Higher Groupoid Actions, Bibundles, and Differentiation Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades „Doctor rerum naturalium“ der Georg-August-Universität Göttingen im Promotionsprogramm PhD School of Mathematical Sciences (SMS) der Georg-August University School of Science (GAUSS) vorgelegt von arXiv:1512.04209v1 [math.DG] 14 Dec 2015 Du Li aus Hunan, China Göttingen, 2014 Betreuungsausschuss Prof. Dr. Chenchang Zhu (Mathematisches Institut) Prof. Dr. Ralf Meyer (Mathematisches Institut) Mitglieder der Prüfungskommission Referentin: Prof. Dr. Chenchang Zhu Korreferent: Prof. Dr. Ralf Meyer Weitere Mitglieder der Prüfungskommission: Dr. Christian Blohmann (Max-Planck-Institut für Mathematik) Prof. Dr. Karl-Henning Rehren (Institut für Theoretische Physik) Prof. Dr. Thomas Schick (Mathematisches Institut) Prof. Dr. Max Wardetzky (Institut für Num. und Angew. Mathematik) Tag der mündlichen Prüfung: 15 Juli 2014 Abstract In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids; we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids; we use inner Kan fibrations over the interval to model bibundles of higher groupoids. In particular, we show that our definitions given by the simplicial method agree with those given by the categorification approach to actions, principal bundles, and bibundles of 2-groupoids. In addition, we use the simplicial technique to prove a theorem on differentiation of higher Lie groupoids, which shows that the differentiation functor sends a higher Lie groupoid to a higher Lie algebroid. iii Acknowledgments I would first like to express my sincere gratitude to my advisor Chen- chang Zhu for her encouragement, guidance, and inspiration during my graduate studies. I am also greatly indebted to Ralf Meyer for his advice and assistance in writing this thesis. My special thanks go to my M.S. supervisor at Peking University, Liu Zhangju, for supporting me in pursuing a Ph.D. career in Göttingen. I am very thankful to the Courant Research Centre “Higher Order Structures in Mathematics” and Mathematisches Institut at Georg-August- Universität Göttingen, which supported my study during the last four years. I am glad to acknowledge the following persons for helpful discussions, suggestions, and comments: Christian Blohmann, Liu Bei, Weiwei Pan, Christopher L. Rogers,Zheng Jiguang. I am indebted to the following persons for useful discussions via email, at Mathematics Stack Exchange, or at MathOverflow: David Carchedi, David Roberts, Chris Schommer-Pries, and Laura Scull about groupoids; Zhen Lin Low and Michał R. Przybyªek about extensive categories. My appreciation also go to Suliman Albandik, Martin Callies, Malte Dehling, Rohit Holkar, George Nadareishvili for their willingness to share their mathematical knowledge with me. I am grateful to Tathagata Banerjee, Qi Zhi, Wu Xiaolei and Xiao Guohui for their linguistic help and other suggestions. I thank Nguyen Thi Thu Huong and Sutanu Roy for sharing their LATEX templates with me. I wish to thank Nan Xi for the continued encouragement. I also thank all of my friends for their help during my stay in Göttingen. Finally, I would like to thank my parents and sisters for their love, understanding, and support. Without them, I would never have been able to continue my education. This dissertation is dedicated to the memory of my dear maternal grandmother. v Contents Abstract iii Acknowledgments v 0 Introduction 1 0.1 Simplicial sets and categories . .1 0.2 2-Groupoids . .2 0.3 Actions of 2-groupoids . .2 0.4 Bibundles of 2-groupoids . .3 0.5 Differentiation of higher Lie groupoids . .3 0.6 Structure of the thesis . .4 0.7 Categorical conventions . .5 1 Lie Groupoids 7 1.1 Definition and first examples . .7 1.1.1 Examples . .9 1.2 Constructions with Lie groupoids . 10 1.2.1 Induced Lie groupoids . 10 1.2.2 Strong pullbacks . 11 1.2.3 Weak pullbacks . 11 1.3 Weak equivalences and generalised morphisms . 12 1.3.1 Weak equivalences . 12 1.3.2 Generalised morphisms . 15 1.3.3 Anafunctors . 17 1.4 Groupoid actions and HS bibundles . 17 1.4.1 Lie groupoid actions . 18 1.4.2 Principal bundles . 19 1.4.3 HS bibundles . 20 1.4.4 2-Pullbacks in the 2-category of HS bibundles . 22 1.5 Comparison of 2-categories of Lie groupoids . 23 2 Simplicial Sets 25 2.1 Preliminaries on presheaves . 25 2.2 Simplicial sets . 26 2.3 Lifting properties . 28 2.3.1 Arrow category . 29 vii Contents 2.3.2 Technical lemmas . 29 2.3.3 Proof of Lemma 2.19 . 31 2.4 Lifting properties of simplicial sets . 32 2.5 Skeleton and coskeleton functors . 34 2.6 Collapsible extensions . 36 2.6.1 Join of simplicial sets . 36 3 Higher Groupoid Actions 39 3.1 Singleton Grothendieck pretopology . 39 3.1.1 Internal groupoids . 40 3.2 The Hom functor . 41 3.3 Higher groupoids in a category with pretopology . 42 3.4 Higher groupoid Kan fibrations . 44 3.4.1 Composition of Kan fibrations . 46 3.4.2 The fibre of a Kan fibration . 46 3.4.3 Pullbacks of Kan fibrations . 47 3.5 Higher groupoid actions . 48 3.6 Higher principal bundles . 50 3.6.1 Acyclic fibrations . 50 3.6.2 Higher principal bundles . 52 4 Higher Groupoid Bibundles 55 4.1 Bimodules between categories and cographs . 55 4.1.1 Bimodules and profunctors . 55 4.1.2 Cographs of bimodules . 56 4.1.3 Bimodules and categories over the interval . 56 4.2 Bibundles of Lie groupoids via simplicial manifolds . 57 4.2.1 Action groupoids in the cogragh . 58 4.3 Augmented simplicial sets and bisimplicial sets . 59 4.3.1 Augmented simplicial sets . 59 4.3.2 Augmented bisimplicial sets . 60 4.4 Augmented bisimplicial objects . 62 4.4.1 Extensive categories . 62 4.4.2 Augmented bisimplicial objects . 63 4.4.3 Colored Kan conditions . 64 4.5 Cographs of higher groupoid morphisms . 66 4.6 Bibundles of higher groupoids . 68 5 Actions of 2-Groupoids 71 5.1 2-groupoids . 71 5.1.1 Categorified groupoids . 71 5.1.2 A one-to-one correspondence . 73 5.1.3 Geometric nerve of a 2-category . 74 5.2 Actions of Lie 2-groupoids via categorification . 74 5.3 From the simplicial picture to categorification . 76 5.3.1 The action morphism . 76 viii Contents 5.3.2 The associator . 77 5.3.3 The unitor . 78 5.4 Finite data of Kan fibrations . 78 5.4.1 Reconstruction of higher dimensions . 79 5.5 From categorification to the simplicial picture . 81 5.5.1 Dimensions 0 and 1 . 81 5.5.2 Dimension 2 . 82 5.6 Principal 2-bundles . 84 6 Bibundles of 2-Groupoids 87 6.1 Categorified bibundles between 2-groupoids . 87 6.2 From categorification to simplicial picture . 89 6.2.1 The dimensions 0, 1, 2 . 89 6.2.2 Dimension 3 . 92 6.2.3 Dimension 4 . 95 6.2.4 Higher dimensions . 97 6.3 From simplicial to categorification picture . 97 6.3.1 The groupoid of bigons . 97 6.3.2 The action morphisms . 99 6.3.3 The associators . 100 6.3.4 The unitors . 102 6.3.5 Principality . 102 6.4 Weak equivalences of 2-groupoids . 102 6.4.1 Weak acyclic fibrations . 103 6.5 Examples . 106 7 Composition of 2-Groupoid Bibundles 109 7.1 Preparations on 2-categories . 109 7.1.1 Remarks on the construction and 2-coends . 109 7.1.2 The pasting theorem of 2-categories . 110 7.1.3 Fundamental groupoids of 2-groupoids . 111 7.2 The composition of 2-groupoid bibundles . 112 7.2.1 Dimension 1 . 113 7.2.2 Dimension 2 . 113 7.2.3 Dimension 3 . 115 7.2.4 Dimension 4 . 119 7.3 Functoriality of the bundlisation . 120 7.4 Unitality of the composition . 123 7.5 Associativity of the composition . 125 8 Differentiation of Higher Lie Groupoids 129 8.1 Basics on supermanifolds . 129 8.1.1 Basic definitions . 129 8.1.2 Functor of points . 131 8.1.3 Symbolic notation . 131 8.1.4 Tangent vectors . ..