing the model category with respect to the weak-equivalences is called the homotopy category and the morphisms between two objects X and Y in the homotopy category are given by the morphisms between Q(R) and R(Y ) in the original category up to homotopy. We will not provide the definition and properties of model categories, as this would require a lengthy section. We instead refer the reader to [32] for a nice account on the topic. As we already mentioned, we are interested in dg categories up to quasi-equivalence. In [75] a model structure in the category of small dg categories is introduced for which the weak-equivalences are the quasi-equivalences. First, we introduce the projective model structure on cochain complexes. This appears in [32, §2.3]. Theorem 1.79. There is a model category in the category of complexes C(k) such that:

Weak equivalences W are the quasi-isomorphisms; • Fibrations are the (degree-wise) surjective morphisms. • Moreover, the homotopy category Ho C k C k W 1 for this model category is the ( ( )) = ( )[ − ] derived category D(Mod(k)). Remark 1.80. It is a well-known fact that weak-equivalences and fibrations (resp. cofibrations) determine cofibrations (resp. fibrations) completely. Hence the data of Theorem 1.79 above is enough to determine a model structure on C(k). An extensive analysis of the cofibrations in this model structure can be also found in [32, §2.3]. Now, we introduce the standard model structure on the category of small dg categories [75]. The model structure is as follows: Theorem 1.81. The category dgcat of small dg categories admits a model structure where

Weak equivalences W are the quasi-equivalences; • A dg functor F : A B is a fibration if it satisfies: • −→ – For every A, A A the induced map A A, A B F A , F A is a fi- 0 ( 0) ( ( ) ( 0)) bration in C(k) ∈for the projective module structure,−→ i.e. it is a degree-wise surjective morphism; Chapter 1. Preliminary results 27

– For any A A and any isomorphism v : H 0 F A B in H 0 B there ( )( ) 0 ( ) 0 0 exists an isomorphism∈ A B in H (A) such that H−→(F )(u) = v . −→ Remark 1.82. By Remark 1.80 we know that weak-equivalences and ﬁbrations determine coﬁbrations completely and hence the data of Theorem 1.81 above determines completely a model structure on dgcat.

We denote by Hqe the homotopy category Ho(dgcat) of small dg categories. Given a dg functor F : A B, we denote by [F ] its image in Hqe and, as it is usually done in the literature, we−→ denote by [ , ] = Hqe( , ) the set of morphisms in Hqe. Observe 0 0 0 that given an element f [A−,B−], it induces− − a functor H (f ) : H (A) H (B). ∈ −→ The following result about ﬁbrant and coﬁbrant objects in dgcat will be useful for us in Chapter5.

Theorem 1.83. Consider dgcat with the standard model structure deﬁned in Theo- rem 1.81. Then, the following hold:

Any object in dgcat is fibrant; • There exists a cofibrant replacement Q : dgcat dgcat such that the natural • morphism Q(A) A is the identity on objects.−→ In addition, if A is cofibrant in dgcat and A, A−→A then A A, A is cofibrant in C k for the projective model 0 ( 0) ( ) structure. ∈

The homotopy category Hqe is deeply analysed in [79]. In particular, Hqe has a monoidal structure for which we also refer the reader to [14]. The tensor product of dg categories can be derived into a functor L deﬁned: ⊗ ⊗ L A B = Q(A) B, (1.26) ⊗ ⊗ which preserves quasi-equivalences and hence induces a functor between the homotopy categories. The inner hom of in the category of small dg categories was given by the category of dg functors. However,⊗ a derived version of this functor is not immediately at our disposal. Nevertheless, the derived tensor product also has an inner hom due to Toën that we denote by RHom and is constructed as follows: L op Consider the derived dg category D(B A ), which is a large dg category. A bimod- L op ule F dgMod(B A ) induces a functor⊗ F : A dgMod(B), and it is called right 0 0 quasi-representable∈ ⊗ provided that the induced functor−→ H (F ) : H (A) D(B) factors 0 0 0 L op through a functor H (F ) : H (A) H (B). We denote by RHom(A,B−→) D(B A ) the small full dg subcategory of right−→ quasi-representable bimodules.⊆ This category⊗ 28 Chapter 1. Preliminary results is not small, but essentially small, and hence can still be considered as an element of Hqe (see [79]). We have thus an adjunction L [A B,C] = [A,RHom(B,C)] ⊗ ∼ in Hqe. This, in particular induces an equivalence L RHom(A B,C) = RHom(A,RHom(B,C)) ⊗ ∼ in Hqe. The relation between the morphisms in Hqe and the inner homs is provided by the following formula 0 [A,B] Iso(H (RHom(A,B))) (1.27) ∼= where by Iso we denote the operation of taking the isomorphism classes. In particular, given an element f RHom(A,B), we will denote by [f ]iso the induced element in [A,B] via the equivalence∈ (1.27).

1.4.3 Pretriangulated dg categories

Dg categories were introduced, as we already mentioned, in order to have a close enough setting to that of triangulated categories but where the difﬁculties mentioned in §1.3.5 disappear. This “enhancing” role will be played by the so called pretriangulated dg categories. In this section we give the deﬁnition and basic properties of pretriangulated dg categories and we formally introduce the notion of dg enhancement for a triangulated category. We additionally state some results on the existence and uniqueness of dg enhancements.

Pretriangulated dg categories were introduced and studied in [5]. We also refer the reader to [22, §2.4] for a nice overview. The notion of dg enhancement also goes back to [5] and we further refer the reader to [56] and [15] for the results on uniqueness and existence of enhancements given at the end of the section. A nice survey on the topic can also be found in [13]. Deﬁnition 1.84. Let A be a small dg category. A twisted complex over A is given by A ((Ei )i Z,qi ,j : E j Ei ), where (Ei )i Z is a family of elements in with Ei = 0 for almost ∈ → ∈ P all i Z and qi j are morphisms of degree i j + 1 such that d qi ,j + k qi ,k qk,j = 0. ∈ − ◦ The category Tw(A) of twisted complexes over A forms a dg category as follows. Consider C E ,q ,C E ,q two twisted complexes, then one deﬁnes = (( i ) i ,j ) 0 = (( i0) i0,j ) n a k Tw(A)(C ,C 0) = A(Er , Es0) (1.28) k+r s =n − Chapter 1. Preliminary results 29 and for each f A E , E k , the differential is given by ( r s0) ∈ X k(r m+1) d f = dA f + (qm,s f + ( 1) − f qr,m ). (1.29) m ◦ − ◦

Deﬁnition 1.85. We denote by PreTr A the dg category of twisted complexes of A , ( ) ⊕ where A is the dg category obtained from A by formally adding all ﬁnite direct sums ⊕ 0 of its elements. Denote by Tr(A) = H (PreTr(A)). Remark 1.86. Observe that given A a dg category, we have a fully-faithful dg functor

A PreTr(A) (1.30) −→ which sends A A to the twisted complex (A,0). ∈ One can easily see that given a dg functor F : A A we have a natural dg functor 0 −→ PreTr(F ) : PreTr(A) PreTr(A0) (1.31) −→ that extends F and that induces a functor

Tr(F ) : Tr(A) Tr(A0) (1.32) −→ between the homotopy categories.

Definition 1.87. Let A be a dg category. Let C = ((Ei )i ,qi ,j ) PreTr(A). For any n Z, th ∈ ∈ we define the n shift of C and denote it by C [n] to be the twisted complex ((Fi )i , ri ,j ) where Fi = Ei +n and ri ,j = qi +n,j +n for all i , j Z. ∈ Definition 1.88. Let A be a dg category. Let C E ,q and C E ,q be two = (( i ) i ,j ) 0 = (( i0) i ,j ) elements in PreTr A and take f f : E E PreTr A C ,C a closed (i.e. ( ) = ( i ,j j i0) ( )( 0) d (f ) = 0) morphism of degree 0 between them.−→ We∈ define the cone of f to be the twisted complex ((Ei00),qi00,j ) given by

Ei00 = Ei Ei0 1 (1.33) ⊕ + and qi ,j fi ,j qi00,j = . (1.34) 0 qi0,j

Observe that we have a diagram of closed morphisms of degree 0

f C C 0 cone(f ) C [1] (1.35) −→ −→ −→ in PreTr A , where C cone f and cone f C 1 are the natural morphisms ( ) 0 ( ) ( ) [ ] induced by E E −→E and E E E−→ respectively. In particular, this i i i0+1 i i +1 i +1 induces a corresponding−→ ⊕ diagram of⊕ morphisms−→ in Tr(A). 30 Chapter 1. Preliminary results

Theorem 1.89. The category Tr(A) has a triangulated structure with the set of distinguished triangles given by those corresponding to the cone diagrams in PreTr(A) as in (1.35) above.

Now consider the dg category dgMod(A) of right A-dg modules as deﬁned above. Every element M dgMod(A) comes endowed with a natural notion of shift. More ∈ precisely, we put F [n](A) = F (A)[n] for all A A and all n Z. Analogously, given a morphism f : M N closed of degree 0,∈ i.e. f C(A)(M∈,N ), we have a natural notion of cone of f ,−→ namely cone(f )(A) = cone(f (A))∈, where this last cone is the cone complex of a morphism of complexes. Observe then that given a closed morphism f : M N of degree 0, we have the diagram of closed morphisms of degree zero −→ M N cone(f ) M [1] (1.36) −→ −→ −→ and this induces a corresponding diagram in H(A).

Theorem 1.90. Let A be a dg category. Then the category H(A) is triangulated with distinguished triangles given by those induced from the cone diagrams in dgMod(A). Theorem 1.91. There is a fully-faithful dg functor

F : PreTr(A) dgMod(A) (1.37) −→ deﬁned by E ,q M , where M A L A E , A i , and for any n , given an (( i ) i j ) ( ) = i Z ( i )[ ] Z element f L 7→A E , A n i , its differential∈ is given by ∈ ( i )i i Z ( i ) [ ] ∈ ∈ n X d ((fi )i ) = (dA(fi ) ( 1) fk qk,i )i . (1.38) − − k ◦ In particular, this functor preserves cones and shifts. 0 In addition, H (F ) identiﬁes Tr(A) with a full triangulated subcategory of H(A).

It follows that Yoneda embedding A dgMod(A) factors through the functor F : PreTr(A) dgMod(A). −→ −→ Observe that PreTr(A) is basically obtained from A by formally adding all ﬁnite sums, shifts and cones so that its homotopy category is triangulated with the inherited structure.

Deﬁnition 1.92. A dg category A is called pretriangulated if and only if for every object A, the shifts A[n] in PreTr(A) dgMod(A) are homotopically equivalent to a representable A-dg module and for⊆ every closed morphism of degree 0 f in A, we have that cone(f ) is homotopically equivalent to a representable A-dg module. Chapter 1. Preliminary results 31

0 Proposition 1.93. Let A be a pretriangulated category. Then H (A) is a triangulated category with the shifts and distinguished triangles induced from the ones that A inherits from PreTr(A). More precisely, the Yoneda embedding

A , PreTr(A) , dgMod(A) −→ −→ 0 identiﬁes H (A) with a triangulated subcategory of H(A). Theorem 1.94. Let A be a pretriangulated dg category, then the fully faithful functor A , PreTr(A) is a homotopy equivalence and the induced equivalence between the −→ 0 = 0 homotopy categories H (A) ∼ Tr(A) is an exact functor, i.e. H (A) and Tr(A) are equivalent as triangulated categories.−→

We have hence the following characterization of pretriangulated dg categories.

0 Proposition 1.95. A dg category A is pretriangulated if and only if H (A) comes with a natural structure of triangulated category via the Yoneda embedding.

This characterization appears sometimes as a deﬁnition in the literature. Neverthe- less, it is always useful to keep in mind that pretriangulated categories come endowed with the notions of shift and cone of a closed morphism of degree 0. We now introduce the notion of dg enhancement.

Deﬁnition 1.96. Let T be a triangulated category. A dg enhancement of T is a pair 0 0 (A,H (A) T) where A is a pretriangulated dg category and H (A) T is an equivalence of triangulated→ categories. →

Deﬁnition 1.97. Let T be a triangulated category. We say T is algebraic if it has a dg enhancement.

Remark 1.98. Different equivalent deﬁnitions of algebraic triangulated category can be found in the literature. We have chosen this particular one from [70] or [13, 15], as it is the most useful for our purposes.

From given algebraic categories, one can obtain other algebraic categories by considering subcategories or Verdier quotients:

Theorem 1.99. A full subcategory of an algebraic category is algebraic. A Verdier quotient (with small Hom-sets) of an algebraic category with respect to a full algebraic subcategory is again algebraic.

Now we proceed to state what we mean by uniqueness of dg enhancements: 32 Chapter 1. Preliminary results

Deﬁnition 1.100. Let T be an algebraic triangulated category. We say T has a unique (dg) enhancement if, given A,H 0 A T and A ,H 0 A T two enhancements ( ( ) ) ( 0 ( 0) ) of T, there exists an equivalence F : A→ A in Hqe. → 0 −→ There are examples of triangulated categories that lack dg enhancements (e.g the homotopy category of spectra, see [45]). Additionally, it could happen that a triangulated category does have enhancements but they are not unique, namely there exist two different dg categories with homotopy categories equivalent to the given triangulated category but that are not equivalent as dg categories. An example of such a situation can be found in [24]. Nevertheless, for many of the triangulated categories we are interested in there exists an enhancement and it is unique.

Theorem 1.101 ([15, Thm A]). Let C be a Grothendieck category. Then D(C) has a unique enhancement.

Corollary 1.102. Let X be an algebraic stack, then the derived category D(Qch(X )) has a unique enhancement.

In [15] we can also ﬁnd other instances of triangulated categories of geometrical origin which also have a unique dg enhancement. In particular, we have the following:

Proposition 1.103. Let X be a noetherian concentrated algebraic stack with quasi- ﬁnite afﬁne diagonal and enough perfect coherent sheaves. Then Perf(X ) has a unique enhancement.

Proposition 1.104. Let X be a noetherian scheme with enough locally free sheaves, b then D (Qch(X )) has a unique enhancement.

1.4.4 Quotient of dg categories

In [41], Keller gives a construction of the quotient of small dg categories. Drinfeld develops Keller’s work further in [22]. We will not explain here how the construction of the Keller-Drinfeld quotient is performed. Instead, we refer the reader to the mentioned references for the details.

In [74], Tabuada further gives three different universal properties of Keller-Drinfeld quotient for three different environments in which the quotient can be thougth of. We state here the universal property of the quotient in Hqe, which will be the one interesting for our purposes. Chapter 1. Preliminary results 33

Theorem 1.105. Given a small dg category A and a full dg subcategory B A, the dg quotient is a dg category A/B together with a morphism Q : A A/B⊆in Hqe 0 0 0 that annihilates B (i.e. the induced functor H (Q) : H (A) H (−→A/B) sends ele- 0 ments in H (B) to contractible objects) and such that composition−→ with Q induces an equivalence RHom(A/B,C) RHomB(A,C) (1.39) ∼= in Hqe, where RHomB(A,C) is the full subcategory of RHomB(A,C) consisting of the right quasi-representable bimodules that annihilate B.

The main feature of the quotient of small dg categories is the fact that it is compatible with Verdier’s quotient of triangulated categories. More precisely:

Theorem 1.106. Given a small dg category A and a full subcategory B A, we have that the induced functor Tr(Q) : Tr(A) Tr(A/B) induces an equivalence:⊆ −→ Tr(A)/Tr(B) Tr(A/B). (1.40) ∼= In particular, if A and B are pretriangulated, we have that:

0 0 0 H (A/B) H (A)/H (B). (1.41) ∼= 34 Chapter 1. Preliminary results CHAPTER 2

Tensor product of linear sites and Grothendieck categories

The content on this chapter is based on [53] which is joint work with Wendy Lowen and Boris Shoikhet. We refer the reader to §1.1 for definitions and results concerning Grothendieck categories and linear Grothendieck topologies. A Grothendieck category C is a cocomplete abelian category with a generator and exact filtered colimits. Grothendieck categories are large abelian categories of great relevance. They play an important role in noncommutative algebraic geometry, where they are used as models for noncommutative spaces since the work of Artin, Stafford, Van den Bergh and others ([2], [3], [73]). In algebraic geometry, one of the most basic operations to be performed with schemes X and Y is taking their product scheme X Y . For affine schemes Spec(A) and Spec(B), this corresponds to taking the tensor product× A B of the underlying rings. Our goal in this chapter is to define ⊗ a tensor product C D for arbitrary Grothendieck categories C and D, such that for rings A and B we have

Mod(A) Mod(B) = Mod(A B). (2.1) ⊗ As was originally shown in the Gabriel-Popescu theorem [65], Grothendieck categories are precisely the localizations of module categories. One way of seeing this, is by describing localizations of the category Mod(A) of modules over a ring A by means of data on A, so called Gabriel topologies. In the Gabriel-Popescu theorem, the endomorphism ring of a generator of C is endowed with such a Gabriel topology. Using the language of linear topologies on linear categories a (see §1.1), more generally one can characterize linear functors a C which induce an equivalence −→ C Sh(a, a) Mod(a), where a is a certain topology on a and Sh(a, a) is the cate- ∼= gory of linearT ⊆ sheaves on a withT respect to this topology [51] (see TheoremT 1.27). Our approach to the deﬁnition of a tensor product of Grothendieck categories consists of the following steps:

35 36 Chapter 2. Tensor product of linear sites and Grothendieck categories

(i) First, we deﬁne the tensor product of linear sites (a, a) and (b, b) to be given T T by (a b, a b) for a certain tensor product topology a b on the standard tensor⊗ productT T of linear categories a b. T T ⊗ (ii) Next, we show that the deﬁnition

Sh(a, a) Sh(b, b) = Sh(a b, a b) (2.2) T T ⊗ T T is a good deﬁnition for Grothendieck categories, as it is independent of the particular sites chosen in the sheaf category representations (up to equivalence of categories).

Step (i) is carried out in §2.1. The topologies a and b naturally give rise to two “one- T T sided” topologies 1 and 2 on a b, and we put a b equal to the supremum of 1 and 2 in the latticeT of topologiesT ⊗ on a b (DeﬁnitionT T 2.12). We further describe theT correspondingT operations between localizing⊗ Serre subcategories, as well as between strict localizations. In particular, we show that

Sh(a b, a b) = Sh(a b, 1) Sh(a b, 2). (2.3) ⊗ T T ⊗ T ∩ ⊗ T For compatible localizing Serre subcategories in the sense of [12], it is well known that their supremum is described by the Gabriel product, and using this description it is easily seen that the infimum of compatible strict localizations is simply their intersection. However, the general case is more subtle and our analysis is based upon the construction of a localizing hull (Proposition 2.2) for full subcategories closed under subquotients. This eventually leads to the proof of (2.3) in complete generality. An application of our constructions to the strict localizations and localizing Serre subcategories corresponding to the linear sites associated to Quillen exact categories, recovers the constructions from [36], which inspired §2.1.8. Step (ii) is based upon an analysis of the functoriality of our tensor product of sites, which is carried out in §2.2. An alternative approach making use of the already established tensor product of locally presentable categories going back to Kelly [42, 43] will be discussed in §2.4.4. Since the functoriality properties established in §2.2 are of independent interest in the context of noncommutative geometry, we present a complete proof of step (ii) without reference to local presentablility, thus reflecting our own initial approach to the subject. A detailed discussion of the relation with the tensor product of locally presentable categories is contained in §2.4.1. We refer the reader to §1.2 for a brief summary on the basic theory of locally presentable categories. As analysed in §1.1.2, the classical notions of continuous and cocontinuous functors from [76] have their linear counterparts, and we show that these types of functors are Chapter 2. Tensor product of linear sites and Grothendieck categories 37 preserved by the tensor product of sites. Our main interest goes out to a special type of functors φ : (a, a) (b, b) between sites, which we call LC functors (the letters stand for “LemmeT de−→ comparaison”).T Roughly speaking, φ satisfies (LC) (Definition 2.19) if:

1. φ is generating with respect to b; T 2. φ is fully faithful up to a; T 3. 1 . a = φ− b T T The technical heart of the chapter is the proof that our tensor product preserves LC functors (Proposition 2.30). Both the generating condition (1) and the fullness part of condition (2) are preserved separately. However, the faithfulness part is only preserved in combination with fullness (Lemma 2.28). This extends the situation for rings: surjections of rings are preserved under tensor product, injections are not (unless some ﬂatness is assumed), but isomorphisms are obviously preserved by any functor hence also by tensoring.

The importance of LC functors φ : (a, a) (b, b) lies in the fact that they induce T −→ T equivalences of categories Sh(b, b) = Sh(a, a). In addition, given a Grothendieck T ∼ T category C, any two representations of C as C = Sh(a, a) and C = Sh(a , a ) can be ∼ ∼ 0 0 related through a roof of LC functors. This easily yieldsT independence ofT(2.2) from the choice of sheaf category representations (Proposition 2.31).

In §2.3, we define the tensor product C D for arbitrary Grothendieck categories C and D by formula (2.2) for arbitrary representations C = Sh(a, a) and D = Sh(b, b) ∼ T ∼ T (Definition 2.32). We apply the tensor product to Z-algebras and schemes. In [6], [83], Z-algebras are used as a tool to describe noncommutative deformations of projective planes and quadrics. They are closely related to the graded algebras turning up in projective geometry, but better suited for the purpose of algebraic deformation. In particular, under some finiteness conditions, they allow nice categories of “quasicoherent modules” [64],[73]. A (positively graded) Z-algebra is a linear category a with Obj(a) = Z and a(n,m) = 0 unless n m. In [20], Z-algebras a are endowed with a ≥ certain tails topology tails and the category Sh(a, tails) is proposed as a replacement for the category of quasicoherentT modules, whichT exists in complete generality. We thus investigate the tensor product of two arbitrary tails sites (a, a) and (b, b) and show the existence of a cocontinuous functor T T

∆ : ((a b)∆, tails) (a b, a b) (2.4) ⊗ T −→ ⊗ T T from the natural diagonal Z-algebra (a b)∆ a b consisting of the objects (n,n) for ⊗ ⊆ ⊗ n Z to the tensor site (Proposition 2.35). For a Z-algebra a, the degree of an element ∈ 38 Chapter 2. Tensor product of linear sites and Grothendieck categories in a(n,m) is n m and we say that a is generated in degree 1 if every element can be written as a linear− combination of products of elements of degree 1 (Deﬁnition 2.36). If a and b are generated in degree 1, then the functor ∆ from (2.4) is actually an LC functor (Theorem 2.41). When applied to projective schemes X and Y , by looking at the Z-algebras associated to deﬁning graded algebras which are generated in degree 1, we obtain the following formula (Theorem 2.43):

Qch(X ) Qch(Y ) = Qch(X Y ). (2.5) × Formula (2.5) is expected to hold in greater generality, at least for schemes and suitable stacks, which we hope to prove in future work. In §2.4.1, we discuss the relation of our tensor product with other tensor products of categories in the literature. The existence of a tensor product of locally presentable categories goes back to [43], [42] and features in [1], [11], [16], [18]. It is well known that Grothendieck categories are locally presentable. For locally α-presentable Grothendieck categories, we use canonical sheaf representations in terms of sites of α-presentable objects in order to calculate our tensor product, and we show that it coincides with the tensor product as locally presentable categories. In particular, the tensor product is again locally α-presentable. As a special case, we observe that locally finitely presentable Grothendieck categories are preserved under tensor product. In contrast, the stronger property of local coherence, which imposes the category of finitely presented objects to be abelian, is not preserved under tensor product, as is already seen for rings. Hence, one can view the tensor product of Grothendieck categories as a solution, within the framework of abelian categories, to the non-existence, in general, of the Deligne tensor product of small abelian categories. Indeed, it was shown by López Franco in [50] that the Deligne tensor product of abelian categories A and B from [21] exists precisely when the finitely cocomplete tensor product A fp B ⊗ is abelian, and this is the case precisely when the tensor product Lex(A) Lex(B) is locally coherent. As suggested to us by Henning Krause, we further examine the situation in terms of an α-Deligne tensor product of α-cocomplete abelian categories, showing that every tensor product of Grothendieck categories is accompanied by a parallel α-Deligne tensor product of its categories of α-presented objects for sufficiently large α. Our tensor product of Grothendieck categories can be seen as a k-linear counterpart to the product of Grothendieck topoi which is described by Johnstone in [34, 35] and its relation with the tensor product of locally presentable categories is to some extent parallel to Pitts’ work in [63]. We should note however that unlike in the case of topoi, working over Mod(k) rather than over Set, the tensor product does not describe a 2- categorical product, but instead introduces a 2-categorical monoidal structure. This Chapter 2. Tensor product of linear sites and Grothendieck categories 39 structure will be studied further in Chapter3. Moreover, the functoriality properties we prove open up the possibility of describing a suitable monoidal 2-category of Grothendieck categories as a bilocalization of a monoidal 2-category of sites at the class of LC functors. This will be addressed in Chapter4. This idea applies equally well to the Set-based setup.

2.1 Tensor product of linear sites

Throughout, let k be a commutative ground ring. For two k-linear categories a, b, we put a b = a k b the k-linear tensor product. Recall that a b is the k-linear category with Obj⊗ a ⊗b Obj a Obj b and a b A, B , A , B ⊗a A, A b B, B for all ( ) = ( ) ( ) (( ) ( 0 0)) = ( 0) k ( 0) A, B , A , B⊗ a b. The× starting point⊗ for our quest for a tensor product⊗ between ( ) ( 0 0) Grothendieck∈ abelian⊗ categories is the requirement that for module categories Mod(a) and Mod(b), we should have

Mod(a) Mod(b) = Mod(a b). (2.6) ⊗ If we want to extend this principle to localizations of module categories, we should ﬁnd a way of associating, to given localizations of Mod(a) and Mod(b), a new localization of Mod(a b). In this section, we detail three natural ways of doing this, based upon the following⊗ three isomorphic posets associated to the localization theory of Mod(c) for a linear category c (see §2.1.3):

1. The poset T of linear topologies on c;

2. The poset W of localizing Serre subcategories of Mod(c); op 3. The opposite poset L of the poset L of strict localizations of Mod(c).

More precisely, taking c = a b: ⊗

1. To topologies a on a and b on b, we associate “one-sided” topologies 1 T T T (induced by a) and 2 (induced by b) on a b, and we put a b = 1 2 in T (see §2.1.4T ). T T ⊗ T T T ∨ T

2. To localizing Serre subcategories Wa Mod(a) and Wb Mod(b), we associate ⊆ ⊆ the localizing Serre subcategories W1,W2 Mod(a b) of objects which are in ⊆ ⊗ Wa (resp. Wb) in the ﬁrst (resp. second) variable, and we put WaWb = W1 W2 in W (see §2.1.5). An explicit description is based upon the construction of∨ the localizing hull for subcategories closed under subquotients from §2.1.2. 40 Chapter 2. Tensor product of linear sites and Grothendieck categories

3. To strict localizations La Mod(a) and Lb Mod(b), we associate the strict ⊆ ⊆ localizations L1,L2 Mod(a b) of objects which are in La (resp. Lb) in the ⊆ ⊗ ﬁrst (resp. second) variable, and we put La Lb = L1 L2 in L (see §2.1.6). ∧ Using the relation between W and L, one sees that actually La Lb = L1 L2. ∩

From the order theoretic deﬁnitions of a b, Wa Wb and La Lb, we conclude that in order to establish that they correspondT T under the isomorphisms between T , op W and L , it sufﬁces to establish the claim for a, Wa and La (and similarly for b, Wb and Lb). This is done in §2.1.7. T T

An application to Quillen exact categories recovers notions from [36] which inspired our deﬁnitions, as discussed in §2.1.8.

2.1.1 The poset of linear sites

We will use the terminology and notations from §1.1. Let k be a commutative ground ring and let a be a small k-linear category. Note that the intersection of a collection of topologies on a remains a topology, and a can be endowed with the discrete topology for which every sieve is covering. Hence, for an arbitrary cover system on a, there exists a smallest (or coarsest) topology top on a with top. If Ris localizing, an explicit description of top is available〈R〉 (see R ⊆ 〈R〉 R upglue 〈R〉 Proposition 1.12), namely we have that top = . Consequently, the poset T 〈R〉 R of topologies on a ordered by inclusion is a complete lattice with infi i = i i and T ∩ T supi i = i i top. T 〈∪ T 〉

2.1.2 The poset of localizing subcategories

Let C be a Grothendieck category. Recall that a localizing Serre subcategory (localizing subcategory for short) W C is a full subcategory closed under subquotients, extensions and coproducts. It follows⊆ in particular that a localizing subcategory W is closed under colimits. As the intersection of localizing subcategories is again such, for every full subcategory H C there is a smallest localizing subcategory H loc with H H loc, the localizing hull⊆ of H. In particular, the poset W of localizing〈 subcate-〉 ⊆ 〈 〉 gories of C is a complete lattice with infi Wi = i Wi and supi Wi = i Wi loc. In this section we give an explicit description of H loc∩when H is closed under〈∪ subquotients.〉 〈 〉 Deﬁnition 2.1. Consider H Obj(C) and C C. An ascending ﬁltration of C consists ⊆ ∈ of an ordinal α and a collection of subobjects (Mβ )β α of C such that M0 = 0, i j ≤ ≤ implies Mi M j , Mβ = γ<β Mγ if β is a limit ordinal, and Mα = C . An ascending ⊆ ∪ Chapter 2. Tensor product of linear sites and Grothendieck categories 41

filtration (Mβ )β α of C is called an H-filtration provided that Mβ+1/Mβ H for all β < α, and in this≤ case C is called H-filtered. ∈

Proposition 2.2. For H C closed under subquotients, H loc is the full subcategory of all H-ﬁltered objects. ⊆ 〈 〉

Proof. Suppose first that H W with W localizing. Consider an object C C with ⊆ ∈ H-filtration (Mβ )β α. We show by transfinite induction that every Mβ W. The ≤ ∈ statement is true for M0 = 0. Suppose Mβ W. For Mβ+1 we have an exact sequence ∈ 0 Mβ Mβ+1 Mβ+1/Mβ 0 so since W is closed under extensions we −→ −→ −→ −→ have Mβ+1 W. For a limit ordinal β, we have Mβ W since W is closed under colimits. ∈ ∈ Next we prove that the full subcategory of H-filtered objects is closed under coproducts and extensions. Note that in this part of the proof we do not use that H is closed under subquotients. L Consider a coproduct C = i I Ci of H-filtered objects. We may safely assume that the coproduct is indexed by successor∈ ordinals, i.e. we can write C L C for = γ+1<α γ+1 an ordinal α. We put Cα = 0. We inductively define an ascending filtration (Dβ )β α of ≤ C with D0 = 0. For a successor ordinal γ + 1 α, we put Dγ+1 = Dγ Cγ+1 and for a ≤ ⊕ limit ordinal β α we put Dβ = γ<β Dγ. Note that Dα = C . ≤ ∪ β 1 By assumption, every C with β 1 < α has an H-filtration M + for some β+1 + ( γ )γ αβ+1 ≤ ordinal αβ+1. By transfinite induction on α we construct for every Dβ with β α ≤ an H-filtration refining the chosen H-filtrations of the Dγ with γ < β. We have the H β filtration (D0)0 for D0 = 0. Suppose a -filtration (Pγ )γ θβ is chosen for Dβ with θβ ≤ some ordinal. For a successor ordinal β, we have Dβ+1 = Dβ Cβ+1. We consider ⊕ the ordinal sum θβ+1 = θβ + αβ+1. We join the two H-filtrations together into an H- β+1 β+1 β β+1 β+1 filtration Pγ γ θ with Pγ Pγ for γ θβ and P Dβ Mγ for γ αβ 1. ( ) β+1 = θβ +γ = + ≤ ≤ ⊕ ≤ For a limit ordinal , we put . We construct an H-filtration P β β α θβ = γ<β θγ ( ζ )ζ θβ ≤ ∪ β γ ≤ of Dβ . For ζ < θβ , there exists γ < β with ζ < θγ, and we put Pζ = Pζ . This is well defined by construction. We further put P β D . We then have that P α is an θβ = β ( ζ )ζ θα H-filtration of C L C . ≤ = γ+1<α γ+1 Next, consider an exact sequence

0 / C / C / C / 0 0 f g 00 in C and H-ﬁltrations (Mβ )β α of C and (Nβ )β α of C . For the ordinal sum 0 0 00 00 , we obtain an ascending≤ ﬁltration P ≤ of C with P f M for α = α0 + α00 ( γ)γ α γ = ( γ) γ α0 ≤ ≤ 42 Chapter 2. Tensor product of linear sites and Grothendieck categories

1 1 and Pα γ = g (Nγ) for γ α . Note that we have Pα = f (Mα ) = g (N0) as desired 0+ − 00 0 0 − since the sequence is exact.≤ In addition, we have g 1 N g 1N N N H. − ( γ+1)/ − γ = γ+1/ γ ∼ ∈ To ﬁnish the proof, we show that the full subcategory of H-ﬁltered objects is closed under subquotients. Consider an exact sequence

0 / C / C / C / 0 0 f g 00 in C and an H-filtration (Mβ )β α of C . In first place we obtain an ascending filtration f 1 M of C and≤ observe that for each we have a canonical ( − ( β ))β α 0 β < α monomorphism f ≤1 M f 1 M M M . Hence, since H is closed under − ( β+1)/ − ( β ) β+1/ β subobjects, this is an H-filtration of−→C . Similarly, we obtain an ascending filtra- 0 tion g M of C such that for each we have a canonical epimorphism ( ( β ))β α 00 β < α ≤ Mβ+1/Mβ g (Mβ+1)/g (Mβ ). Hence, since H is closed under quotient objects, this is an H-filtration−→ of C . 00 Corollary 2.3. In the lattice W of Serre localizing subcategories of C, we have that supi Wi = i Wi loc is given by the full subcategory of ( i Wi )-filtered objects. 〈∪ 〉 ∪

Proof. Observe that i Wi is closed under subquotients. Then the statement is a direct consequence of∪ Proposition 2.2.

Remark 2.4. Recall that two (localizing) Serre subcategories W1, W2 are compatible if W1 W2 = W2 W1 for the Gabriel product ∗ ∗ W1 W2 = C C W1 W1,W2 W2, 0 W1 C W2 0 . ∗ { ∈ | ∃ ∈ ∈ −→ −→ −→ −→ } and in this case W1 W2 is the smallest (localizing) Serre subcategory containing W1 and W2. Note that in∗ general, by Proposition 2.2 we have

W1 W2 W1 W2 loc, (2.7) ∗ ⊆ 〈 ∪ 〉 and we have equality if and only if W1 and W2 are compatible.

To end this section we describe the relation with orthogonal complements. We call an object C left orthogonal to an object D and D right orthogonal to C (notation 0 1 C D ) provided that ExtC(C ,D ) = 0 = ExtC(C ,D ). For a full subcategory H C, we obtain⊥ the following full subcategories of C: ⊆

H C C H C H H ; ⊥ = • { ∈ | ⊥ ∀ ∈ } H C C C H H H ; ⊥ = • { ∈ | ⊥ ∀ ∈ } Chapter 2. Tensor product of linear sites and Grothendieck categories 43 which are called the right orthogonal complement and the left orthogonal complement of H respectively. Note that in the literature, different variations upon the terminology are used involving either Ext0 or Ext1. The following statement for Ext1- orthogonals is known as Eklof’s lemma, see for instance [62, Lemma 6.3.1] in the context of Grothendieck categories. The corresponding statement for Ext0-orthogonals is trivially veriﬁed. Hence, in our terminology, we have:

Proposition 2.5. For a full subcategory H C, the left orthogonal H is closed under ⊥ H-ﬁltered objects. ⊆ ⊥ Corollary 2.6. Let H C be a full subcategory closed under subquotients. We have H H . ⊆ ( loc)⊥ = ⊥ 〈 〉 Proof. This follows from Propositions 2.2 and 2.5.

Corollary 2.7. For localizing subcategories W , we have sup W W . ( i )i ( i i )⊥ = i ⊥i ∩ Proof. This follows from Corollaries 2.3 and 2.6.

2.1.3 Equivalent approaches to localization

Let a be a linear category and let C be a Grothendieck category. Recall that a strict localization L C is a full subcategory which is closed under adding isomorphic objects, for which⊆ the inclusion functor i : L C has an exact left adjoint a : C L. Consider the following posets, ordered by inclusion:−→ −→

1. The poset T of linear topologies on a;

2. The poset W of localizing Serre subcategories of C;

3. The poset L of strict localizations of C.

It is well known that the data in (2) and (3) are equivalent, and for C = Mod(a) all three types of data are equivalent. Let us brieﬂy recall the isomorphisms involved. We have an order isomorphism between T and W , and dualities between T and L and between W and L respectively (the duality between W and L holds for arbitrary C). We use the following notations. For T , W and L are the associated T T localizing subcategory and the associated localization.T ∈ For W W , W and L are T the associated topology and the associated localization. For L∈ L, TL and WL are the associated topology and localizing subcategory. ∈ T We describe the involved constructions: 44 Chapter 2. Tensor product of linear sites and Grothendieck categories

Consider T . We have that L = Sh(a, ), which is a localization of Mod(a) • (see TheoremT ∈ 1.24). A module FT Mod(Ta) is called a null presheaf if for all x F A there exists R A such∈ that for all morphism f : A A in R A ( ) ( ) 0 ( 0) we∈ have F (f )(x ) = 0. Then∈ T W is the full subcategory of null presheaves.−→ T Consider W W . A subobject R a( , A) is in W(A) if and only if we have • a , A R W∈. We have L W .⊆ − T ( )/ W = ⊥ − ∈ Consider L L and let #: Mod(a) L be an exact left adjoint of the inclu- • sion L Mod∈ (a). We have W = Ker−→(#) = F Mod(a) #(F ) = 0 and a sieve ⊆ { ∈ | } r : R a( , A) is in L if and only if #(r ) is an isomorphism. −→ − T Let us ﬁrst consider the duality between W and L in an arbitrary Grothendieck category C. We obtain:

Proposition 2.8. Given a collection of strict localizations (Li )i in L, we have that infi Li = i Li . ∩

Proof. It sufﬁces that i Li is a strict localization, which follows from Corollary 2.7 L W ∩ W after writing i = ⊥i for the corresponding localizing subcategories i .

Next consider the order isomorphism between T and W for C = Mod(a). Since it respects suprema, we have:

Proposition 2.9. W sup W and sup . supi i = i i supi Wi = i Wi T T T T

2.1.4 The tensor product topology

Consider linear sites (a, a) and (b, b). In this section we deﬁne a topology = a b on c = a b, called theT tensor productT topology. For modules M ModT (Ta) andT ⊗ ∈ N Mod(b), we obtain M N Mod(a b) with (M N )(A, B) = M (A) k N (B). ∈ ⊗ ∈ ⊗ ⊗ ⊗ Consider objects A a, B b and covering sieves R a(A) and S b(B). We have a( , A) b( , B) = c( ∈,(A, B∈)) and we thus obtain a canonical∈ T morphism∈ T − ⊗ − − φR,S : R S c( ,(A, B)). ⊗ −→ − We deﬁne the tensor product sieve of R and S to be

R S = Im(φR,S ). n Concretely, any element in R S A , B can be written as P α β with α R A ( )( 0 0) i =1 i i i ( 0) and S B . Consider the following cover systems on a b: ⊗ ∈ βi ( 0) ∈ ⊗ Chapter 2. Tensor product of linear sites and Grothendieck categories 45

a = R b( , B) R a, B b ; •R { − | ∈ T ∈ } b = a( , A) S S b, A a ; •R { − | ∈ T ∈ } = R S R a,S b . •R { | ∈ T ∈ T } A direct check provides us the following two results. Lemma 2.10. Consider A, A a and B, B b and let R A and S B be cov- 0 0 a( ) b( ) ering sieves. Let h Pn f g∈ a A , A ∈b B , B c ∈A T, B , A, B be∈ T a morphism = i =1 i i ( 0 ) k ( 0 ) = (( 0 0) ( )) in c. We have that: ⊗ ∈ ⊗ n 1 n 1 1 ( i 1 fi− R) ( i 1gi− S) h − (R S). (2.8) ∩ = ∩ = ⊆ Lemma 2.11. Consider objects A a and B b, a covering sieve R a(A), and for every morphism h Pn f g ∈R b , B ∈A , B with A a and ∈B T b a covering = i =1 i i ( ( ))( 0 0) 0 0 sieve R A . Then we have:⊗ ∈ − ∈ ∈ h a( 0) ∈ T (R (R f 1)f ) b( , B) (R b( , B)) (Rh b( , B 0))h . (2.9) ◦ ⊗ − ⊆ − ◦ − Deﬁnition 2.12. The tensor product topology = a b on a b is the smallest topology containing a and b, that is T T T ⊗ R R a b = a b top. T T 〈R ∪ R 〉 The tensor product site of (a, a) and (b, b) is T T (a, a) (b, b) = (a b, a b) = (c, ). T T ⊗ T T T Proposition 2.13.

up up 1. The cover systems 1 = a and 2 = b are topologies. T R T R up 2. The cover systems and 1 2 are upclosed and localizing. R T ∪ T 3. The topoplogy is the smallest topology containing . T R upglue upglue 4. We have = and = ( a b) . T R T R ∪ R up Proof. The cover systems 1, 2, 1 2 and are localizing by Lemma 2.10 and 1 and 2 are topologies byT furtherT T invoking∪ T LemmaR 2.11. This proves (1) and (2). T T We prove (3). Trivially a b so it remains to show . For R S (A, B ), we consider R b , B R ∪RA,⊆B R. For every h Pn fR ⊆g T R b , B∈ RA , B , ( ) a( ) = i =1 i i ( ( ))( 0 0) observe that we have− a∈ R, A S h 1 R S , so R S ⊗ ∈A, B by− the glueing ( 0) − ( ) ( ) property. − ⊆ ∈ T Now (4) follows trivially from (3) and the deﬁnition of tensor product site. 46 Chapter 2. Tensor product of linear sites and Grothendieck categories

In the lattice T of topologies on a b, we have ⊗ a b = 1 2. (2.10) T T T ∨ T

2.1.5 Tensor product of localizing subcategories

Consider linear categories a and b with c = a b and full subcategories Wa Mod(a) ⊗ ⊆ and Wb Mod(b). ⊆ Consider the following full subcategories of Mod(c):

W1 = F Mod(c) F ( , B) Wa B b ; • { ∈ | − ∈ ∀ ∈ } W2 = F Mod(c) F (A, ) Wb A a . • { ∈ | − ∈ ∀ ∈ } Proposition 2.14. If Wa (resp. Wb) is closed under extensions, coproducts, subobjects or quotient objects, then so is W1 (resp. W2).

Proof. This is a direct consequence of the fact that limits and colimits are deﬁned point-wise in module categories.

We deﬁne the tensor product of localizing subcategories Wa and Wb to be

Wa Wb = W1 W2 loc. 〈 ∪ 〉 More precisely, in the lattice W of localizing subcategories in Mod(c), we thus have

Wa Wb = W1 W2. (2.11) ∨

2.1.6 Tensor product of strict localizations

Let a, b and c = a b be as above. Consider strict localizations ia : La , Mod(a) ⊗ −→ and ib : Lb , Mod(b) and denote by #a : Mod(a) La and #b : Mod(b) Lb the corresponding−→ exact left adjoints. Consider the following−→ full subcategories−→ of Mod(c):

L1 = F Mod(c) F ( , B) La B b ; • { ∈ | − ∈ ∀ ∈ } L2 = F Mod(c) F (A, ) Lb A a . • { ∈ | − ∈ ∀ ∈ } The natural functors Chapter 2. Tensor product of linear sites and Grothendieck categories 47

#1 : Mod(c) L1 : F (#1(F ) : (A, B) (#a(F ( , B)))(A)); • −→ 7−→ 7−→ − #2 : Mod(c) L2 : F (#2(F ) : (A, B) (#b(F (A, )))(B)) • −→ 7−→ 7−→ − are easily seen to be exact left adjoints of the inclusions i1 : L1 Mod(c) and −→ i2 : L2 Mod(c) respectively. −→ We deﬁne the tensor product localization of La and Lb to be

La Lb = L1 L2, ∩ which is a strict localization by Proposition 2.8. In the lattice L of strict localizations of Mod(c), we thus have La Lb = L1 L2. (2.12) ∧

2.1.7 Relation between the three tensor products

Let a, b and c = a b be as before. Suppose a is endowed with a topology a, a localizing ⊗ T subcategory Wa and a strict localization La (with left adjoint #a : Mod(a) L) which correspond as in §2.1.3, an similarly b is endowed with a topology b−→, a localizing T subcategory Wb and a strict localization Lb (with left adjoint #b : Mod(b) L) which −→ also correspond. Our aim in this section is to show that = a b, W = Wa Wb T T T and L = La Lb (with left adjoint # : Mod(c) L) correspond as well. −→ Let us ﬁrst look at the relation between W and L. We ﬁrst note the following:

Proposition 2.15. The localizing subcategory W1 (resp. W2) and the strict localization L1 (resp. L2) correspond under the isomorphism between W and L.

Proof. Consider Wa = Ker(#a) and Wb = Ker(#b) and let W1, W2 be as deﬁned in §2.1.5. By direct inspection, we have Ker(#1) = W1 and Ker(#2) = W2.

Corollary 2.16. The localizing subcategory Wa Wb and the strict localization La Lb correspond under the isomorphism between W and L.

Proof. This follows from Proposition 2.15 and equations (2.11) and (2.12).

Next we look at the relation between and W. Again, we must ﬁrst establish the relation between 1 and W1 and betweenT 2 and W2. T T Proposition 2.17. The topology 1 and the localizing subcategory W1 (resp. the topology 2 and the localizing subcategoryT W2) correspond under the isomorphism between T andT L. 48 Chapter 2. Tensor product of linear sites and Grothendieck categories

Proof. It suffices to show the following inclusions: (1) 1 W1 , (2) W1 W 1 . T ⊆ T ⊆ T We first prove (1). Given R a(A) and B b, we consider the covering sieve ∈ T ∈ R b( , B) 1(A, B). The exact sequence − ∈ T R a( , A) a( , A)/R 0 −→ − −→ − −→ gives rise, by tensoring with b( , B), to the exact sequence − R b( , B) c( ,(A, B)) (a( , A)/R) b( , B) 0. ⊗ − −→ − −→ − ⊗ − −→ Hence we have that F = c( ,(A, B))/(R b( , B)) (a( , A)/R) b( , B). It suffices ∼= to show that F W . Now− we have Z −a , A R − W and⊗ for− every B b, 1 = ( )/ a 0 F , B Z b ∈B , B is a colimit of objects in− W , hence∈ it is itself in W as∈ de- ( 0) = ( 0 ) a a sired.− ⊗

For (2), consider F W1 and x F (A, B) = F ( , B)(A). Since F ( , B) Wa, there exists R such that∈ for all f R∈A we have that− − ∈ a ( 0) ∈ T ∈

F (f 1) : F (A, B) F (A0, B) : x 0. ⊗ −→ 7−→ It now sufﬁces to consider R b( , B) 1(A, B). Observe that for every element h P f g P 1 g f 1− R∈ T b , B A , B , we have F h x 0 as = i i i = i ( i )( i ) ( ( ))( 0 0) ( )( ) = desired. ⊗ ⊗ ⊗ ∈ −

Corollary 2.18. The topology a b and the localizing subcategory Wa Wb correspond under the isomorphismT betweenT T and W .

Proof. This follows from Proposition 2.17 and equations (2.10) and (2.11).

2.1.8 Exact categories

The following setup from [36] inspired our deﬁnitions. Let a be an exact category in the sense of Quillen, and let Lex(a) Mod(a) be the op full subcategory of left exact functors, i.e. a-modules F : a Mod⊆(k) which send conﬂations −→ 0 K D C 0 −→ −→ −→ −→ to short exact sequences

0 F (C ) F (D ) F (K ). −→ −→ −→ Chapter 2. Tensor product of linear sites and Grothendieck categories 49

The category a can be endowed with the single deflation topology, for which a sieve is covering if and only if it contains a deflation for the exact structure. The category of sheaves for this topology is precisely Lex(a), and the corresponding localizing Serre subcategory Wa is the category of weakly effaceable modules, that is, modules M Mod a such that for every x M A there exists a deflation A A for which ( ) ( ) 0 M ∈A M A maps x to zero. ∈ −→ ( ) ( 0) −→ Now let a and b be two exact categories. In [36, §2.6] the authours introduce the full op op category Lex(b,a ) Mod(b a ) of bimodules M for which every M ( , A) Lex(b) op op and every M (B, ) ⊆Lex(a )⊗for the natural “dual” exact structure on a− . With∈ the definition from §−2.1.6∈ , we thus have

op op Lex(b,a ) = Lex(b) Lex(a ) and additionally, in [36, Prop. 2.22], the relation with the localizing Serre subcategory WbWaop from Corollary 2.16 was shown using the description of the Gabriel product. op In [36], it is argued that Lex(b,a ) is the correct bimodule category to consider between exact categories, where one looks at bimodules contravariant in B b and covariant in A a. In particular, it is shown that over a ﬁeld k, Hochschild cohomology∈ of a in the sense∈ of [38], and of Lex(a) in the sense of [55] is equal to

n n HH a Ext op 1 ,1 . ( ) = Lex(a,a )( a a) It is not clear how this approach could be extended to more general sites, as it makes essential use of the existence of a natural “dual site” for the site associated to an exact category, which is not readily available for more general sites.

2.2 Functoriality of the tensor product of linear sites

Let a and b be k-linear categories as before. Let us return to the starting point for our quest for a tensor product between Grothendieck abelian categories, namely the requirement that Mod(a) Mod(b) = Mod(a b). (2.13) ⊗ Using the 2-categorical structure of the category Cat(k) of k-linear categories, functors and natural transformations, it is not hard to see that can be defined based upon (2.13) for module categories C. A module category C is intrinsically characterized by the existence of a set of finitely generated projective generators, and different choices of generators give rise to Morita equivalent linear categories. For Morita bimodules M between a and a and N between b and b , it is easily seen that the 0 0 50 Chapter 2. Tensor product of linear sites and Grothendieck categories tensor product M N given by M N A, B , A , B M A, A N B, B defines (( ) ( 0 0)) = ( 0) k ( 0) a Morita bimodule⊗ between a b and⊗ a b . Our aim in this section⊗ is to develop the 0 0 necessary tools in order to extend⊗ this situation⊗ from module categories to arbitrary Grothendieck categories. Rather than focussing on bimodules, we first focus on functors between sites. The underlying idea is that any equivalence between sheaf categories can be represented by a roof of LC functors between sites, where an LC functor is a particular kind of functor which induces an equivalence between sheaf categories. Roughly speaking, an LC functor φ : (a, a) (c, c) is generating with respect to , fully faithfull up to , and has 1 T −→(DefinitionT 2.19). The main c a φ− c = a result of thisT section is that LC functorsT are preservedT T under tensor product of sites (Proposition 2.30).

2.2.1 LC functors

Consider k-linear categories a and c endowed with cover systems a and c respectively. Suppose a and c are localizing systems on the respectiveT categories.T We point the reader toT §1.1.1T for the definitions of continuous and cocontinuous functors φ : (a, a) (c, c). T −→ T Next we recall some special conditions (see [52, §2.5]), which are a “relativization” of the conditions with the same name already exposed in §1.1.3 (see [52, §1]). Definition 2.19. Consider a k-linear functor φ : a c. −→ 1. Suppose c is endowed with a cover system c. We say that φ : a (c, c) satisfies T −→ T

(G) if for every C c there is a covering family (φ(Ai ) C )i for c. ∈ −→ T 2. Suppose a is endowed with a cover system a. We say that φ : (a, a) c satisﬁes T T −→

(F) if for every c : A A in c there exists a covering family a : A A φ( ) φ( 0) i i for and f : A −→A with c a f ; −→ a i i 0 φ( i ) = φ( i ) (FF) if forT every a : A −→ A in a with a 0 there exists a covering family 0 φ( ) = −→ ai : Ai A for a with aai = 0. −→ T 3. Suppose a and c are endowed with cover systems a and c respectively. We T T say that φ : (a, a) (c, c) satisﬁes T −→ T (LC) if φ satisﬁes (G) with respect to c, (F) and (FF) with respect to a, and we further have 1 , i.e.T A R a , A R T A , a = φ− c a( ) = ( ) φ( ) c(φ( )) T T T { ⊆ − | ∈ T } where if R = ri , we have that φ(R) = φ(ri ) (see [52, §2.5]). 〈 〉 〈 〉 Chapter 2. Tensor product of linear sites and Grothendieck categories 51

We have the following “Lemme de comparaison” (see [76], [51], [52]):

Theorem 2.20. If φ : (a, a) (c, c) satisﬁes (LC) and c is a topology, then a is a T −→ T T T topology, φ is continuous and φs : Sh(c, c) Sh(a, a) is an equivalence of categories. T −→ T Remark 2.21. In particular, by [52, Lem. 2.15], we have that φ is also cocontinuous s and an easy check shows that φe = φ , which is the quasi-inverse of φs . ∗ ∼ The following lemma, which is easily proven by induction, will be used later on:

Lemma 2.22. Suppose the functor φ : (a, a) (c, c) satisﬁes (F) and a is a topology and consider morphisms c : A T −→A forTi 1,...,n. There existsT a collec- i φ( ) φ( 0) = −→ tion of morphisms (hj : A j A)j J with hj a and a collection of morphisms g : A A such−→ that c∈ h 〈 〉g ∈ Tfor all i 1,...,n and j J. ( i j j 0)1 i n,j J i φ( j ) = φ( i j ) = −→ ≤ ≤ ∈ ∈

2.2.2 Tensor product of functors

Let a, b, c and d be k-linear categories and φ : a c and ψ : b d be k-linear functors. Consider the tensor product functor −→ −→

φ ψ : a b c d. ⊗ ⊗ −→ ⊗ Notice that we have that φ ψ = (1 ψ)(φ 1) for ⊗ ⊗ ⊗ φ 1 : a b c b ⊗ ⊗ −→ ⊗ and 1 ψ : c b c d. ⊗ ⊗ −→ ⊗ Suppose a, b, c and d are cover systems on the respective categories. T T T T Proposition 2.23. Suppose all cover systems are localizing. If φ and ψ are continuous, then so is φ ψ. ⊗ Proof. We have to look at the functor : Mod c d Mod a b . According (φ ψ)∗ ( ) ( ) ⊗ ⊗ −→ ⊗ to §2.1.6, §2.1.7 we have F Sh(c d, c d) if and only if F ( , ) is a sheaf in both variables for and respectively.∈ ⊗ ItT readilyT follows that − − F F , c d (φ ψ)∗ = (φ( ) ψ( )) is a sheaf in theT ﬁrstT variable for a as soon as φ is continuous,⊗ and a sheaf− in the− second variable for b as soon as Tψ is continuous. T The following is easy to check: 52 Chapter 2. Tensor product of linear sites and Grothendieck categories

Lemma 2.24. Consider A a, B b, and sieves R a( , A) and S b( , B). As sieves on (φ(A),ψ(B)), we have ∈ ∈ ⊆ − ⊆ −

φ(R) ψ(S) = (φ ψ)(R S). ⊗ Proposition 2.25. If φ and ψ are cocontinuous, then so is φ ψ. ⊗ Proof. Since cocontinuous functors are stable under composition, it suffices to consider ψ = 1 : (b, b) (b, b). We show that T −→ T φ 1 : (a b, a b) (c b, c b) ⊗ ⊗ T T −→ ⊗ T T is cocontinuous. By [52, Lem. 2.12], it suffices to show that φ 1 is cocontinuous ⊗ up with respect to the localizing cover systems La,b = R S R a,S b and up { | ∈ T ∈ T } Lc,b = T S T c,S b . Thus, consider T S with T c(φ(A)) and S b(B ). By the{ assumption| ∈ T there∈ T exists} R a with φR T . Consequently,∈ T by Lemma∈ T 2.24 ∈ T ⊆ we have (φ 1)(R S) T S as desired. ⊗ ⊆ Lemma 2.26. Suppose the functor φ (resp. ψ) satisfies (G) with respect to c (resp. T d). Then the functor φ ψ satisfies (G) with respect to c d. T ⊗ T T

Proof. Consider (C ,D ) c d. There exists R = fi : φ(Ai ) C c(C ) and ∈ ⊗ 〈 −→ 〉 ∈ T S = g j : ψ(Bj ) D d(D ). It is easily seen that 〈 −→ 〉 ∈ T R S = fi g j : (φ ψ)(Ai , Bj ) (C , B) . 〈 ⊗ ⊗ −→ 〉

Lemma 2.27. Suppose the functor φ (resp. ψ) satisﬁes (F) with respect to a (resp. b). T T Then the functor φ ψ satisﬁes (F) with respect to a b. ⊗ T T

Proof. We prove it ﬁrst for the case in which ψ = 1 : (b, b) (b, b). n T −→ T Consider a morphism h P c b : φ A , B φ A , B . We proceed by = i =1 i i ( ( ) ) ( ( 0) 0) induction. ⊗ −→

By hypothesis, we have a collection (aα : Aα A)α of morphisms in a with a A (and hence a 1 A, B )−→ and a collection a : A A α a( ) α B a b( ) ( α0 α 0)α such〈 〉 ∈ that T 1 a b 〈 c⊗ b〉 ∈ T aT 1 . −→ (φ )( α0 1) = ( 1 1) (φ( α) B ) ⊗ ⊗ ⊗ ◦ ⊗ Suppose now we have a collection (aα : Aα A)α of morphisms in a such that −→ aα a(A) (and hence aα 1B a b(A, B )) and for i 1,...,m 1 with m n 〈and〉 ∈, T there exists g i : A〈 ⊗ A 〉such ∈ T thatT ∈ { − } ≤ α α α 0 −→ i (φ 1)(gα bi ) = (ci bi ) (φ(aα) 1B ). ⊗ ⊗ ⊗ ◦ ⊗ Chapter 2. Tensor product of linear sites and Grothendieck categories 53

We show that the same holds for i = m. To this end, we consider

(cm bm ) (φ(aα) 1B ) = cm φ(aα) bm . ⊗ ◦ ⊗ ⊗ For c a : A A , since by hypothesis satisﬁes (F) there is a collection m φ( α) φ( α) φ( 0) φ a α : Aα A with−→a α A and there exist morphisms f α : Aα A with ( β β α)β β a( α) β β 0 α −→ α 〈 〉 ∈ T α −→α φ(fβ ) = cm φ(aα)φ(aβ ). Hence, the collection of compositions (aαaβ : Aβ A)α,β α α −→ are such that aαaβ a(A) by the glueing property (thus aαaβ 1B a b(A, B )). 〈 〉 ∈ T i α 〈 ⊗ 〉 ∈ T αT For i 1,...,m 1 , we have (φ 1)((gα bi ) (aβ 1B )) = (ci bi ) (φ(aαaβ ) 1B ) by ∈ { − } ⊗ ⊗ ◦ ⊗α ⊗ ◦ ⊗α the induction hypothesis. For m, we have (φ 1)(fβ bm ) = (cm bm ) (φ(aαaβ ) 1B ) as desired. ⊗ ⊗ ⊗ ◦ ⊗ Now, observe that the same argument applies for 1 ψ : c b c d. ⊗ ⊗ −→ ⊗ Consider now a morphism h Pn c d : A , B A , B . Then, = i 1 i i (φ( ) ψ( )) (φ( 0) ψ( 0)) = ⊗ −→ combining both results, one can see that there exist covering families (aα : Aα A)α −→ in a(A) and (bβ : Bβ B)β in b(B) and for all 1 i n and all α,β collections f i T: A A and g i :−→B B suchT that ≤ ≤ α α 0 β β 0 −→ −→ n X i i (φ ψ)( fα gβ ) = h (φ ψ)(aα bβ ). ⊗ i =1 ⊗ ◦ ⊗ ⊗

But it is easy to see that aα bβ = aα bβ a b(A, B), which concludes the argument. 〈 ⊗ 〉 〈 〉 〈 〉 ∈ T T

Lemma 2.28. Suppose the functor φ (resp. ψ) satisﬁes (F) and (FF) with respect to a T (resp. b). Then the functor φ ψ satisﬁes (F) and (FF) with respect to a b. T ⊗ T T

Proof. We ﬁrst prove the statement for the case in which ψ = 1 : (b, b) (b, b). T −→ T Consider a morphism h Pn a b : A, B A , B such that = i 1 i i ( ) ( 0 0) = ⊗ −→ n X 0 = (φ 1)(h) = φ(ai ) bi . ⊗ i =1 ⊗ Let c be a collection of generators of the k-module c A , A such that ( λ)λ Λ (φ( ) φ( 0)) ∈ 1,...,n Λ and ci = φ(ai ) for i 1,...,n . Put bλ = 0 for λ Λ 1,...,n . We thus have{ 0 } ⊆P c b c A , ∈A { b B} , B . According to∈ 26\{, Lem. 6.4} , there = λ Λ λ λ (φ( ) φ( 0)) ( 0) [ ] exists a family∈ b¯ ⊗ of∈ elements b¯ ⊗b B, B , and a family of elements ( j )j J j ( 0) (κλ,j )λ Λ,j J ∈ ∈ ∈ ∈ κλ,j k that contains only ﬁnitely many non-zero elements, such that the following hold:∈ 54 Chapter 2. Tensor product of linear sites and Grothendieck categories

P ¯ 1. bλ = j κλ,j bj for all λ Λ; ∈ n 2.0 P κ c P κ φ a P κ c for all j J . = λ λ,j λ = i =1 i ,j ( i ) + λ Λ 1,...,n λ,j λ ∈ \{ } ∈ Using property (F) for φ, we will ﬁrst realize the right hand side of (2) as being in the image of φ up to a covering. Let Λ0 Λ be the subset that contains those λ’s ⊆ for which there exists j J with κλ,j = 0. Hence Λ0 is ﬁnite. By Lemma 2.22, there exists a collection h : A∈ A with6 h A and g : A A such that ( σ σ )σ Σ σ a( ) λ,σ σ 0 −→ ∈ 〈 〉 ∈ T −→ cλφ(hσ) = φ(gλ,σ) for λ Λ0 and σ Σ. Further, we may clearly suppose that ∈ ∈ gi ,σ = ai hσ (2.14) for i = 1,...,n. Hence, for j J and σ Σ, from (2) we obtain: ∈ ∈ X X 0 = κλ,j cλφ(hσ) = φ( κλ,j gλ,σ). λ λ

σ σ Using (FF) for φ, for every σ Σ we obtain a collection hω : Aω Aσ for ω Ωσ σ ∈ −→ ∈ with hω a(Aσ) such that for every ω Ωσ 〈 〉 ∈ T ∈ X σ 0 = κλ,j gλ,σhω . (2.15) λ

σ σ Now consider the collection hσhω : Aω A for σ Σ and ω Ωσ. By the glueing σ −→ σ∈ ∈ property we have hσhω a(A) and hence hσhω 1B a b(A, B). We claim that h becomes zero〈 on this〉 ∈ T covering sieve of〈(A, B).⊗ We have〉 ∈ T T

n n X X X ¯ h = ai bi = ( κi ,j ai ) bj . i =1 ⊗ j i =1 ⊗ We compute n σ X X σ ¯ h (hσhω 1) = ( κi ,j ai hσhω bj ). ◦ ⊗ j J i =1 ⊗ ∈ Consider the expressions

X X σ ¯ x = ( κλ,j gλ,σhω bj ) j J λ Λ ⊗ ∈ ∈ and

X X σ ¯ X σ X ¯ y = ( κλ,j gλ,σhω bj ) = (gλ,σhω κλ,j bj ). j J λ Λ 1,...,n ⊗ λ Λ 1,...,n ⊗ j J ∈ ∈ \{ } ∈ \{ } ∈ Chapter 2. Tensor product of linear sites and Grothendieck categories 55

Using equation (2.14), we clearly have

σ x = h(hσhω 1) + y. ⊗ By equation (2.15), we have x = 0. By deﬁnition and by condition (2) above, for P ¯ λ Λ 1,...,n , we have 0 = bλ = j J κλ,j bj so also y = 0. We conclude that ∈ \{σ } ∈ h(hσhω 1) = 0 as desired. ⊗ Observe that the same argument applies for 1 ψ : c b c d. ⊗ ⊗ −→ ⊗ Consider now a morphism h Pn a b : A, B A , B such that = i 1 i i ( ) ( 0 0) = ⊗ −→ n X 0 = (φ ψ)(h) = φ(ai ) ψ(bi ). ⊗ i =1 ⊗

Then, combining both results, there exist covering families (aα : Aα A)α in −→ a(A) and (bβ : Bβ B)β in b(B) such that h (aα bβ ) = 0. But we have that T −→ T ◦ ⊗ aα bβ = aα bβ a b(A, B), which concludes the argument. 〈 ⊗ 〉 〈 〉 〈 〉 ∈ T T Lemma 2.29. If the functors φ : (a, a) (c, c) and ψ : (b, b) (d, d) both satisfy T −→ T T −→ T (LC), then so does the functor φ ψ : (a b, a b) (c d, c d). ⊗ ⊗ T T −→ ⊗ T T Proof. By Lemmas 2.26, 2.27, and 2.28, φ ψ satisfies (G), (F) and (FF). We have 1 and 1 , and it remains⊗ to show that a = φ− c b = ψ− d T T T T 1 a b = (φ ψ)− ( c d). T T ⊗ T T To prove the inclusion 1 , it suffices to look at a sieve a b (φ ψ)− ( c d) T T ⊆ ⊗ T T R S with φ(R) c and ψ(S) d. It immediately follows from Lemma 2.24 that ∈ T ∈ T (φ ψ)(R S) c d. ⊗ ∈ T T For the other inclusion, by [52, Prop. 2.16], it suffices to show that

φ ψ : (a b, a b) (c b, c b) ⊗ ⊗ T T −→ ⊗ T T is cocontinuous. By [52, Lem. 2.15], φ and ψ are cocontinuous, whence it follows by Proposition 2.25 that φ ψ is cocontinuous as desired. ⊗ We can summarize the results obtained above as follows:

Proposition 2.30. If the functors φ : (a, a) (c, c) and ψ : (b, b) (d, d) both satisfy (G) (resp. (F), resp. (F) and (FF)T ,−→ resp. (LC)T ), then so doesT the−→ functorT φ ψ : (a b, a b) (c d, c d). ⊗ ⊗ T T −→ ⊗ T T 56 Chapter 2. Tensor product of linear sites and Grothendieck categories

2.3 Tensor product of Grothendieck categories

Based on the previous sections, we are finally in position to define the tensor product of Grothendieck categories C = Sh(a, a) and D = Sh(b, b) to be given by ∼ T ∼ T C D = Sh(a b, a b). ⊗ T T This will be done in in §2.3.1 below. Functoriality of the tensor product of sites from §2.2.2 ensures that C D is well-defined up to equivalence of categories. An alternative approach to proving well-definedness is available in terms of the already existing tensor product of locally presentable categories. It will be analysed in next section (see §2.4.4). The remainder of this section is devoted to an application of our tensor product to Z-algebras and schemes. In §2.3.2 we provide a nice realization of the tensor product of Z-algebras, while in §2.3.3, we show that given projective schemes X and Y we have Qch(X ) Qch(Y ) = Qch(X Y ). (2.16) × We actually show this result holds true for noncommutative projective schemes, and our proof is actually based upon the results in §2.3.2. Here, we use Z-algebras as models for noncommutative schemes following [6], [83], [73], [20], and to a Z-algebra a we can associate a certain category Qch(a) which replaces the quasicoherent sheaves, and which is obtained as a linear sheaf category with respect to a certain topology. For two Z-algebras a and b generated in degree 1, there is a naturally associated diagonal Z-algebra (a b)∆, for which we show that ⊗ Qch(a) Qch(b) = Qch((a b)∆). (2.17) ∼ ⊗ The relation between (2.16) and (2.17) is provided by graded algebras (generated in degree 1), which on the one hand are used to represent schemes through the Proj construction, and which on the other hand give rise to associated Z-algebras.

2.3.1 Tensor product of Grothendieck categories

Let C be a k-linear Grothendieck category and let (a, a) be a k-linear site. We say T that a k-linear functor u : (a, a) C satisﬁes (LC), or is an LC morphism provided T −→ that u : (a, a) (C, C) satisﬁes (LC) where C is the topology of jointly epimorphic sieves. ThatT is,−→R T C if and only if L T C C is an epimorphism. C( ) (f :C f C ) R f Observe that in this∈ case T properties (G), (F) and −→(FF)∈take the−→ form in which they were presented in §1.1.3. The general Gabriel-Popescu theorem (Theorem 1.27) states that Chapter 2. Tensor product of linear sites and Grothendieck categories 57 for u 1 , we have that u is an LC morphism if and only if is a topology and a = − C a T T T u gives rise to an equivalence C Sh(a, a) (see [51]). −→ T Consider k-linear Grothendieck categories C and D.

Proposition 2.31. Consider LC morphisms u : (a, a) C, u : (a , a ) C and 0 0 0 v : b, D, v : b , D. There exists an equivalenceT −→ of categoriesT −→ ( b) 0 ( 0 b ) T −→ T 0 −→

Sh(a b, a b) = Sh(a0 b0, a b ). ⊗ T T ∼ ⊗ T 0 T 0 Proof. Consider c C be the full subcategory with ⊆

Obj(c) = u(A) A a u 0(A) A0 a0 . { | ∈ } ∪ { | ∈ } Similarly, consider d D be the full subcategory with ⊆

Obj(d) = v (B) B b v 0(B) B 0 b0 . { | ∈ } ∪ { | ∈ } Observe that both c and d are small categories. Put i 1 for the inclusion c = − C i : c C and the canonical topology on C andT put T j 1 for the in- C d = − D clusion−→j : d D and the canonical topologyT D on D.T Consequently,T the induced functors−→u¯ : a, c, , u¯ : a , a T c, , v¯ : b, d, , ( a) ( c) 0 ( 0 0) ( c) ( b) ( d) v¯ : b , d, areT all−→ LC morphisms.T T By Proposition−→ T 2.30,T it follows−→ thatT 0 ( 0 b ) ( d) T 0 −→ T u¯ v¯ : (a b, a b) (c d, c d) and u¯ v¯ : (a b , a b ) (c d, c d) 0 0 0 0 0 0 are⊗ LC morphisms,⊗ T T and−→ in⊗ particularT T we have⊗ equivalences⊗ T of categoriesT −→ ⊗ T T

Sh(a b, a b) = Sh(c d, c d) = Sh(a0 b0, a b ). ⊗ T T ∼ ⊗ T T ∼ ⊗ T 0 T 0

Thanks to Proposition 2.31, we can now make the following deﬁnition:

Deﬁnition 2.32. Consider Grothendieck categories C and D. The tensor product CD is the following Grothendieck category, deﬁned up to equivalence of categories: given arbitrary LC morphisms u : (a, a) C and v : (b, b) D, we put T −→ T −→ C D = Sh(a b, a b). ⊗ T T

Consider Grothendieck categories C = Mod(a) and D = Sh(b, b). In this case we have T C D = F Mod(a b) F (A, ) D for all A a . { ∈ ⊗ | − ∈ ∈ } 58 Chapter 2. Tensor product of linear sites and Grothendieck categories

op From the equivalence Mod(a b) = Funk (a ,Mod(b)), we deduce an equivalence ⊗ ∼ op Mod(a) D Funk (a ,D). ∼= Let R S be a ring homomorphism between coherent commutative rings. In [54], given−→ an S-linear abelian category A, an abelian R-deformation of A is by definition an R-linear abelian category B, flat in an appropriate sense, such that FunR (S,B) A ∼= as S-linear categories. Hence, if A and B are Grothendieck categories, this formula can be reinterpreted as Mod(S) R B A, ∼= relating the deformation theory of Grothendieck categories to the tensor product in Definition 2.32.

2.3.2 Tensor product of Z-algebras

Recall that a Z-algebra is a linear category a with Obj(a) = Z. We further suppose that a is positively graded, that is a(n,m) = 0 for n < m. Following [20], we consider the sieves a( ,m) n a( ,m) for n m Z with − ≥ ⊆ − ≥ ∈ ¨ a(l ,m) if l n a(l ,m) n = ≥ ≥ 0 otherwise and we consider the tails localizing system

up Ltails = a( ,m) n n m { − ≥ | ≥ } and the tails topology upglue tails = Ltails . T Remark 2.33. In many cases of interest, we have Ltails = tails. This is the case for a T noetherian Z-algebra or for a connected, ﬁnitely generated Z-algebra in the sense of [20].

For a,b positively graded Z-algebras, we deﬁne the diagonal Z-algebra c = (a b)∆ with ⊗ c(n,m) = (a b)((n,n),(m,m)) = a(n,m) b(n,m). ⊗ ⊗ There is a corresponding fully faithful functor

∆ : c a b : n (n,n). −→ ⊗ 7−→ Chapter 2. Tensor product of linear sites and Grothendieck categories 59

Let La, Lb, Lc denote the tails localizing systems on a, b and c respectively, and let a, b, c denote the corresponding tails topologies. Further, consider the following coverT T systemT on a b: ⊗ up La b = R S R La,S Lb ⊗ { | ∈ ∈ } A direct check shows that:

Lemma 2.34. The cover system La b is localizing and upclosed and we have ⊗ upglue a b = La b . T T ⊗ Proposition 2.35. The functor ∆ : (c, c) (a b, a b) is cocontinuous and we have T −→ ⊗ T T 1 ∆− ( a b) c. T T ⊆ T Proof. The second claim follows from the first one by [52, Prop. 2.16] According to [52, Lem. 2.12], it suffices to prove the statement for Lc on c and La b on a b. Hence, consider m c and ∆(m) = (m,m) a b, and a covering sieve ⊗ ⊗ ∈ ∈ ⊗ a b L R = ( ,m) n1 ( ,m) n2 a b(m,m). − ≥ − ≥ ∈ ⊗ For n = max(n1,n2), consider S = c( ,m) n Lc. We have − ≥ ∈ a b a b S(l ) = (l ,m) n (l ,m) n (l ,m) n1 (l ,m) n2 = R(l ,l ) ≥ ⊗ ≥ ⊆ ≥ ⊗ ≥ so ∆S R as desired. ⊆ In order to improve upon Proposition 2.35, we look at generation of Z-algebras in the sense of [20]. Definition 2.36.

1. A linear category a is generated by subsets X A, A a A, A if every element ( 0) ( 0) of a can be written as a linear sum of products of elements⊆ in X .

2. A Z algebra a is generated in certain degrees D N if it is generated by X with ⊆ X (n,m) = ∅ unless n m D . − ∈ 3. A Z-algebra a is ﬁnitely generated if it is generated by X such that for all m the set S X m d ,m is ﬁnite. d N ( + ) ∈ 4.A -algebra a is connected if a(n,n) k for all n. Z ∼= 60 Chapter 2. Tensor product of linear sites and Grothendieck categories

We make the following observation:

Proposition 2.37. Consider Z-algebras a and b and put c = (a b)∆. ⊗

1. If a is generated by Xa and b is generated by Xb, then c is generated by Xc with Xc(n,m) = Xa(n,m) Xb(n,m). × 2. If a and b are generated in degrees D (resp. ﬁnitely generated, resp. connected), then so is c.

Remark 2.38. It was shown in [20] that for a connected, ﬁnitely generated Z-algebra a, we have Ltails = tails. T Lemma 2.39. Suppose the Z-algebras a and b are generated in degree 1. Then, the functor ∆ : c (a b, a b) satisﬁes (G). −→ ⊗ T T

Proof. Consider (m1,m2) a b. Suppose for instance that m2 m1. Consider a b ∈ ⊗ ≥ the cover ( ,m1) m2 ( ,m2) a b(m1,m2). We claim that this cover is gen- − ≥ − ∈ T T erated by the morphisms x 1 a(m2,m1) b(m2,m2) from the diagonal element ⊗ ∈ ⊗ ∆(m2) = (m2,m2) to (m1,m2). Indeed, consider an element a b a(l1,m1) b(l2,m2) with l m , by the hypothesis on a we can write a Pk a⊗a ∈for certain⊗ elements 1 2 = i =1 i0 i00 a a m≥ ,m and a a l ,m . Hence, we have a b Pk a 1 a b , which i0 ( 2 1) i00 ( 1 2) = i =1( i0 )( i00 ) concludes∈ the argument.∈ ⊗ ⊗ ⊗

Lemma 2.40. Suppose the Z-algebras a and b which are generated in degree 1. Then we have that 1 . c ∆− ( a b) T ⊆ T T Proof. Observe that trivially satisfies (F) and (FF) with respect to 1 as ∆ ∆− ( a b) T T it is a fully faithful functor. Additionally, ∆ satisfies (G) with respect to a b by Lemma 2.39. It follows by 52, Thm. 2.13 that the cover system 1 Tis a topol-T [ ] ∆− ( a b) ogy. Hence, to prove the desired inclusion, it suffices to show that LT T 1 . c ∆− ( a b) ⊆ T T Consider S = c( ,m) n Lc(m). We show that ∆S a b(m,m). Now ∆S is generated by the morphisms− ≥ ∈ in a(l ,m) b(l ,m) for l ∈n. T We claimT that ⊗ ≥ ∆S = a( ,m) n b( ,m) n . − ≥ − ≥ Indeed, let a b be an element in a(l1,m) b(l2,m) with l1,l2 n. Assume, for instance, that⊗l l . Then, by the hypothesis⊗ on b, we can write ≥b Pk b b for 2 1 = i =1 i0 i00 b ≥ b Pk bi0 (l1,m) and bi00 (l2,l1) and hence a b = i 1(a bi0)(1 bi00) ∆S. ∈ ∈ ⊗ = ⊗ ⊗ ∈ The main result of this section is the following: Chapter 2. Tensor product of linear sites and Grothendieck categories 61

Theorem 2.41. Consider Z-algebras a and b which are generated in degree 1. The functor : c, a b, satisﬁes (LC). In particular, we have 1 ∆ ( c) ( a b) c = ∆− ( a b) and T −→ ⊗ T T T T T Sh(a, a) Sh(b, b) = Sh(a b, a b) = Sh(c, c). T T ⊗ T T ∼ T Proof. This follows from Proposition 2.35 and Lemmas 2.39 and 2.40.

2.3.3 Quasicoherent sheaves on projective schemes

Next we apply the results of §2.3.2 to graded algebras and schemes. A graded algebra A A is an algebra A L A with 1 A and multiplication de- = ( n )n N = n N n 0 ∈ ∈ ∈ termined by An Am An+m . Such an algebra has an associated Z-algebra a(A) ⊗ −→ with a(A)(n,m) = An m . The algebra A is generated in degrees D N (resp. finitely − ⊆ generated, resp. connected) if and only if the associated Z-algebra a(A) is. Now if A is a finitely generated, connected graded algebra, the category Gr(A) of graded A-modules has a localizing subcategory Tors(A) of torsion modules, and one obtains the quotient category Qgr(A) = Gr(A)/Tors(A). By Serre’s theorem [71], if A is commutative with associated projective scheme Proj(A), we have Qch(Proj(A)) = Qgr(A). The category Qgr(A) has been generalized to certain classes of Z-algebras in [73], [83], [64] and in [20], the category Sh(a, tails) is introduced as a further generalization to T arbitrary Z-algebras. In particular, for a finitely generated connected graded algebra A, we have Qgr(A) = Sh(a(A), tails) (2.18) ∼ T

Remark 2.42. Provided (2.18), one notices that Theorem 2.41 above is a particular case of [67, Thm 3.5]. Observe that the proof of the theorem in loc.cit. does not rely on linear topologies. Nevertheless, we expect that an alternative proof along the lines of that of Theorem 2.41 is also available.

Next we turn to tensor products. For two graded algebras A and B, the cartesian product A cart B is deﬁned by (A cart B)n = An Bn . We clearly have × × ⊗ a(A cart B) = (a(A) a(B))∆. (2.19) × ⊗ Theorem 2.43.

1. For two graded algebras A and B which are connected and ﬁnitely generated in degree 1, we have Qgr(A) Qgr(B) = Qgr(A cart B). × 62 Chapter 2. Tensor product of linear sites and Grothendieck categories

2. For two projective schemes X and Y , we have

Qch(X ) Qch(Y ) = Qch(X Y ). × Proof. We ﬁrst prove (1). Put a = a(A), b = a(B). According to (2.18) and Theorem 2.41, we have

Qgr(A) Qgr(B) = Sh(a, tails) Sh(b, tails) = Sh((a b)∆, tails) ∼ T T ∼ ⊗ T and by (2.19) and (2.18) , the category on the right hand side is isomorphic to Qgr(A cart B). Now we prove (2). It sufﬁces to write X Proj(A) and Y Proj(B) for ∼= ∼= connected× graded algebras generated in degree 1. Then it follows from (1) and the fact that Proj(A) Proj(B) = Proj(A cart B). × × Remark 2.44. We expect the formula Qch(X )Qch(Y ) = Qch(X Y ) to hold in greater generality, at least for schemes and suitable stacks. An approach× to the proof would go through studying the compatibility with our tensor product with descent. We hope to carry this research further in the near future.

2.4 Relation with other tensor products

Our tensor product of Grothendieck categories is in close relation with two well- known tensor products of categories. In this section we analyse those relations. The first one is the tensor product of locally presentable categories. It is well-known that every Grothendieck category is locally presentable. In §2.4.1 we prove that taking our tensor product of two Grothendieck categories coincides with taking their tensor product as locally presentable categories. In particular, the class of locally α-presentable Grothendieck categories for a fixed cardinal α is stable under our tensor product. This applies, for example, to the class of locally finitely presentable Grothendieck categories. This should be contrasted with the more restrictive class of locally coherent Grothendieck categories, which is not preserved, as is already clear from the ring case. The second one is Deligne’s tensor product of small abelian categories. In §2.4.2, for small abelian categories A and B with associated Grothendieck categories Lex(A) and Lex(B) of left exact modules, based upon [50] the tensor product Lex(A) Lex(B) is shown to be locally coherent precisely when the Deligne tensor product of A and B exists, and in this case the Deligne tensor product is given by the abelian category of finitely presented objects in Lex(A) Lex(B). Following a suggestion by Henning Krause, in §2.4.3 we define an α-version of the Deligne tensor product which is shown Chapter 2. Tensor product of linear sites and Grothendieck categories 63 to underlie any given tensor product of Grothendieck categories, as long as we choose α sufficiently large.

2.4.1 Tensor product of locally presentable categories

As mentioned in §1.2, local presentability of categories is classically considered in a non-enriched context [27], for which enriched analogues exist [43]. Recall that, in the case of k-linear categories, i.e. categories enriched over Mod(k), the classical and the enriched notions of local presentability coincide. Nevertheless, for the constructions considered in this section it is essential to work enriched over Mod(k). All categories and constructions in this section are thus to be understood in the k-linear sense. Consider a locally presentable k-linear category C. We point the reader to §1.2 for the definition and basic properties of locally presentable categories. Recall in particular that if C is locally α-presentable, its full subcategory of α-presented objects Cα is small, α-cocomplete (i.e. closed under α-small colimits) and it is obtained as the closure of any category of α-presentable strong generators under α-small colimits (Theorem 1.36). Observe that the notion of local presentability is a generalization to bigger cardinals of the notion of locally finitely presentable k-linear category, which is obtained as the particular case with . In that case we write fp C C 0 . α = 0 ( ) = ℵ ℵ It is well known that Grothendieck categories are locally presentable (see for example [9, Prop 3.4.16]). Let’s fix some notations for the rest of the section. Consider k-linear categories A, B and C. We denote by Cont(A,B) (resp. by Contα(A,B)) the category of k-linear continuous (resp. α-continuous) functors from A to B, i.e. functors preserving all (existing) limits (resp. α-small limits). We denote by Cont(A,B;C) (resp. by Contα(A,B;C)) the category of functors A B C which are k-linear and continuous (resp. α- continuous) in each variable,× or analogously,−→ the k-linear functors A B C which are continuous (resp. α-continuous) in each variable. ⊗ −→

In a similar way, the categories Cocont(A,B), Cocontα(A,B), Cocont(A,B;C) and Cocontα(A,B;C) are deﬁned with limits replaced by colimits. In the following theorem a tensor product of locally presentable categories is described.

Theorem 2.45 ([11, Lem. 2.6, Rem. 2.7], [16, §2], [18, Cor. 2.2.5]). Consider locally presentable k-linear categories A and B.

1. The category Cocont(A,B) of k-linear cocontinuous functors is also a locally presentable k-linear category. 64 Chapter 2. Tensor product of linear sites and Grothendieck categories

2. There exists a locally presentable k-linear category A LP B such that for every cocomplete k-linear category C there is a natural equivalence of categories:

Cocont(A LP B,C) Cocont(A,B;C) Cocont(A,Cocont(B,C)) ∼= ∼= op 3. In (2) we can take A LP B = Cont(A ,B).

For a small α-cocomplete k-linear category c, we put

op Lexα(c) = Contα(c ,Mod(k)) Mod(c). ⊆ For small α-cocomplete k-linear categories c and d, we put

op op Lexα(c,d) = Contα(c ,d ;Mod(k)) Mod(c d). ⊆ ⊗ c c For α = 0, we obtain the familiar categories Lex( ) = Lex 0 ( ) of left exact (that ℵ c d c d ℵ is, ﬁnite limit preserving) modules and Lex( , ) = Lex 0 ( , ) of modules that are left exact in both variables. ℵ The following results are a summary of Theorems 1.36 and 1.39 and Proposition 1.38. We refer to [43, Thm. 7.2, 7.3 + §7.4 & Thm. 9.9] for the proves.

α The category Lexα(c) is locally α-presentable, and we have (Lexα(c)) = c. In • ∼ addition, the category Lexα(c) is the α-free cocompletion of c: every object in it can be written as an α-ﬁltered colimit of c-objects. Further observe that, for any cocomplete k-linear category D, we have

Cocont(Lexα(c),D) = Cocontα(c,D). (2.20)

Conversely, for a locally α-presentable k-linear category C, we have • α C Lexα(C ). (2.21) ∼=

Consequently, one also obtains a natural α-cocomplete tensor product for small α-cocomplete k-linear categories c and d [43], [42], given by

c α d = (Lexα(c) LP Lexα(d))α. ⊗ This α-cocomplete tensor product satisﬁes the following universal property for every small α-cocomplete k-linear category e:

Cocontα(c α d,e) = Cocontα(c,d;e). (2.22) ⊗ ∼ Chapter 2. Tensor product of linear sites and Grothendieck categories 65

c d c d c d For small ﬁnitely cocomplete categories and , we denote fp = 0 . ⊗ ⊗ℵ The following alternative description of the tensor product of locally presentable categories is useful for our purpose. It appears for example in [16]; we provide a proof for the convenience of the reader.

Proposition 2.46. For locally α-presentable k-linear categories C and D, we have an equivalence α α C LP D Lexα(C ,D ). ∼=

Proof. We have equivalences

op C LP D = Cont(C ,D) op op = Cocont(C,D ) ∼ α op op = Cocont(Lexα(C ),D ) ∼ α op op = Cocontα(C ,D ) ∼ α op α op = Contα((C ) ,Contα((D ) ,Mod(k))) ∼ α α Lexα(C ,D ), ∼= where we have used (2.21) in the third and ﬁfth steps, (2.20) in the fourth step, and the α op fact that limits are computed pointwise in Contα((D ) ,Mod(k)) in the last step.

Next we turn our attention to (k-linear) Grothendieck categories. The following result combines Theorem 1.27 and Remark 1.28 with Theorem 1.39:

Theorem 2.47. Let C be a locally α-presentable Grothendieck category and consider α the inclusion uC : C C. The canonical functor −→ α C Mod(C ) : C C(u( ),C ) −→ 7−→ − factors through an equivalence of categories

α 1 α C Sh(C , uC− C) = Lexα(C ). −→ T We can now prove the main result of this section:

Theorem 2.48. For Grothendieck categories C and D, we have an equivalence of categories C D C LP D. ∼= 66 Chapter 2. Tensor product of linear sites and Grothendieck categories

Proof. Let α be a regular cardinal for which both C and D are locally α-presentable. By Theorem 2.47, we have α α 1 1 α α C D = Sh(C D , uC− C uD− D) = Lexα(C ,D ) α α ⊗ T T since Lexα(C ,D ) describes the intersection of the two one-sided sheaf categories following Theorem 2.47. This ﬁnishes the proof by Proposition 2.46.

Remark 2.49. The way in which the tensor product LP of locally presentable categories is deﬁned through a universal property, makes it well-deﬁned up to equivalence of categories. As an alternative to our independent approach to the tensor product of Grothendieck categories based upon functoriality, one can show in the spirit of Proposition 2.46 that

Sh(a, a) LP Sh(b, b) = Sh(a b, a b). T T ⊗ T T We point the reader to §2.4.4 below for a more detailed explanation. Corollary 2.50. The subclass of Grothendieck k-linear categories within the class of locally presentable k-linear categories is closed under the tensor product LP.

2.4.2 Relation with Deligne’s tensor product

In [21], Deligne defined a tensor product for abelian categories through a universal property. This tensor product is known to exist only under additional assumptions on the categories. Recall that a Grothendieck category C is locally coherent if it is locally finitely presentable and fp(C) is abelian. This defines a 1-1 correspondence between locally coherent Grothendieck categories on the one hand and small abelian categories on the other hand, the inverse being given by A Lex(A). For small abelian categories A and B, according to §2.4.1 we have 7−→

Lex(A) Lex(B) = Lex(A,B). Since the tensor product of coherent rings is not necessarily coherent (see for instance [50, Ex. 21]), the tensor product of locally coherent Grothendieck categories is not necessarily locally coherent. We can complete [50, Thm. 18] as follows: Theorem 2.51. For small abelian categories A and B, the following are equivalent:

1. Deligne’s tensor product of A and B exists;

2. The tensor product A fp B is abelian; ⊗ 3. The tensor product Lex(A) Lex(B) is locally coherent.

In this case, Deligne’s tensor product equals A fp B = fp(Lex(A) Lex(B)). ⊗ Chapter 2. Tensor product of linear sites and Grothendieck categories 67

2.4.3 The α-Deligne tensor product

As suggested to us by Henning Krause, we define an α-version of the Deligne tensor product for a cardinal α and we show that every tensor product of Grothendieck categories is accompanied by a parallel α-Deligne tensor product of its categories of α-presented objects for sufficiently large α. Definition 2.52.

1. Let A and B be α-cocomplete abelian categories. An α-Deligne tensor product of A and B is an α-cocomplete abelian category A α B with a functor • A B A α B which is α-cocontinuous in each variable and induces equivalences⊗ −→ • Cocontα(A α B,C) = Cocontα(A,B;C) • ∼ for every α-cocomplete abelian category C.

2. Let A and B be abelian categories. If it exists, we deﬁne the modiﬁed α-Deligne tensor product to be α α A˜αB = Lex(A) α Lex(B) . • • A B A B A B A B Note that for α = 0, we have ˜ 0 = 0 = , where denotes Deligne’s (original) tensor product.ℵ •ℵ •ℵ • •

The following is proven along the lines of [37, Prop. 6.1.13], using the description of Lexα(A) Indα(A) as “ind-completion” in terms of α-ﬁltered colimits. ∼= Lemma 2.53. Let A be a small α-cocomplete abelian category. The category Lexα(A) is abelian.

The following analogue of [50, Lem. 17] is proven along the same lines, based upon Lemma 2.53.

Lemma 2.54. Suppose for small α-cocomplete abelian categories A and B, the α- Deligne tensor product A α B exists. The category Lexα(A α B) is characterized by the following universal property• for cocomplete abelian categories• C:

Cocont(Lexα(A α B),C) = Cocontα(A,B;C). (2.23) • The following replacement of [50, Thm. 18] is proven along the same lines. For α = 0, note that the second part of condition (1) is automatically fulﬁlled. ℵ Theorem 2.55. For α-cocomplete abelian categories A and B, the following are equivalent: 68 Chapter 2. Tensor product of linear sites and Grothendieck categories

1. The α-Deligne tensor product A α B exists and Lexα(A α B) is abelian; • ⊗ 2. The α-cocomplete tensor product A α B is abelian. ⊗

In this case, we have A α B = A α B. • ⊗

Proof. If A α B is abelian, it obviously satisﬁes the universal property of A α B and ⊗ • further, Lexα(A α B) is abelian by Lemma 2.53. Conversely, suppose A α B exists ⊗ • and Lexα(A α B) is abelian. The categories Lexα(A α B) and Lexα(A α B) have the ⊗ • ⊗ categories A α B and A α B as respective categories of α-presented objects, whence • ⊗ it sufﬁces to show that Lexα(A α B), being cocomplete and abelian by assumption, has the universal property of Lemma⊗ 2.54. But this is clearly the case by (2.20) and (2.22).

We recall the following:

Proposition 2.56 ([46, Cor. 5.2]). Let C be a Grothendieck category. There exists a cardinal α such that C is locally β-presentable and Cβ is abelian for β α. ≥ Whereas the tensor product of two Grothendieck categories cannot be related to the Deligne tensor product in general, it can always be related to an α-Deligne tensor product in the following way:

Proposition 2.57. Let C and D be Grothendieck categories. There exists a cardinal α β β such that for β α the β-Deligne tensor product C β D exists and we have ≥ • β β β (C D) = C β D . ∼ • Proof. It sufﬁces to note that by Proposition 2.56, we can choose α such that for β α β β β ≥ the categories C, D and C D are locally β-presentable and C , D and (C D) are β β β abelian. Hence, we have C β D (C D) by Theorem 2.48, and thus the desired ∼= isomorphism holds by Theorem⊗ 2.55.

As a special case, whereas two small abelian categories do not necessarily have a Deligne tensor product, they do have a modiﬁed α-Deligne tensor product for sufﬁciently large α:

Corollary 2.58. Let A and B be small abelian categories. There exists a cardinal α such that for β α the modiﬁed β-Deligne tensor product A˜β B exists and we have ≥ • β (Lex(A,B)) = A˜β B. ∼ • Chapter 2. Tensor product of linear sites and Grothendieck categories 69

2.4.4 Relation with the tensor product of topoi

In Theorem 2.48 we have shown that the tensor product of Grothendieck categories is a special instance of the tensor product of locally presentable linear categories, using special linear site presentations of the categories. This raises the natural question whether, if one takes the tensor product of locally presentable categories as starting point, there is a shorter route to the tensor product of Grothendieck categories than the one we followed. First one may note that in order to obtain an abstract tensor product of Grothen- dieck categories, it suffices to prove Corollary 2.50 directly. As an anonymous referee of the article [53] in which this chapter is based suggested, one can prove along the lines of [50, Cor. 15] that the tensor product of locally presentable categories preserves the Grothendieck property. However, this does not bring us any closer to the concrete expressions of the tensor product in terms of arbitrary representations in terms of linear sites, which is the main aim of the current chapter. Secondly one may note that, after proposing our concrete formula for the tensor product of Grothendieck categories using linear sites, it suffices to show that this formula satisfies the universal property of the tensor product of locally presentable linear categories in order to show at once that our formula leads to a good definition, and that Theorem 2.48 holds. This approach indeed works, and is based upon the possibility to write down an analogous formula to (2.20), with regard to a linear site (a, ). Precisely, for any cocomplete k-linear category D we have T Cocont(Sh(a, ),D) = Cocont (a,D) (2.24) T T where the right hand side denotes the category of linear functors F : a D whose induced colimit preserving functor Fˆ : Mod(a) D sends inclusions−→ of covers R a( , A) to isomorphisms in D. Rather than−→ spelling out the proof of the universal⊆ property− for the tensor product in Definition 2.32, we refer the reader to [63] where the parallel reasoning is performed for topoi over Set. The tensor product of Grothendieck categories which we have introduced can be seen as a linear counterpart to the product of Grothendieck topoi which is described by Johnstone in [34], [35]. In [63], Pitts shows that the product of Grothendieck topoi is a special instance of the (Set-based) tensor product of locally presentable categories, using the universal property.

Unlike in the case of topoi, working over Mod(k) rather than over Set, our tensor product does not describe a 2-categorical product, but instead introduces a 2-categorical monoidal structure on linear topoi. The natural way of defining a product (Grothendieck) topology as the smallest topology making the projections 70 Chapter 2. Tensor product of linear sites and Grothendieck categories continuous is not available in the linear setup, as there are no projection morphisms to the factors of the tensor product. Hence our definition of the product site in §2.1.4 is necessarily somewhat ad hoc. Further, we should note that the establishment of the correct formula for the tensor product does not automatically yield the tangible functoriality properties for linear sites which we have proven. With our motivation coming from noncommutative geometry, it is precisely the flexibility in choosing appropriate sites, and the possibility to view certain functors of geometric origin as induced by natural morphisms of sites, which is of greatest interest to us. The notion of LC morphism (Definition 2.19) which we prove in Proposition 2.30 to be stable under the tensor product of linear sites, is more restrictive than a morphism inducing an equivalence on the level of sheaf categories, and so this result cannot be deduced a posteriori from the existence of the tensor product satisfying the universal property. In fact, the class of LC morphisms opens up the interesting possibility (see for example the proof of Proposition 2.31) to describe the “category of Grothendieck categories” up to equivalence as a bicategory of fractions, obtained from the category of linear sites by inverting LC morphisms. This will be carried out in Chapter4. CHAPTER 3

The tensor product of Grothendieck categories as a ﬁltered bicolimit

In Chapter2 we have constructed a tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We further have shown in §2.4.1 that the tensor product of Grothendieck categories is an instance of the tensor product LP of locally presentable categories. In particular, the 2-category of locally presentable categories endowed with the tensor product LP is a closed symmetric monoidal bicategory in the sense of [31] (see [11, Lem 2.7], [42, §6.5], [1, Exerc 1.l]). The inner hom is given by the cocontinuous functors. Most precisely, given A, B and C locally presentable categories, we have the universal property

Cocont(A LP B,C) Cocont(A,Cocont(B,C)), ∼= in the 2-category of locally presentable categories. Observe that from the universal property one can easily deduce the associativity, symmetry and functoriality with respect to cocontinuous functors of LP (up to equivalence of categories). This immediately endows the tensor product of Grothendieck categories with the same nice properties. On the other hand, as we have seen in §2.4.1, §2.4.2 and §2.4.3, the tensor product of k-linear locally presentable categories and the tensor product of k-linear Grothendieck categories are closely related to two tensor products of small categories, namely Deligne tensor product (and its α-version developed in §2.4.3) and Kelly’s tensor product of α-cocomplete k-linear categories, for any regular cardinal α. This relation is provided by taking α-presentable objects and free α-cocompletions. From both small tensor products, Kelly’s tensor product α of α-cocomplete categories is α the best-behaved. In particular, the category Cat (k⊗) of small α-cocomplete k-linear categories endowed with α is proved in [42, §6.5] to be a closed symmetric monoidal bicategory in the sense of ⊗[31]. The inner hom is given by the α-cocontinuous functors.

71 72 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

α More precisely, given a, b and c in Cat (k), we have the universal property

Cocontα(a α b,c) = Cocontα(a,Cocontα(b,c)) ⊗ ∼ α in Cat (k). Let C and B be two Grothendieck categories and given any regular cardinal α, denote by Cα, Bα their respective small k-linear categories of α-presentable objects. Our aim of this chapter is first, to show that the tensor product CB of Grothendieck categories can be obtained as a filtered bicolimit in the sense of [23] of the small α α categories C α B with α varying in the regular cardinals, and second, to explicitely describe, by means⊗ of this construction, the functoriality of the tensor product of Grothendieck categories. We also discuss how, with this explicit construction at hand, functoriality, associativity and symmetry can be shown without reference to the universal property on the level of the large categories. Alternatively, one could focus on all linear site representations (instead of only those given by the subcategories of presentable objects) in order to obtain a similar bicolimit construction of the tensor product. More precisely, given C and B two Grothendieck k-linear categories, one could consider all the linear site representations (c, c) C, T −→ (d, d) B given by LC morphisms. Then, the tensor product C B could be recoveredT −→ as the filtered bicolimit, in the sense of [23], of the family of small categories (c d) where (c, c) (resp. (d, d)) varies in the family of linear site representations of C (resp.⊗ B). UsingT the roof argumentT on the proof of Proposition 2.31, one can easily observe that this family is indeed filtered with morphisms given by tensor products of LC morphism f g : (c d, c d) (c d , c d ), where here denotes the tensor 0 0 0 0 product of topologies⊗ ⊗ fromT §2.1T .−→ By means⊗ T of thisT bicolimit construction, we could recover the functoriality of the tensor product of Grothendieck categories from the functoriality of the tensor product of sites. However, the construction of this bicolimit entails a rather more technical description than the construction of the bicolimit using the tensor products of the subcategories of locally presentable objects above. This is due to the fact that in this case the indexing category is more complicated, and the indexing category for the tensor product is obtained as the product of the indexing categories of the individual categories. Hence, we will only develop the bicolimit construction in terms of the subcategories of locally presentable objects, as this is enough for our purposes of proving functoriality, associativity and symmetry of the tensor product of Grothendieck categories. Nevertheless, it is important to keep in mind that a filtered bicolimit construction is possible considering all representations, as this allows us to use the functoriality of sites developed in Section 2.2 in order to compute tensor products of functors between Grothendieck categories when dealing with concrete examples. Chapter 3. The tensor product of Grothendieck categories as a filtered bicolimit 73

The chapter is organised as follows. In first place we recall the definition of 2-filtered bicolimit of categories from [23] and we provide its description when the indexing category is an ordinary filtered category induced by a directed poset. Additionally, we prove in Proposition 3.4 that, for this particular type of indexing categories, the bicolimit of k-linear categories remains k-linear. Based on the construction of the filtered bicolimit from §3.1, we prove in §3.2 that any locally presentable k-linear category is the bicolimit of its small k-linear subcategories Cα with α taking values in the totally ordered set of regular cardinals. Hence the statement holds for Grothendieck k-linear categories. Relying on the representation of any locally presentable category as a bicolimit and the properties of LC morphisms, we are in the position in §3.3 to, given two Grothendieck categories C and B, express their tensor product CB also a as a bicol- α α imit, namely as the bicolimit of the tensor products (C α B )α where α takes values again in the totally ordered set of regular cardinals. This⊗ is done in Theorem 3.13. This result allows us to translate the functoriality, associativity and symmetry of the tensor product of α-cocomplete categories (for each regular cardinal α) to the corresponding nice properties of the tensor product of Grothendieck categories. This is carried out in §3.4.

3.1 Generalities on the 2-ﬁltered bicolimit of categories

Let Cat denote the 2-category of small categories with 1-morphisms given by functors and 2-morphisms given by natural transformations. Given a pseudofunctor F : A Cat where A is a filtered category, we can consider → Grothendieck’s construction of the colimit category limA(F ) [77, Exposé VI]. In [23] a suitable generalization to the 2-categorical world−→ is provided, and referred to as 2-filtered bicolimit. We first fix some notations for the rest of the chapter. Given a bicategory, we denote by the vertical composition of 2-morphisms and by the horizontal composition of 2-morphisms.• In particular, given a diagram ◦ f g

A Idf B α C ⇓ ⇓ f h in a bicategory C, we denote by α f to the horizontal composition α Idf . ◦ ◦ 74 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

Now, let’s recall some important definitions for the rest of the chapter. Let F,G : A B be two pseudofunctors between 2-categories A and B. Recall that a pseudonatural−→ transformation φ : F G is given by a family of 1-morphisms ⇒ (φA : F (A) G (A))A Obj A and a family (φf : φ(A ) F (f ) G (f ) φ(A)) f :A A A of ( ) 0 ( 0) invertible−→ 2-morphisms∈ with the corresponding coherence◦ ⇒ laws. ◦ → ∈ Given two pseudonatural transformations φ,ψ : F G , a morphism of pseudonatural transformations r between φ and ψ is a modification,⇒ that is a family of 2-morphisms (rA : φA ψA)A A such that (G (f ) rA) φf = ψf (rA F (f )) for all 0 f : A A in A. We denote⇒ by∈ Psnat F,G the category◦ • of pseudonatural• ◦ transfor- 0 ( ) mations−→ between F and G , with morphisms given by the modifications (see [23] or [49]). The notion of 2-filtered 2-category is introduced in [23, §2] as a suitable generalisa- tion in the 2-categorical realm of the classical notion of filtered category. In particular, as it is already mentioned in the introduction of [23], any (1-)category considered as a trivial 2-category is 2-filtered if and only if it is filtered as an ordinary category. Throughout this chapter we will always use an indexing category which is of this latter type, hence we can safely avoid going through the technicalities of the definition of 2-filtered 2-category.

We are now in position to define the 2-filtered bicolimit [23, Thm. 1.19]. Definition 3.1. Given a pseudofunctor F : I Cat, where I is a 2-filtered 2-category1, the 2-filtered bicolimit of F is a category B →together with a pseudonatural transformation F B from F to the constant 2-functor I Cat taking the value B such that, for every category⇒ C, it induces via composition an→ isomorphism of categories

Cat(B,C) = Psnat(F,C) (3.1) between the category of functors B C and the category of pseudonatural transformations between the pseudofunctor→F and the constant 2-functor taking the value C.

Remark 3.2. Note that such a category is uniquely determined up to a unique equivalence. We will denote it by L(F ), following the notations from [23]. Remark 3.3. Observe that the original deﬁnition (see [23, Thm. 1.19]) only considers 2-ﬁltered bicolimits of F with F a strict 2-functor. For our purposes we need to consider the more general situation in which F is a pseudofunctor.

1The bicolimit and its construction actually work when the indexing category is a pre 2-filtered 2- category, which is a weaker notion than that of 2-filtered 2-category, as pointed out in the introduction of [23]. Chapter 3. The tensor product of Grothendieck categories as a filtered bicolimit 75

The main result of [23] is, given a 2-functor F : I Cat, the construction of the bicolimit L(F ) in an intrinsic way in terms of the 2-functor−→ F . One can observe that, when the indexing category is just an ordinary filtered (1-)category the construction is greatly simplified. In particular, we will be focusing on ordinary filtered categories given by a directed poset. For this choice of indexing category, one can easily extend the construction from [23] to the case in which F is a pseudofunctor by means of a slight generalization of the results explained in loc.cit.2 We flesh out below the construction of the bicolimit for our particular situation, i.e. when

F is not necessarily a strict 2-functor but a pseudofunctor, • the indexing category is an ordinary filtered category given by a directed poset. • Consider I an ordinary filtered category induced by a directed poset (in particular, between two objects there is at most one morphism). Consider a pseudofunctor F : I Cat. We define the following category L(F ): → Description of the objects:

Objects of L(F ) are pairs (x , A) where A I and x F (A). ∈ ∈ Description of the morphisms:

First we describe the class of premorphisms. Consider two objects (x , A) and • (y, B).A premorphism (x , A) (y, B) consists of an object C I with C A, B and a morphism f : F (A →C )(x ) F (B C )(y ) in F (C ).∈ We will use≥ the ≤ → ≤ notation fC : (x , A) (y, B). →

Two pre-morphisms fC1 ,gC2 : (x , A) (y, B) are said to be homotopical if • → there exists an object C C1,C2 such that F (C1 C )(f ) F (C2 C )(g ) in ∼= F (C )(F (A C )(x ), F (B C≥)(y )). ≤ ≤ ≤ ≤ Morphisms in L(F ) between two objects are given by premorphisms between • those two objects modulo homotopy.

Given two morphisms [fD1 ] : (x , A) (y, B), [gD2 ] : (y, B) (z ,C ), the composi- • → → I tion [gD2 fD1 ] is given by [(F (D2 D )(g ) F (D1 D )(f ))D ] for D such that D D1,D◦2. Observe that it is well-deﬁned.≤ ◦ ≤ ∈ ≥ 2The general construction for F a pseudofunctor should be possible in full generality, without restrictions of the indexing category, just by readjusting the notion of homotopy in [23, 1.5(iii)]. 76 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

The category L(F ) fulfills the universal property (3.1) above. Fix now k a commutative ring. We are interested in k-linear categories and filtered bicolimits of k-linear categories. More explicitely, we are interested in 2-filtered bicolimits of pseudofunctors F : I Cat that take values in the 2-category Cat(k) of small k-linear categories with k-linear−→ functors and k-linear natural transformations, that is, in functors F : I Cat that factor through the forgetful functor Cat(k) Cat. In general, the 2-filtered−→ bicolimit L(F ) under these hypothesis will not necessarily−→ be k-linear, as it is also the case for the classical Grothendieck construction. However, for nice choices of the indexing category I and the functor F , this will hold true. In particular, it is true for our case of interest.

Proposition 3.4. Let I be a category given by a directed poset. Take F : I Cat a functor that factors through the forgetful functor Cat(k) Cat. Then, L→(F ) is a k-linear category. −→

Proof. Consider (x , A),(y, B ) L(F ). The class L(F )((x , A),(y, B )) has a natural structure of k-module induced from∈ the k-linear structure of the values of F . Indeed, consider two morphisms [fC1 ],[gC2 ] : (x , A) (y, B) and an element λ k, we deﬁne → ∈ [fC1 ] + λ[gC2 ] = [F (C1 C )(f ) + λF (C2 C )(g )] for C C1,C2. This is easily seen to be well-deﬁned. The fact≤ that it provides≤ a k-module structure≥ is directly deduced from the fact that, by hypothesis, for each object C I, F (C ) is a k-linear category and for each morphism C D in I, the functor F (C ∈D ) : F (C ) F (D ) is k-linear. ≤ ≤ −→ In addition, one can easily show that the composition is k-linear. To show this, consider morphisms f , g : x , A y, B , f , g : y, B z ,C and elements [ C1 ] [ C2 ] ( ) ( ) [ D01 ] [ D0 2 ] ( ) ( ) , k. We have that → → λ λ0 ∈

[fD0 ] ([fC1 ] + λ[gC2 ]) 1 ◦ = [(F (D1 D )(f 0) (F (C1 D )(f ) + λF (C2 D )(g )))D ] ≤ ◦ ≤ ≤ = [(F (D1 D )(f 0) F (C1 D )(f ) + λ(F (D1 D )(f 0) F (C2 D )(g ))D ] ≤ ◦ ≤ ≤ ◦ ≤ = [fD0 ] [fC1 ] + λ[fD0 ] [fC1 ], 1 ◦ 1 ◦ where D C1,C2,D1. Similarly, one proves that ≥

([fD0 ] + λ[gD0 ]) [fC1 ] = [fD0 ] [fC1 ] + λ[gD0 ] [fC1 ]. 1 2 ◦ 1 ◦ 2 ◦ Hence the composition is k-linear as desired.

Remark 3.5. Observe that one could define a k-linear 2-filtered bicolimit by replacing in Definition 3.1 above Cat by Cat(k) and the category Psnat(F,C) by its k-linear analogue. Notice then that, given a pseudofunctor F as in Proposition 3.4 above, we Chapter 3. The tensor product of Grothendieck categories as a filtered bicolimit 77 have that the 2-filtered bicolimit L(F ) coincides with the k-linear 2-filtered bicolimit of F .

3.2 Locally presentable categories as bicolimits of small categories

In this section we prove, based upon §3.1, that every locally presentable k-linear category can be written as a 2-filtered bicolimit of small categories. α Fix a universe U. Let α be a U-small regular cardinal. Denote by Cat (k) the bicategory of α-cocomplete U-small k-linear categories with α-cocontinuous k-linear α functors and k-linear natural transformations. For a Cat (k), the category Lexα(a) is defined with respect to the category U-Mod(k) of U-small∈ k-modules. By definition, a U-locally presentable category is a k-linear category with U-small colimits and a U-small set of α-presentable strong generators for some U-small regular cardinal α. Let V be a larger universe such that for all U-small cardinal α, all the categories Lexα(a) α with a Cat (k) are V-small, and so is the category K given by the totally ordered class of∈U-small regular cardinals. Observe that given a U-locally-α-presentable category C, its subcategory of α-presentable objects Cα is essentially U-small and hence α we can still consider it as an element in Cat (k). In the rest of the chapter, we will omit the universes U and V from our notations and terminology.

Theorem 3.6. Let C be a category which is a union of full subcategories indexed by a directed poset. Then, C is the filtered bicolimit of that family of subcategories. More S precisely, if C is a category such that C = i I Ci , where Ci C are full subcategories, I ∈ is a directed poset and Ci Cj if and only if i j , then C ⊆is a filtered bicolimit of the ⊆ ≤ family (Ci )i I . ∈ Proof. Denote by I the filtered category given by the directed poset I , and denote by ιi ,j : Ci , Cj the natural embeddings for i j . We define the 2-functor −→ ≤ FC : I Cat (3.2) → to be given by FC(i ) = Ci for every i I and FC(i j ) = ιi ,j : Ci , Cj for every morphism i j in I. ∈ ≤ −→ ≤ We build a functor φ : L(FC) C → defined as follows:

φ(x ,i ) = x Ci C for every (x ,i ) L(FC); • ∈ ⊆ ∈ 78 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

φ([fk ]) = f Ck (ιi ,k (x ),ιj,k (y )) = C(x , y ) for every [fk ] : (x ,i ) (y, j ) in L(FC). • ∈ ∼ −→ S Observe that this is well-deﬁned. As C = i I Ci , one trivially has that this functor ∼ ∈ is essentially surjective, and because ιi ,j are fully faithful for all i , j I, one easily deduces that the functor is fully faithful. ∈

Remark 3.7. Assume C is k-linear, and hence Ci is k-linear for all i I and so are the fully faithful functors ιi ,j : Ci Cj for i j in I. Then we have that∈ both I and FC ⊆ ≤ are in the hypothesis of Proposition 3.4 above, hence L(FC) is k-linear. Observe that the functor φ deﬁned in the proof above is as well k-linear, and thus C and L(FC) are equivalent as k-linear categories.

As indicated above, we denote by K the category obtained from the totally ordered set of regular cardinals. Recall that given C a locally presentable k-linear category and α K, we denote by Cα the full k-linear subcategory of C consisting of the α- presentable∈ objects. Recall additionally that Cα is an α-cocomplete category and that if α β, we have a fully faithful embedding Cα Cβ (see §1.2). Also recall that S ≤ α ⊆ C = α K C (Proposition 1.34). ∼ ∈ Corollary 3.8. Let C be a locally presentable k-linear category. Then C is a ﬁltered α bicolimit of its family of subcategories of locally presentable objects (C )α, where α varies in K.

Proof. It follows from the discussion above by applying Theorem 3.6 and Remark 3.7.

Remark 3.9. Observe that the statement is true for Grothendieck k-linear categories, as they are an instance of locally presentable k-linear categories.

3.3 The tensor product of Grothendieck categories as a ﬁltered bicolimit

In this section we analyse the tensor product of Grothendieck categories from Chap- ter2 in terms of the realization of Grothendieck categories as filtered bicolimits provided by Corollary 3.8. Recall there is a well-defined notion of tensor product of α-cocomplete k-linear α categories. In particular, the bicategory Cat (k) of α-cocomplete k-linear categories as defined above is, together with this tensor product, a closed monoidal bicategory [42, §6.5]. More precisely, we have the following: Chapter 3. The tensor product of Grothendieck categories as a filtered bicolimit 79

Deﬁnition 3.10. Given a,b two α-cocomplete k-linear categories, there exists an- α other α-cocomplete k-linear category a α b and a k-linear functor ua,b : a b a α b which is α-cocontinuous in each variable,⊗ such that, for every α-cocomplete⊗ →k-linear⊗ category c, composition with ua,b induces an equivalence

Cocontα(a α b,c) = Cocontα(a,b;c) = Cocontα(a,Cocontα(b,c)) (3.3) ⊗ ∼ ∼ α in Cat (k).

α Consider a, b Cat (k). We know that a α b is the closure under α-small colimits of the image of∈ the composition ⊗

Y R a b , Mod(a b) Lexα(a,b) ⊗ −→ ⊗ −→ where the functor Y : a b , Mod(a b) is the Yoneda embedding and the func- ⊗ −→ ⊗ tor R : Mod(a b) Lexα(a,b) is the left adjoint to the canonical embedding ⊗ −→ Lexα(a,b) , Mod(a b) (see [42],[43], [50]). −→ ⊗ In addition, we know that given locally α-presentable categories A,B, we have that α α ALP B = Lexα(A ,B ) is α-locally presentable and its subcategory of α-presentable α α objects is given by A α B (see Theorem 2.55 and Proposition 2.57). ⊗ Let A, B be two locally presentable k-linear categories and choose the smallest regular cardinal κ such that both are locally κ-presentable. Observe that, for all regular cardinals α β, we have a canonical morphism ≤ α α β β fα,β : A α B A β B (3.4) ⊗ → ⊗ that makes the diagram

A B ια,β ια,β Aα Bα ⊗ Aβ Bβ ⊗ ⊗u uAα,Bα Aβ ,Bβ (3.5) f α α α,β β β A α B A β B . ⊗ ⊗ commutative. Indeed, fα,β is deﬁned as the image via the universal property (3.3) in α α β β Cocontα(A α B ,A β B ) of the composition ⊗ ⊗ A B u ια,β ια,β Aβ ,Bβ α α ⊗ β β β β A B A B A β B , ⊗ −−−−→ ⊗ −−−−→ ⊗ which is α-cocontinuous in each variable. 80 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

First observe that, as a direct consequence of Theorem 3.6, we have that A LP B α is the ﬁltered bicolimit of the family ((A LP B) )α K and that from α κ we have α α α ∈ ≥ that (A LP B) = A α B . However, it is not directly obvious how to translate the properties of the tensor⊗ product of locally presentable categories to the bicolimit, as α α β β it is not clear at ﬁrst sight whether the fully faithful functors A α B , A β B ⊗ −→ ⊗ with β α κ are isomorphic to the canonical functors fα,β induced by the fully faithful≥ functors≥ in both variables. We will now provide a positive answer in the context of Grothendieck categories.

Theorem 3.11. Let A, B be two Grothendieck k-linear categories and choose the smallest regular cardinal κ such that both A and B are locally κ-presentable. Then, for all α,β K such that β α κ the canonical functor ∈ ≥ ≥ α α β β fα,β : A α B A β B ⊗ → ⊗ deﬁned in (3.4) is fully faithful. In particular, fα,β is isomorphic to the canonical α β inclusion (A B) (A B) . ⊆ Proof. Consider a locally κ-presentable Grothendieck k-linear category C. Take α,β κ two regular cardinals with α β. If we endow Cα and Cβ with the topologies induced≥ by the canonical topology in≤C via the natural inclusions, we have that the C α β fully faithful functor ια,β : C , C is an LC morphism and hence −→ C α β (ια,β )s : C = Lexα(C ) Lexβ (C ) = C −→ is an equivalence. Consequently, the diagram

Cα Cβ (3.6) C s (ι ) α α,β β C Lexα(C ) Lexβ (C ) C ∼= ∼=

C s is commutative, and the functor (ια,β ) is an equivalence (see §1.1.2 and §2.2.1). We know that the tensor product of LC morphisms remains an LC morphism (Propo- A B sition 2.30), hence for all β α κ, we have that ια,β ια,β is an LC morphism with the tensor product of the induced≥ ≥ topologies. Consequently,⊗ the functor

A B α α β β (ια,β ια,β )s : A B = Lexα(A ,B ) Lexβ (A ,B ) = A B ⊗ ∼ −→ ∼ A B s is an equivalence with quasi-inverse given by (ια,β ια,β ) . ⊗ Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit 81

We thus have a diagram

A B ια,β ια,β Aα Bα ⊗ Aβ Bβ ⊗ ⊗ Y Y f α α α α α,β β β β β Mod(A B ) A α B A β B Mod(A B ) ⊗ ⊗ ⊗ ⊗ R R A B s (ια,β ια,β ) α α ⊗ β β Lexα(A ,B ) Lexβ (A ,B )

= ∼ ∼= A B A B, where the upper square and two squares on the sides are commutative diagrams, and so is the bigger square. Hence the lower square is also commutative. Observe that the vertical arrows in that lower square are fully faithful and the lower horizontal arrow is an equivalence. Consequently, fα,β is fully-faithful as desired. In addition, because of A B s the commutativity of the outer diagram, the fact that (ια,β ια,β ) is colimit preserving ⊗ and fα,β is α-colimit preserving, one can see that fα,β is actually isomorphic to the α β natural embedding (A B) (A B) . ⊆ We can now deﬁne

FA,B : K Cat (3.7) → the pseudofunctor given by

α α FA,B(α) = FA(α) α FB(α) = A α B for every α K; • ⊗ ⊗ ∈ FA,B(α β) = fα,β for every morphism α β in K; • ≤ ≤ with the notations from §3.2 above.

Remark 3.12. Observe that FA,B is a pseudofunctor and not a strict 2-functor.

Theorem 3.13. Given A and B two Grothendieck k-linear categories, one has that

A B = L(FA,B).

Proof. By Proposition 3.4, L(FA,B) is a k-linear category. In addition, we know that α α there exists a κ K such that A B is α-presented with A α B the small full ∈ ⊗ 82 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit subcategory of α-presentable objects, for every α κ. Consider κ the smallest regular cardinal with such property. We build a k-linear≥ functor

φ : L(FA,B) A B → α α as follows. For an object (x ,α) L(FA,B), we put φ(x ,α) = x A α B A B ∈ ∈ ⊗ ⊆ if α κ and φ(x ,α) = (fα,κ)(x ) Aκ κ Bκ A B otherwise. For a morphism ≥ ∈ ⊗ ⊆ [gγ] : (x ,α) (y,β), we define φ([gγ]) as follows: → If α,β κ, we put • ≥ φ([gγ]) = g Aγ γ Bγ(fα,γ(x ), fβ,γ(y )) ∈ ⊗ A B(x , y ). ∼= Observe that in this case γ κ holds. ≥ If α < κ and β κ, we put • ≥ φ([gγ]) = g Aγ γ Bγ(fα,γ(x ), fβ,γ(y )) ∈ ⊗ A B(fα,κ(x ), y ). ∼= Observe that in this case γ κ holds. ≥ If α κ and β < κ, we put • ≥ φ([gγ]) = g Aγ γ Bγ(fα,γ(x ), fβ,γ(y )) ∈ ⊗ A B(x , fβ,κ(y )). ∼= Observe that in this case γ κ holds. ≥ If α,β < κ and γ κ, we put • ≥ φ([gγ]) = g Aγ γ Bγ(fα,γ(x ), fβ,γ(y )) ∈ ⊗ A B(fα,κ(x ), fβ,κ(y )). ∼= If α,β,γ < κ, we put: • φ([gγ]) = fγ,κ(g ) Aκ κ Bκ(fα,κ(x ), fβ,κ(y )) ∈ ⊗ A B(fα,κ(x ), fβ,κ(y )). ∼= Observe this functor is well-defined and k-linear. In addition, we have that [ α [ α [ α α A B = (A B) = (A B) = A α B . ∼ α K ∼ α κ α κ ⊗ ∈ ≥ ≥ Consequently, the functor is essentially surjective. We also have that all the transition functors fα,β are fully-faithful for β α κ by Theorem 3.11 above, hence one can conclude that the functor is fully-faithful≥ ≥ as desired. Chapter 3. The tensor product of Grothendieck categories as a filtered bicolimit 83

3.4 The tensor product of Grothendieck categories: Functoriality, associativity and symmetry

In this section, based upon Theorem 3.13 above and the properties of the tensor product of α-cocomplete categories, we prove that the tensor product of Grothendieck categories is functorial with respect to cocontinuous functors, associative and symmetric up to equivalence of categories. Consider A, B two locally presentable categories and a regular cardinal α. A functor F : A B is said to have rank α if it preserves α-ﬁltered colimits [8, §5.5]. It is trivial to see−→ that if a functor F has rank α, then it has rank β for every β α. We say a functor has rank if there exists a regular cardinal α such that it has rank≥ α. We have the following useful proposition.

Proposition 3.14 ([8, Prop. 5.5.6]). Let G : A B a functor between locally presentable categories. If G has a left adjoint, then−→ G has a rank.

The following is easy to show, but we provide a proof for the convenience of the reader.

Proposition 3.15. Let F : A B a cocontinuous functor between locally presentable β β categories. Then, there exists−→ a regular cardinal α such that F (A ) B for every β α. ⊆ ≥

Proof. By the dual of the Special Adjoint Functor Theorem [7, Thm. 3.3.4], we have that F has a right adjoint G . In particular, by Proposition 3.14, G has rank. Fix the smallest α such that G has rank α. Then, given an element C Aα, we have that ∈ B(F (C ),colimi Di ) = A(C ,G (colimi Di )) = A(C ,colimi G (Di )) = colimi A(C ,G (Di )) = colimi (F (C ),Di )

α where colimi Di is any α-ﬁltered colimit in B. Hence F (C ) B as desired. ∈ Remark 3.16. Given F : A B as in the proposition, note that the restriction- β β −→ corestriction Fβ : A B of F is β-cocontinuous for all β α. → ≥ Consider Grothendieck k-linear categories A,B,C and D and cocontinuous functors F : A C and F : B D. Take the smallest regular cardinal for which both 0 κ −→ −→ 84 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

F and F preserve -presentable objects for every . We deﬁne a pseudonatural 0 α α κ transformation ≥ φ : FA,B C D, ⇒ where FA,B is deﬁned as in (3.7), as follows. For each α K, we put ∈

If α < κ, we deﬁne φα as the natural composition • α α κ κ κ κ A α B A κ B C κ D C D; ⊗ → ⊗ → ⊗ →

If α κ, we deﬁne φα as the natural composition • ≥ α α α α A α B C α D C D. ⊗ → ⊗ → For each morphism α β in K, we put the invertible natural transformations ≤ φα β : φβ FA,B(α β) φα to be the natural ones induced by the universal properties≤ involved.◦ ≤ ⇒ This construction immediately provides the desired functoriality of the tensor product of Grothendieck categories with respect to cocontinuous functors. Indeed, we have the following:

Definition 3.17. Given cocontinuous functors F : A C and F : B D as above, 0 we define F F : A B C D to be the functor associated→ to the pseudonatural→ 0 → transformation FA,B C D above via the universal property of L(FA,B). ⇒ Remark 3.18. Note that F F is also cocontinuous. The filtered nature of the bi- 0 colimit plays an important role in the proof. Roughly it can be done as follows. Consider colimi Xi the colimit of a small family of objects in A B. Then, we can choose a regular cardinal α such that colimi Xi is an α-small colimit, all the Xi are α- presentable and F and F preserve -presentable objects. Then we can see colim X 0 α i i α α as an element in A α B and we have that ⊗

F F 0(colimi Xi ) = Fα α Fα0(colimi Xi ) = colimi (Fα α Fα0(Xi )) = colimi (F F 0(Xi )), ⊗ ⊗ where we have used that F F : Aα Bα Cα Dα preserves -small colimits α α α0 α α α ⊗ ⊗ −→ ⊗ by the universal property of α. ⊗ Now, we proceed to prove the associativity and the symmetry of the tensor product of Grothendieck categories by using the ﬁltered bicolimit construction we have provided in §3.3. Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit 85

Consider Grothendieck k-linear categories A,B and C. Deﬁne now

F(A,B),C : K Cat −→ α α α by F(A,B),C(α) = (A α B ) α C with the natural transition functors ⊗ ⊗ α α α β β β F(A,B),C(α β) : (A α B ) α C (A β B ) β C . ≤ ⊗ ⊗ −→ ⊗ ⊗ Analogously, we put

FA,(B,C) : K Cat −→ α α α with FA,(B,C)(α) = A α (B α C ) with the natural transition functors ⊗ ⊗ α α α β β β F(A,(B,C)(α β) : A α (B α C ) A β (B β C ). ≤ ⊗ ⊗ −→ ⊗ ⊗ In a similar fashion to Theorem 3.13, one can show that

LF A,B ,C (A B) C, ( ) ∼= and analogously LFA, B,C A (B C). ( ) ∼= α We know that for all regular cardinal α the category Cat (k) of α-cocomplete small categories with the tensor product α is a closed monoidal symmetric bicategory in the sense of [31]. In particular, we have⊗

(a α b) α c = a α (b α c) ⊗ ⊗ ∼ ⊗ ⊗ for all α-cocomplete categories a,b,c.

Consequently, there is a canonical isomorphism F A,B ,C(α) FA, B,C (α) for each α, ( ) ∼= ( ) and it behaves functorially. We thus have

F A,B ,C FA, B,C . (3.8) ( ) ∼= ( ) We are already in position to provide the desired associativity for the tensor product of Grothendieck categories:

Proposition 3.19. Let A,B and C be Grothendieck categories, then, there exists an equivalence (A B) C A (B C). ∼=

Proof. It follows from applying ﬁltered bicolimits to (3.8). 86 Chapter 3. The tensor product of Grothendieck categories as a ﬁltered bicolimit

The argument to prove the symmetry of the tensor product of Grothendieck categories is analogous. Consider two Grothendieck categories A and B. As the monoidal α bicategory (Cat (k), α) is symmetric, we have ⊗ a α b = b α a ⊗ ∼ ⊗ for all α-cocomplete categories a,b. Thus, reasoning as above, we have a canonical isomorphism

FA,B FB,A, (3.9) ∼= where FA,B and FB,A are deﬁned as in (3.7).

Proposition 3.20. Let A,B be Grothendieck categories. Then, there exists an equivalence A B B A. ∼= Proof. It follows from applying ﬁltered bicolimits to (3.9). CHAPTER 4

Localization of sites with respect to LC morphisms

Throughout the chapter k will be a commutative ring. Let Grt denote the 2-category of k-linear Grothendieck categories with cocontinuous (that is, colimit preserving) k-linear functors and k-linear natural transformations. Let Site denote the 2-category of k-linear sites with k-linear continuous morphisms of sites and k-linear natural transformations1. The aim of this chapter is to study the relation between Grt and Site on a bicategorical level.

Given a linear site (a, a) we can naturally associate to it a Grothendieck category, T namely its category of sheaves Sh(a, a). In addition, given a continuous morphism T f : (a, a) (b, b) between sites, it induces naturally a colimit preserving functor s T −→ T f : Sh(a, a) Sh(b, b) between the corresponding categories of sheaves (see §1.1.2). InT particular,−→ LCT morphisms between two sites induce equivalences between the corresponding categories of sheaves (Theorem 2.20 + Remark 2.21). From the Gabriel-Popescu theorem, it follows that every Grothendieck category can be realised as a category of sheaves on a site (see §1.1.3). Moreover, every cocontinuous functor between two categories of sheaves can be obtained as a “roof” of functors coming from continuous morphisms between sites, where the “reversed arrows” are equivalences induced by LC morphisms (see Theorem 4.14 below).2 These observations make it natural to view Grt as a kind of “localization” of Site at the class of LC morphisms. In this chapter we make this idea precise by using the localization of bicategories with respect to a class of 1-morphisms developed by Pronk in [68] and further analysed by Tommasini in the series of papers [82, 80, 81]. Our main result is the following:

1We address size issues in §4.2. 2We have implicitely used this roof construction in the proof of Proposition 2.31.

87 88 Chapter 4. Localization of sites with respect to LC morphisms

Theorem 4.1 (Theorem 4.21). There exists a pseudofunctor

Φ : Site Grt −→ which sends LC morphisms to equivalences in Grt, such that the pseudofunctor

˜ 1 Φ : Site[LC− ] Grt −→ induced by Φ via the universal property of the bicategory of fractions is an equivalence of bicategories.

The chapter is structured as follows.

First, in §4.1, we revisit some basic notions and results from [68] and [82] on bicategories and their localizations with respect to classes of 1-morphisms, which we will refer to as bilocalization from now on. In §4.2 we show that the natural map that assigns to each site its category of sheaves and to each continuous morphism between sites the induced cocontinuous functor between the categories of sheaves extends to a pseudofunctor

Φ : Site Grt. (4.1) −→ In addition, the class LC of LC morphisms admits a calculus of fractions in Site and 1 hence we have a bilocalization Site[LC− ]. This is done in §4.3. In particular, Φ sends LC morphisms to equivalences in Grt, and hence, by the universal property of the bilocalization we obtain a pseudofunctor from the bilocalization to Grt: ˜ 1 Φ : Site[LC− ] Grt. (4.2) −→ In §4.4, based on [81], we show that the pseudofunctor Φ : Site Grt fulfills the nec- −→˜ 1 essary and sufficient conditions for the induced pseudofunctor Φ : Site[LC− ] Grt to be an equivalence of bicategories, which finishes the proof of Theorem 4.1−→. Our interest in representing Grt as a localization of Site comes from our aim to analyse the monoidal structure in Grt induced by the tensor product of Grothendieck categories. Recall from Chapter2 that the tensor product of Grothendieck categories is defined in terms of a tensor product of linear sites. In particular, the fact that LC morphisms are closed under the tensor product of sites is key to the proof of the independence of the tensor product of Grothendieck categories from the sheaf representations chosen. Combining Theorem 4.1 with this fact makes it natural to think that the monoidal structure on Site could be transferred to Grt via the bilocalization, Chapter 4. Localization of sites with respect to LC morphisms 89 following the same principle as the monoidal localization of ordinary categories from [19]. This is briefly addressed in §4.5. We are convinced that the arguments presented in this chapter can be, with some work, suitably adapted to the set-theoretical setup of topos theory. The key point is to determine the parallel notion of LC morphism in the set-theoretical realm, for which the results in [17, §2] are of special relevance. Hence, if we denote by Topoi the 2-category of Grothendieck topoi with colimit preserving functors and natural transformations and by GrSite the 2-category of Grothendieck sites with continuous morphisms and natural transformations, we can state the corresponding analogue of Theorem 4.1 above: Theorem 4.2. There exists a pseudofunctor

Φ : GrSite Topoi −→ which sends LC morphisms to equivalences, such that the pseudofunctor ˜ 1 Φ : GrSite[LC− ] Topoi −→ induced by Φ via the universal property of the bicategory of fractions is an equivalence of bicategories.

4.1 Bicategories of fractions

In this section we introduce the main notions and results on localizations of bicategories from [68] and [82]. In general we will follow the notations and terminology from [68] with the exception that, following a more standard terminology, we will call pseudofunctor what in [68] is called homomorphism of bicategories. We will follow the same conventions as in Chapter3 for the notations of the horizontal ( ) and vertical ( ) composition of 2-morphisms. ◦ • Let’s recall some important deﬁnitions for the rest of the chapter. Deﬁnition 4.3. Given a 1-morphism f : A B in a bicategory C, we say it is an equivalence (or internal equivalence in the−→ terminology of [82]) if there exists another 1-morphism g : B A and a pair of 2-isomorphisms α : IdA g f and β : f g IdB satisfying the−→ triangle identities, i.e. the compositions ⇒ ◦ ◦ ⇒ f α β f f ◦ f g f ◦ f ; ◦ ◦ α g g β g ◦ g f g ◦ g ◦ ◦ 90 Chapter 4. Localization of sites with respect to LC morphisms are the identity on f and on g respectively.

Deﬁnition 4.4. A pseudofunctor Φ : A B between two bicategories is −→

essentially surjective on objects if and only if for all B Obj(B) there exists an • A Obj(A) such that there is an equivalence Φ(A) B ∈in B; ∈ ' essentially full if for all A, A Obj A the functor 0 ( ) • ∈

ΦA,A : A(A, A0) B(Φ(A),Φ(A0)) 0 −→ is essentially surjective ;

fully faithful on 2-morphisms if for all A, A Obj A the functor 0 ( ) • ∈

ΦA,A : A(A, A0) B(Φ(A),Φ(A0)) 0 −→ is fully faithful;

an equivalence of bicategories if it is essentially surjective on objects, essentially • full and fully faithful on 2-morphisms.

In [68, §2] a localization theory for bicategories along a class of 1-morphisms is developed generalizing the well-known localization of (1-)categories in the Gabriel- Zisman sense. In particular, the bicategory of fractions in loc.cit. is deﬁned and constructed by means of a right calculus of fractions. Observe that one could analogously develop the theory for a left calculus of fractions, as it is done in the (1-)categorical case. More precisely, a class of (1-)morphisms admits a left calculus of fractions if and only if the same class of (1-)morphisms in the opposite (bi)category admits a right calculus of fractions. Recall that the opposite bicategory (or transpose bicategory in the terminology of [4]) is given by reversing the 1-morphisms and keeping the direction of the 2-morphisms. In our case, we will be interested in a left calculus of fractions, hence we introduce the analogous results from [68] for a left calculus of fractions.

Deﬁnition 4.5. [68, §2.1] Let C be a bicategory. We say a class W of 1-morphisms on C admits a left calculus of fractions if it satisﬁes:

LF1 All equivalences belong to W;

LF2 W is closed under composition of 1-morphisms; Chapter 4. Localization of sites with respect to LC morphisms 91

LF3 Every solid diagram A f

C w B in C with w W can be completed to a square ∈ A v D α f g

C w B

where α is a 2-isomorphism and v W; ∈ LF4 (1) Given two morphisms f ,g : B A, a morphism w : B B in W and a 0 2-morphism : f w g w−→, there exists a morphism−→v : A A in W α 0 and a 2-morphism◦β :⇒v f◦ v g such that v α = β w ; −→ ◦ ⇒ ◦ ◦ ◦ (2) if α is an isomorphism, we require β to be an isomorphism too; and (3) given another pair v : A A in W and : v f v g satisfying 0 0 β 0 0 0 condition (1), there exist 1-morphisms−→ u, u : A ◦ A⇒with◦u v , u v 0 0 00 0 0 in W and a 2-isomorphism : u v u v such−→ that the diagram◦ ◦ ε 0 0 ◦ ⇒ ◦ u β u v f ◦ u v g ◦ ◦ ◦ ◦ ε f ε g ◦ ◦ u v f u v g 0 0 u 0 0 0 β 0 ◦ ◦ ◦ ◦ ◦ is commutative;

LF5 W is closed under 2-isomorphisms.

Remark 4.6. The ﬁrst axiom can be weakened as is done in [82].

Deﬁnition 4.7. [68, §2] Given a category C and a class of 1-morphisms W in C ad- mitting a left calculus of fractions, a bilocalization of C along W is a pair C W 1 , , ( [ − ] Ψ) where C W 1 is a bicategory and : C C W 1 is a pseudofunctor such that: [ − ] Ψ [ − ] −→ 1. Ψ sends elements in W to equivalences;

2. Composition with Ψ gives an equivalence of bicategories

1 Hom(C W− ,D) HomW(C,D) −→ 92 Chapter 4. Localization of sites with respect to LC morphisms

for each bicategory D, where Hom denotes the bicategory of pseudofunctors (see [4, §8]) and HomW its full sub-bicategory of elements sending W to equivalences.

Observe that, in particular, C W 1 is unique up to equivalence of bicategories 68, [ − ] [ §3.3].

In 68, §2 a detailed construction for C W 1 , is provided for a right calculus [ ] ( [ − ] Ψ) of fractions and in [82] a simpliﬁed version of this construction is provided, less dependent of the axiom of choice. By inverting the direction of 1-morphisms one gets the analogous construction of the bilocalization for a left calculus of fractions.

4.2 The 2-category of Grothendieck categories and the 2-category of sites

Fix a universe U. For a U-small k-linear site (a, a), the category Sh(a, a) is defined with respect to the category U-Mod(k) of U-smallT k-modules. Let SiteT denote the 2-category of U-small k-linear sites with k-linear continuous morphisms of sites and k-linear natural transformations. By definition, a U-Grothendieck category is a k- linear abelian category with a U-small set of generators, U-small colimits and exact U- small filtered colimits. Let V be a larger universe such that all the categories Sh(a, a) are V-small and let Grt denote the 2-category of k-linear V-small U-GrothendieckT categories. Up to equivalence, Grt is easily seen to be independent of the choice of V. In the rest of the chapter, we will omit the universes U and V from our notations and terminology. Remark 4.8. Observe that these are actually enriched 2-categories, more precisely k-linear 2-categories in the sense of [29, Def 2.4 & 2.5]. Remark 4.9. Observe that equivalences (see Definition 4.3 above) in Grt are just the cocontinuous functors which are equivalences of categories in the usual sense, while equivalences in Site are just the continuous morphisms of sites which are equivalences of categories in the usual sense. Notation 4.10. We denote by LC the family of LC morphisms in Site (see Defini- tion 2.19).

The 2-category Grt is related to Site in a natural way. Indeed, we deﬁne a pseudofunctor Φ : Site Grt (4.3) −→ as follows: Chapter 4. Localization of sites with respect to LC morphisms 93

Given a site (a, a), we deﬁne • T Φ(a, a) = Sh(a, a), T T which is a Grothendieck category;

Given a continuous map between two sites f : (a, a) (b, b), we deﬁne • T −→ T f s Φ(f ) = Sh(a, a) Sh(b, b), T T which is colimit preserving;

Given two continuous morphisms f ,g : (a, a) (b, b) and a natural trans- • formation α : f g , we define a natural transformationT −→ T ⇒ s s s Φ(α) = α : f g (4.4) ⇒ as follows. For any F Sh(b, b) and any A a, we have the following morphism: ∈ T ∈ (αs )F (A) := F (αA) : gs (F )(A) = F (g (A)) F (f (A)) = fs (F )(A) −→ which is k-linear and natural in A and F and hence it defines a natural transfor- s mation αs : gs fs . We define α as the natural transformation corresponding ⇒ to αs via the natural adjunctions. More precisely, for all F Sh(b, b) and all ∈ T G Sh(a, a) we have a composition: ∈ T (αs )F Sh(a, a)(G ,gs (F )) ◦ − Sh(a, a)(G , fs (F )) T T ∼= ∼= s s Sh(b, b)(g (G ), F ) Sh(b, b)(f (G ), F ) T T where the vertical functors are the adjunctions. Observe this composition is natural in F and G . Consequently, there is an induced 2-morphism f s g s , and this is the 2-morphism we denote by αs . ⇒

One can easily check these data indeed deﬁne a pseudofunctor:

Given any two sites (a, a),(b, b) the map • T T Φa,b : Site((a, a),(b, b)) Grt(Sh(a, a),Sh(b, b)) T T −→ T T induced by Φ is a functor. Indeed, consider a continuous morphism f : a b, s s trivially the 2-morphism Idf : f f is mapped to Idf s : f f . Now−→ cr ⇒ ⇒ 94 Chapter 4. Localization of sites with respect to LC morphisms

tonsider α : f g and β : g h and their vertical composition β α : f h. One has that ⇒ ⇒ • ⇒

αs βs G (A) = G (αA) G (βA) = G ((α β)A) = (β α)s G (A) • ◦ • • for all G Sh(b) and all A a. Hence, by adjunction, Φa,b preserves compositions. ∈ ∈

Let a = (a, a) be a site and consider its identity morphism Ida. One has that • T (Ida)s = IdSh(a) is the identity functor of Sh(a). Hence, by adjunction,

s IdSh a (Ida) , ( ) ∼= which gives us the unitor of Φ. Consider now two continous morphisms f : a b and g : b c in Site. By deﬁnition, we have that −→ −→ s s s (g f ) = g f , ◦ ∼ ◦ which provides the associator of Φ.

It can be readily seen, using the fact that adjoints are unique up to unique • isomorphism, that the unitor and associator of Φ fulﬁll the corresponding coherence axioms.

Observe that Φ sends LC morphisms to equivalences. This is a direct consequence of the Lemme de comparaison. Hence, if LC admits a left calculus of fractions in Site, we will get, by the universal property of bilocalizations, a pseudofunctor

˜ 1 Φ : Site[LC− ] Grt. (4.5) −→ Remark 4.11. Recall from Remark 4.8 that Site and Grt are k-linear 2-categories, and observe that Φ is also a k-linear pseudofunctor. While in this chapter we only need the bilocalization to exist as an ordinary bicategory, it is possible to show that in this case the bilocalization automatically satisﬁes the universal property of an “enriched bilocalization” (and in particular, the induced functor Φ˜ is automatically k-linear), where we use the term in analogy with the enriched localizations from [85].

4.3 Bilocalization of the 2-category of sites with respect to LC morphisms

In this section we prove that LC admits a left calculus of fractions in Site. Chapter 4. Localization of sites with respect to LC morphisms 95

First, we fix the following notations. Given a site (a, a), it will usually be denoted T simply by a for the sake of brevity. We will denote by ia : Sh(a) , Mod(a) the natural −→ inclusion and by #a : Mod(a) Sh(a) the corresponding sheafification functor. The indexes will be omitted if the−→ site we are working with is clear from the context. Furthermore, given an object A a we will denote hA = a( , A) the corresponding ∈# − representable presheaf and by hA = #(a( , A)) its sheafification. We adopt these notations in order to simplify the formulas− that will appear further in the chapter. We first prove the following result, which will be useful for further steps.

Proposition 4.12. Consider a morphism of sites f : (a, a) (b, b) satisfying (G) with respect to b and (F), (FF) with respect to a. Then,T the following−→ T are equivalent: T T

1. The morphism f satisﬁes (LC) (with respect to a and b); T T 2. The morphism f is continuous and cocontinuous.

Proof. We have already seen that every LC morphism is continuous and cocontinuous (see Theorem 2.20 and Remark 2.21). The converse is a direct consequence of [52, Prop 2.15] and the linear counterpart of [76, Exposé iii, Prop. 1.6].

Remark 4.13. Let f : (a, a) (b, b) be an LC morphism. As it is continuous and cocontinuous, one can considerT −→ theT induced functors between the corresponding sheaf categories both as a continuous and as a cocontinuous morphism. An easy check shows that those are related as follows:

s fe = f , (4.6) ∗ ∼ and hence

fe fs . (4.7) ∗ ∼= Making essential use of LC morphisms we are able recover any colimit preserving functor between Grothendieck categories as being induced by a roof of continuous morphisms of linear sites. More precisely:

Theorem 4.14 (Roof theorem). Let (a, a) and (b, b) be linear sites and consider a T T cocontinuous functor F : Sh(a, a) Sh(b, b). Then, there exist a subcanonical site T −→ T (c, c) and a diagram T c f w (4.8) a b, 96 Chapter 4. Localization of sites with respect to LC morphisms where f is a continuous morphism and w is an LC morphism, such that

F Sh(a, a) Sh(b, b) T T (4.9) f s w˜ ∗ Sh(c, c) T is a commutative diagram up to isomorphism.

This theorem is a slight generalization of [72, Tag 03A2] in the linear setting, where the result is provided for geometric morphisms between Grothendieck topoi (i.e. adjunctions of functors between Grothendieck topoi where the left adjoint is left exact). Our version focuses on the left adjoints, or equivalently on the colimit preserving functors, without requiring them to be left exact (i.e. without requiring the adjunction to be a geometric morphism). Observe that we call LC morphism what in [72] is called special cocontinuous functor (see Proposition 4.12). The proof can be obtained along the lines of [72, Tag 032A]. We will just provide, for convenience of the reader, the construction of the site (c, c) and the morphisms f and w , as these constructions will be frequently used throughoutT the chapter.

Take c to be the full k-linear subcategory of Sh(b, b) with the following set of objects T # # Obj(c) = hB B b F (hA) A a. (4.10) { } ∈ ∪ { } ∈ We endow it with the topology c induced from the canonical topology in Sh(b, b). T T Then we deﬁne f : a c as the composition F #a Ya and w : b c as the composition #b Yb. −→ ◦ ◦ −→ ◦ We now proceed to prove that LC admits a left calculus of fractions in Site. Lemma 4.15. Condition LF1 holds for LC in Site.

Proof. Take an equivalence f Site(a,b), and denote by g : b a its quasi-inverse. Then, it is easy to see that the induced∈ functor −→

fs : Sh(b) Sh(a) −→ is an equivalence of Grothendieck categories, with quasi-inverse given by gs . We prove that f belongs to LC. Property (G) follows from the fact that f is essentially surjective, and properties (F) and (FF) follow immediately from the fact that f is fully-faithful. By Proposition 4.12, it only remains to prove that f is cocontinuous. One can easily see that

f ∗ : Mod(b) Mod(a) −→ Chapter 4. Localization of sites with respect to LC morphisms 97 is also an equivalence with quasi-inverse given by g : Mod a Mod b . Hence we ∗ ( ) ( ) have that f f g by unicity of adjoint functors. Observe then−→ that = ! = ∗ ∗ ∼ ∼ f ia = g ∗ ia = ib gs , ∗ ◦ ∼ ◦ ◦ which implies that f Sh(a) takes values in Sh(b) Mod(b). Consequently f is cocontinuous. ∗| ⊆

Lemma 4.16. Condition LF2 holds for LC in Site.

Proof. The composition of continuous morphisms is again continuous, and the analogous statement is true for cocontinuous morphisms. So we only have to see that the composition of two LC morphisms again fulﬁlls properties (G), (F) and (FF) and we conclude by Proposition 4.12. Consider v : a b and w : b c two LC morphisms between sites. −→ −→ Property (G) for w v follows immediately from the fact that both v and w have property (G) and that◦ the composition of covers is again a cover. We now prove property (F). Consider a morphism

c : (w v )(A) (w v )(A0) ◦ −→ ◦ in c. As w has property (F), we know there exist a collection (ri : Bi v (A))i I of morphisms in b with r v A and a collection of morphisms b−→: B v∈ A i b( ( )) i i ( 0) such that 〈 〉 ∈ T −→ c w (ri ) = w (bi ). (4.11) ◦ b Now, as v has property (G), we can ﬁnd covering families (si j : v (Ai j ) Bi )j Ji in for all i . Consider the morphisms r s : v A v A and b s : v−→A ∈ v A i i j ( i j ) ( ) i i j ( i j ) ( 0) for each i I and each j Ji . As◦v has property−→ (F), we know◦ there exist−→ collections of morphisms∈ ti j k : A∈i j k Ai j k K and t : A Ai j such that ( ) i ,j ( i0 j k 0i j k k Ki0,j −→ ∈ −→ } ∈ ti j k k K and t belong to a, and collections ai j k : Ai j k A k K and i ,j i0 j k k Ki0,j ( ) i ,j 〈a 〉: A∈ A〈 〉 ∈ in a, such thatT −→ ∈ ( i0 j k 0i j k 0)k Ki0,j −→ ∈ ri si j v (ti j k ) = v (ai j k ) ◦ ◦ bi si j v (ti0 j k ) = v (ai0 j k ) ◦ ◦ for all i , j,k. Consider now the intersection of the two covering sieves ti j k and ti0 j k in a for each i amd each j Ji and take a covering family ui j k : Ai j〈 k 〉 Ai j 〈 〉 ( )k Ki00,j generating this sieve. Then∈ we have that −→ ∈

ri si j v (ui j k ) = v (a¯i j k ) ◦ ◦ ¯ bi si j v (ui j k ) = v (a¯i j k ) ◦ ◦ 98 Chapter 4. Localization of sites with respect to LC morphisms for morphisms a¯ : A A and a¯ : A A . Observe that the family i j k i j k i j k i j k 0 formed by the compositions−→ri si j v ui j k −→ is a cover because it is ( ( ))i I ,j Ji ,k Ki00,j ◦ ◦ ∈ ∈ ∈ a composition of covers (note that v ui j k is a covering family of v Ai j in ( ( ))k Ki00,j ( ) ∈ b because ui j k is a covering family of Ai j in a and v is an LC morphism). ( )k Ki00,j ∈ Consequently a¯i j k is a cover in a of A, because v is an LC morphism ( )i I ,j Ji ,k Ki00,j ∈ ∈ ∈ and v a¯i j k is a cover on b. Hence precomposing with w si j v ui j k ( ( ))i I ,j Ji ,k Ki00,j ( ( )) in both terms∈ of (∈4.11∈) we have that: ◦

c w (ri ) w (si j v (ui j k )) = w (bi ) w (si j v (ui j k )) ◦ ◦ ◦ ◦ ◦ for all i I , j Ji ,k Ki00,j . Observe that the ﬁrst term is equal to c (w v )(a¯i j k ) and ∈ ∈ ∈ ¯ ◦ ◦ the second term is equal to (w v )(a¯i j k ). This proves (F) for w v . ◦ ◦ To conclude, we prove property (FF). Consider a morphism a : A A in a such 0 that (w v )(a) = 0. As w has property (FF), there is a collection−→ of morphisms ◦ (ri : Bi v (A))i I in b such that ri b(v (A)) and −→ ∈ 〈 〉 ∈ T v (a) ri = 0 (4.12) ◦ b for all i I . As v has property (G), there exists a cover (si j : v (Ai j ) Bi )j Ji in for ∈ −→ b∈ each i I . Consider the cover (ri si j : v (Ai j ) v (A))i I ,j Ji of v (A) in given by ∈ ∈ the composition.∈ As v has property◦ (F), for each−→i I and each j Ji there exist a a ∈ ∈ cover (ti j k : Ai j k Ai j )k Ki ,j in and a family of morphisms a¯i j k : Ai j k A such that: −→ ∈ −→ ri si j v (ti j k ) = v (a¯i j k ). (4.13) ◦ ◦ Then, precomposing in both terms of (4.12) with si j v (ti j k ), one has that ◦ v (a) ri si j v (ti j k ) = v (a a¯i j k ) = 0. ◦ ◦ ◦ ◦

Eventually, as v has property (FF), we know that for every i I , j I j and k Ki ,j a ∈ ∈ ∈ there exists a cover (ui j kl : Ai j kl Ai j k )l Li ,j,k in such that: −→ ∈ a a¯i j k ui j kl = 0 ◦ ◦ But the family (a¯i j k ui j kl : Ai j kl A)i j kl is a cover because it is a composition of ◦ −→ b covers (observe in (4.13) that, as v is LC, (v (ti j k )k Ki ,j is a cover in and hence so is ∈ a (v (a¯i j k ))i I ,j Ji ,k Ki ,j and thus (a¯i j k )i I ,j Ji ,k Ki ,j is a cover in ). Hence we conclude the argument.∈ ∈ ∈ ∈ ∈ ∈

Lemma 4.17. Condition LF3 holds for LC in Site. Chapter 4. Localization of sites with respect to LC morphisms 99

Proof. Assume we have a solid diagram b

w c a f in Site with w LC. We have to prove that it can be completed to a square ∈ g b d α w v c a f where α is an invertible 2-morphism and v LC. ∈ Consider the following morphism of Grothendieck categories

fs w˜ Sh(a) Sh(c) ∗ Sh(b) induced by f and w , whose left adjoint is given by f s w˜ : Sh b Sh a . ∗ ( ) ( ) ◦ −→ We apply the roof theorem (Theorem 4.14) to this latter morphism:

Consider d the linear site deﬁned as the full subcategory of Sh(a) with objects h # f s w˜ h # and endow it with the topology induced by the canonical A A a ( ∗)( B ) B b topology{ } ∈ ∪{ in Sh◦(a). Then} consider∈ the following morphisms: g b d

v a deﬁned by v # Y and g f s w˜ # Y . = a a = ( ∗) b b ◦ ◦ ◦ ◦ Now, as w is an LC morphism by hypothesis, and thus continuous and cocontinuous, we have the following chain of invertible 2-morphisms (see §1.1.2 for the properties of continuous and cocontinuous functors): s g w = f w˜ ∗ #b Yb w ◦ s ◦ ◦ ◦ ◦ = f #c w ∗ ib #b Yb w s ◦ ◦ ◦ ◦ s◦ ◦ = f #c w ∗ ib w #c Yc s ◦ ◦ ◦ ◦ s ◦ ◦ = f #c ic ws w #c Yc s ◦ ◦ ◦ ◦ ◦ ◦ = f #c Yc ∼ ◦ ◦ = #a Ya f ◦ ◦ = v f ◦ 100 Chapter 4. Localization of sites with respect to LC morphisms

Hence, if we take α to be this invertible 2-morphism, we conclude the argument.

Lemma 4.18. Condition LF5 holds for LC in Site.

Proof. Consider two morphisms v, w : a b in Site such that w LC. Assume we also have an invertible 2-morphism α : v −→w . We want to prove that∈ v also belongs to LC. ⇒

Pick B b an object. By hypothesis, there exists a cover (ri : w (Ai ) B)i and we can consider∈ the associated cover −→

αAi ri (v (Ai ) w (Ai ) B) −→ −→ by composing with the isomorphism αAi given by the invertible 2-morphism (recall that any isomorphism generates a covering sieve in a topology, the corresponding representable one). This proves property (G) for v . Consider now a morphism b : v A v A in b. Then we can take the morphism ( ) ( 0) 1 −→ αA b α : w (A) w (A ), and as property (F) holds for w , we have that there 0 −A 0 exists◦ a◦ cover s : A −→ A and morphisms a : A A in a, such that ( i i )i i i 0 −→ −→ 1 αA b α−A w (si ) = w (ai ) 0 ◦ ◦ ◦ for all i . Consequently:

1 1 1 v (ai ) = α−A w (ai ) αAi = α−A αA b α−A w (si ) αAi = b v (si ), 0 ◦ ◦ 0 ◦ 0 ◦ ◦ ◦ ◦ ◦ which proves that property (F) holds for v . Take now a : A A in a such that v a 0. This implies that w a 0 and hence 0 ( ) = ( ) = −→ there exists a cover (ti : Ai A)i such that a ti = 0 for all i , which proves that property (FF) holds for v . −→ ◦ It remains to prove that v 1 . We have that w 1 by hypothesis, hence a = − b a = − b T T T T given a covering sieve ri : Ai A of A in a, we have that ri a(A) if and only if 〈 −→ 〉 〈 〉 ∈ T w (ri ) : w (Ai ) w (A) b(w (A)). 〈 −→ 〉 ∈ T On the other hand, it is easy to see that w (ri ) b(w (A)) if and only if 〈 〉 ∈ T 1 α−A w (ri ) αAi = v (ri ) : v (Ai ) v (A) b(v (A)), 〈 ◦ ◦ −→ 〉 ∈ T because αAi ,αA are isomorphisms. This concludes the argument.

Lemma 4.19. Condition LF4 holds for LC in Site. Chapter 4. Localization of sites with respect to LC morphisms 101

Proof. First we prove (1) holds. Given two continuous morphisms f ,g : b a in Site, an LC morphism w : b b and a 2-morphism : f w g w , we−→ have to 0 α prove that there exists an LC morphism−→ v : a a and a 2-morphism◦ ⇒ ◦ : v f v g 0 β such that v α = β w . −→ ◦ ⇒ ◦ ◦ ◦ Consider the morphisms

f w g w Sh a ( )s Sh b and Sh a ( )s Sh b ( ) ◦ ( 0) ( ) ◦ ( 0) between the corresponding sheaf categories, with respective left adjoints given by f w s : Sh b Sh a and g w s : Sh b Sh a . We now perform a similar ( ) ( 0) ( ) ( ) ( 0) ( ) construction◦ to−→ that on the roof theorem.◦ Take−→ the full subcategory a of Sh a with 0 ( ) # s # s # Obj(a0) = hA A a (f w ) (hB ) B b (g w ) (hB ) B b , { } ∈ ∪ { ◦ 0 } 0∈ 0 ∪ { ◦ 0 } 0∈ 0 and endow it with the topology given by the restriction of the canonical topology in Sh a . In particular, a is subcanonical. We construct the following roofs ( ) 0 a a 0 0 rf v rg v

b a b a 0 0 s s where v = #a Ya, rf = (f w ) #b Yb and rg = (g w ) #b Yb . One can easily see, 0 0 0 0 following the◦ same arguments◦ ◦ as in◦ the roof theorem,◦ that◦ v ◦is an LC morphism and s s s s that (f w ) v˜ (rf ) and (g w ) v˜ (rg ) . We have chosen this special a in ∼= ∗ ∼= ∗ 0 order to◦ have the same◦ site on the◦ top of both◦ roofs, but the reasoning to prove that these roofs behave as the usual roof construction is not affected by this enlargement of the top category, as it remains to be small. Now that v is constructed, we proceed to build the 2-morphism β : v f v g . ◦ ⇒ ◦ s s s First observe that given α : f w g w we have α : (f w ) (g w ) the induced 2-morphism described in (4.4◦). In⇒ particular,◦ one has that:◦ ⇒ ◦

h # # αA # HomSh b (h , F ) − ◦ HomSh b (h , F ) ( 0) (g w )(A) ( 0) (f w )(A) ◦ ◦

∼= ∼= F (αA ) F ((g w )(A)) F ((f w )(A)) ◦ ◦ (4.14) ∼= ∼= # (αs )F # HomSh(a)(hA,(g w )s (F )) ◦ − HomSh(a)(hA,(f w )s (F )) ◦ ◦ = = ∼ s ∼ (α )h# s # − ◦ A s # HomSh(b )((g w ) (hA), F ) HomSh(b )((f w ) (hA), F ) 0 ◦ 0 ◦ 102 Chapter 4. Localization of sites with respect to LC morphisms is a commutative diagram. Now observe that:

s v f = #a Ya f = f #b Yb, ◦ ◦ ◦ s ◦ ◦ v g = #a Ya g = g #b Yb. ◦ ◦ ◦ ◦ ◦ But notice that, as w is an LC morphism, w s : Sh b Sh b is an equivalence with ( 0) ( ) quasi-inverse given by w : Sh b Sh b . Hence we−→ have that s ( ) ( 0) −→ s v f = (f w ) (ws #b Yb) ◦ ∼ ◦ s ◦ ◦ (4.15) v g = (g w ) (ws #b Yb). ◦ ∼ ◦ ◦ ◦ We deﬁne β as the composition

= s s = v f ∼ (f w ) ws #b Yb (g w ) ws #b Yb ∼ v g (4.16) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ s where the second 2-morphism is given by α (ws #b Yb). ◦ ◦ ◦ Let’snow check that v α = β w . First observe that have the following commutative diagram ◦ ◦ # # h g w B h f w B ( )( 0) βw (B ) ( )( 0) ◦ 0 ◦

∼= ∼= s # s # g w ws h s f w ws h ( ) w (B ) α # ( ) w (B ) 0 ( )ws (h ) 0 ◦ ◦ ◦ w (B ) ◦ ◦ ◦ 0 ∼= ∼= s # s # (g w ) (hB ) s (f w ) (hB ) 0 (α )h# 0 ◦ B ◦ 0

h # h # , g w B # f w B ( )( 0) h ( )( 0) ◦ αB ◦ 0 where the last commutative square comes from the commutative diagram (4.14) above. It is easily seen that the vertical compositions are the identity. Hence we have # that βw B = h = v (αB ) for all B b , which concludes the argument. ( 0) αB 0 0 0 0 ∈ We prove now (2). Assume α is an invertible 2-morphism. Then so are αs and s s α , and hence α (ts w ) is also an invertible 2-morphism. As β is obtained from s ◦ ◦ α (ws v ) via pre- and postcomposing (vertically) with invertible 2-morphisms, we conclude◦ ◦ the argument. Finally, we prove (3). Assume there exists another v : a a in LC and another 0 0 2-morphism : v f v g with v w . We have−→ to prove that there exist β 0 0 0 0 α = β 0 ◦ ⇒ ◦ ◦ ◦ Chapter 4. Localization of sites with respect to LC morphisms 103

morphisms u, u : a a such that u v LC and u v LC, and an invertible 0 0 00 0 0 2-morphism : u v −→u v such that◦ the∈ following diagram◦ ∈ ε 0 0 ◦ ⇒ ◦ u β u v f ◦ u v g ◦ ◦ ◦ ◦ ε f ε g (4.17) ◦ ◦ u β u v f 0 0 u v g 0 0 ◦ 0 0 ◦ ◦ ◦ ◦ commutes. Consider the equivalence v s v : Sh a Sh a , whose quasi-inverse is given 0 s ( 0) ( 0) by v s v . We consider the associated◦ roof construction−→ for v s v : s0 s0 ◦ ◦ a 00 u u 0

a a , 0 0

# s # s with a = h A a v v (h ) A a , u = #a Ya and u = (v v ) #a Ya . In 00 A 0 0 s0 A 0 0 0 0 0 s0 0 0 particular,{u is0 } in∈LC∪and { as◦ so was0 } v∈by assumption,◦ so it follows from◦ ◦ Lemma◦ 4.16 that u v belongs to LC. ◦ Now let’s construct ε. For each A a we have: ∈ s s s s u v = #a Ya v = v #a Ya = (v vs0 v 0 ) #a Ya = (v vs0)(#a Ya v 0) = u 0 v 0. ◦ 0 ◦ 0 ◦ ◦ ◦ ∼ ◦ ◦ ◦ ◦ ∼ ◦ 0 ◦ 0 ◦ ◦ Let’s denote this composition of invertible 2-morphisms by : u v u v . Observe ε 0 0 ﬁrst that from Lemma 4.18 above, it follows that u v also belongs◦ ⇒ to LC◦ . To ﬁnish 0 0 the argument, it remains to check that the diagram◦ (4.17) above is commutative. Evaluating in any object B b we obtain the commutative diagram (4.18). ∈ Observe that the left vertical composition in the diagram (4.18) equals εf (B) and the right vertical composition equals εg (B), which concludes our argument. 104 Chapter 4. Localization of sites with respect to LC morphisms

u(βB ) u v f (B) u v g (B) ◦ ◦ ◦ ◦ # hβ h # B h # v f (B) v g (B) ◦ ◦ = β w s = ∼ ( )w h# ∼ s # ◦ s ( B ) s # (v f w ) (ws (hB )) (v g w ) (ws (hB )) ◦ ◦ ◦ ◦ v α s ( )w h# s # ◦ s B s # (v f w ) (ws (hB )) (v f w ) (ws (hB )) ◦ ◦ s s ◦ ◦ = (v vs ) (v α) # = ∼ 0 0 ws h ∼ v s v v f w s w h # ◦ ◦ ◦ B v v v f w s w h # ( s0) ( 0 ) ( s ( B )) ( s0) ( 0 ) ( s ( B )) ◦ ◦ ◦ ◦ s s ◦ ◦ ◦ ◦ (v vs ) (β w ) # 0 0 ws h v s v v f w s w h # ◦ ◦ ◦ B v s v v g w s w h # ( s0) ( 0 ) ( s ( B )) ( s0) ( 0 ) ( s ( B )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ = v s v β s = ∼ ( s0) h0 # ∼ v s v v f s h # ◦ ◦ B v s v v g s h # ( s0) ( 0 ) ( B ) ( s0) ( 0 ) ( B ) ◦ ◦ ◦ ◦ ◦ ◦ v s v h # ( s0) β v s v h # ◦ ◦ B0 v s v h # ( s0)( (v f )(B)) ( s0)( (v g )(B)) ◦ 0◦ ◦ 0◦ u (β ) u v f B 0 B0 u v g B 0 0 ( ) 0 0 ( ) ◦ ◦ ◦ ◦

(4.18) Chapter 4. Localization of sites with respect to LC morphisms 105

Finally, we are in the position to prove the following.

Proposition 4.20. LC admits a left calculus of fractions in Site.

Proof. The statement follows from Lemmas 4.15 to 4.19 above.

Consequently, we can localize Site with respect to LC and obtain the bilocalization 1 Site[LC− ].

4.4 The 2-category of Grothendieck categories as a bilocalization of the 2- category of sites

In this section we prove the main result of the chapter.

Theorem 4.21. There exists a pseudofunctor

Φ : Site Grt −→ which sends LC morphisms to equivalences in Grt, such that the pseudofunctor

˜ 1 Φ : Site[LC− ] Grt −→ induced by Φ via the universal property of the bicategory of fractions is an equivalence of bicategories.

Let C be a bicategory and W a class of 1-morphisms in C that admits a calculus of left fractions. Given a bicategory D and a pseudofunctor Φ : C D sending 1-morphisms that belong to W to equivalences in D, we have that Φ induces−→ a pseudofunctor ˜ : C W 1 D by the universal property of bilocalizations. A characteri- Φ [ − ] zation of the pseudofunctors−→ Φ such that Φ˜ is an equivalence of bicategories (in the case of a right bicategory of fractions) is provided in [81] . The characterization makes use of the right saturation of a class of morphisms introduced in [80]. We formulate below an analogue for a left calculus of fractions.

Deﬁnition 4.22. Let W be a class of 1-morphisms in the bicategory C. The (left) saturation Wsat of W is the class of all 1-morphisms f : A B in C, such that there exists a pair of objects C ,D C and a pair of morphisms g−→: B C and h : C D , such that both g f and h ∈g belong to W. −→ −→ ◦ ◦ We say that W is (left) saturated if W = Wsat. 106 Chapter 4. Localization of sites with respect to LC morphisms

In analogy to [80, Rem 2.3] in the case of right saturation, we have the following statement for the left saturation.

Proposition 4.23. If the class of morphisms W admits a left calculus of fractions, then W Wsat. ⊆ Proposition 4.24 ([81, Thm 0.4]). Let C be a bicategory and W a class of 1-morphisms in C that admits a calculus of left fractions. Given a bicategory D and a pseudofunctor Φ : C D sending 1-morphisms that belong to W to equivalences in D, we have that −→ induces an equivalence of bicategories ˜ : C W 1 ∼= D if and only if: Φ Φ [ − ] −→ B1 Φ is essentially surjective on objects;

= B2 Given objects C1,C2 C and an equivalence e : Φ(C2) ∼ Φ(C1), there exits an object C3 C, a pair of∈ morphisms w1 : C1 C3 in W and−→w2 : C2 C3 in Wsat, an equivalence∈ e : C C in D and−→ a pair of invertible−→2-morphisms 0 Φ( 3) Φ( 1) δ1,δ2 as follows −→ e Φ(C2) δ2 Φ(w2) ⇓ e 0 Φ(C3) Φ(C1). Φ(w1) δ1 ⇓ Φ(C1) IdΦ(C1)

B3 Given objects C C and D D and a morphism f : Φ(C ) D , there exists an ∈ ∈ −→ object C C, a morphism g : C C in C, an equivalence e : C ∼= D in 0 0 Φ( 0) D and an∈ invertible 2-morphism−→α : f e Φ(g ). −→ ⇒ ◦ B4 Given objects C ,C C, two 1-morphisms f , f : C C in C and two 2- 0 1 2 0 morphisms , : f ∈ f such that , there−→ exits an object C C γ1 γ2 1 2 Φ(γ1) = Φ(γ2) 00 and a 1-morphism w⇒: C C in W such that w w . ∈ 0 00 γ1 = γ2 −→ ◦ ◦ B5 Given objects C ,C C, a pair of morphisms f , f : C C and a 2-morphism 0 1 2 0 : f f , then,∈ there is an object C C, a morphism−→ w : C C in W α Φ( 1) Φ( 2) 00 0 00 and a 2-morphism⇒ β : w f w f such that∈ Φ w α ψΦ Φ β−→ψΦ 1, 1 2 ( ) = w,f2 ( ) ( w,f1 )− where ψΦ denotes the associator◦ ⇒ of◦ the pseudofunctor◦ Φ. • •

Remark 4.25. We need to use this set of necessary and sufficient conditions from [81] as the set of sufficient conditions provided by [68, Prop 24] is not satisfied in our case. Chapter 4. Localization of sites with respect to LC morphisms 107

We are now in the position to prove Theorem 4.21.

Proof. Recall from §4.2 that we have a pseudofunctor Φ : Site Grt that sends LC morphisms to equivalences. Hence by the universal property−→ of the bilocalization, we have an induced pseudofunctor ˜ 1 Φ : Site[LC− ] Grt. (4.19) −→ Consequently, if we prove that the pseudofunctor Φ : Site Grt satisﬁes properties B1 to B5 from Proposition 4.24, we conclude the argument.−→ This is done in Lemma 4.26 below.

Lemma 4.26. The pseudofunctor Φ : Site Grt satisﬁes properties B1 to B5 in Proposition 4.24 above. −→

Proof. We know every Grothendieck category can be realised as a category of sheaves on a site, hence Φ is essentially surjective on objects, which proves B1.

= We now prove property B2. Let a1,a2 be two sites and e : Sh(a2) ∼ Sh(a1) an equivalence in Grt. We apply the roof theorem to the functor e . Let−→a3 be the site with objects # # Obj a h a e h a Sh a , ( 3) = A1 A1 1 ( A2 ) A2 2 ( 1) { } ∈ ∪ { } ∈ ⊆ and the topology induced by the canonical topology in Sh(a1). We have the roof construction a3 w2 w1

a2 a1, where w1 = #a1 Ya1 and w2 = e #a2 Ya2 . By the roof theorem, w1 is an LC morphism. On the other hand,◦ we have that◦ ◦

s s e = (Þw1)∗ (w2) = (w1)s (w2) , ∼ ◦ ∼ ◦ where the ﬁrst step is given by the roof theorem and the second by Remark 4.13. Observe that (w1)s is an equivalence because w1 is an LC morphism. Then, as e is s an equivalence by hypothesis, (w2) is also and equivalence and hence so is (w2)s . In summary, we have a morphism w2 : a2 a3 with a3 subcanonical and such that −→ (w2)s is an equivalence. Then, it follows from [51, Cor 4.5] that w2 is an LC morphism. Hence, in particular, as LC LCsat by Proposition 4.23, we have that w2 LCsat. Consider now the equivalence⊆ ∈

∼= e 0 : Sh(a3) Sh(a1), −→ 108 Chapter 4. Localization of sites with respect to LC morphisms

given by e w ∗. Then we can then choose to be the isomorphism: 0 = (Þ1) δ2 s e = e 0 (w2) = e 0 Φ(w2) ∼ ◦ ◦ given by the roof decomposition of e . On the other hand, we have that

s s e w w ∗ w w w Id , 0 Φ( 1) = (Þ1) ( 1) = ( 1)s ( 1) = Sh(a1) ◦ ◦ ∼ ◦ ∼ where the second step follows from Remark 4.13, and the last from the fact that w1 is s LC, and hence (w1) is an equivalence with quasi-inverse given by (w1)s . Then, we can denote by δ1 the horizontal composition of this chain of invertible 2-morphisms, which concludes the argument. We now proceed to prove B3. Fix a site b, a Grothendieck category A and a 1- morphism f : Sh(b) A in Grt. We choose a site a such that we have an equivalence = −→ e : A ∼ Sh a . Consider the morphism e f : Sh b Sh a in Grt and its associ- 0 ( ) 0 ( ) ( ) ated roof−→ decomposition ◦ −→ c g w

b a.

We then have an equivalence w˜ e : A ∼= Sh a ∼= Sh c . Choose e a quasi-inverse 0 ( ) ( ) 00 ∗◦ =−→ −→ of e and consider e e w˜ : Sh c ∼ A which is a quasi-inverse of w˜ e . Then, 0 = 00 ∗ ( ) ∗ 0 by the roof theorem, we◦ have that e−→f w˜ g s and hence, by postcomposing◦ 0 ∼= ∗ with e on both the left and the right hand◦ side,◦ we have an invertible 2-morphism 00 s f = e g = e Φ(g ), which ﬁnishes the argument. ∼ ◦ ◦ We now prove B4. Fix two sites a,b, two continuous morphisms f1, f2 : a b and a pair of 2-morphisms γ1,γ2 : f1 f2 such that −→ ⇒s s s s (γ1) = (γ2) : (f1) (f2) . ⇒ # Take c the site with objects hB B b with the topology induced by the canonical { } ∈ topology in Sh(b) and w = #b Yb : b c the corresponding LC morphism. Observe s s that γ # γ # for all◦A a. This−→ implies, applying the commutative diagram (( 1) )hA = (( 2) )hA # # ∈ (4.14), that h h for all A a. Hence, we have that w γ1 A w γ2 A for all (γ1)A = (γ2)A ( ) = ( ) A a and natural in A, which concludes∈ the argument. ◦ ◦ ∈ Finally, we prove property B5. Consider two sites a,b, two continuous morphisms s s f1, f2 : a b and a 2-morphism α : f1 f2 . Consider the site c with objects # −→ ⇒ hB B b Sh(b) and the topology induced by the canonical topology in Sh(b) and the { } ∈ ⊆ LC morphism w = #b Yb : b c. Take β : w f1 w f2 the 2-morphism given by ◦ −→ ◦ ⇒ ◦ # Y s α a a s ◦ ◦ w f1 = #b Yb f1 = f1 #a Ya = f2 #a Ya = #b Yb f2 = w f2. ◦ ◦ ◦ ◦ ◦ ⇒ ◦ ◦ ◦ ◦ ◦ Chapter 4. Localization of sites with respect to LC morphisms 109

Then, we have that

s s s w α s s w f1 ◦ w f2 ◦ ◦ ψS ψS 1 w,f1 = = ( w,f2 )− ∼ S 1 w s S ∼ (ψw,f )− ( α) ψw,f s 2 • ◦ • 1 s (w f1) (w f2) ◦ ◦

s s s (α #a Ya) s s (f1 #a Ya) ◦ ◦ (f2 #a Ya) ◦ ◦ ◦ ◦ is a commutative diagram of 2-morphisms. But observe that the composition

# Y s s s s (α a a) s s ◦ ◦ (w f1) = (f1 #a Ya) = (f2 #a Ya) = (w f2) ◦ ◦ ◦ ⇒ ◦ ◦ ◦ is just β s by deﬁnition, hence

s S s S 1 w α = ψw,f β (ψw,f )− , ◦ 2 • • 1 which concludes the argument.

4.5 Monoidal bilocalization

Let C be a category and W a class of morphisms which admits a calculus of fractions in the sense of Gabriel-Zisman [28]. Then, as it is proven in [19], if C has a symmetrical monoidal structure such that W is closed under tensoring, then the localization C W 1 has a monoidal structure such that the localization functor C C W 1 is a [ − ] [ − ] monoidal functor. This is what in [19] is referred to as monoidal localization−→ . It is reasonable to believe that an analogous result for monoidal bicategories and bilocalizations holds true, and we plan to return to this topic in the future. In this section, we briefly sketch a possible application of the main result of this chapter given that we have a satisfactory theory of monoidal bilocalization available. In Chapter2, we have defined a tensor product of linear sites, which is seen to define a symmetric monoidal structure on the bicategory Site

: Site Site Site : ((a,Ta),(b,Tb)) (a b,Ta Tb). × −→ 7−→ ⊗ We further have shown that the class of LC morphisms is closed under , hence we obtain an induced bi-pseudofunctor

˜ 1 1 1 1 : Site[LC− ] Site[LC− ] = (Site Site)[(LC LC)− ] Site[LC− ] × × × −→ 110 Chapter 4. Localization of sites with respect to LC morphisms which a general theory of monoidal bilocalization would yield to deﬁne a monoidal structure in the bicategorical sense. This structure could then be transferred to the equivalent bicategory Grt (in a non-canonical way). CHAPTER 5

Tensor product of well-generated pretriangulated categories

Well-generated triangulated categories (see §1.3.3) are a very important class of triangulated categories. They were introduced by Neeman as a generalization of compactly generated triangulated categories. In particular, Brown representability holds for them and they are well-behaved in terms of localization theory (see §1.3.4). Among examples of interest, it is well-known that the derived category of a Grothendieck abelian category is well-generated [60, Thm 0.2], which provides us with a large class of examples of geometrical origin, namely the derived categories of (quasi-coherent sheaves of) schemes and algebraic stacks. As it happens with Grothendieck abelian categories, triangulated categories and their enhancements (see §1.4.3) are used as models for noncommutative spaces, as an approach to derived geometry in noncommutative settings (see for example [5], [44], among many other works). Our aim for this chapter is to deﬁne a tensor product of well-generated pretriangulated dg categories in order to provide, in the same spirit as in the case of the tensor product of Grothendieck categories, a suitable notion of “product of (derived) noncommutative spaces”.

In [66] a triangulated version of Gabriel-Popescu theorem is proven, which allows us to identify well-generated algebraic triangulated categories as the triangulated analogues to Grothendieck abelian categories. More precisely, just like Grothendieck categories are the localizations of module categories, we have that well-generated algebraic triangulated categories are the localizations (with respect to localizing subcategories generated by a set of objects) of derived categories of small dg categories. In particular, this result can be enhanced to the dg level, more concretely, pretri- 0 angulated dg categories C with H (C) well-generated are Drinfeld dg quotients of derived dg categories of a small dg category with respect to a localizing subcategory

111 112 Chapter 5. Tensor product of well-generated pretriangulated categories generated (in the triangulated sense) by a set of objects. In the dg world, dg topologies are not quite the right tool in order to perform localization on the derived level. However, a dg counterpart of Pitt’s approach to the tensor product of topoi [63] combined with our description of the tensor product of Serre localizing subcategories from §2.1.5 leads to a natural notion of tensor product of well-generated pretriangulated dg categories. It is our expectation for this tensor product to be compatible with Lurie’s tensor product of presentable infinity categories [57]. However, we will not analyse this question here, as infinity categories are not in the scope of this thesis. Contrarily, the relation between the tensor product of Grothendieck categories and the tensor product of well-generated pretriangulated categories is subtler due to flatness issues. It is our intention to investigate this topic further in the future. The structure of the chapter is as follows. Our starting point will be the homotopy category of dg categories Hqe developed by Tabuada and Toën. As shown in [79], Hqe has a monoidal structure given by the derived tensor product of dg categories L and this monoidal structure is closed with its internal hom (denoted by RHom⊗) given by the right quasi-representable bimodules. The first part of the chapter, contained in §5.1,§5.2 and §5.3, will be focused on studying the subcategory of Hqe formed by the homotopically-cocomplete pretrian- pt gulated categories, that will be denoted by Hqec . In §5.1, we consider the restriction of the internal hom to RHomc(A,B), that is, to bimodules inducing coproduct-preserving functors on the level of homology. In pt particular, we will show that RHomc is internal in Hqec . In §5.2, we introduce the notion of cpt dg quotient of a homotopically Cocomplete 0 PreTriangulated dg category C with respect to a class of objects N H (C), which can be characterized by the following replacement of (2.24) ⊆

RHomc(C/N,D) RHomc,N(C,D), (5.1) ∼= where the right hand side denotes the subcategory of RHomc(C,D) consisting of the 0 0 bimodules for which the induced cocontinuous functor H (C) H (D) sends N to zero. −→ We deﬁne a dg site as a small dg category a along with a localizing thick subcategory W D(a) of the derived category and a dg topos as any dg category equivalent (in Hqe) to D⊆(a)/W for some dg site (a,W). Observe that by the triangulated Gabriel-Popescu Chapter 5. Tensor product of well-generated pretriangulated categories 113 theorem, dg topoi are precisely the pretriangulated dg categories whose homotopy category is well-generated. Note as well that they are large categories, as so is D(a). Finally, in §5.3, we proceed to analyse the behaviour of RHom in the two variable setting. In particular we analyse both cocontinuity and the cpt quotient in the two variables setting.

In the second part of the chapter we focus on the subclass of Hqewg given by the pretriangulated categories with well-generated homotopy category, which will be called well-generated dg categories. In §5.4, the well-generated tensor product between well-generated dg categories A and B is deﬁned, if it exists, as the unique homotopically cocomplete pretriangulated dg category A B satisfying the following universal property with respect to cocomplete dg categories C:

RHomc(A B,C) RHomc(A,RHomc(B,C)). (5.2) ∼=

The rest of the chapter is devoted to prove the existence of the well-generated tensor product and to provide different constructions of it in terms of the different realizations of well-generated dg categories. In order to do so, we first study the localization theory of well-generated dg categories. In parallel with Grothendieck categories, we can approach localization of well-generated pretriangulated categories not only via localizing subcategories generated by sets but also via well-generated strict localizations (which are “enhancements of Bousfield localizations”). This will be introduced in §5.5 and the equivalence between both approaches will be made explicit. Next, we analyse the relation of the well-generated tensor product and the cpt quotient of well-generated dg categories. As analysed in §5.5, the cpt quotient of a well-generated dg category at a localizing subcategory generated by a set of objects remains well-generated. In particular, we show that if the tensor product of two well- generated dg categories exists, so does the well-generated tensor product between any such cpt quotients of them. This is done in §5.6. In §5.7, we can finally prove the existence of the well-generated tensor product. In order to do so, we first prove that the well-generated tensor product of derived dg categories exists. Then, in view of Gabriel-Popescu theorem, a combination of this result with those of §5.6 provides us the desired existence of the tensor product. The proof of Porta’s Gabriel-Popescu theorem makes essential use of a very special type of Bousfield localizations of derived categories of small dg categories, namely the so called α-continuous derived categories. Given a homotopically α-cocomplete 114 Chapter 5. Tensor product of well-generated pretriangulated categories

small dg category a, its α-continuous derived category Dα(a) can be deﬁned, roughly, as the full subcategory of D(a) with objects given by the dg functors whose homology preserves α-small coproducts. In particular, the Gabriel-Popescu theorem can be re-stated as saying that a triangulated category is algebraic and well-generated if and only if there exists a regular cardinal α such that it is triangle equivalent to Dα(a) for a certain homotopically α-cocomplete dg category a. In particular, we can enhance this result and talk about α-continuous derived dg categories, which will be denoted by Dα(a). It is then natural to ask whether there is a nice realization of the tensor product of well-generated dg categories (or dg topoi) in terms of α-continuous derived dg categories. To do so, we follow two steps:

1. Given two homotopically α-cocomplete categories a, b, one can deﬁne the α-cpt tensor product of a and b, if it exists, as the homotopically α-cocomplete L small dg category a α b characterized by the universal property ⊗ RHomα(a b,c) RHomα(a,RHomα(b,c)), (5.3) ∼=

where c is a homotopically α-cocomplete smal dg category and where RHomα denotes the quasi-representable bimodules which are α-cocontinuous at the level of homology. We prove its existence for any two homotopically α-cocomplete pretriangulated dg categories and provide a construction. This will be done in §5.9.

2. We show that given a and b two α-continuous derived dg categories, we have the following equivalence

L Dα(a) Dα(b) = Dα(a α b) (5.4) ∼ ⊗ of large dg categories, which provides the desired interpretation of the well- generated tensor product. This is contained in §5.10.

We refer the reader to §1.3 and §1.4 for the basic theory on triangulated and dg categories and the notations that will be used in the rest of the chapter.

5.1 Hom and tensor of dg categories

Throughout k is a commutative ground ring.

Fix U a universe. Recall that the category U-dgcatk of U-small dg categories over k has a standard model structure, due to Tabuada, with the class of weak-equivalences Chapter 5. Tensor product of well-generated pretriangulated categories 115 given by the quasi-equivalences [75]. We point the reader to §1.4.2 for a brief account on the properties of this model structure. Our starting point will be the homotopy category U-Hqe obtained by inverting quasi-equivalences in U-dgcatk . Observe that, given a U-small dg category b, we have that the dg derived category D(b), taken with respect to the category of U-small dg k-modules U-dgMod(k), is no longer U-small. We will ﬁx another universe V, with U V, such that for all U-small dg category b, the category D(b) is V-small. We will keep⊆ this choice of universes in sections §5.1, §5.2 and §5.3, where our constructions will depend on the choice of both universes. From §5.4 on, we will be able, when dealing with well-generated pretriangulated dg categories, to address size issues such as our constructions will only depend on the initial universe U. This will be explained in due time. For the moment, and as a convention, when the universe does not need to be made precise because the statement just works for any ﬁxed universe, we will simply talk about small categories. When we need to make the universes precise, we will always indicate them. Let a,b,c be small dg categories. Our initial setup will be the closed symmetric monoidal structure in Hqe provided by Toën in [79, §6]. In particular, we have the adjunction L [a b,c] [a,RHom(b,c)]. (5.5) ∼= ⊗ L between the derived tensor product a b and Toën’s internal RHom(b,c) in Hqe. A description of the dg category RHom⊗(b,c) can be given in terms of right quasi- representable bimodules. We refer the reader to §1.4.2 for a brief description based on [79, §4,6]. Recall that RHom(b,c) as constructed in loc.cit. is actually not small, but essentially small. It will nevertheless still be considered as an element in Hqe. 0 Let b be a small pretriangulated dg category (see §1.4.3). This means that H (b) has a canonical triangulated structure. More precisely, the image of the functor 0 0 0 H (Yb) : H (b) H (dgMod(b)) −→ induced by the dg Yoneda embedding Yb : b dgMod(b) is a triangulated subcategory. Recall that pretriangulated dg categories−→ have a notion of cone of closed 0 morphisms of degree 0 and of shift that induce the triangulated structure of H (b) and that present them as dg categories very close to categories of dg modules. We have the following well-known result: Proposition 5.1. Consider b and c two small pretriangulated dg categories. Then for 0 0 0 every F RHom(b,c), the induced functor H (F ) : H (b) H (c) is exact. ∈ −→ The following result is also well-known. We provide a proof for the convenience of the reader: 116 Chapter 5. Tensor product of well-generated pretriangulated categories

Lemma 5.2. Consider dg categories b and c. If c is a pretriangulated dg category, then so is RHom(b,c).

L op Proof. The derived dg category D(c b ) is a pretriangulated dg category because we 0 L op L op 0 have that H (D(c b )) = D(c b⊗ ). It is thus enough to prove that H (RHom(b,c)) L op is a triangulated subcategory⊗ of⊗D(c b ). ⊗ L op Take F RHom(b,c) and consider its shift F [1] in D(c b ). We have that the 0 0 induced H∈ (F [1]) : H (b) D(c) is given by ⊗ −→ 0 0 (H (F )[1])(B) = (H (F )(B))[1]

0 0 for all B H (b). As H (c) is a triangulated subcategory of D(c) by hypothesis and 0 0 0 0 H (F )(B)∈ H (c) for all B b by assumption, (H (F )(B))[1] H (c) for all B B. We can thus conclude∈ that F [∈1] RHom(b,c). ∈ ∈ ∈ Similarly, consider an exact triangle

F F 0 F 00 F [1] −→ −→ −→ in D c L bop , with F, F RHom b,c . Observe, by the argument above, that also ( ) 0 ( ) F [1] RHom⊗ (b,c). Then, we∈ have that ∈ 0 0 0 0 H (F )(B) H (F 0)(B) H (F 00)(B) (H (F )(B))[1] −→ −→ −→ 0 0 is an exact triangle in D(c) for all B H (b). But as by hypothesis H (c) is triangulated in D c , H 0 F B H 0 c for∈ all B H 0 b . Hence we can conclude that ( ) ( 00)( ) ( ) ( ) F RHom b,c . ∈ ∈ 00 ( ) ∈ Let α be a (U-small) regular cardinal. We say that a dg category b is U-cocomplete (resp. α-cocomplete) it it is closed under all U-small (resp. α-small) coproducts. Observe that U-cocomplete dg categories are generally not U-small. In the ﬁrst three sections of this chapter we will work with V-small U-cocomplete dg categories. In contrast, we can work, and will do so, with α-cocomplete U-small dg categories. Similarly, a dg category b is said to be homotopically U-cocomplete (resp. homo- 0 topically α-cocomplete) if H (b) is closed under all U-small coproducts (resp. under all α-small coproducts). Observe once more that homotopically U-cocomplete dg categories are not U-small, while we can safely talk about U-small homotopically α-cocomplete dg categories. We will work along the three ﬁrst sections of this chapter with V-small homotopically U-cocomplete dg categories and with U-small homotopically α -cocomplete dg categories. Chapter 5. Tensor product of well-generated pretriangulated categories 117

Remark 5.3. One can readily see that (α-)cocompleteness implies homotopically (α-)cocompleteness.

Let B and C be homotopically cocomplete V-small dg categories. A right quasi- L op representable bimodule F dgMod(C B ) is called cocontinuous if the func- 0 0 0 tor H (F ) : H (B) H (C∈) induced by⊗ F preserves coproducts. We denote by −→ RHomc(B,C) RHom(B,C) the full dg category of cocontinuous bimodules. ⊆ Similarly, let b and c be homotopically α-cocomplete U-small dg categories. We say L op that a right quasi-representable bimodule F dgMod(c b ) is α-cocontinuous if the 0 0 0 induced functor H (F ) : H (b) H (c) preserves∈ α-small⊗ coproducts. We denote −→ by RHomα(b,c) RHom(b,c) the full dg subcategory of α-cocontinuous bimodules. ⊆ Lemma 5.4. Consider V-small dg categories B and C.

1. If C is homotopically cocomplete, then RHom(B,C) is cocomplete. In particular, if C is cocomplete, then so is RHom(B,C).

2. If B and C are homotopically cocomplete, then RHomc(B,C) is cocomplete. In particular, if B and C are cocomplete, then so is RHomc(B,C).

L op Proof. We prove (1). We know that D(C B ) is cocomplete. Assume C is homotopi- L op cally cocomplete. We prove that RHom⊗(B,C) is closed in D(C B ) under U-small coproducts. ⊗ ` L op Let Fi be a U-small family of objects in RHom(B,C) and take i Fi in D(C B ). Observe{ } that its induced coproduct functor ⊗ a Fi : B D(C) i −→ is given by the point-wise coproduct and that taking H 0 commutes with coproducts. Hence we have that

0 a a 0 0 H ( Fi ) = H (Fi ) : H (B) D(C), i i −→

0 0 which is point-wise as well. As the image of each H (Fi ) is contained in H (C), and 0 0 ` 0 H (C) is cocomplete by assumption, the image of H ( i Fi ) also lies in H (C). Hence ` i Fi RHom(B,C), which concludes the argument. ∈ We prove (2). Assume ﬁrst that B,C are homotopically cocomplete. Then, by part (1), we know that RHom(B,C) is cocomplete. It is then enough to show that RHomc(B,C) is closed in RHom(B,C) under coproducts. 118 Chapter 5. Tensor product of well-generated pretriangulated categories

Take Fi a U-small family of objects in RHomc(B,C) and consider their coprod- ` { } uct i Fi : B D(C) in RHom(B,C), which is point-wise as we saw in the proof −→ 0 ` ` 0 of part (1), and recall that we also have H ( i Fi ) = i H (Fi ). By assumption, 0 0 0 H (Fi ) : H (B) H (C) is cocontinuous for all i . Then we have that −→ 0 0 H (B) ! H (B) !! 0 a a a 0 a H Fi Bj = (H (Fi )) Bj i j i j 0 H (C) a a 0 = (H (Fi ))(Bj ) i j 0 H (C) a a 0 = (H (Fi ))(Bj ) j i 0 H (C) a 0 a = H Fi (Bj ). j i

` Consequently i Fi RHomc(B,C), which ﬁnishes the argument. ∈ The analogous results holds for the α-case:

Lemma 5.5. Consider U-small dg categories b and c.

1. If c is homotopically α-cocomplete, then RHom(b,c) is α-cocomplete. In particular, if c is α-cocomplete, then so is RHom(b,c).

2. If b and c are homotopically α-cocomplete, then RHomα(b,c) is α-cocomplete. In particular, if both b and c are cocomplete (resp. α-cocomplete), then so is RHomα(b,c).

Proof. Observe that the same argument of the proof of Lemma 5.4 with the obvious modiﬁcations from “small” to “α-small” can be applied.

Lemma 5.6. Consider V-small dg categories B and C.

1. If C is homotopically cocomplete and pretriangulated, then RHom(B,C) is cocomplete and pretriangulated. In particular this holds when C is cocomplete and pretriangulated. Chapter 5. Tensor product of well-generated pretriangulated categories 119

2. If B and C are homotopically cocomplete and pretriangulated, then RHomc(B,C) is cocomplete and pretriangulated.

In particular this holds when B and C are cocomplete and pretriangulated.

Proof. Part (1) is a direct consequence of Lemma 5.2 and Lemma 5.4. We now prove statement (2). Assume B,C are homotopically cocomplete and pretriangulated. We know by Lemma 5.4 that RHomc(B,C) is cocomplete. It remains to prove that it is also pretriangulated. Observe that it is enough to show that 0 0 H (RHomc(B,C)) is a triangulated subcategory of H (RHom(B,C)). Take F RHomc(B,C) and consider its shift F [1] when seen in the triangulated cat- 0 egory H ∈(RHom(B,C)). We prove that F [1] is homotopically cocontinuous. Indeed, we have that

0 0 H (B) ! H (B) !! 0 a 0 a (H (F )[1]) Bi = H (F ) Bi [1] i i 0 H (C) ! a 0 = H (F )(Bi ) [1] i 0 H (C) a 0 = (H (F )(Bi )[1]) i 0 H (C) a 0 = (H (F )[1])(Bi ), i where in the ﬁrst and last equalities we use the fact that triangulated structure in 0 L op H (RHom(B,C)) is inherited from the canonical one in D(C B ) (see Lemma 5.2), in the second equality we use that H 0 commutes with coproducts⊗ and in the third equality we use that shifts commute with coproducts. Now consider an exact triangle

F F 0 F 00 F [1] −→ −→ −→ in H 0 RHom B,C , where F, F H 0 RHom B,C . Given a family B of elements ( ( )) 0 ( c( )) i 0 in H (B), for all i we have the exact∈ triangle { }

H 0 F B H 0 F B H 0 F B H 0 F B 1 (5.6) ( )( i ) ( 0)( i ) ( 00)( i ) ( )( i )[ ] 120 Chapter 5. Tensor product of well-generated pretriangulated categories

0 in H (C). Observe now that we have the following diagram with rows exact triangles:

0 0 0 0 H (B) H (B) H (B) H (B) H 0 F ` B H 0 F ` B H 0 F ` B H 0 F ` B 1 ( ) i ( 0) i ( 00) i ( ) i [ ] i i i i

0 0 0 0 H (C) H (C) H (C) H (C) ` H 0 F B ` H 0 F B ` H 0 F B ` H 0 F B 1 ( )( i ) ( 0)( i ) ( 00)( i ) ( )( i )[ ] i i i i where the exact triangle below is the coproduct of the family of exact triangles from (5.6) above, and the vertical equalities are given because both H 0 F and H 0 F are ( ) ( 0) cocontinuous by hypothesis. By the axioms of triangulated categories, we have that

0 0 H (B) ! H (C) 0 a a 0 H (F 00) Bi = H (F 00)(Bi ) i ∼ i for all family B of elements in H 0 B . Hence F RHom B,C , which concludes i ( ) 00 c( ) the argument.{ } ∈

Observe that part (2) of the previous lemma also holds for the α-case. Namely:

Lemma 5.7. Let b and c be two U-small dg categories. If b and c are homotopically α-cocomplete and pretriangulated, then RHomα(b,c) is α-cocomplete and pretriangulated. In particular this holds when b and c are α-cocomplete and pretriangulated.

Proof. One can apply the analogous argument of the proof of Lemma 5.6 in the “α-small” case to conclude.

pt pt We denote by V-Hqec V-Hqe (resp. U-Hqeα U-Hqe) the full subcategory of homotopically U-cocomplete⊆ (resp. homotopically⊆α-cocomplete) pretriangulated V- small (resp. U-small) dg categories. Observe that by Lemma 5.6 (resp. by Lemma 5.7), pt pt we have that RHomc (resp. RHomα) are internal in V-Hqec (resp. in U-Hqeα ).

5.2 Quotient of cocomplete pretriangulated dg categories

0 Let b,c be small dg categories and let Σ be a set of morphisms in H (b). We denote by RHomΣ(b,c) RHom(b,c) the full dg subcategory of bimodules F for which ⊆ Chapter 5. Tensor product of well-generated pretriangulated categories 121

0 0 0 H (F ) : H (b) H (c) sends the morphisms in Σ to isomorphisms. A dg category of fractions b −→1 is deﬁned by the following universal property in U Hqe: [Σ− ] − 1 RHom(b[Σ ],c) RHomΣ(b,c) (5.7) − ∼= (see [79, Cor. 8.8]). Let B,C be pretriangulated V-small dg categories and let N be a class of objects in 0 H (B). L op Deﬁnition 5.8. A right quasi-representable bimodule F dgMod(C B ) anni- 0 0 0 hilates N if the induced functor H (F ) : H (B) H (C) annihilates∈ N⊗ (§1.4.4), i.e. sends the objects in N to contractible objects−→ (objects whose identity morphism 0 vanishes in H (C)).

We denote by RHomN(B,C) RHom(B,C) the full dg subcategory of quasi-representable bimodules that annihilate⊆ N. Assume additionally that both B and C are homotopically cocomplete. We denote by RHomc,N(B,C) RHomc(B,C) the full dg subcategory of cocontinuous quasi-representable bimodules⊆ that annihilate N. Deﬁnition 5.9. A cpt dg quotient B/N is a homotopically cocomplete pretriangulated pt dg category with the following universal property in V-Hqec :

RHomc(B/N,C) RHomc,N(B,C). (5.8) ∼= Remark 5.10. Recall that there exists a notion of a quotient of a dg category along a dg subcategory which was introduced by Keller in [41] and analysed further by Drinfeld in [22]. We point the reader to §1.4.4 for a very brief summary on the topic. In particular, given A a small dg category and B A a dg subcategory, for all small dg categories C, one has that ⊆

RHom(A/B,C) RHomB(A,C). (5.9) ∼= One can actually check, for example by means of Keller’s construction, that if A is 0 0 homotopically cocomplete and H (B) is closed in H (A) under coproducts, then A/B is also homotopically cocomplete and the canonical quasi-representable bimodule “A A/B” is cocontinuous. Observe then, that for all homotopically cocomplete dg−→ categories C, the quasi-equivalence (5.9) restricts to a quasi-equivalence

RHomc(A/B,C) RHomc,B(A,C). (5.10) ∼= 0 Hence if we take N as above to be a triangulated subcategory of H (B) and we denote by C its natural enhancement as a dg subcategory of B, we have that the cpt dg quotient B/N coincides with the Keller-Drinfeld dg quotient B/C. 122 Chapter 5. Tensor product of well-generated pretriangulated categories

We provide the following deﬁnition, inspired by Gabriel-Popescu theorem for triangulated categories [66].

Deﬁnition 5.11. A dg site is a pair (a,N) given by a U-small dg category a and a localizing subcategory N D(a) generated by a set of objects. ⊆ Deﬁnition 5.12. We say that a V-small dg category C is a dg topos if there exists a dg site (a,N) such that C D(a)/N. ∼=

5.3 Two variables setting

In this section we introduce two-variable versions of the previously introduced functor categories. From now on, we will make implicit use of the fact that for every homotopically U- cocomplete pretriangulated V-small dg category, we can pick a coﬁbrant replacement pt in V-Hqec and this coﬁbrant replacement is the identity on objects (see §1.4.2 and more precisely Theorem 1.83). pt L Let A,B,C V-Hqec . Consider F RHom(A B,C). Observe that the mod- L op L op ule F dgMod∈ (C A B ) with evaluations∈ ⊗F (A, B,C ) gives rise to a bimod- ∈ ⊗ ⊗ L op ule FA = F ( , A, ) dgMod(C B ) for every A A and similarly to a bimodule − − ∈ L ⊗op ∈ FB = F ( , , B) dgMod(C A ) for every B B, and according to (5.5) these are all right− quasi-representable.− ∈ ⊗ ∈

L Deﬁnition 5.13. We call F RHom(A B,C) right cocontinuous provided that every FB is cocontinuous, left cocontinuous∈ ⊗provided that every FA is cocontinuous, and bicocontinuous provided that it is left and right cocontinuous.

L L We denote by RHomc,c(A B,C) RHom(A B,C) the full dg subcategory of bicocontinuous modules. ⊗ ⊆ ⊗ Given a regular cardinal α, the notions of left-, right- and bi-α-cocontinuous are L L deﬁned similarly. We denote by RHomα,α(A B,C) RHom(A B,C) the full dg subcategory of bi-α-cocontinuous quasi-representable⊗ ⊆ modules.⊗

Consider NA a class of objects in A and NB a class of objects in B. With the same L notations as above, we say F RHom(A B,C) biannihilates NA-NB provided that every FA annihilates NB and F∈B annihilates⊗ NA. We denote by

L L RHomNA,NB (A B,C) RHom(A B,C) ⊗ ⊆ ⊗ the full dg subcategory of quasi-representable modules that biannihilate NA-NB. Chapter 5. Tensor product of well-generated pretriangulated categories 123

A L B C A L B C Similarly, we denote by RHom(c,NA),(c,NB)( , ) RHom( , ) the full dg subcategory of bicocontinuous quasi-representable⊗ ⊆ modules that⊗ biannihilate NA-NB.

Lemma 5.14. 1. For homotopically cocomplete V-small dg categories A, B and C, we have: L RHomc,c(A B,C) = RHomc(A,RHomc(B,C)) (5.11) ⊗ ∼ 2. For homotopically α-cocomplete U-small dg categories a, b and c, we have:

L RHomα,α(a b,c) = RHomα(a,RHomα(b,c)) (5.12) ⊗ ∼ 3. For homotopically cocomplete V-small dg categories A, B and C and sets of objects NA in A and NB in B, we have:

A L B C A B C RHom(c,NA),(c,NB)( , ) = RHomc,NA ( ,RHomc,NB ( , )) (5.13) ⊗ ∼ L Proof. We prove part 1. Consider F RHomc,c(A B,C). We have then that the ∈ ⊗ induced bimodules FA RHomc(B,C) and FB RHomc(A,C) for all A A and all B B. Now, to each∈ such F we associate the∈ quasi-representable bimodule∈ F dgMod∈ RHom B,C L Aop with F A : F for all A A, which is easily seen 0 ( c( ) ) 0( ) = A to be∈ cocontinuous by using⊗ that FB is cocontinuous for all∈ B B. Hence we can L ∈ deﬁne a functor RHomc(A B,C) RHomc(A,RHomc(B,C)) deﬁned in objects as just described and in morphisms⊗ −→ in the natural way. This functor is a dg functor and it is easily seen to be a quasi-equivalence. Observe that a similar argument proves part 2. A LB C A LB C We prove part 3. Observe that RHom(c,NA),(c,NB)( , ) RHomc,c( , ) is a ⊗ ⊆ ⊗ dg subcategory and RHomc,NA (A,RHomc,NB (B,C)) is equivalent to a dg subcategory of RHomc(A,RHomc(B,C)). It is then enough to see that the quasi-equivalence (5.11) constructed above restricts to a quasi-equivalence (5.13). But this follows from the A LB C fact that for all quasi-respresentable module F RHom(c,NA),(c,NB)( , ) we have that FA is cocontinuous and annihilates NA for∈ all A A and FB is cocontinuous⊗ and annihilates NB for all B B. ∈ ∈

5.4 Well generated pretriangulated dg categories

Well-generated triangulated categories in the sense of Neeman [61] form a very important class of triangulated categories. We point the reader to §1.3.3. Not only do they enjoy very nice properties (concerning for example localizations, or Brown 124 Chapter 5. Tensor product of well-generated pretriangulated categories representability), but they also appear naturally in many contexts. From amongst geometrical examples of well-generated triangulated categories, one of the most relevant are derived categories of Grothendieck categories.

Porta shows in [66] that in the triangulated world, well-generated algebraic triangulated categories play the analogous role to the one that Grothendieck categories play in the abelian world, in the sense that they fulﬁll a triangulated version of the well-known Gabriel-Popescu theorem for Grothendieck categories. More precisely, the Gabriel-Popescu theorem for triangulated categories [66, Thm 7.2] states that a triangulated category is well-generated and algebraic if and only if it can be realised as a Verdier quotient of D(a) with respect to a localizing subcategory generated by a set of objects, for a certain small dg category a. In particular, we will be interested in the pretriangulated dg version of well-generated algebraic triangulated categories:

Deﬁnition 5.15. A pretriangulated dg category A is called well-generated if the ho- 0 0 motopy category H (A) is a well-generated triangulated category (observe that H (A) is already algebraic, as it has an enhancement).

Remark 5.16. Note that every well-generated pretriangulated category is homotopically cocomplete (but not necessarely cocomplete). Actually, the question already posed in [66] of whether for any homotopically cocomplete dg category there exists a cocomplete dg category quasi-equivalent to it is still open. Remark 5.17. From now on, when dealing with well-generated pretriangulated dg categories, we will usually omit the term pretriangulated for the sake of brevity, as it will not lead to confussion.

One has that the Gabriel-Popescu theorem for triangulated categories can be enhanced (see for example [15, Thm 2.8]). The statement goes as follows. Any well- generated pretriangulated dg category B can be written as dg quotient of D(a) with respect to the enhancement of a localizing subcategory of D(a), for some small dg category a. Observe that for any small dg category a, D(a) is cocomplete and pretriangulated, and hence any dg quotient of D(a) with respect ot an enhancement of localizing subcategory is actually a cpt dg quotient (see 5.9 above). We can rephrase this as follows:

Theorem 5.18. Well-generated pretriangulated dg categories are precisely the dg topoi.

Proof. This follows directly from the deﬁnition of dg topos (Deﬁnition 5.12) and the discussion above. Chapter 5. Tensor product of well-generated pretriangulated categories 125

We address now, as indicated in §5.1, the universe choices that will hold for the rest of the chapter.

Fix a universe U. For a U-small k-linear site (a,Wa), the category D(a)/Wa is defined with respect to the category U-dgMod(k) of U-small k-modules. By definition, a U-well-generated dg category is a pretriangulated dg category whose homotopy category is closed under U-small coproducts and is well-generated by a U-small set of generators. Let V be a larger universe such that all the categories D(a)/Wa are V- small and let Hqewg denote the full subcategory of V-Hqe given by the V-small U-well- generated dg categories. Up to equivalence, Hqewg is easily seen to be independent of the choice of V. In the rest of the chapter, we will omit the universes U and V from our notations and terminology. In view of the picture for Grothendieck categories, we give the following definition.

Deﬁnition 5.19. Let A and B be well-generated dg categories. A well-generated tensor product of A and B is deﬁned as a well-generated dg category A B such that pt the following universal property holds (in Hqec ) for all C Hqewg: ∈ RHomc(A B,C) RHomc(A,RHomc(B,C)). (5.14) ∼=

The rest of the chapter will be devoted to prove that this tensor product exists. We will further provide different constructions of it for the different available realizations of well-generated categories. In order to do so, a careful study of the localization theory of well-generated dg categories will be required. This will be carried out in the following sections.

5.5 Localization of well-generated dg categories

Porta’s triangulated Gabriel-Popescu theorem provides us a characterization of well- generated algebraic triangulated categories in terms of Verdier quotients with respect to localizing subcategories generated by a set of objects. We know, that this is equivalent to performing Bousﬁeld localization with kernel generated by a set of objects (see §1.3.4). Our characterization of well-generated dg categories relies on an enhancement of the Verdier quotient. In this section we introduce an enhancement of Bousﬁeld localizations, which will be useful for further steps. More precisely, we discuss two different approaches to the localization of well-generated dg categories, namely:

Localizing subcategories generated by a set; • 126 Chapter 5. Tensor product of well-generated pretriangulated categories

Strict localizations; • and we prove they are equivalent. This is basically an enhancement of well-known results from localization theory of well-generated triangulated categories (see §1.3.4).

5.5.1 Localizing subcategories generated by a set

Let B be a well-generated small dg category. Consider a localizing subcategory 0 W H (B) generated by a set. ⊆ 0 Observe that in particular H (B) is localizing as a subcategory of itself and it is generated by a set (as it is well-generated). In addition, the intersection of localizing subcategories generated by a set is again such (see [33, Lem 3.2]). Consequently, for 0 every full triangulated subcategory H H (B) there is a smallest localizing subcate- 0 gory H containing H. In particular, the⊆ poset of localizing subcategories of H (B) 〈 〉 is a complete lattice with infi Wi = i Wi and supi Wi = i Wi . ∩ 〈∪ 〉 0 Definition 5.20. Consider H H (B) and B B.A filtration of B consists of a countable collection X of objects⊆ in H 0 B with∈ X 0 and maps x : X X ( i )∞i =0 ( ) 0 = i i i +1 for all i 0 such that hocolim X B . A filtration X of B is called an H-filtration−→ ( i ) = ( i )∞i =0 ≥ if the cone of each xi : Xi Xi +1 belongs to H and in this case B is called H-filtered. −→ 0 Proposition 5.21. Let W be a localizing subcategory of H (B) generated by a set. Then, 0 there exists a set N generating W (i.e. W = N ) such that X H (B) belongs to W if and only if it is N-filtered, where N is the class〈 〉 of small coproducts∈ of elements in N.

Proof. By Theorem 1.63, we know we can take a regular cardinal α such that W and 0 H (B) are both α-compactly generated. In particular, the class of α-compact objects α α W = W B is essentially small (see Proposition 1.55). Take N to be set of objects in W consisting∩ of taking for each isomorphic class of Wα a representative. We have that W = N . By applying [61, Lemma B.1.3], we know that every X W is N-ﬁltered. 0 On the other〈 〉 hand, as W is localizing, every N-ﬁltered object X in H∈ (B) belongs to W (see Remark 1.53), which concludes the argument.

To ﬁnish this section we describe the relation with orthogonal complements. We point the reader to §1.3.2 for the deﬁnitions and the notations that we use here regarding orthogonal complements in triangulated categories.

Proposition 5.22. Let W be a localizing subcategory of B generated by a set N, i.e. W N . Then we have that W N . = ⊥ = ⊥ 〈 〉 Chapter 5. Tensor product of well-generated pretriangulated categories 127

Proof. We have that N W, hence W N . On the other hand, we have that ⊥ ⊥ N N and N is⊆ easily seen to be⊆ a localizing (hence triangulated) subcat- ⊥( ⊥) ⊥( ⊥) egory⊆ 61, Lem 9.1.12 . Hence we have that W N N , and applying right [ ] = ⊥( ⊥) orthogonals, we obtain that N W , which concludes〈 〉 ⊆ the argument. ⊥ ⊥ ⊆

5.5.2 Strict localizations

Let A, B be two small dg categories. In the literature, the elements of the category 0 H (RHom(A,B)) are usually called quasi-functors between A and B (see, for example [40]). Given F RHom(A,B), we denote also by F the same element seen in 0 H (RHom(A,B)) and∈ we will refer to it as the underlying quasi-functor of F .

Deﬁnition 5.23. Let A,B be two dg categories and consider F RHom(A,B) and G RHom(B,A). We say that F is a quasi-left adjoint of G (or∈G is a quasi-right adjoint∈ of F ) and we denote it by F H 0 G if and only if a 0 0 0 0 H (F ) H (G ) : H (A) H (B) a is an adjoint pair.

Remark 5.24. This deﬁnition can be seen to be equivalent to the following deﬁnition: F is a quasi-left adjoint of G if and only if the underlying quasi-functor of F is left adjoint to the underlying quasi-functor of G in the sense of adjoint pairs of quasi-functors from [30]. In particular, then there exist morphisms IdA G F in 0 op 0 −→ ◦ op H (RHom(A,A)) D(A A ) and F G IdB in H (RHom(B,B)) D(B B ), called the unit and⊆ counit⊗ of the adjunction◦ −→ respectively. ⊆ ⊗

Deﬁnition 5.25. Consider A,B be two well-generated dg categories. We say that a bimodule G RHom(A,B) is a dg strict localization if the associated dg functor G : A D(B)∈is quasi-fully faithful and there exists a quasi-representable bimodule −→ F RHom(B,A) such that F H 0 G . Observe then that the counit F G IdA is an 0 isomorphism∈ in H (RHom(Aa,A)). ◦ −→

Remark 5.26. Observe that A,B are pretriangulated dg categories, a dg strict localization G RHom(A,B) with quasi-left adjoint F , automatically induces a Bousﬁeld 0 localization∈ functor of H (A), given by

0 0 0 0 0 H (F ) H (G ) : H (A) H (B) H (A). ◦ −→ −→ Moreover, F is cocontinous, i.e. it belongs to RHomc(B,A). 128 Chapter 5. Tensor product of well-generated pretriangulated categories

Remark 5.27. Given a strict localization G RHom(A,B) between two well-generated dg categories A and B, we have that G induces∈ a quasi-equivalence of A with a full well-generated dg subcategory of B, more precisely the one with objects in the essential image of G . Hence from now on, when given a strict localization G RHom(A,B), we will always assume that A is a well-generated dg subcategory of B∈ and G is the actual dg functor given by the natural embedding. We will say then that A B is a well-generated strict localization. ⊆

5.5.3 Equivalent approaches to localization

When we restrict to the world of well-generated triangulated categories, there is a nice correspondence between localizing subcategories and Bousﬁeld localization (see §1.3.4). This result can be easily enhanced to the dg realm. In particular, for a well-generated dg category B, there is a poset isomorphism between:

0 1. The poset Wdg of localizing subcategories of H (B) generated by a set, ordered by inclusion;

op 2. The opposite poset (Ldg) of the poset Ldg of well-generated localizations of B, ordered by inclusion.

The poset isomorphism is described similarly:

0 1. Let W be a localizing subcategory of H (B) generated by a set. In particular, we have that W H 0 B has a left adjoint and hence gives rise to a localization ⊥ ( ) functor ⊆ 0 0 H (B) W⊥ H (B), −→ −→ the composition W H 0 B H 0 B W is an equivalence and W is well- ⊥ , ( ) ( )/ ⊥ generated (see Corollary→ 1.64 and→ Theorem 1.63).

Denote by LW the full dg subcategory of B obtained as an enhancement of W H 0 B via the natural enhancement of H 0 B . We have then that ⊥ ( ) ( ) LW is a⊆ well-generated dg category. In addition, the composition of quasi- representable bimodules “LW , B B/W” is an equivalence in Hqe inducing the equivalence W H→0 B →W of triangulated categories. Hence ⊥ ( )/ → there exists a bimodule F RHomc(B,LW) such that the induced functor H 0 F : H 0 B W is the∈ left adjoint of W H 0 B and Ker H 0 F W. ( ) ( ) ⊥ ⊥ ( ) ( ( )) = Consequently,−→F is a quasi-left adjoint to the embedding⊆ LW B, and thus LW B is a well-generated strict localization. ⊆ ⊆ To each W Wdg we assign the so constructed LW Ldg. ∈ ∈ Chapter 5. Tensor product of well-generated pretriangulated categories 129

2. Let L B be a well-generated strict localization with quasi-left adjoint de- 0 0 noted⊆ by F . In particular, Ker(H (F )) is a localizing subcategory of H (B). In 0 addition, as L B is cocontinuous, by [47, Thm 7.4.1] we have that Ker(H (F )) is well-generated,⊆ and hence, in particular, it is generated by a set. We put 0 WL = Ker(H (F )).

We assign to L Ldg the so constructed WL Wdg. ∈ ∈

5.6 Tensor products and quotients

Consider A,B,C Hqewg and suppose A B exists. By Lemma 5.14 above and the ∈ universal property of , we have a quasi-equivalence

L RHomc(A B,C) = RHomc,c(A B,C), (5.15) ∼ ⊗ for every well-generated dg category C. Hence there exists, corresponding to the identity quasi-representable module on the left hand side by taking C = A B, a L canonical bicocontinuous quasi-representable module RHomc,c(A B,A B). We will denote the induced functor at the level of homotopy⊗ ∈ by ⊗

0 L 0 H 0 : H (A B) H (A B), ⊗ ⊗ −→ 0 instead of our usual notation H ( ). Let XA A and XB B be classes of objects. We deﬁne the class ⊗ ⊆ ⊆

XA H 0 XB = XA H 0 XB XA XA, XB XB (5.16) ⊗ { ⊗ | ∈ ∈ } 0 of objects in H (A B). Remark 5.28. Let C be a dg category. Observe that taking a class of objects in C is the 0 0 same as taking a class of objects in H (C) as Obj(H (C)) = Obj(C).

In ﬁrst place, let’s analyse the relation of the well-generated tensor product and the pt cpt quotient in Hqec .

0 0 Proposition 5.29. Consider classes NA H (A) and NB H (B) of objects. The class ⊆ ⊆ 0 NA Cl NB = NA H 0 B A H 0 NB H (A B) (5.17) ⊗ ∪ ⊗ ⊆ is such that

RHom A B,C RHom A L B,C . (5.18) c,NAClNB ( ) = (c,NA),(c,NB)( ) ∼ ⊗ 130 Chapter 5. Tensor product of well-generated pretriangulated categories

Proof. We have the quasi-equivalence

L RHomc(A B,C) = RHomc,c(A B,C) ∼ ⊗ from (5.15) given by composition with the canonical bicocontinuous quasi-repre- L sentable bimodule between A B and A B. Then it is enough to see that this quasi-equivalence restricts⊗ to a quasi-equivalence⊗ (5.18). Consider F RHom A B,C . Then F RHom A L B,C is trivially c,NAClNB ( ) c,c( ) bicocontinuous∈ and biannihilates (NA-NB). ◦⊗ ∈ ⊗ A L B C On the other hand, given any G RHom(c,NA),(c,NB)( , ), we have that ∈ ⊗ 0 0 H (G ) = H (F ) H 0 ∼ ◦ ⊗ for some F RHomc(A B,C). Consequently, for every object B B, we have that 0 ∈ 0 ∈ H (F )(NA H 0 B ) = H (G )(NA, B ) is contractible and, similarly, for every object A A, 0 ⊗ ∼ 0 0 ∈ H (F )(A H 0 NB) H (G )(A,NB) is contractible. Thus we have that H (F ) annihilates ∼= N N⊗ , therefore F RHom A B,C as desired. A Cl B c,NAClNB ( ) ∈

We will call NA Cl NB the tensor product of classes of objects NA and NB. Let B, C be cocomplete pretriangulated categories and let N be a class of objects 0 in B. Let N H (B) be the smallest localizing subcategory containing N. For 〈 〉 ⊆ 0 0 0 F RHomc(B,C), the induced H (F ) : H (B) H (C) is exact and cocontinuous. 0 0 As∈ a consequence, Ker(H (F )) is a localizing subcategory−→ of H (B). It follows that RHomc,N(B,C) = RHomc, N (B,C) and consequently 〈 〉 B/N = B/ N . (5.19) ∼ 〈 〉 Hence, when considering quotients of a cocomplete pretriangulated dg category B, 0 we can assume they are taken with respect to localizing subcategories of H (B). In particular, we will be interested in cpt quotients of well-generated dg categories with respect to localizing subcategories generated by a set. The main reason is that, as shown in §5.5, Hqewg is closed under these type of quotients. 0 0 Let A, B be two well-generated dg categories and WA H (A) and WB H (B) localizing subcategories generated by sets. We know by Theorem⊆ 1.63 that⊆ we can 0 0 choose a regular cardinal α, such that H (A), H (B), WA and WB are α-compactly generated. In addition, by Proposition 1.55, we know that the class of α-compact objects in an α-compactly generated category is essentially small, i.e. there exist only a set of isomorphism classes of α-compact objects. Let’s consider the sets 0 0 of generators GA, GB of H (A) and H (B) formed by taking one representative of Chapter 5. Tensor product of well-generated pretriangulated categories 131

each isomorphism class of α-compact objects, and analogously Na, Nb the sets of generators of Wa, Wb formed by taking one representative of each isomorphism class of α-compact objects.

Lemma 5.30. With the notations above, we have that:

WA Cl WB = NA H 0 GB GA H 0 NB . 〈 〉 〈 ⊗ ∪ ⊗ 〉

Hence WA Cl WB is generated by a set of objects. 〈 〉

Proof. By deﬁnition we have that

WA Cl WB = NA H 0 B A H 0 NB . 〈 〉 〈〈 〉 ⊗ ∪ ⊗ 〈 〉〉

As it is a localizing subcategory and it trivially contains NA H 0 GB GA H 0 NB, we ⊗ ∪ ⊗ have that NA H 0 GB GA H 0 NB WA Cl WB . 〈 ⊗ ∪ ⊗ 〉 ⊆ 〈 〉 In order to prove the other inclusion, consider an element X NA H 0 B. If ∈ 〈 〉 ⊗ it belonged to A H 0 NB , we argue analogously. By [61, Lem B.1.3], there exist a countable family⊗ N〈 〉` N of small coproducts of elements of N and a i = ji ji i A countable family G{ ` G of} small coproducts of elements in G such that k = lk lk k B { }

X = hocolimi Ni H 0 hocolimk Gk . ∼ ⊗

As H 0 is bicocontinuous, it preserves coproducts and homotopy colimits in both variables⊗ and hence aa 0 X = hocolimi hocolimk (Nji H Glk ), ji lk ⊗ which is an element of NA H 0 GB GA H 0 NB . Hence we also have an inclusion 〈 ⊗ ∪ ⊗ 〉 WA WB NA H 0 GB GA H 0 NB which concludes the proof. 〈 〉 ⊆ 〈 ⊗ ∪ ⊗ 〉 Theorem 5.31. We have

A B A B = . (5.20) WA WB WA Cl WB 〈 〉

0 Proof. The subcategory WA Cl WB H (A B) is a localizing subcategory gen- 〈 〉 ⊆ erated by a set of objects as proved in Lemma 5.30. Hence, A B/ WA Cl WB is a 〈 〉 132 Chapter 5. Tensor product of well-generated pretriangulated categories well-generated dg category. If we show that it satisﬁes the universal property (5.14), we conclude our argument. For any well-generated dg category C, we have:

A B A B RHomc( ,C) = RHomc( ,C) WA Cl WB ∼ WA Cl WB 〈 〉 A B C RHomc,WA ClWB ( , ) ∼= A L B C = RHom(c,WA),(c,WB)( , ) ∼ ⊗ RHomc,WA (A,RHomc,WB (B,C)) ∼= A B = RHomc( ,RHomc( ,C)), ∼ WA WB where the ﬁrst equivalence comes from (5.19) above, the second and last equivalences are given by the universal property of the cpt dg quotient, the third one is given by Proposition 5.29 and the third one by Lemma 5.14.

Corollary 5.32. Let A,B be two well-generated dg categories. If the tensor product AB exists, so does the well-generated tensor product between any two cpt dg quotients of A, B with respect to localizing subcategories generated by a set of objects.

Proof. This is a direct consequence of Theorem 5.31.

5.7 Tensor product of well-generated dg categories

In this section we show that the well-generated tensor product exists and we provide a construction in terms of dg sites. We will proceed as follows. We will show that the well-generated tensor product of derived dg categories exists and it is again a derived dg category. This result will allow us, using Theorem 5.18, to approach the construction of the tensor product for arbitrary well-generated dg categories making essential use of Corollary 5.32 above. We will start ﬁrst with some considerations on the two variable setting.

The fact that the coﬁbrant replacement Q in dgcatk can be taken to be the identity on objects, permits to deﬁne a canonical functor

L iB : a a b = a Q(b) : A (A, B) −→ ⊗ ⊗ 7−→ for all B b (see [79, §4]). ∈ One can then consider the induced dg functor

L (iB )∗ : dgMod(a b) dgMod(a) : F F iB = F ( , B), ⊗ −→ 7−→ ◦ − Chapter 5. Tensor product of well-generated pretriangulated categories 133 sometimes called restriction of scalars. This dg functor has a left adjoint

L (iB )! : dgMod(a) dgMod(a b), −→ ⊗ sometimes also called extension of scalars. Moreover, i preserves acyclic dg mod- ( B )∗ ules, hence it induces an exact functor

L (iB )∗ : D(a b) D(a) ⊗ −→ In addition, the left derived functor

L L(iB )! : D(a) D(a b). −→ ⊗ is a left adjoint for i . This can be found for example in 56, §1 . Observe our ( B )∗ [ ] notations for the restriction and extension of scalars functors follow the convention we used in the linear categories (see §1.1.2) while in loc.cit. another convention is used.

Lemma 5.33. Let a and b be small dg categories and consider an object B b. Then L ∈ we have that the functor L(iB )! = b( , B). ∼ − ⊗ −

Proof. Observe that L(iB )! is a left adjoint, and hence it preserves coproducts. There- fore, it is fully determined by its value on the representables. Consider a module L F D(a b). Then, for any object A a we have that ∈ ⊗ ∈ L D(a b)(L(iB )!(a( , A)), F ) = D(a)(a( , A),(iB )∗(F )) ⊗ − ∼ − D(a)(a( , A), F ( , B)) ∼= 0 − − H (F (A, B)) ∼= L L D(a b)(a b(( , ),(A, B)), F ) ∼= ⊗L ⊗ −L− = D(a b)(a( , A) b( , B), F ), ∼ ⊗ − ⊗ − where the first equivalence is given by the adjunction L i i , the second by ( B )! ( B )∗ definition of i , the third and the fourth by definition of thea morphisms in derived ( B )∗ categories (see [39, §4]) and the last one can be readily seen using Theorem 1.83. As L this holds for all F D(a b), we conclude by Yoneda lemma. ∈ ⊗ The following result along the lines of the results in §5.1, will be relevant for further steps.

Proposition 5.34. Let b be a small dg category and C a well-generated dg category. Then, RHom(b,C) is a well-generated dg category. 134 Chapter 5. Tensor product of well-generated pretriangulated categories

Proof. As C is a well-generated dg category, by Porta’s Gabriel-Popescu theorem, there exists a small dg category c such that C is a strict localization of D(c), that is, there exists a quasi-fully faithful functor i : C D(c) which has a cocontinuous −→ quasi-left adjoint F RHomc(D(c),C). ∈ Observe that, by [79, Cor 6.6], the morphism

RHom(b,C) RHom(b,D(c)) −→ induced by the dg functor i is quasi-fully faithful as well. op L In addition, observe that RHom(b,D(c)) = D(b c) as a direct consequence of op ∼ the fact that D(c) RHom(c ,D(k)) (see [79, §7]). Denote⊗ by ∼= op L op L iB : c b c = Q(b ) c : C (B,C ) −→ ⊗ ⊗ 7−→ the evaluation functor at B bop as discussed above. ∈ We have a quasi-fully faithful morphism

op L j : RHom(b,C) D(b c). −→ ⊗ op L As both RHom(b,C) and D(b c) are pretriangulated by Lemma 5.2, one has that 0 0 H (j ) is a fully-faithful functor.⊗ In particular, observe that the image of H (j ) is given by X D bop L c such that i X X B, Im H 0 i D c . ( ) ( B )∗( ) = ( ) ( ( )) ( ) ∈ ⊗ − ∈ ⊆ Then, the natural bimodule F RHom D bop L c ,RHom b,C given by 0 c( ( ) ( )) ∈ ⊗ op L F 0 : D(b c) RHom(b,C) : X (F 0(X ) : (B,C ) F ((iB )∗(X ))(C )) ⊗ −→ 7−→ 7−→ op L can be easily seen to be a quasi-left adjoint of j : RHom(b,C) D(b c). −→ ⊗ We thus have an enhanced Bousﬁeld localization of the well-generated dg category D bop Lc . Consequently, if we prove that Ker H 0 F is generated by a set of objects, ( ) ( ( 0)) we can⊗ conclude that RHom(b,C) is well-generated (see §5.5). 0 Let’s denote by W = Ker(H (F )), which is a localizing subcategory of D(c). Denote then by W be the subcategory of D bop Lc of the bimodules X such that X B, W 0 ( ) ( ) for all B bop. Observe that, just by deﬁnition,⊗ we have that Ker H 0 F W .− Notice∈ ( ( 0)) = 0 that W is∈ also localizing. 0 As D(c) is well-generated, we have by Theorem 1.63 that W is well-generated as β β well. Hence we can choose a regular cardinal β such that W = W D(c) , which is essentially small, generates W. Consider now the class of objects ∩

op L op L op β N = b ( , B) N D(b c) B b ,N W . { − ⊗ ∈ ⊗ | ∈ ∈ } Chapter 5. Tensor product of well-generated pretriangulated categories 135

Observe that N is essentially small. We are going to prove that W N . 0 = 〈 〉 We ﬁrst prove that N W . As W is localizing, it is enough with showing that 0 0 N W . Let’s take X 〈 bop〉 ⊆ , B L N N and observe that 0 = ( ) ⊆ − ⊗ ∈ op L X (B 0, ) = b (B 0, B) N . − ⊗ We claim that X B , W. Indeed, N k 0 L N W where k 0 D Mod k ( 0 ) = [ ] [ ] ( ( )) denotes the complex− concentrated∈ in degree 0⊗ with ∈k in the 0-term.∈ In addition, k 0 generates D k , hence bop B , B D k can be written as a homotopy colimit [ ] ( ) ( 0 ) ( ) L of coproducts of shifts of k[0]. As in∈D(k) commutes with homotopy colimits, we can conclude. ⊗ We now prove that W N . Observe that it is enough with showing the inclusion 0 N W . Indeed, if we⊆ 〈 take〉 left orthogonals, we have that ⊥ ( 0)⊥ 〈 〉 ⊆ W0 ⊥(W0)⊥ ⊥( N ⊥) = N , ⊆ ⊆ 〈 〉 〈 〉 where the ﬁrst inclusion comes from 61, Prop 9.1.12 because W is localizing and [ ] 0 the last equality comes from Theorem 1.61 because N is a localizing subcategory generated by a set of objects of a well-generated triangulated〈 〉 category. Recall from Proposition 5.22 that N N . Let’s consider X N . Then, we have ⊥ = ⊥ ⊥ that 〈 〉 ∈ op L op L 0 = D(b c)(b ( , B) N , X ), op L ⊗ − ⊗ for all b ( , B) N N. Hence, by Lemma 5.33, we have that − ⊗ ∈ op L op L 0 = D(b c)(b ( , B) N , X ) = D(c)(N , X (B, )) ⊗ − ⊗ ∼ − for all bop , B L N N. Hence, X B, Wβ W for all B bop. Now, if we ( ) ( ) ( )⊥ = ⊥ consider any− W⊗ W ∈, we have that − ∈ ∈ 0 ∈ D(c)(W (B, ), X (B, )) = 0 − − for all B bop. Consequently, X W . This concludes the argument. ( 0)⊥ ∈ ∈ Now, we provide the following extension of the well-known derived Morita theory by Toën [79, Thm 7.2(1)]. Proposition 5.35. Let b be a small dg category and C a well-generated dg category. We have RHomc(D(b),C) RHom(b,C). (5.21) ∼= in Hqewg. 136 Chapter 5. Tensor product of well-generated pretriangulated categories

Proof. If C = D(c), the theorem reduces to derived Morita theory. In order to provide the proof for C an arbitrary well-generated dg category, we will mimic the proof of derived Morita theory as provided in [14] (see Corollary 4.2 in loc.cit.). Consider C a well-generated dg category. In particular, there exists a small dg category c and bimodules a RHomc(D(c),C), i RHom(C,D(c)) such that a H 0 i ∈ 0 ∈ a and hence in particular, i a = IdC H (RHom(C,C)). This implies that [a i ]iso = IdC 0 ∼ in Iso(H (RHom(C,C))) ◦[C,C]. ∈ ◦ ∼= First, we prove that the map

[D(b),C]c [b,C] : f f [Yb] −→ 7→ ◦ is a bijection, where [ , ]c indicates the subset of morphisms in Hqe such that the induced morphism between− − the homotopy categories is cocontinuous. Take f , f D b ,C such that f Y f Y . Hence, by composing with i , 0 [ ( ) ]c [ b] = 0 [ b] [ ]iso we have i ∈ f Y i f ◦Y D b◦,D c . Applying now derived Morita [ ]iso [ b] = [ ]iso 0 [ b] [ ( ) ( )]c theory, we have◦ that◦ i f ◦ i◦ ∈f , and composing with a , we have that [ ]iso = [ ]iso 0 [ ]iso f a i f a i ◦f f , which◦ proves injectivity. = [ ]iso = [ ]iso 0 = 0 ◦ ◦ ◦ ◦ Next, consider g [b,C]. Then, [i ]iso g [b,D(c)] and by derived Morita theory, ∈ ◦ ∈ there exists f [D(b),D(c)]c such that f [Yb] = [i ]iso g . Consider now [a]iso f , which ∈ ◦ ◦ ◦ belongs to [D(b),C]c. Then, [a]iso f [Yb] = [a i ]iso g = g , which proves surjectivity. ◦ ◦ ◦ ◦ Now, deﬁne D b L a,C as the subset of D b L a,C consisting of elements f [ ( ) ]0c [ ( ) ] 0 such that H (f )( , A⊗) is cocontinuous for all A ⊗a. Then, we have the following − ∈ diagram induced by the Yoneda embedding Yb : b D(b): −→

[a,RHomc(D(b),C)] [a,RHom(b,C)]

= ∼ ∼= L L (5.22) D(b) a,C c0 a b,C ⊗ ⊗ = ∼= ∼ [Yb] [D(b),RHom(a,C)]c −◦ [b,RHom(a,C)],

L where the vertical arrows are induced by the -RHom in Hqe. As RHom(a,C) is well- generated by by Proposition 5.34, we have that⊗ the horizontal arrow is a bijection. Hence, applying Yoneda lemma in Hqe we can conclude.

We are now in the position to prove the desired result. Chapter 5. Tensor product of well-generated pretriangulated categories 137

Proposition 5.36. Consider small dg categories a and b. In Hqewg, we have

L D(a) D(b) = D(a b). (5.23) ∼ ⊗ Proof. For a well-generated dg category C, we have

L L RHomc(D(a b),C) = RHom(a b,C) ⊗ ∼ ⊗ RHom(a,RHom(b,C)) ∼= RHomc(D(a),RHomc(D(b),C)) ∼= where the ﬁrst and the last equivalences are given by (5.21) and the second one is by the ( L - RHom)-adjunction in Hqe. ⊗ We are ﬁnally in the position to prove the existence of the well-generated tensor product.

Theorem 5.37. Let A, B be two well-generated dg categories such that A D(a)/Wa ∼= and B D(b)/Wb for small dg categories a, b with Wa D(a) and Wb D(b) localizing ∼= subcategories generated by a set of objects. Then, the well-generated⊆ tensor⊆ product of A and B exits and it is given by

L A B = D(a b)/ Wa Cl Wb . (5.24) ∼ ⊗ 〈 〉 In particular, A B is independent of the chosen realizations.

Proof. We have A D(a)/Wa and B D(b)/Wb with Wa and Wb localizing subcate- ∼= ∼= gories generated by a set of objects. By Proposition 5.36 we know that D(a)D(b) exists L and equals D(a b). Then, by Corollary 5.32, we have that AB = D(a)/WaD(b)/Wb ⊗ L ∼ exists and it is given by D(a b)/ Wa Cl Wb , and it is obviously independent of the realizations chosen, as it fulﬁls⊗ the〈 universal〉 property.

Definition 5.38. Given two dg sites (a,Wa) and (b,Wb), we define the tensor product of dg sites as L (a,Wa) (b,Wb) = (a b, Wa Cl Wb ). (5.25) ⊗ 〈 〉 Remark 5.39. The reader can observe that a tensor product of homotopically cocomplete pretriangulated dg categories can be defined analogously to the tensor product of well-generated dg categories. Nevertheless, we are, so far, not able to extend the results obtained for the well-generated tensor product to such a general tensor product. The main obstacle to do so is the generalization of 5.35 above for the bigger class of homotopically cocomplete pretriangulated dg categories. 138 Chapter 5. Tensor product of well-generated pretriangulated categories

5.8 Localization theory of well-generated dg categories and tensor product

By Gabriel-Popescu theorem, we know that well-generated dg categories are precisely the cpt quotients of dg categories with respect to localizing subcategories generated by a set. Equivalently, as it has been analysed in §5.5, well-generated dg categories are precisely the well-generated strict localization of derived dg categories. In this section we focus on the localization theory for derived dg categories and we analyse its relation with the well-generated tensor product. More precisely, we provide a deﬁnition of tensor product of localizing subcategories of derived dg categories generated by a set and a deﬁnition of a tensor product of well-generated strict localizations of derived dg categories. The rest of the section is devoted to prove that both notions of tensor product correspond and give rise to the well-generated tensor product.

5.8.1 Tensor product of localizing subcategories

Let a, b be two small dg categories and consider the derived dg categories D(a) and D(b). Let Wa D(a) and Wb D(b) be localizing subcategories generated by sets of L objects. We deﬁne⊆ one-sided⊆ localizing subcategories of D(a b) as follows: L ⊗ W1 := F D(a b) F ( , B) Wa B b { ∈ ⊗L | − ∈ ∀ ∈ } (5.26) W2 := F D(a b) F (A, ) Wb A a { ∈ ⊗ | − ∈ ∀ ∈ } Assume Wa D(a) and Wb D(b) are respectively generated by the sets NA and ⊆ ⊆ NB. We choose GD(a) = a( , A) A a as a set of generators of D(a) and respectively { − } ∈ GD(b) = b( , B) B b as a set of generators of D(b). { − } ∈ By Lemma 5.30, we have that

Wa Cl Wb = Na H 0 GD(b) GD(a) H 0 Nb . 〈 〉 〈 ⊗ ∪ ⊗ 〉

L Lemma 5.40. The subcategories W1 and W2 of D(a b) deﬁned above are generated by sets of objects. More precisely, we have that ⊗

W1 = Na H 0 GD(b) ; 〈 ⊗ 〉 (5.27) W2 = GD(a) H 0 Nb . 〈 ⊗ 〉 In order to prove this lemma, we ﬁrst provide an explicit description of the quasi- L L representable bimodule between D(a) D(b) and D(a) D(b) D(a b) (see ∼= §5.6). ⊗ ⊗ ⊗ Chapter 5. Tensor product of well-generated pretriangulated categories 139

Lemma 5.41. Let a and b be small dg categories and consider the canonical bimodule L L RHomc,c(D(a) D(b),D(a b)). Then, given F D(a),G D(b), we have that: ⊗ ∈ ⊗ ⊗ ∈ ∈ L (F H 0 G )(A, B) = F (A) G (B) (5.28) ⊗ ⊗ in D(k) for all and A a, B b. ∈ ∈

Proof. Recall that given c a small dg category, representables c( ,C ) C c form a set of generators of D(c). Consequently, we have that F (resp. G{ )− can be} ∈ written as a homotopy colimit of direct sums of shifts of representables in D(a) (resp. in D(b)). Because H 0 is bicocontinuous and triangulated in each variable, and thus commutes with homotopy⊗ colimits, direct sums and shifts in both variables, then F H 0 G can be hence written as a homotopy colimit of directs sums of shifts of elements⊗ of the L form a( , A) H 0 b( , B) in D(a b). − ⊗ − ⊗ L L Now recall that is just the image of the identity in RHomc(D(a b),D(a b)) via the chain of quasi-equivalences⊗ ⊗ ⊗

L L L L RHomc(D(a b),D(a b)) RHom(a b,D(a b)) ∼= ⊗ ⊗ ⊗ ⊗ L RHom(a,RHom(b,D(a b))) ∼= ⊗ L RHomc(D(a),RHomc(D(b),D(a b)) ∼= L L ⊗ = RHomc,c(D(a) D(b),D(a b)) ∼ ⊗ ⊗ deﬁned above (see (5.21), (5.5) and (5.11)). On the other hand, observe that the L L identity in RHomc(D(a b),D(a b)) gets mapped under the ﬁrst quasi-equivalence L L (5.21) to the Yoneda embedding⊗ ⊗a b D(a b). Therefore, when restricted to the representables, one just has that⊗ −→ ⊗

L a( , A) H 0 b( , B) = (a b)( ,(A, B)). − ⊗ − ⊗ − Now observe that

L (a b)( ,(A, B)) (A0, B 0) = (Q(a) b)((A0, B 0),(A, B)) ⊗ − ⊗ = Q(a)(A0, A) b(B 0, B) (5.29) L ⊗ = a(A0, A) b(B 0, B), ⊗ where Q denotes the coﬁbrant replacement functor in dgcat, which can be chosen such that Q(a) a is the identity on objects (Theorem 1.83). In addition, also by Theorem 1.83−→, we have that the induced Q a A , A a A , A is a coﬁbrant ( )( 0 ) ( 0 ) replacement for a A , A in C k . −→ ( 0 ) ( ) 140 Chapter 5. Tensor product of well-generated pretriangulated categories

L Recall that coproducts, cones and shifts are point-wise in D(a b), and hence the evaluation of F 0 G at any point A , B reduces to compute⊗ the homotopy H ( 0 0) colimit of coproducts⊗ of shifts of elements of the form a A , A L b B , B . But as L ( 0 ) ( 0 ) is bicocontinuous in D(k) and applying again that coproducts,⊗ cones and shifts are⊗ L point-wise in D(a b), we obtain that ⊗ L (F H 0 G )(A0, B 0) = F (A0) G (B 0) ⊗ ⊗ for all A , B a L b and we conclude. ( 0 0) ∈ ⊗

We are in position now to prove Lemma 5.40, whose proof is very similar to that of Proposition 5.34 above:

L Proof. Take W1 D(a b) as deﬁned in (5.26) above. ⊆ ⊗ First, we prove that Na H 0 GD(b) W1. Let’s take X = Na b( , B) Na H 0 GD(b). By Lemma 5.41 we have⊗ that X ,⊆B N L b B , B . One⊗ − can easily∈ ⊗ see then ( 0) = a( ) ( 0 ) that X , B W . Indeed, we have− that N − N⊗ L k 0 W where k 0 D k ( 0) a a = a( ) [ ] a [ ] ( ) denotes− the complex∈ concentrated in degree 0 with− ⊗k in the∈ 0-term. In addition,∈ k 0 generates D k , hence b B , B D k can be written as a homotopy colimit of [ ] ( ) ( 0 ) ( ) L coproducts of shifts of k[0]. As in∈ D(k) commutes with homotopy colimits, we ⊗ can conclude. Hence, we have that Na H 0 GD(b) W1. 〈 ⊗ 〉 ⊆ Now we prove that W1 Na H 0 GD(b) . Observe that it is enough with proving that N 0 G W .⊆ Indeed, 〈 ⊗ if we take〉 left orthogonals, we have that a H D(b) ⊥ ( 1)⊥ 〈 ⊗ 〉 ⊆

W1 ⊥(W)⊥ ⊥( Na H 0 GD(b) ⊥) = Na H 0 GD(b) , ⊆ ⊆ 〈 ⊗ 〉 〈 ⊗ 〉 where the ﬁrst inclusion comes from [61, Prop 9.1.12] because W1 is localizing and the last equality comes from Theorem 1.61 because Na H 0 GD(b) is a localizing subcategory generated by a set of a well-generated category.〈 ⊗ 〉

Recall from Proposition 5.22 that N 0 G N 0 G . Let’s consider a H D(b) ⊥ = ( a H D(b))⊥ X N 0 G . Then, we have that〈 ⊗ 〉 ⊗ ( a H D(b))⊥ ∈ ⊗ L 0 = D(a b)(N H 0 b( , B), X ), ⊗ ⊗ − for all N H 0 b( , B ) Na H 0 GD(b). Hence, by Lemma 5.41 and Lemma 5.33, we have that ⊗ − ∈ ⊗

L 0 = D(a H 0 b)(N H 0 b( , B), X ) = D(a H 0 b)(N b( , B), X ) = D(c)(N , X ( , B)) ⊗ ⊗ − ∼ ⊗ ⊗ − ∼ − Chapter 5. Tensor product of well-generated pretriangulated categories 141

for all N 0 b , B N 0 G . Thus X , B N W for all B b. Now, if H ( ) a H D(b) ( ) ( a)⊥ = ⊥a we consider⊗ any−W ∈W1,⊗ we have that − ∈ ∈ ∈ D(b)(W ( , B), X ( , B)) = 0 − − for all B b. Consequently, ∈ L 0 = D(a b)(W, X ), ⊗ and hence X W . This concludes the argument. ( 1)⊥ ∈ Analogously, one can prove that W2 = GD(a) H 0 Nb . 〈 ⊗ 〉

We have that W1,W2 are localizing subcategories generated by sets of objects. We now put:

Deﬁnition 5.42. The tensor product of localizing subcategories Wa Wb is given by

W1 W2 = W1 W2 ∨ 〈 ∪ 〉 L in the poset W of localizing subcategories of D(a b) generated by a set of objects. ⊗ We have then the following relation between the tensor product of localizing subcategories and the tensor product of classes.

Proposition 5.43. With the notations above, we have that

Wa Wb = Wa Cl Wb (5.30) 〈 〉

Proof. As a direct consequence of 5.40, we have that

Wa Wb = W1 W2 = Na H 0 GD(b) GD(a) H 0 Nb = Wa Cl Wb , 〈 ∪ 〉 〈〈 ⊗ 〉 ∪ 〈 ⊗ 〉〉 〈 〉 as desired.

Corollary 5.44. We have that

L D(a) D(b) D(a b) = ⊗ . (5.31) Wa Wb ∼ Wa Wb 142 Chapter 5. Tensor product of well-generated pretriangulated categories

5.8.2 Tensor product of strict localizations

Let a, b be two small dg categories and consider the derived dg categories D(a) and D(b). Let La D(a) and Lb D(b) be well-generated strict localizations with quasi-left ⊆ ⊆ L adjoints Fa and Fb respectively. Consider the following full subcategories of D(a b): ⊗ L L L1 = F D(a b) | F ( , B) La B b D(a b); • { ∈ ⊗ − ∈ ∀ ∈ } ⊆ ⊗ L L L2 = F D(a b) | F (A, ) Lb A a D(a b). • { ∈ ⊗ − ∈ ∀ ∈ } ⊆ ⊗ L L The natural bimodules F1 RHomc(D(a b),L1) and F2 RHomc(D(a b),L2) given by ∈ ⊗ ∈ ⊗

L F1 : D(a b) L1 : X (F1(X ) : (A, B) Fa(X ( , B))(A)); • ⊗ −→ 7−→ 7−→ − L F2 : D(a b) L2 : X (F2(X ) : (A, B) Fb(X (A, ))(B)); • ⊗ −→ 7−→ 7−→ − L can be easily seen to be quasi-left adjoints for the inclusions i1 : L1 D(a b) and L −→ ⊗ i2 : L2 D(a b) respectively. We have thus that L1and L2 are strict localizations L of D(a−→b). Additionally,⊗ observe that ⊗ 0 L 0 Ker(H (F1)) = F D(a b) | F ( , B) Ker(H (Fa)) B b , { ∈ ⊗ − ∈ ∀ ∈ } and analogously

0 L 0 Ker(H (F2)) = F D(a b) | F (A, ) Ker(H (Fb)) A a . { ∈ ⊗ − ∈ ∀ ∈ } 0 0 As Ker(H (Fa)) and Ker(H (Fb)) are by hypothesis generated by a set, we have, by 0 0 Lemma 5.40 above, that Ker(H (F1)) and Ker(H (F2)) are also generated by a set of objects. Hence F1 H 0 i1 and F2 H 0 i2 are well-generated strict localizations of L D(a b). a a ⊗ Deﬁnition 5.45. The tensor product of well-generated strict localizations La Lb is given by

L1 L2 ∧ L in the poset L of well-generated strict localizations of D(a b). ⊗ Proposition 5.46. We have that

L1 L2 = L1 L2 (5.32) ∧ ∩ Chapter 5. Tensor product of well-generated pretriangulated categories 143

Proof. Observe we have that:

L L W W 1 2 = ( L1 L2 )⊥ ∧ ∨ W W = L1 L2 ⊥ 〈 ∪ 〉 = W⊥L W⊥L 1 ∩ 2 = L1 L2 ∩ where the ﬁrst and last equalities are given by the isomorphism of posets described in §5.5.3, the second by the description of the poset of localizing subcategories generated by a set and the third by Proposition 5.22.

5.8.3 Relation between the tensor product in both approaches

Let a, b be two small dg categories. Suppose we have a localizing subcategory Wa D(a) generated by a set and a well-generated strict localization La D(a) (with ⊆ ⊆ quasi-adjoint Fa RHomc(D(a),La)) which correspond under the isomorphism of posets described∈ in Section 5.5. Similarly, suppose we have a localizing subcategory Wb D(b) generated by a set and a well-generated strict localization Lb D(b) (with ⊆ ⊆ quasi-adjoint Fb RHomc(D(b),Lb)) which also correspond under the isomorphism ∈ of posets. Our target is to prove that Wa Wb and La Lb correspond as well.

Proposition 5.47. The localizing subcategory W1 (resp. W2) and the well-generated strict localization L1 (resp. L2) correspond under the isomorphism between Wdg and Ldg.

0 0 Proof. Consider Wa = Ker(H (Fa)) and Wb = Ker(H (Fb)) and let W1 and W2 be de- 0 ﬁned as in §5.8.1. Then observe that, just by deﬁnition, we have that Ker(H (F1)) = W1 0 and Ker(H (F2)) = W2.

Corollary 5.48. The localizing subcategory Wa Wb generated by a set and the well- generated strict localization La Lb correspond under the isomorphism between W and L.

Proof. This is a consequence of Proposition 5.47, Deﬁnition 5.42 and Deﬁnition 5.45. 144 Chapter 5. Tensor product of well-generated pretriangulated categories

5.9 Tensor product in terms of α-continuous derived categories

In order to prove the Gabriel-Popescu theorem for triangulated categories [66, Thm 7.2], Porta makes essential use of both equivalent approaches to localization of well- generated triangulated categories. The proof is actually based on a particular and very interesting type of well-generated strict localizations of derived categories D(a) of homotopically α-cocomplete small dg categories a, namely the so called α-continuous derived category Dα(a). Deﬁnition 5.49. Let a be a small homotopically α-cocomplete small dg category. The α-continuous derived category Dα(a) is deﬁned as the full subcategory of D(a) with objects given by the dg functors X such that for every α-small family of objects Ai i I the canonical morphism { } ∈ H 0 ! n a Y n H (X ) Ai H (X )(Ai ) (5.33) i −→ i

H 0 ` 0 is invertible for all n Z, where Ai denotes the coproduct taken in H (a). ∈ i

In addition, one can give an equivalent deﬁnition of Dα(a) as a localization of D(a) with respect to a localizing subcategory W generated by a set. This is done in [66, §6] and in particular three different sets of generators for W are provided. Hence we have that Dα(a) D(a) is actually a well-generated strict localization of triangulated categories. But actually,⊆ more can be said. In particular, we have the following very nice statement.

Theorem 5.50 ([66, Thm 6.4]). Let a be a homotopically α-cocomplete small dg category. Then Dα(a) is α-compactly generated by the images of the free dg modules a( , A) A a. { − } ∈ The α-continuous derived category is the key point used in [66] for the proof of the Gabriel-Popescu theorem for triangulated categories and one could actually rephrase the statement by saying that a triangulated category T is algebraic and α-compactly generated if and only if there exists a homotopically α-cocomplete small dg category a such that T Dα(a) as a triangulated category. ∼= Observe that this whole machinery can be naturally enhanced to the dg level. Denote by Dα(a) the natural enhancement of Dα(a) via the enhancement D(a) of D(a). We will call Dα(a) the α-cocontinuous derived dg category of a. It is a well- generated strict localization of D(a), α-compactly generated by the images of the Chapter 5. Tensor product of well-generated pretriangulated categories 145

dg modules a( , A) A a. In addition, given a dg category A, we have that A is α- compactly generated{ − } if∈ and only if there exists a homotopically α-cocomplete small dg category a such that A Dα(a) in Hqe. ∼= We provide the description of the tensor product of well-generated dg categories when we realise them as α-continuous dg categories. We will use in particular the description of the tensor product of well-generated strict localizations of derived dg categories provided in Section 5.8.2.

Proposition 5.51. Let a,b be two homotopically α-cocomplete small dg categories and consider their respective α-cocontinuous dg derived categories Dα(a),Dα(b). Then we have that: L Dα(a) Dα(b) = Dα,α(a b) ⊗ L L where Dα,α(a b) denotes the full dg subcategory of D(a) D(b) = D(a b) formed ⊗ ⊗ by the bimodules F such that F (A, ) Dα(b) for all A a and F ( , B) Dα(b) for all B b. − ∈ ∈ − ∈ ∈

Proof. This follows form Proposition 5.46.

5.10 From small to large dg categories

Fix α a regular cardinal. Given a homotopically α-cocomplete small dg category a, we can associate to it the dg category Dα(a) which is α-compactly generated, as we have pointed out in §5.9 above. Suppose now we have a,b two homotopically α-cocomplete small dg categories. L We know that Dα(a) Dα(b) = Dα,α(a b) is a well-generated dg category, and hence, there exists a regular cardinal β and a⊗ homotopically β-cocomplete small dg category L c such that Dα,α(a b) = Dβ (c). It is reasonable to ask the following questions: ⊗ ∼

Can we ﬁnd such a c with β = α? Or in other words, is the tensor product of • α-compactly generated dg categories again α-compactly generated?

Can c be found in terms of the provided a and b? •

The answer to both questions is afﬁrmative (see Proposition 5.56 and Corollary 5.57 below). Showing this will be the main goal of this section. 146 Chapter 5. Tensor product of well-generated pretriangulated categories

5.10.1 Tensor product of homotopically α-cocomplete dg categories

We can define a homotopically α-cocomplete tensor product in Hqeα. Definition 5.52. Let a and b be α-homotopically cocomplete dg categories. A homotopically α-cocomplete tensor product of a and b is defined as a homotopically L α-cocomplete small dg category a α b such that the following universal property ⊗ holds in Hqeα: L RHomα(a α b,c) = RHomα(a,RHomα(b,c)). (5.34) ⊗ ∼ Remark 5.53. Observe that for α = 0, as triangulated categories are in particular additive categories, we have that: ℵ

Hqe Hqe; 0 = • ℵ a b a b RHom 0 ( , ) = RHom( , ); • ℵ and hence a L b a L b. 0 = • ⊗ℵ ⊗ Theorem 5.54. Let α be a regular cardinal and a,b homotopically α-cocomplete small L dg categories. Then, a α b exists. ⊗ Proof. The proof consists basically in mimicking the construction of the tensor product of α-cocomplete k-linear categories following [42, §6.5], or [43, §10] and [50, §2.4] for the concrete case of α = 0. ℵ L L Consider the Yoneda embedding Y : a b D(a b) and denote by a the quasi- L ⊗ −→ L ⊗ left adjoint of the inclusion Dα,α(a b) D(a b). Recall that a is cocontinuous, L ⊗ ⊆L ⊗ and hence a RHomc(D(a b),Dα,α(a b)). ∈ ⊗ ⊗ L L Consider the bimodule a Y RHom(a b,Dα(a b)) given by the composition of bimodules. We prove that◦ a∈ Y is α-bicocontinuous.⊗ ⊗ Observe that a Y is α- bicocontinuous if and only if ◦ ◦

L 0 0 L Dα,α(a b)(H (a) H (Y )( , ), X ) : a b D(k) ⊗ ◦ − − ⊗ −→ L preserves α-small coproducts in both variables for all X Dα,α(a b), where we put L 0 L ∈ ⊗ Dα,α(a b) = H (Dα,α(a b)). But observe that ⊗ ⊗ L 0 0 L 0 0 Dα,α(a b)(H (a) H (Y )( , ), X ) = D(a b)(H (Y )( , ), X ) = H (X )( , ) ⊗ ◦ − − ⊗ − − − − L which preserves α-small colimits in both variables because X Dα,α(a b). L ∈ ⊗ L Denote by G the set of representables in D(a b). We know that Dα,α(a b) is -compactly generated by G F G α. Denote by⊗ d D a L b the enhancement⊗ α 0 = ( ) α,α( ) 〈 〉 ⊆ ⊗ Chapter 5. Tensor product of well-generated pretriangulated categories 147 of G via the natural enhancement D a L b of D a L b . In particular, observe 0 α,α( ) α,α( ) that d is an essentially small dg category which⊗ is homotopically⊗ α-cocomplete, and hence can be considered as an element in Hqeα. Consider the corestriction to d of the bimodule a Y above ◦ Ψ : a L b d ⊗ −→ which remains trivially α-bicocontinuous. We claim that d satisﬁes the universal L property of a α b with universal α-bicocontinuous bimodule given by Ψ. ⊗ Consider c a homotopically α-cocomplete small dg category.

First, observe that any F RHomα(d,c) induces, via composition with Ψ, an α- ∈ L bicocontinuous bimodule F Ψ RHomα,α(a b,c). L ◦ ∈ ⊗ L Given G RHomα,α(a b,c), consider the associated dg functor G : a b D(c). L Observe that,∈ because G⊗ is quasi-representable, we have that for all (A⊗, B)−→a b ∈ ⊗ exists a C(A,B) c such that G (A, B ) = Yc(C(A,B)) Dα(c). We can then deﬁne a bimodule ∈ ∼ ∈ F RHomα(d,c), by putting ∈ F (Ψ(A, B)) = G (A, B), and requiring it to be α-cocontinuous in Dα(c). This determines the bimodule F completely up to quasi-isomorphism, as every element in d can be obtained from the image by Ψ of representables by taking shifts in both directions, cones, α-small coproducts and direct factors. In addition, observe that F is right quasi-representable because c is homotopically α-cocomplete and hence the corestriction Yc : c Dα(c) is α-cocontinuous, as it can be readily deduced from the proof of [66, Thm−→ 6.4]. We L extend morphisms between objects in RHomα,α(a b,c) to morphisms between ⊗ the corresponding objects in RHomα(d,c) accordingly. It is then trivial to see that 0 F Ψ = G in H (RHomα(d,c)), which proves quasi-essential surjectivity. ◦ ∼ Consider now the morphism RHom d,c F, F RHom F , F in C k . α( )( 0) α,α( Ψ 0 Ψ) ( ) Observe that any : F F is uniquely determined−→ up to quasi-isomorphism◦ ◦ by φ 0 A, B : F A, B −→F A, B , which proves quasi-fully faithfulness. Hence φ(Ψ( )) (Ψ( )) 0(Ψ( )) −→ we have that, for all c Hqeα, the dg functor ∈ Ψ L RHomα(d,c) −◦ RHomα,α(a b,c) (5.35) ⊗ induces a quasi-equivalence of homotopically α-cocomplete small dg categories.

By Lemma 5.14, we can conclude that RHomα(d,c) RHomα(a,RHomα(b,c)) in ∼= Hqeα, which ﬁnishes the argument. 148 Chapter 5. Tensor product of well-generated pretriangulated categories

We have now an analogous statement to Proposition 5.35, whose proof is also very similar.

Proposition 5.55. Let b be a homotopically α-small dg category and C a well-generated dg category. In Hqewg, we have

RHomc(Dα(b),C) RHomα(b,C). (5.36) ∼=

Proof. First recall that we have an strict localization a H 0 i :“Dα(b) D(b)”. One can a easily see that a Y RHomα(b,Dα(b)) is α-cocontinuous and that it is isomorphic ◦ ∈ to the correstriction of the Yoneda embedding Yb : b Dα(b). Hence, we have that −→ i a Yb = Yb : b Dα(b). Then, one can easily show that the map ◦ ◦ ∼ −→ [Dα(b),C]c [b,C]α : f f [a]iso [Yb] −→ 7−→ ◦ ◦ is a bijection, where [ , ]c (resp. [ , ]α) indicates the subset of morphisms in Hqe such that the induced− morphism− − between− the homotopy categories is cocontinuous (resp. -cocontinuous). Indeed, given f a Y f a Y , then, by α [ ]iso [ b] = 0 [ ]iso[ b] derived Morita theory, as both f a and f ◦ a are◦ cocontinuous,◦ we have that [ ]iso 0 [ ]iso f a f a . Consequently,◦ ◦ [ ]iso = 0 [ ]iso ◦ ◦

f = f [a]iso [i ]iso = f 0 [a]iso [i ]iso = f 0, ◦ ◦ ◦ ◦ which proves injectivity.

Next, consider g [b,C]α. Then, by derived Morita theory, there is an element ∈ f [D(b),C]c such that f [Yb] = g . We are going to show that f factors through ∈ ◦ 0 [a]iso : D(b) Dα(b). Indeed, by the description of the kernel of H (a) provided in [66, §6] and−→ the universal property of the cpt dg quotient (5.8), f factors through [a]iso if and only if 0 a 0 a H (f )( Yb(Bi )) = H (f )(Yb( Bi )), i ∼ i ` 0 where i Bi is seen in H (b), for all α-small coproducts. But this condition is readily seen to be satisfied taking into account that f is cocontinuous and f [Yb] = g is α-cocontinuous, and hence ◦ f = t [a]iso. ◦ In addition, t is also cocontinuous (by the universal property of the cpt quotient (5.8)), that is t [Dα(b),C]c. ∈ Now, observe that t [a]iso [Yb] = f [Yb] = g , ◦ ◦ ◦ Chapter 5. Tensor product of well-generated pretriangulated categories 149 which proves surjectivity. Now, we define D b L a,C as the subset of D b L a,C consisting of ele- [ α( ) ]0c [ α( ) ] 0 ments f such that H (f ⊗)( , A) is cocontinuous for all A⊗ a. We define as well a L b,C as the subset of −a L b,C consisting of elements∈f such that H 0 f , A [ ]0α ( )( ) is α⊗-cocontinuous for all A a⊗. Then, we have the following diagram induced by− the ∈ morphism [a]iso [Yb] : b Dα(b): ◦ −→

[a,RHomc(Dα(b),C)] [a,RHomα(b,C)]

∼= ∼= D b L a,C a L b,C (5.37) α( ) c0 α0 ⊗ ⊗ = ∼ ∼= [a]iso [Yb] [Dα(b),RHom(a,C)]c −◦ ◦ [b,RHom(a,C)]α ,

5.10.2 Relation with the tensor product of well-generated dg categories

Proposition 5.56. Let a, b be two homotopically α-cocomplete small dg categories. Then, we have that L Dα(a) Dα(b) = Dα(a α b) (5.38) ⊗ in Hqewg.

Proof. We have that:

L RHomc(Dα,α(a b),C) = RHomc(Dα(a),RHomc(Dα(b),C)) ⊗ ∼ RHomα(a,RHomα(b,C)) ∼= L RHomα(a b,C) ∼= α ⊗ L = RHomc(Dα(a α b),C) ∼ ⊗ for every well-generated dg category C, where the ﬁrst equivalence follows from Proposition 5.51, the second and the last equivalences from Proposition 5.55 and the third equivalence follows from the deﬁnition of the α-cpt tensor product. 150 Chapter 5. Tensor product of well-generated pretriangulated categories

Corollary 5.57. The tensor product of two α-compactly generated dg categories is again α-compactly generated.

Proof. The theorem follows from the triangulated Gabriel-Popescu theorem and Proposition 5.56 above. Bibliography

[1] Jiˇrí Adámek and Jiˇrí Rosický. Locally presentable and accesible categories. En- glish. London Mathematical Society Lecture Note 189. Cambridge Books Online. Cambridge University Press, 1994.

[2] Michael Artin, John Tate, and Michel Van den Bergh. “Some Algebras Associated to Automorphisms of Elliptic Curves”. In: The Grothendieck Festschrift: A Collec- tion of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck. Ed. by Pierre Cartier et al. Boston, MA: Birkhäuser Boston, 2007, pp. 33–85.

[3] Michael Artin and James J. Zhang. “Noncommutative projective schemes”. English. In: Adv. Math. 109.2 (1994), pp. 228–287.

[4] Jean Bénabou. “Introduction to bicategories”. In: Reports of the Midwest Cate- gory Seminar. Springer, Berlin, 1967, pp. 1–77.

[5] Alexey I. Bondal and Mikhail M. Kapranov. “Framed triangulated categories”. In: Mat. Sb. 181.5 (1990), pp. 669–683.

[6] Alexey I. Bondal and Alexander E. Polishchuk. “Homological properties of associative algebras: the method of helices”. In: Izv. Ross. Akad. Nauk Ser. Mat. 57.2 (1993), pp. 3–50.

[7] Francis Borceux. Handbook of categorical algebra. 1. Vol. 50. Encyclopedia of Mathematics and its Applications. Basic category theory. Cambridge University Press, Cambridge, 1994, pp. xvi+345. [8] Francis Borceux. Handbook of categorical algebra. 2. Vol. 51. Encyclopedia of Mathematics and its Applications. Categories and structures. Cambridge University Press, Cambridge, 1994, pp. xviii+443. [9] Francis Borceux. Handbook of categorical algebra. 3. Vol. 52. Encyclopedia of Mathematics and its Applications. Categories of sheaves. Cambridge University Press, Cambridge, 1994, pp. xviii+522.

151 152 Bibliography

[10] Francis Borceux and Carmen Quinteiro. “A theory of enriched sheaves”. English. In: Cah. Topol. Géom. Différ. Catég. 37.2 (1996), pp. 145–162.

[11] Martin Brandenburg, Alexandru Chirvasitu, and Theo Johnson-Freyd. “Reﬂex- ivity and dualizability in categoriﬁed linear algebra”. In: Theory Appl. Categ. 30 (2015), pp. 808–835.

[12] José Luis Bueso, Pascual Jara, and Alain Verschoren. Compatibility, stability, and sheaves. English. Vol. 185. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1995, pp. xiv+265. [13] Alberto Canonaco and Paolo Stellari. “A tour about existence and uniqueness of dg enhancements and lifts”. In: J. Geom. Phys. (2016).

[14] Alberto Canonaco and Paolo Stellari. “Internal Homs via extensions of dg functors”. In: Adv. Math. 277 (2015), pp. 100–123.

[15] Alberto Canonaco and Paolo Stellari. “Uniqueness of dg enhancements for the derived category of a Grothendieck category”. In: J. Eur. Math. Soc. (JEMS) (2016).

[16] Giovanni Caviglia and Geoffroy Horel. “Rigidiﬁcation of higher categorical structures”. In: Algebr. Geom. Topol. 16.6 (2016), pp. 3533–3562.

[17] Claudia Centazzo and Enrico M. Vitale. “A classiﬁcation of geometric morphisms and localizations for presheaf categories and algebraic categories”. In: J. Algebra 303.1 (2006), pp. 77–96.

[18] Alex Chirvasitu and Theo Johnson-Freyd. “The fundamental pro-groupoid of an afﬁne 2-scheme”. In: Appl. Categ. Structures 21.5 (2013), pp. 469–522.

[19] Brian Day. “Note on monoidal localisation”. In: Bull. Austral. Math. Soc. 8 (1973), pp. 1–16.

[20] Olivier De Deken and Wendy Lowen. “Abelian and derived deformations in the presence of Z-generating geometric helices”. English. In: J. Noncommut. Geom. 5.4 (2011), pp. 477–505.

[21] Pierre Deligne. “Catégories tannakiennes”. French. In: The Grothendieck Festschrift, Vol. II. Vol. 87. Progr. Math. Birkhäuser Boston, Boston, MA, 1990, pp. 111–195.

[22] Vladimir Drinfeld. “DG quotients of DG categories”. English. In: J. Algebra 272.2 (2004), pp. 643–691.

[23] Eduardo J. Dubuc and Ross Street. “A construction of 2-ﬁltered bicolimits of categories”. In: Cah. Topol. Géom. Différ. Catég. 47.2 (2006), pp. 83–106. Bibliography 153

[24] Daniel Dugger and Brooke Shipley. “A curious example of triangulated- equivalent model categories which are not Quillen equivalent”. In: Algebr. Geom. Topol. 9.1 (2009), pp. 135–166.

[25] Samuel Eilenberg and Saunders MacLane. “General theory of natural equivalences”. In: Trans. Amer. Math. Soc. 58 (1945), pp. 231–294.

[26] David Eisenbud. Commutative algebra. Vol. 150. Graduate Texts in Mathemat- ics. With a view toward algebraic geometry. Springer-Verlag, New York, 1995, pp. xvi+785. [27] Peter Gabriel and Friedrich Ulmer. Lokal präsentierbare Kategorien. Lec- ture Notes in Mathematics, Vol. 221. Springer-Verlag, Berlin-New York, 1971, pp. v+200. [28] Peter Gabriel and Michel Zisman. Calculus of fractions and homotopy theory. English. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 35. Berlin- Heidelberg-New York: Springer-Verlag, x, 168 p. (1967). 1967.

[29] Nora Ganter and Mikhail Kapranov. “Representation and character theory in 2-categories”. In: Adv. Math. 217.5 (2008), pp. 2268–2300.

[30] Francesco Genovese. “Adjunctions of Quasi-Functors Between DG-Categories”. In: Applied Categorical Structures (2016), pp. 1–33.

[31] Robert Gordon, A. John Power, and Ross H. Street. “Coherence for tricategories”. English. In: Mem. Am. Math. Soc. 558 (1995), p. 81.

[32] Mark Hovey. Model categories. English. Providence, RI: American Mathematical Society, 1999, pp. xii + 209. [33] Srikanth B. Iyengar and Henning Krause. “The Bousﬁeld lattice of a triangulated category and stratiﬁcation”. In: Math. Z. 273.3-4 (2013), pp. 1215–1241.

[34] Peter T. Johnstone. Sketches of an elephant: a topos theory compendium. Vol. 1. English. Vol. 43. Oxford Logic Guides 43. The Clarendon Press, Oxford University Press, New York, 2002, pp. xxii+468+71. [35] Peter T. Johnstone. Sketches of an elephant: a topos theory compendium. Vol. 2. English. Vol. 44. Oxford Logic Guides 43. The Clarendon Press, Oxford University Press, Oxford, 2002, i–xxii, 469–1089 and I1–I71.

[36] Dmitry Kaledin and Wendy Lowen. “Cohomology of exact categories and (non-) additive sheaves”. In: Adv. Math. 272 (2015), pp. 652–698.

[37] Masaki Kashiwara and Pierre Schapira. Categories and sheaves. English. Vol. 332. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006, pp. x+497. 154 Bibliography

[38] Bernhard Keller. “Derived invariance of higher structures on the Hochschild complex”. Preprint on webpage at https :/ /webusers. imj - prg. fr / ~bernhard.keller/publ/dih.pdf. [39] Bernhard Keller. “Deriving DG categories”. English. In: Ann. Sci. École Norm. Sup. (4) 27.1 (1994), pp. 63–102.

[40] Bernhard Keller. “On differential graded categories”. English. In: International Congress of Mathematicians. Vol. II. Eur. Math. Soc., Zürich, 2006, pp. 151–190.

[41] Bernhard Keller. “On the cyclic homology of exact categories”. In: J. Pure Appl. Algebra 136.1 (1999), pp. 1–56.

[42] Gregory Maxwell Kelly. “Basic concepts of enriched category theory”. In: Repr. Theory Appl. Categ. 10 (2005). Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714], pp. vi+137. [43] Gregory Maxwell Kelly. “Structures deﬁned by ﬁnite limits in the enriched context. I”. In: Cahiers Topologie Géom. Différentielle 23.1 (1982). Third Colloquium on Categories, Part VI (Amiens, 1980), pp. 3–42.

[44] Maxim Kontsevich. “Homological algebra of mirror symmetry”. English. In: Proceedings of the international congress of mathematicians, ICM ’94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser, 1995, pp. 120–139.

[45] Henning Krause. “Derived categories, resolutions, and Brown representability”. English. In: Interactions between homotopy theory and algebra. Vol. 436. Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 101–139.

[46] Henning Krause. “Deriving Auslander’s formula”. In: Doc. Math. 20 (2015), pp. 669–688.

[47] Henning Krause. “Localization theory for triangulated categories”. English. In: Triangulated categories. Vol. 375. London Math. Soc. Lecture Note Ser. Cam- bridge Univ. Press, Cambridge, 2010, pp. 161–235.

[48] Henning Krause. “On Neeman’s well generated triangulated categories”. In: Doc. Math. 6 (2001), pp. 118–123.

[49] Tom Leinster. Basic bicategories. English. Version 1. Oct. 4, 1998. arXiv: 9810017v1. [50] Ignacio López Franco. “Tensor products of ﬁnitely cocomplete and abelian categories”. English. In: J. Algebra 396 (2013), pp. 207–219.

[51] Wendy Lowen. “A generalization of the Gabriel-Popescu theorem”. English. In: J. Pure Appl. Algebra 190.1-3 (2004), pp. 197–211. Bibliography 155

[52] Wendy Lowen. “Linearized topologies and deformation theory”. In: Topology Appl. 200 (2016), pp. 176–211.

[53] Wendy Lowen, Julia Ramos González, and Boris Shoikhet. “On the tensor product of linear sites and Grothendieck categories”. In: Int. Math. Res. Not. IMRN (2017).

[54] Wendy Lowen and Michel Van den Bergh. “Deformation theory of abelian categories”. English. In: Trans. Amer. Math. Soc. 358.12 (2006), 5441–5483 (elec- tronic).

[55] Wendy Lowen and Michel Van den Bergh. “Hochschild cohomology of abelian categories and ringed spaces”. In: Adv. Math. 198.1 (2005), pp. 172–221.

[56] Valery A. Lunts and Dmitri O. Orlov. “Uniqueness of enhancement for triangulated categories”. In: J. Amer. Math. Soc. 23.3 (2010), pp. 853–908.

[57] Jacob Lurie. Derived Algebraic Geometry II: Noncommutative Algebra. English. Version 5. Sept. 19, 2007. arXiv: math/0702299. [58] Saunders Mac Lane. Categories for the working mathematician. Second. Vol. 5. Graduate Texts in Mathematics. Springer-Verlag, New York, 1998, pp. xii+314. [59] J. Peter May. “The additivity of traces in triangulated categories”. In: Adv. Math. 163.1 (2001), pp. 34–73.

[60] Amnon Neeman. “On the derived category of sheaves on a manifold.” eng. In: Documenta Mathematica 6 (2001), pp. 483–488.

[61] Amnon Neeman. Triangulated categories. Vol. 148. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2001, pp. viii+449. [62] Marco A. Pérez. Introduction to Abelian model structures and Gorenstein homological dimensions. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016, pp. xxv+343. [63] Andrew Pitts. “On product and change of base for toposes”. In: Cahiers Topolo- gie Géom. Différentielle Catég. 26.1 (1985), pp. 43–61.

[64] Alexander Polishchuk. “Noncommutative proj and coherent algebras”. In: Math. Res. Lett. 12.1 (2005), pp. 63–74.

[65] Nicolae Popesco and Pierre Gabriel. “Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes”. French. In: C. R. Acad. Sci. Paris 258 (1964), pp. 4188–4190.

[66] Marco Porta. “The Popescu-Gabriel theorem for triangulated categories”. In: Adv. Math. 225.3 (2010), pp. 1669–1715. 156 Bibliography

2 [67] Dennis Presotto. Z -algebras as noncommutative blow-ups. English. Version 1. arXiv: 1701.00413. [68] Dorette A. Pronk. “Etendues and stacks as bicategories of fractions”. In: Com- positio Math. 102.3 (1996), pp. 243–303.

[69] Daniel G. Quillen. Homotopical algebra. 1967. [70] Stefan Schwede. Topological triangulated categories. English. Version 1. Jan. 4, 2012. arXiv: 1201.0899. [71] Jean-Pierre Serre. “Faisceaux algébriques cohérents”. French. In: Ann. of Math. (2) 61 (1955), pp. 197–278.

[72] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia. edu. 2014. [73] J. Toby Stafford and Michel Van den Bergh. “Noncommutative curves and noncommutative surfaces”. English. In: Bull. Amer.Math. Soc. (N.S.) 38.2 (2001), pp. 171–216.

[74] Gonçalo Tabuada. “On Drinfeld’s dg quotient”. English. In: J. Algebra 323.5 (2010), pp. 1226–1240.

[75] Gonçalo Tabuada. “Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories”. In: C. R. Math. Acad. Sci. Paris 340.1 (2005), pp. 15– 19.

[76] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture Notes in Mathematics, Vol. 269. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P.Deligne et B. Saint-Donat. Springer-Verlag, Berlin-New York, 1972, pp. xix+525. [77] Théorie des topos et cohomologie étale des schémas. Tome 2. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P.Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 270. Berlin: Springer-Verlag, 1972, pp. iv+418. [78] Bertrand Toën. “Lectures on dg-categories”. English. In: Topics in algebraic and topological K -theory. Vol. 2008. Lecture Notes in Math. Springer, Berlin, 2011, pp. 243–302.

[79] Bertrand Toën. “The homotopy theory of dg-categories and derived Morita theory”. In: Invent. Math. 167.3 (2007), pp. 615–667. Bibliography 157

[80] Matteo Tommasini. Some insights on bicategories of fractions II - Right satu- rations and induced pseudofunctors between bicategories of fractions. English. Version 2. arXiv: 1410.5075v2. [81] Matteo Tommasini. Some insights on bicategories of fractions III - Equivalences of bicategories of fractions. English. Version 2. arXiv: 1410.6395v2. [82] Matteo Tommasini.“Some insights on bicategories of fractions: representations and compositions of 2-morphisms”. In: Theory Appl. Categ. 31 (2016), Paper No. 10, 257–329.

[83] Michel Van den Bergh. “Noncommutative quadrics”. English. In: Int. Math. Res. Not. IMRN 17 (2011), pp. 3983–4026.

[84] Jean-Louis Verdier. “Des catégories dérivées des catégories abéliennes”. In: Astérisque 239 (1996). With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis, xii+253 pp. (1997). [85] Harvey Wolff. “V-fractional categories.” English. In: Cah. Topologie Géom. Différ. Catégoriques 16 (1976), pp. 149–168.