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 COVER DESIGN: 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, asso- ciativity 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 prod- uct............................................... 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 cate- gories....................................... 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.