Categories of spaces built from local models Low Zhen Lin Trinity Hall University of Cambridge 20th June 2016 This dissertation is submitted for the degree of Doctor of Philosophy 獻僅 給 以 我 此 ⽗ 論 母 ⽂ This dissertation is dedicated to my parents. Abstract Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known examples are manifolds and schemes. The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces. The theory developed here will be illustrated with reference to examples, including the aforementioned manifolds and schemes. For concreteness, consider the passage from commutative rings to schemes. There are three main steps: first, one identifies a distinguished class of ring homo- morphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these distinguished homomorphisms; and finally, one embeds the opposite of the category of commutative ringsin an ambient category in which one can glue (the formal duals of) commutative rings along (the formal duals of) distinguished homomorphisms. Traditionally, the ambient category is taken to be the category of locally ringed spaces, but fol- lowing Grothendieck, one could equally well work in the category of sheaves for the large Zariski site—this is the so-called ‘functor of points approach’. A third option, related to the exact completion of a category, is described in this thesis. The main result can be summarised thus: categories of spaces built from local models are extensive categories with a class of distinguished morphisms, sub- ject to various stability axioms, such that certain equivalence relations (defined relative to the class of distinguished morphisms) have pullback-stable quotients; moreover, this construction is functorial and has a universal property. 君⼦於其⾔︐無所苟⽽已矣。故君⼦名之必可⾔也︐⾔之必可⾏也。刑罰不中︐則民無所措⼿⾜。禮樂不興︐則刑罰不中︔事⾔ 不名 不 成 不 順 ︐ 正 ︐ 則 ︐ 則 禮 則 事 樂 ⾔ 不 不 不 成 興 順 ︔ ︔ ︔ If names be not correct, language is not in accordance with the truth of things. If language be not in accordance with the truth of things, affairs cannot be carried on to success. When affairs cannot be carried on to success, proprieties and music will not flourish. When proprieties and music do not flourish, punishments will not be properly awarded. When punishments are not properly awarded, the people do not know how to move hand or foot. Therefore a superior man considers it necessary that the names he uses may be spoken ︽ 論 語 ‧ ⼦ 路 ︾ appropriately, and also that what he speaks may be carried out appropriately. What the superior man requires is just that in his words there may be nothing incorrect. Analects, Book XIII, translated by James Legge Contents Preface xi Introduction xiii Context .................................. xiii Summary ................................. xiv Outline .................................. xix Guide for readers ............................. xxi Chapter I. Abstract topology 1 1.1. The relative point of view ..................... 1 1.2. Local properties of morphisms . 13 1.3. Regulated categories ........................ 31 1.4. Exact quotients ........................... 40 1.5. Strict coproducts .......................... 61 Chapter II. Charted objects 77 2.1. Sites ................................ 77 2.2. Ecumenae ............................. 89 2.3. Extents ............................... 102 2.4. Functoriality ............................ 117 2.5. Universality ............................ 138 ix Contents Chapter III. Specificities 145 3.1. Compactness ............................ 145 3.2. Discrete fibrations . 153 3.3. Manifolds ............................. 161 3.4. Topological spaces . 166 3.5. Schemes .............................. 174 Appendix A. Generalities 199 a.1. Presheaves ............................. 199 a.2. Coverages ............................. 209 a.3. Sheaves ............................... 223 Bibliography 233 Index 235 x Preface Declaration of originality This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the preface and spe- cified in the text. It is not substantially the same as any that I have submitted,or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other university or similar institution except as declared in the preface and specified in the text. I further state that no substan- tial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma, or other qualification at the University of Cambridge or any other university or similar institution except as declared in the preface and specified in the text. xi Preface Acknowledgements I am very grateful to my supervisor, Peter Johnstone, for taking me as a student and allowing me to chart my own course through the last three and a half years. I would also like to thank all the other members of the category theory research group, past and present, whose company I have had the pleasure of having: Mar- tin Hyland, Julia Goedecke, Olivia Caramello, Ignacio López Franco, Nathan Bowler, Rory Lucyshyn-Wright, Guilherme Lima de Carvalho e Silva, Achilleas Kryftis, Christina Vasilakopoulou, Tamara von Glehn, Sori Lee, Filip Bár, Paige North, Enrico Ghiorzi, Adam Lewicki, Sean Moss, Nigel Burke, Georgios Char- alambous, and Eric Faber. Their presence contributed to a stimulating working environment in Cambridge, and I am glad to have been part of it. In this context, I must also mention Jonas Frey and Ohad Kammar, whose regular participation in the category theory seminar surely warrants their inclusion in the preceding list as honorary members of our research group. The internet has been a great boon to researchers all over the world. I have benefited from the insights gleaned from electronic correspondence with fellow mathematicians: Michael Shulman, Aaron Mazel-Gee, Emily Riehl, Qiaochu Yuan, Lukas Brantner, Geoffroy Horel, David White, Martin Brandenburg, and many others besides—primarily users of 푛Forum, MathOverflow, and Mathe- matics StackExchange—who are too numerous to list. In addition, TEX Stack- Exchange has been a very helpful resource for resolving TEXnical difficulties. The research presented in this thesis was conducted mostly during the course of a Ph.D. studentship funded by the Cambridge Commonwealth, European and International Trust and the Department of Pure Mathematics and Mathematical Statistics. I gratefully acknowledge their financial support. Last, but not least, I thank all my friends and family for their moral support, particularly Saran Tunyasuvunakool and Kathryn Atwell, whose friendship I have been fortunate to enjoy over the past seven and a half years. There are no words to express the profundity of my gratitude to my parents. I dedicate this thesis to them. xii Introduction Context Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known examples are manifolds and schemes. In fact, manifolds comprise a whole family of examples: after all, there are smooth manifolds, topological manifolds, complex analytic manifolds, manifolds with boundaries, manifolds with corners, etc. All of these notions of manifold are defined in roughly the same way, namely, as topological spaces equipped with a covering family of open embeddings of local models such that certain regularity conditions are satisfied. On the other hand, the traditional defin- ition of scheme is much more involved: because the local models are not given directly as geometric objects, one has to first find a suitable geometric incarnation of the local models, which is highly non-trivial. Another manifold-like notion is the notion of sheaf. Indeed, a sheaf on a topo- logical space is a topological space obtained by gluing together open subspaces of the base space. This heuristic can be made precise and remains valid for Gro- thendieck’s notion of sheaf on a site: a site is a category equipped with a notion of covering, and a sheaf on a site can be regarded as a formal colimit of a diagram in the base. Thus, it should come as no surprise that manifold-like objects can be represented by sheaves on the category of local models, in the sense that there is a fully faithful functor from the category of manifold-like objects to the category of sheaves. There is an alternative definition of scheme based on the aforementioned sheaf representation: this is called ‘the functor-of-points approach to algebraic geo- xiii Introduction metry’ and can be found in e.g. [Demazure and Gabriel, 1970]. The great advant- age of the definition in terms of functors over the traditional one in terms oflocally ringed spaces is that one no longer needs to explicitly model affine schemes— the local models—as geometric objects; instead, one can just glue together affine schemes formally. In exchange, one has to give up a certain sense of concreteness, but it is precisely the high level of abstraction that makes the functor-of-points approach so amenable to generalisation: one can mimic the functorial definition of ‘scheme’ in