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Universit´eParis VII - Denis Diderot Ecole´ Doctorale de Science Math´ematiquesde Paris Centre These` de doctorat Discipline : Math´ematique pr´esent´eepar Simon Henry Des topos `ala g´eom´etrienon commutative par l'´etudedes espaces de Hilbert internes dirig´eepar Alain Connes Soutenue le 25 septembre 2014 devant le jury compos´e de : M. Alain Connes Coll`egede France/IHES directeur M. Alain Proute´ Universit´eParis 7 examinateur M. Jean Renault Universit´ed'Orl´eans examinateur M. Georges Skandalis Universit´eParis 7 examinateur M. Isar Stubbe Univ. du Littoral-C^oted'Opale rapporteur M. Steve Vickers University of Birmingham rapporteur Institut de Math´ematiques de UPMC Jussieu Ecole Doctorale de Sciences 5, rue Thomas Mann Math´ematiquesde Paris Centre 75 013 Paris 4 place Jussieu 75252 Paris Cedex 05 Boite courrier 290 2 Remerciement Mes remerciements vont tout d'abord `aAlain Connes pour avoir accept´ede diriger cette th`ese.Il a ´et´epour moi un mentor incroyablement talentueux, disponible et g´en´ereuxqui m'a ´enorm´ement appris aussi bien sur la g´eom´etrie non-commutative et les math´ematiques en g´en´eralque sur l'´ecriturescientifique. Je le remercie pour toutes les discusions qui ont jalonn´emon apprentissage des math´ematiqueset guid´ema d´ecouverte de la recherche. Je tiens aussi `ale re- mercier tout particuli`erement pour m'avoir soutenu et encourag´equand, apr`es quelques mois de th`ese,j'ai d´ecid´ede m'aventurer dans un nouveau sujet de recherche qu'on aurait pu croire ´eloign´ede ses propres centres d'int´er^etscienti- fiques. Cette th`esen'aurait jamais pu voir le jour sans son soutien et sa capacit´e `am'avoir encadrer dans ce nouveau domaine. Je souhaite ensuite remercier Isar Stubbe, d'une part pour le rapport tr`esd´etaill´e qu'il a fait de cette th`ese,mais aussi pour les nombreux et tr`espertinents conseils qu'il m'a donn´esapr`esavoir relu l'une des premi`ereversion de mon travail ainsi pour m'avoir pr´esent´ela communaut´edes cat´egoriciens. I also would like to thanks Steve Vickers for his review of my thesis, and espe- cially for the long and detailed list of correction and suggestion of improvement he made about it, this has largely contributed to improve this thesis and its rea- dability. I also thank him for the very interesting talks we had at the CT2014 in Cambridge. Je remercie aussi bien entendu Alain Prout´e,Jean Rennault et Georges Skan- dalis pour m'avoir fait l'honneur d'accepter d'^etremembres du jury. Je tiens `aremercier tout les membres du projet Alg`ebresd'op´erateurpour m'avoir accueilli `aParis 7. Je remercie aussi Anatole Khelif et le petit groupe de logique cat´egoriquepour l'int´er^etqu'ils ont port´e`ames travaux et pour m'avoir permis de les exposer `aplusieurs reprises. Merci `aIeke Moerdijk de m'accueillir `aNijmegen l'ann´eeprochaine et de me permettre ainsi de continuer les travaux que j'ai entam´esdans cette th`ese. j'adresse un remerciement sp´ecial`aMax Fathi et Alexandre Afgoustidis qui ont relu chacun un chapitre de ma th`eseet corrig´eun nombre effarant de fautes d'anglais. 3 Ces ann´eesde th`ese n'auraient pas ´et´eles m^emessans la petite communaut´e de doctorants de Paris 7, animant les repas et pauses caf´ede grandes discus- sions, math´ematiquesou autres, ainsi que les petits d´ejeunersmath´ematiquesdu mercredi. Je remercie pour cela Alex, Amaury, Arnaud, Baptise, Dragos, Elodie, Ho¨el,Julie, Lukas, Martin, Nicolas, Robert, Sary, Victoria... Je tiens `aremercier plus particuli`erement Alexandre, Fathi, J´er´emy et Martin pour l'excellente am- biance du bureau 747 et les nombreuses discussions, souvent anim´ees,autour du tableau qui m'ont permis d'apprendre beaucoup de math´ematiquesext´erieures aux domaines abord´esdans cette th`ese, et qui m'ont m^emeparfois aid´e`aavan- cer dans mes propres travaux (Martin reconna^ıtrapeut-^etrela proposition 2.7.3 du chapitre 3). Comme, heureusement, ces trois derni`eresann´eesn'ont pas ´et´euniquement rem- plies de math´ematiques,il me faut aussi remercier tous ceux qui ont su m'ar- racher `ama th`ese,mes amis de l'ENS : Igor, Furcy, Roland, C´ecile,J´er´emie, Pierre, Samuel, Mathias, etc. Les 'anciens du C2' : Nicolas, Thomas, Xiao, Syl- vain, S´everine et Louis. Les westies (et ex-westies) : David, Thomas, Emilie (aussi pour toutes ces vacances un peu partout en France), R´emi,Daphn´e,Na- thalie, Nicolas, Tiphaine, Isma¨el, Aur´eliane,Louis et bien-s^urSand et Ludo pour nous avoir communiqu´e`atous leur passion. Merci aussi `aMax, Sylvain, et Bastien pour les voyages `aLi`egeset autres soir´eesbien anim´ees. Enfin, un immense merci `ames parents qui m'ont soutenu et aid´ede maintes et maintes fa¸cons,non seulement pendant ces trois ans mais surtout pendant toute ma vie. Last, but not least, `aCamille, qui me supporte, me soutient et m'encourage depuis le d´ebutde cette th`ese,je ne pourrais compter les raisons que j'ai de te remercier et j'esp`erepouvoir te le rendre pendant encore de longues ann´ees. 4 Des topos `ala g´eom´etrienon commutative par l'´etudedes espaces de Hilbert internes R´esum´e: Cette th`eseest consacr´ee`al'´etudede relations entre la g´eom´etrie non commutative et la th´eoriedes topos, comme deux mod`elesde topologie g´en´eralis´ee.L'outil principal que nous utilisons est l'´etude des champs continus d'espaces de Hilbert sur un topos, d´efinispar l'utilisation de la logique interne du topos. En consid´erant les alg`ebresd'op´erateursborn´essur de tels champs on obtient des C∗-alg`ebresassoci´eesau topos. Dans le chapitre 1 nous ´etudionscette relation par l'interm´ediairedes quantales et dans le cas des topos atomiques. Dans ce cas, la relation avec les alg`ebres d'op´erateurspeut-^etred´ecriteexplicitement et cela procure un mod`elesimple des ph´enom`enesqui apparaissent. Le chapitre 2 d´efinitune notion de th´eorie de la mesure sur les topos et la relie `ala th´eoriedes W ∗-alg`ebres,c'est `adire `ala th´eoriede la mesure non commutative. Inspir´espar les r´esultatsdu chapitre 1 nous d´efinissons une notion de mesure invariante qui apparait comme analogue `ala notion de trace. La classification de ces mesures fait apparaitre un R>0-fibr´eprincipal canonique sur tout topos bool´eenint´egrablelocalement s´epar´e, qui est l'analogue de l'´evolution temporelle des W ∗-alg`ebres(ceci est pr´ecis´e`ala fin du chapitre 2). Dans le chapitre 3, nous d´efinissons et ´etudions les notions d'espaces m´etriques et d'espaces de Banach "localiques". Notre motivation est de pouvoir g´en´eraliser les techniques que nous utilisons pour les topos `ades groupoides topologiques ou localiques, ainsi que d'obtenir une extension de la dualit´eede Gelfand construc- tive conjectur´eepar C.J.Mulvey et B.Banachewski. Nous prouvons aussi que dans un topos satisfaisant une certaine condition g´en´eralisant la paracompa- cit´e,la notion d'espace de Banach localique est ´equivalente `ala notion usuelle d'espace de Banach. 5 From toposes to non-commutative geometry through the study of internal Hilbert spaces Abstract: The goal of this thesis is to study some relations between non- commutative geometry and topos theory, both being seen as generalizations of topological spaces. The main tool we are using is the study of continuous bundles of Hilbert spaces over a topos which are defined as Hilbert spaces in the internal logic of the topos. By looking at the algebras of bounded operators over such Hilbert spaces one can associate C∗-algebras to a topos. In chapter 1 we study this relation through the use of quantales, and in the case of atomic toposes. For such toposes the relation with operator algebras can be described explicitely, and this provides an interesting toy-model for the case of more general toposes. In chapter 2 we focus on measure theoretic aspects. We define a notion of generalized measure class over a topos, and this notion appears to be closely related to the theory of W ∗-algebras. Inspired by the results of chapter 1 we define a notion of invariant measure, which appears to be analogous to the notion of trace on a W ∗-algebra. The classification of such measures gives rise to a canonical R>0-principal bundle on every integrable locally separated boolean topos, which is the analogue of the modular time evolution of W ∗-algebras (this analogy is made precise at the end of the chapter). In chapter 3, we define and study a notion of localic metric spaces and localic Banach spaces. The motivations for such notions are that they allow one to generalize the techniques used on toposes in this thesis to topological and lo- calic groupoids, and that they allows to obtain an extension of the constructive Gelfand duality as conjectured by C.J.Mulvey and B.Banachewski. We also prove that over a topos satisfying a certain technical condition analogous to paracompactness, the notion of localic Banach space is equivalent to the usual notion of Banach space. 6 Contents Introduction (fran¸cais) 9 Introduction (english) 24 Notations, conventions and preliminaries 37 1 Toposes, quantales and C∗-algebras, the atomic case. 40 1 Introduction . 40 2 Toposes, quantales and sup-lattices . 42 2.1 Introduction . 42 2.2 The category sl(T ) of sup-lattices . 44 2.3 Categories enriched over sl .................
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