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Profinite Homotopy Theory

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Profinite Homotopy Theory Master's Thesis Profinite Homotopy Theory Supervisor: Author: Prof. Dr. Ieke Moerdijk Thomas Blom Second reader: Dr. Lennart Meier January 13, 2019 Acknowledgements First and foremost I would like to thank Prof. Dr. Ieke Moerdijk for his supervision and for providing me with many ideas and views on the subject that I probably never would have come up with myself. Secondly, I want to express my gratitude to Dr. Lennart Meier for the discussions we had about my thesis and for being my second reader. I would also like to thank Dr. Geoffroy Horel for taking the time to discuss his work with me when he was in Utrecht. Furthermore, I want to thank Bjarne Kosmeijer and Fiona Middleton for proofreading my thesis and providing me with a lot of helpful commentary. Lastly, I want to mention that I greatly appreciated all the support from my friends and family during the writing of this thesis. ii Contents 1 Introduction 1 1.1 Overview . .3 1.2 Preliminaries . .3 2 Pro-categories 4 2.1 The category of pro-objects . .4 2.1.1 Pro-objects . .5 2.1.2 Reindexing pro-objects and level representations . .9 2.1.3 Cofiltered limits in pro-categories . 14 2.1.4 Pro-categories of categories with finite limits . 18 2.2 Profinite sets and Stone spaces . 21 2.2.1 Cofiltered limits of finite sets . 21 2.2.2 Profinite sets and profinite groups . 22 2.2.3 Profinite G-sets and profinite G-modules . 25 2.3 Profinite completion . 29 2.3.1 General construction of the profinite completion functor . 29 2.3.2 Profinite completion of sets and groups . 31 2.4 Pro-objects in presheaf categories . 33 2.4.1 Diagrams in pro-categories versus pro-objects in diagram categories 34 2.4.2 When the index category has finite Hom-sets . 36 3 Profinite groupoids 40 3.1 Some basic facts on (profinite) groupoids . 41 3.1.1 Basic constructions with groupoids . 41 3.1.2 The nerve of a groupoid . 43 3.1.3 Profinite groupoids . 44 3.2 Profinite groupoids as topological groupoids . 46 3.2.1 Characterizing profinite groupoids among Stone groupoids . 49 3.3 Profinite completion . 51 3.4 Weak equivalences of profinite groupoids . 53 3.4.1 The definition of a weak equivalence . 54 3.4.2 Basic properties of weak equivalences . 55 3.4.3 Weak equivalences are fully faithful and essentially surjective . 58 3.4.4 Level representations of weak equivalences . 60 3.4.5 Comparing weak equivalences to homotopy equivalences . 63 iii 3.5 A fibrantly generated model structure on Gb ................. 65 4 Profinite spaces 71 4.1 Some basic facts on simplicial (profinite) sets . 72 4.1.1 Skeletal and coskeletal simplicial sets . 72 4.1.2 Simplicial homotopy . 74 4.1.3 Profinite completion . 75 4.2 Connectedness and the fundamental groupoid . 76 4.2.1 Connected profinite spaces . 76 4.2.2 The fundamental groupoid of a profinite space . 78 4.3 Principal G-bundles and H1(X; G)...................... 80 4.3.1 BG classifies profinite G-bundles . 81 4.4 Cohomology with local coefficients . 86 4.4.1 Profinite local coefficient systems . 87 4.4.2 Homotopy invariance . 89 4.4.3 Representing cohomology with local coefficients . 92 4.5 A fibrantly generated model structure on Sb .................. 99 4.5.1 Weak equivalences . 100 4.5.2 The model structure . 104 4.5.3 Relation to other model categories . 108 4.6 Coverings of profinite spaces . 109 4.6.1 Finite coverings . 109 4.6.2 Verification of the axioms of a Galois category . 112 4.6.3 Profinite coverings . 117 4.6.4 Connected coverings and closed subgroups of π1 ........... 122 4.6.5 An alternative construction of the fundamental groupoid . 125 4.6.6 Comparing the constructions of the fundamental groupoid . 127 Appendices A Fibrantly generated model categories 131 iv Chapter 1 Introduction Profinite homotopy theory originated from the desire to apply methods of algebraic topol- ogy to algebraic geometry, in particular to assign invariants from algebraic topology to schemes. It was believed that by using such invariants, in particular a good cohomol- ogy theory for schemes, the Weil conjectures could be proved. In SGA1 ([Gro+71]), Grothendieck defined the ´etalefundamental group of a connected and locally Noetherian scheme using finite ´etalecoverings. Etale´ cohomology was developed by Grothendieck and M. Artin, and was published in [Art62] and SGA4 ([GAV72]). Michael Artin and Barry Mazur refined the theory of ´etalecohomology by associating a \pro-homotopy type" to a connected and locally Noetherian scheme in their book Etale homotopy [AM69], called the ´etalehomotopy type of the scheme. A pro-homotopy type is a pro-object in the homotopy category of simplicial sets Ho(S). A pro-object in a category is defined as a functor from a cofiltered category to this category. There exists a good notion of morphism between pro-objects, and they in particular form a category. For these pro-homotopy types, there exists a good notion of the fundamental group and of cohomology, and these turn out to be equal to the ´etalefundamental group (after applying a certain profinite completion) and ´etalecohomology of the scheme, respectively. Pro-homotopy types share many properties with the usual homotopy types of spaces, and allow for the application of methods from algebraic topology to algebraic geometry. The study of the ´etalefundamental group, ´etalecohomology and these ´etalehomotopy types has found many applications. An important application of ´etalecohomology is the proof of the last Weil conjecture by Deligne [Del74]. Remarkable applications of ´etale homotopy theory can be found in the proofs of the Adams conjecture by Friedlander [Fri73] and Sullivan [Sul74]. The motivation for this thesis came from a recent application of `etalehomotopy the- ory, namely the proof by Geoffroy Horel that the group of homotopy automorphisms of the profinite completion of the little 2-discs operad is isomorphic to the profinite Grothendieck-Teichm¨ullergroup [Hor17]. The goal of this thesis is to gain a better un- derstanding of certain parts of this proof. We will provide a brief sketch of Horel's proof, to give some context and motivation for the work done in this thesis. Those readers who are not familiar with operads or the Grothendieck-Teichm¨ullergroup should not worry. Although Horel's article [Hor17] is about profinite completion of operads and the 1 2 CHAPTER 1. INTRODUCTION Grothendieck-Teichm¨ullergroup, both operads and the Grothendieck-Teichm¨ullergroup do not appear anywhere in this thesis, except in the following sketch of Horel's proof. In [Hor17], Horel first defines the profinite completion of an operad over simplicial sets, and of an operad over groupoids. These profinite completions land in the category of so-called \weak operads" over “profinite spaces" and “profinite groupoids", respectively. For both of these categories of \weak operads", there exists a model structure in a natural way. In particular, it is possible to talk about the group of homotopy automorphisms of these (weak) operads. The profinite completion functor for groupoids over simplicial sets is applied to the little 2-discs operad. The little 2-discs operad is weakly equivalent to the operad B(PaB), where PaB is the operad of \parenthesized braids", an operad over groupoids. Here B is the functor which associates to a groupoid A its nerve BA 2 S, applied levelwise to the operad PaB. There is a profinite completion functor both for operads over simplicial sets and for operads over groupoids. It is shown that for the operad PaB, the objects B(P[aB) and B\(PaB) are weakly equivalent, where (c·) denotes the profinite completion functor. In particular, to study the homotopy automorphisms of the profinite completion of B(PaB), one can also first apply the profinite completion functor (for operads over groupoids) to PaB, and then apply the nerve functor B. The homotopy automorphism group of B(P[aB) is equal to that of P[aB. The homotopy automorphism group of the latter operad can be computed, and turns out to be isomorphic to the profinite Grothendieck-Teichm¨ullergroup. In [AM69], Artin and Mazur construct a profinite completion functor, which associates a pro-object in Ho(S)fin to a pro-object in Ho(S). Here Ho(S)fin denotes the full sub- category of Ho(S) of simplicial sets whose homotopy groups are finite. Such a profinite completion functor is often used in applications of ´etalehomotopy theory. However, it is usually desirable to work in a model category when doing homotopy theory, instead of the homotopy category itself, as homotopy categories often behave badly. Working in the pro-category of a homotopy category is therefore undesirable. In Geoffroy Horel's proof, to define the profinite completion of an operad, it is necessary to have a profinite comple- tion functor on the level of simplicial sets. More precisely, a model category is needed in which one can study “profinite spaces", meaning that its homotopy category should, in a certain sense, be the category of profinite homotopy types. Furthermore, there should be a profinite completion functor, which should be a functor from simplicial sets to this model category. Such a model category, together with a profinite completion functor, is constructed in [Qui08]. To define the profinite completion of an operad over groupoids, a good profinite completion functor for groupoids is needed, which maps to a model cat- egory whose objects are “profinite groupoids". Such a model category is constructed by Horel himself in [Hor17, x4]. The aim of this thesis is to understand these “profinite spaces" and “profinite groupoids" and the model categories in which they live, as defined by Quick in [Qui08] and by Horel in [Hor17, x4].
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