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Algant Master Thesis Compactness in Toposes Candidate: Advisor: Mauro Mantegazza Dr. Jaap van Oosten Coadvisors: Prof. Sandra Mantovani Prof. Ronald van Luijk Universita` degli Studi Universiteit di Milano Leiden 30 April 2015 ii Contents Introduction . .v Acknowledgements . viii 1 Toposes, Sites and 2-Categories 1 1.1 Toposes . .3 1.2 Grothendieck toposes . .7 1.3 Grothendieck topologies . 10 1.4 Coherent sites . 20 1.5 2-categories . 31 1.6 Beck-Chevalley conditions . 39 2 Geometry of toposes 45 2.1 Frames . 45 2.2 From spaces to locales . 47 2.3 Maps of locales . 54 2.4 Sheaves of locales . 57 2.5 From locales to toposes . 62 3 Internal structures 69 3.1 Internal category theory . 70 3.2 Limits and colimits of internal diagrams . 77 3.3 Internal categories in a topos . 78 3.4 Internal categories in toposes over a base . 84 4 Special geometric morphisms 91 4.1 Surjections and inclusions . 91 4.2 Bounded geometric morphisms . 95 4.3 Hyperconnected and localic morphisms . 100 5 Compact toposes and proper morphisms 105 5.1 Compactness . 105 5.2 Proper maps . 112 5.3 Pretopos sites and weak BC condition . 123 5.4 Further developments . 130 iii iv CONTENTS A Trees and induction 133 A.1 Pretopologies . 136 A.2 Applications to Section 1.4 . 141 Index of symbols 147 Index 149 Bibliography 153 CONTENTS v Introduction One of the possible interpretations of a topos is as a generalized space. The reason why we can think in these terms is because of the example we are about to give. Consider a topological space X, we can consider the category Sh(X) of sheaves over X and morphisms between them. It turns out that such a cat- egory is a topos and moreover, if the topological space is regular enough (i.e. sober), then the topos Sh(X) alone contains enough categorical information in order to reconstruct back X. We might therefore identify (sober) topo- logical spaces with their topos of sheaves and hence see them as a particular case of a broader concept of space represented by toposes. Broadly speaking, the main objective of this thesis is to translate the classi- cal notion of compactness in topos theoretic terms, that is, to find a suitable definition of compact topos such that in the particular case of Sh(X) it will coincide with the compactness of X as a topological space. Actually the situation is more delicate: rather than picking toposes alone as spaces, we should consider toposes E equipped with a particular morphism (geometric morphism) into another topos S called base topos. In order to understand the reason behind this requirement, we will need to discuss an- other interpretation of toposes, namely the interpretation as a universe for mathematics. In the classical theory, the base topos is the category Set of sets and functions, in fact what we usually do in ordinary mathematics is to describe the objects we are working with using sets and functions. For instance a group G is a set G equipped with two functions e : 1 / G and · : G×G / G, the first selecting the identity and the second describing the multiplication, such that the usual axioms are satisfied. A group though is just a model for a certain theory (Group theory) and it turns out that there is a way to interpret the standard logical formulas in every topos S so that it becomes possible to talk about models of group theory inside S. For example if X is a topological space, a group in Sh(X) is a group sheaf, that is, a sheaf F on X where the families F (U) are groups for every open U and the restriction morphisms are group homomorphisms. Following this principle then, we could just take a topos S, set it as universe for our own mathematics and, working as if it were our category of sets, we could deduce our own interpretation of mathematics. An emblematic exam- ple can be seen in Section 2.4 of [LT], where it is shown the construction of a topos S which is a universe where every real function f : R / R must be continuous. We can introduce a special kind of morphisms called geometric morphisms, which will also provide a way to transfer informations from one topos to another, and if we have a geometric morphism E / S we will think of E as a topos in the universe S and for this reason we can refer to such a topos E as a S-topos. vi CONTENTS Coming back to the idea of generalized space, we can think that a (sober) topological space X is represented by a geometric morphism of the form Sh(X) / Set, for in fact a topological space is considered in the universe of sets. We can then think of Set-toposes as generalized spaces in the universe of sets and more in general, for every topos S, we can interpret S-toposes as generalized spaces in the universe S. We can finally state more precisely the first aim of this thesis, which is to give a good definition of compact S-topos that generalizes the classical compactness (so in particular Sh(X) will be a compact Set-topos iff X is compact). Once we achieve this goal, we start studying some properties and characterizations of S-compact toposes and in particular the properties of those geometric morphisms E / S which render E a compact S-topos, which will be given the name of proper geometric morphisms. For these purposes, we follow the ideas already presented in the book [MV], exposing the theoretical preliminaries required to understand it and filling in the gaps left to the reader. The following is a brief description of the contents of this thesis. In the first chapter we give the definitions of topos and geometric mor- phism and we study some fundamental result about them. We then move to the particular case of Grothendieck toposes, which are a generalization of the notion of "category of sheaves", where instead of considering sheaves over opens of a topological space, we consider sheaves over sites, that is, cat- egories equipped with a notion of coverage. Later on we study some special site with nice properties, for they will be needed in the following chapters. We then move to a brief study of 2-categories which will enable us to talk about the 2-category Geom of toposes and geometric morphisms, and about Beck-Chevalley conditions. The second chapter is of a more geometrical nature: here in fact we anal- yse in a more detailed way the steps that bring us from topological spaces to toposes, passing in particular through locales, which are a generalization of topological spaces where points are no more needed. We study in par- ticular locales and the particularly regular way in which they embed in Geom. The third chapter treats the internalization of category theory in toposes and in toposes over a base. More precisely we describe the internal version of categories, diagrams, limits and colimits and then we study some condi- tions about preservation of filtered internal colimits. In the fourth chapter we come back to geometry, for we deal with some example of geometric morphism, namely surjections, inclusions and their fac- torization system; bounded morphisms and their relation with Grothendieck CONTENTS vii toposes; and the hyperconnected-localic factorization. In the last chapter we finally deal with compactness, studying first the case of a compact Set-topos and then generalizing it to the general case represented by proper geometric morphisms. We then analyse the relation between proper morphisms and the ones described in the previous chapter, along with some other property and characterization of proper geometric morphisms. We finish it by hinting at some other developments of this the- ory. The Appendix contains some technical result about trees and well-founded induction. In particular it also contains some fact about Grothendieck pre- topologies (a special kind of coverage) and some useful result used in this thesis. viii CONTENTS Acknowledgements I wish to thank Dr. Jaap van Oosten for his helpful advices, Prof. Sandra Mantovani for her availability and both of them for their courses in category theory. Along with them I wish to express my gratitude to all those pro- fessors that, through their passion, were able to convey the beauty of their teachings. Thanks to all those friends that I met during my studies, for the ideas and thoughts shared in these years and for the great times (and the less great ones) spent together. I also want to thank my awesome friends from all over the world, encountered during my staying abroad, for thanks to them it really felt like home, and thanks to ALGANT as well for allowing me this opportunity. Special thanks go to Andrea Gagna for the good and bad experiences we shared together, for the inspiring discussions we had, for his availability in time of need and for his everlasting patience. Last but not least I wish to thank my family, without whose encouragement and unconditioned support I would not have made it this far. Chapter 1 Toposes, Sites and 2-Categories In this first chapter we are presenting some of the main tools required to understand the topic of this thesis. We will start by exploring some funda- mental notions of topos theory in the first two sections. After that we will have to go through some more technical result regarding sites (a sort of wide generalization of a space containing the minimal information needed in order to talk about sheaves).
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