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Topos-Theoretic Methods in Noncommutative Geometry

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Faculty of Science Department of Mathematics and Computer Science Topos-theoretic methods in noncommutative geometry Dissertation submitted to obtain the degree of Doctor of Science: Mathematics at the University of Antwerp, to be defended by Jens Hemelaer Supervisor: Prof. dr. Lieven Le Bruyn Antwerp, 2019 Faculteit Wetenschappen Departement Wiskunde en Informatica Topostheoretische methodes in niet-commutatieve meetkunde Proefschrift voorgelegd tot het behalen van de graad van doctor in de wetenschappen: wiskunde aan de Universiteit Antwerpen te verdedigen door Jens Hemelaer Promotor: Prof. dr. Lieven Le Bruyn Antwerpen, 2019 Acknowledgements First of all, I would like to thank my PhD supervisor Lieven Le Bruyn. I didn't know Lieven very well when we first applied for my PhD fellowship, but it couldn't have turned out better for me. I have enjoyed being his PhD student, and I have learned a lot from him. Parts of Chapter 3 and Chapter 5 are based on a joint paper with Lieven Le Bruyn; Chapter 6 is a joint paper with Lieven Le Bruyn and Karin Cvetko-Vah. Lieven and Karin, it has been an honor working with you. I would like to thank the members of the jury for their valuable comments and suggestions. Thanks to Jan Van Geel for making time to already discuss the thesis before the predefense. Some other people who made valuable suggestions regarding my research are Yinhuo Zhang, Julia Ramos Gonz´alez,Pieter Belmans, Aur´elienSagnier, and the anonymous reviewers of the papers. Special thanks go to Pieter Belmans for introducing me to the research activities and the people at the University of Antwerp, while I was still writing my master's thesis. I would also like to thank Jeffrey De Fauw, with whom I did my first steps towards research and Algebraic Geometry six years ago. I am thankful for getting to know my colleagues a little bit better, either during seminars or conferences, or by playing card games during lunch, or by sharing an office. I am grateful to FWO (Research Foundation { Flanders) for granting me a PhD fellowship, so I could do the research leading to this thesis. I would like to thank my parents, my sister Bente and my friends and family, for always being there for me, and for the good times together. And I would like to thank my girlfriend Caroline, for bringing more color to my life. Writing these acknowledgements with you nearby, on our sunny terrace full of flowers, is more than what I could wish for. i Doctoral committee: Prof. dr. Wendy Lowen, University of Antwerp Prof. dr. Sonja Hohloch, University of Antwerp Prof. dr. Fred Van Oystaeyen, University of Antwerp Prof. dr. Jan Van Geel, Ghent University Prof. dr. Karin Cvetko-Vah, University of Ljubljana Prof. dr. Isar Stubbe, Universit´edu Littoral C^oted'Opale Prof. dr. Lieven Le Bruyn, University of Antwerp ii Contents Acknowledgements i Introduction v Bibliographical comments xi Nederlandstalige samenvatting xii 1 Short introduction to topos theory 1 1.1 Basic notions . .1 1.2 Points of a topos and localic reflections . .5 1.3 Examples . .6 1.3.1 Sheaves on a locale . .7 1.3.2 Presheaf toposes . .7 1.3.3 Sets with a monoid action . .8 1.3.4 Slice toposes . .8 1.3.5 Equivariant sheaves . 10 2 Grothendieck topologies on posets 12 2.1 Presheaves on a poset . 13 2.2 Grothendieck topologies versus subsets . 15 2.3 Grothendieck topologies of finite type . 17 2.4 More explicit description . 19 2.5 Cardinalities of sets of Grothendieck topologies . 21 2.5.1 Artinian posets . 23 2.5.2 Other posets . 24 3 The Connes{Consani Arithmetic Site 27 3.1 Preliminaries . 28 3.2 The category of points: alternative proofs . 29 3.2.1 Using ind-categories . 30 3.2.2 Description as topos of equivariant sheaves . 32 3.3 Grothendieck topologies on the big cell . 33 3.3.1 Grothendieck topologies for closed (open) subspaces . 33 3.3.2 Grothendieck topologies of finite type . 34 3.3.3 Grothendieck topologies with enough points . 38 3.4 Subtoposes of the Arithmetic Site . 39 3.4.1 Subtoposes with enough points . 40 3.4.2 Prime set topologies and duality . 41 iii 3.4.3 Equivariant sheaves . 43 3.4.4 Notes about finiteness conditions . 45 4 An arithmetic topos for integer matrices 47 ns 4.1 The topos of M2 (Z)-sets . 49 4.2 Description of the slice topos . 50 4.2.1 The poset of abelian groups Z2 ⊆ M ⊆ Q2 ......... 50 4.2.2 Combinatorial description . 53 4.2.3 Relation to Conway's big picture . 55 ns 4.3 Automorphisms of M2 (Z)-Sets ................... 58 4.4 Alternative: the ax + b monoid . 61 4.5 Applications . 64 1 4.5.1 Relation to Ext (Q; Z).................... 64 4.5.2 Relation to Goormaghtigh conjecture . 67 5 Azumaya toposes 70 5.1 Grothendieck topologies on Azumaya algebras . 72 5.2 Azumaya representation schemes . 79 5.3 Trivializing Grothendieck topologies on Azumaya algebras . 83 5.4 Topos-theoretic points . 85 5.5 Sheaves with action of the projective general linear group . 89 6 What is a noncommutative topos? 98 6.1 Noncommutative Heyting algebras . 101 6.2 Noncommutative subobject classifiers . 103 6.3 Noncommutative Grothendieck topologies . 106 6.4 A class of examples . 109 Bibliography 115 Index 119 Index of symbols 122 iv Introduction In this thesis, we will use topos theory to study the geometry behind some algebraic structures, such as: • the nonzero natural numbers under multiplication (Chapter 3); • the 2 × 2 integer matrices with nonzero determinant, under multiplication (Chapter 4); • the category of Azumaya algebras and center-preserving algebra morphisms between them (Chapter 5). In the introduction, we give a preview of some of the ideas in the thesis, focusing on the second example of 2 × 2 integer matrices. But first: What is a topos? The first step leading to the notion of topos is replacing a topological space X by the category of sheaves on X. Recall that a presheaf F on X consists of a set F(U) for every open subset U ⊆ X, together with maps ρVU : F(U) −! F(V ) (1) whenever V ⊆ U, such that ρWV ◦ ρVU = ρWU . Geometrically, we can interpret s 2 F(U) as some kind of function defined on U. The maps ρVU then correspond to restriction, which is why we will use the shorthand sjV = ρVU (s): (2) S We say that F is a sheaf if for any open cover U = i2I Ui we have an equality F(U) = f(si)i2I : si 2 F(Ui); sijUi\Uj = sjjUi\Uj g: (3) The category Sh(X) of sheaves on X has properties similar to the properties of the category of sets. Moreover, we can reconstruct X from Sh(X) if X is Hausdorff (or more generally, if X is sober). The definition of a sheaf on X only depends on the poset O(X) of open subsets U ⊆ X. We can interpret O(X) as a category, with the open sets as objects and a morphism V ! U whenever V ⊆ U. A presheaf on X is then the same as a functor F : O(X)op −! Sets (4) to the category of sets, and the sheaf property is the same as above. v Expressing sheaves as some kind of functor opens the door to several general- izations. The poset O(X) is a frame: you can take arbitrary unions and finite intersections, and these satisfy ! [ [ V \ Ui = (V \ Ui): (5) i2I i2I It turns out that not every frame can be written as O(X) for some topological space X, but this does not stop us from defining sheaves with respect to arbitrary frames. Again, the category of sheaves on a frame has many good properties, and we can completely reconstruct the frame from the category of sheaves on it. The advantage of this approach is that we can now start thinking about frames geometrically: an arbitrary frame is interpreted as the poset of open subsets of a new notion of space that is (by definition) called a locale. We can define a point of a topological space in terms of a category of sheaves, and this can in turn be used to define points for a locale. The difference between locales and topological spaces is that two open sets in a locale are not necessarily equal if they contain the same points. Some nontrivial locales even have no points at all, which is why the study of locales is sometimes called pointless topology. While topological spaces are related to classical logic, locales can be used to model constructive logic. In order to get to (Grothendieck) toposes, we need to replace the poset O(X) by an arbitrary small category C. A presheaf is then defined as a (covariant) functor F : Cop −! Sets (6) to the category of sets. So for each object C of C there is a set of sections F(C), and for all morphisms f : D −! C there are restriction maps ρf : F(C) !F(D) (7) such that ρf◦g = ρg ◦ ρf . In order to define sheaves, we first have to specify when an object C is covered by a family of morphisms fi : Ci −! C; i 2 I: (8) For some categories, there might be an obvious way to define coverings (for example for the category of open sets of a topological space, as above). In general though, there are many possibilities, each leading to a different definition of sheaves. This is formalized using Grothendieck topologies. A presheaf F is then a sheaf with respect to a certain Grothendieck topology J if F(C) = f(si)i2I : si 2 F(Ci); ρg(si) = ρh(sj) whenever fi ◦ g = fj ◦ hg (9) for each J-covering consisting of morphisms fi : Ci ! C; i 2 I.
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