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Fundamental Bigroupoids and 2-Covering Spaces Fundamental bigroupoids and 2-covering spaces David Michael Roberts Thesis submitted for the degree of Doctor of Philosophy in the School of Mathematical Sciences University of Adelaide December 2009 Contents Abstract v Statement of originality vii Acknowledgements ix Introduction 1 Chapter 1. Internal categories and anafunctors 11 1. Internal categories and groupoids 11 2. Sites and covers 15 3. Weak equivalences 19 4. Anafunctors 22 5. Localising bicategories at a class of 1-cells 28 6. Anafunctors are a localisation 31 Chapter 2. A fundamental bigroupoid for topological groupoids 41 1. Preliminaries on topological groupoids and n-partitions 42 2. Paths and surfaces in topological groupoids 51 3. Thin homotopies and structure morphisms 58 4. 2-tracks 64 5. Compositions and concatenations 71 6. The fundamental bigroupoid 76 7. Calculations and comparisons 96 Chapter 3. Interlude: Pointed topological anafunctors 125 Chapter 4. 2-covering spaces I: Basic theory 135 1. Review of covering spaces 135 2. Locally trivial and weakly discrete groupoids 140 3. 2-covering spaces 147 4. Lifting of paths and surfaces 153 5. 2-covering spaces and bundle gerbes 168 Chapter 5. 2-covering spaces II: Examples 173 1. Some topology on mapping spaces 173 2. The topological fundamental bigroupoid of a space 182 3. The canonical 2-connected cover 196 4. A 2-connected 2-covering space 199 5. Vertical fundamental groupoid 213 iii Appendix A. Bicategories, bigroupoids and 2-groups 217 1. Bicategories 217 2. 2-groups 222 Bibliography 225 iv Abstract This thesis introduces two main concepts: a fundamental bigroupoid of a topolog- ical groupoid and 2-covering spaces, a categorification of covering spaces. The first is applied to the second to show, among other things, that the fundamental 2-group of the 2-covering space is a sub-2-group of the fundamental 2-group of the base. Along the way we derive general results about localisations of the 2-categories of categories and groupoids internal to a site at classes of weak equivalences, construct a topolog- ical fundamental bigroupoid of locally well-behaved spaces, and finish by providing a rich source of examples of 2-covering spaces, including a functorial 2-connected 2-covering space. v vi Statement of originality This thesis contains no material which has been accepted for the award of any other degree or diploma at any other university or other tertiary institution to me and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying, subject to the provisions of the Copyright Act 1968. I also give permission for the digital verions of my thesis to be made available on the web, via the University’s digital research repository, the Library catalogue, the Australasian Digital Theses Program (ADTP) and also through web search engines. David Michael Roberts Adelaide, South Australia December 2009 vii Acknowledgements I must first thank my principal supervisor, Michael Murray, for his patience and for trusting me to pretty much direct my own research. I also thank my supervisor Mathai Varghese for imparting to this one-time physicist a sense of thoroughness and mathematical rigor. His emphasis on the need for existence of nontrivial examples was a major motivation for this thesis. Mathai also helped organise a visit to the Erwin Schr¨odingerInstitute in Vienna which gave me the opportunity to meet my now coauthor Urs Schreiber. The visit to ESI was jointly funded by the Institue and a Cathy Candler Award from AFUW-SA. I must also thank Jim Stasheff for agreeing to join my supervisory panel late in my degree – it was a step into the unknown to ask him and a most pleasant surprise to hear a ‘yes’. Nick Buchdahl, while his time as a supervisor was only a few months at the start of my candidature, always helped me when I turned up with bewildered questions, for which I say merci. The postgraduate students in the School of Mathematical Sciences have formed a wonderful group of friends, and special thanks are due to David Butler and Jono Tuke for being there from the beginning. These two, together with Richard Green, Rongmin Lu, Ryan Mickler, Tyson Ritter and Raymond Vozzo, provided not only listeners to whom I could pour out my sometimes obscure theories, but welcome distractions and opportunities to learn things I would otherwise never have consid- ered. The hosts of the n-category caf´e have also provided a wonderful environment for the reading and discussion of many fascinating topics. I greatly appreciate help with proofreading and suggestions for improvements from Richard Green, Rongmin Lu, Michael Murray, Tyson Ritter, Jim Stasheff and Raymond Vozzo. During the first four and a half years of my candidature I was supported by an Australian Postgraduate Award, and I am also grateful for study leave from my employer, the National Centre for Vocational Education Research, during the final stages of the writing up of this thesis. Lastly, I would like to thank my mother, Shirley Woodhouse, for her constant patience and support throughout my entire time at university, and my fianc´ee,Dani- Marie Breen, for joining me on my journey, and filling it with many wonderful moments. ix Introduction The theory of covering spaces is an indispensible tool in the toolboxes of algebraic and differential topologists. Covering spaces were historically the first examples of fibre bundles and principal bundles. They also make an appearance as locally con- stant sheaves and, when the fibres have a group structure, are used in cohomology as systems of local coefficients. Geometric constructions requiring a vanishing fun- damental group can often be realised by passing to a simply-connected covering space. Whereas covering spaces are intrinsically linked with the homotopy 1-type of a space, there does not as yet exist an analogous topological structure for the 2-type. Our aim is to describe objects, dubbed 2-covering spaces, that fulfill this rˆole,and also provide some tools with which to study them. The fibres of covering spaces are sets, which are 0-types, so it is natural to use ‘bundles’, locally trivial after some fashion, where the fibres are 1-types. Just as covering spaces are not just bundles where the fibres consist of a disjoint union of contractible spaces (which are also 0-types), it is also justifiable to use the ‘most discrete’ representatives of 1-types. For this purpose we use the well-known equivalence groupoids ↔ 1-types so that the fibres of 2-covering spaces are groupoids. Before we can study 2-covering spaces in detail, a number of concepts and ideas from (algebraic) topology need to be reinvented; paths, homotopies, homo- topy groups and lifting of functions all have to be generalised to work in this new setting. ◦ A crucial idea here is categorification, a process of taking a category of inter- esting objects and replacing sets in the definition(s) of said objects by categories or groupoids. Because of the presence of natural transformations (i.e. 2-arrows in Cat), one can, and does, weaken equalities of objects and arrows to isomorphisms. These isomorphisms then need to satisfy axioms of their own. A historical example is the introduction of homology groups by E. Noether and Vietoris to replace the Betti and torsion numbers. An equality of Betti numbers became an isomorphism of free abelian groups and so on. In recent years there has been great interest in extending categorification from algebraic structures, such as groups [Sin75, JS86, BL04], vector spaces [KV94, 1 Bae97], tangles [BL03], monoidal categories [BN96] etc., to geometric and topolog- ical structures, such as homogeneous geometry [nLa] and bundles [Bar06, BS07], principal and otherwise. It should be said that before the categorification program was explicitly described in [BD95], one of the earliest examples were stacks, de- fined in the 1960s by Grothendieck and his school [Gir71, Gro71]. In that case, functors1 O(X)op → Set are replaced by weak 2-functors O(X)op → Gpd. In par- ticular, gerbes, which are a specialisation of stacks, have enjoyed a steady revival since the 1990s most likely due to the treatment of the differential geometry of abelian gerbes by Brylinski [Bry93] and applications to twisted K-theory starting with [BCM+02]. Great progress is being made of late in the study of topological stacks [Noo05], their homotopy type in the broad sense [Noo08b, Ebe09] and covering stacks (i.e. the analogues of covering spaces) [Noo08a]. However, there is no theory that deals with what one might think is a natural generalisation of covering spaces, where the fibre is bumped up a categorical dimension, not just the base and total ‘spaces’. One reason for this may be that when one works with a covering stack, it is locally (that is, pulled back to a stack that is a space) a covering space, which is a well-understood situation. Although recent interest in categorified bundles (called in various places 2- bundles [Bar06, BS07], 2-torsors [Bak07] and categorical torsors [CG01]) has been directed by category theoretic considerations or by interest in possible applica- tions to string theory [Sch05], not much attention has been paid to what one might call the phenomenology of 2-bundles. That is, there is a paucity of examples which can be used as test cases for constructing calculational techniques or hypothesis forming. ◦ The aim of this thesis is to begin to fill this gap with 2-covering spaces. The ‘total space’ of a 2-covering space is necessarily a topological groupoid, a groupoid where the collections of objects and arrows are not just sets but topological spaces and the various maps relating them (source, target, etc.) are all continuous [Ehr59].2 Groupoids in the usual sense are then topological groupoids equipped with the discrete topology on the sets of objects and arrows, and to distinguish these from topological groupoids these shall be called topologically discrete groupoids.
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