Stable homotopy theory of dendroidal sets Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. dr. Th. L. M. Engelen, volgens besluit van het college van decanen in het openbaar te verdedigen op donderdag 23 april 2015 om 12.30 uur precies door Matija Baˇsi´c geboren op 21 augustus 1985 te Zagreb, Kroati¨e Promotor: prof. dr. I. Moerdijk Manuscriptcommissie: prof. dr. N.P. Landsman (vz) dr. C. Berger (Universit´ede Nice-Sophia Antipolis, Frankrijk) dr. J.J. Guti´errezMarin prof. dr. K. Hess (Ecole´ Polytechnique F´ed´eralede Lausanne, Zwitserland) dr. U. Schreiber (Eduard Cechˇ Institute, Tsjechische Republiek) ISBN { 978-94-6259-634-4 Gedrukt door: Ipskamp Drukkers, Enschede Contents Contents i 1 Introduction 1 1.1 Algebraic structures in topology . .1 1.1.1 Homotopy invariants . .1 1.1.2 Homotopy invariant algebraic structures . .2 1.1.3 Loop spaces and infinite loop spaces . .3 1.1.4 Operads . .5 1.2 Homotopy theory . .8 1.2.1 Homotopy theory in various categories . .8 1.2.2 Axiomatic homotopy theory . 10 1.2.3 A categorical definition of simplicial sets . 11 1.2.4 Homotopy theory of simplicial sets . 12 1.2.5 Higher categorical point of view . 14 1.2.6 Simplicially enriched categories . 15 1.3 Dendroidal sets . 17 1.3.1 The category Ω of trees . 17 1.3.2 Dendroidal sets and operads . 19 1.3.3 The operadic model structure . 20 1.3.4 Homotopy theory of homotopy coherent algebras . 21 1.4 Main results and the organization of the thesis . 23 1.4.1 The dendroidal group{like completion . 23 1.4.2 An elementary construction of the stable model structure . 25 1.4.3 Homology of dendroidal sets . 26 1.4.4 Overview of the chapters . 28 2 Background on axiomatic homotopy theory 29 2.1 Categorical preliminaries and the definition of a model category . 29 2.1.1 Conventions about sets and categories . 29 2.1.2 Localization of categories . 32 2.1.3 Factorization and lifting properties . 33 i 2.1.4 Quillen model structures . 34 2.2 Examples of homotopy theories . 36 2.2.1 Small categories . 36 2.2.2 Chain complexes . 36 2.2.3 Topological spaces . 37 2.2.4 Simplicial sets . 38 2.2.5 Operads . 41 2.3 Further properties of model categories . 46 2.3.1 Left and right homotopy . 46 2.3.2 The construction of the homotopy category . 47 2.3.3 Detecting weak equivalences . 49 2.3.4 Cofibrantly generated model categories . 52 2.3.5 Quillen functors . 53 2.3.6 Model structures on categories of diagrams . 55 2.3.7 Simplicial model categories and function complexes . 57 2.3.8 Left Bousfield localization . 61 3 Dendroidal sets 63 3.1 The formalism of trees . 63 3.1.1 Definition of a tree . 63 3.1.2 Operads associated with trees and the category Ω . 64 3.1.3 Elementary face and degeneracy maps . 66 3.2 Dendroidal sets . 68 3.2.1 Relating dendroidal sets to simplicial sets and operads . 68 3.2.2 A tensor product of dendroidal sets . 69 3.2.3 Normal monomorphisms and normalizations . 74 3.2.4 Dendroidal Kan fibrations . 76 3.2.5 Homotopy theories of dendroidal sets . 79 4 Combinatorics of dendroidal anodyne extensions 81 4.1 Elementary face maps . 82 4.1.1 Planar structures and a total order of face maps . 82 4.1.2 Combinatorial aspects of elementary face maps . 83 4.2 The method of canonical extensions . 85 4.3 The pushout-product property . 90 5 Construction of the stable model structure 101 5.1 Homotopy of dendroidal sets . 101 5.2 Stable weak equivalences . 103 5.3 Stable trivial cofibrations . 105 5.4 The stable model structure . 110 ii 5.5 Quillen functors from the stable model structure . 114 6 Dendroidal sets as models for connective spectra 117 6.1 Fully Kan dendroidal sets and Picard groupoids . 118 6.2 The stable model structure . 121 6.3 Equivalence to connective spectra . 123 6.4 Proof of Theorem 6.2.3, part I . 129 6.5 Proof of Theorem 6.2.3, part II . 136 6.6 Proof of Theorem 6.2.3, part III . 138 7 Homology of dendroidal sets 145 7.1 The sign conventions . 145 7.2 The unnormalized chain complex . 148 7.3 The normalized chain complex . 151 7.4 The equivalence of the unnormalized and the normalized chain complex . 152 7.5 The associated spectrum and its homology . 154 7.6 The homology of A1 .............................. 156 7.7 Acyclicity argument . 157 Bibliography 159 Summary 163 Samenvatting 165 Curriculum vitae 167 Acknowledgments 169 iii Chapter 1 Introduction The focus of this thesis is on dendroidal sets and their stable homotopy theory. In this introductory chapter we would like to motivate the reader for this topic and present our main results. We will first give a short overview of the basic principles of algebraic topology and, in particular, explain what is understood under the term homotopy theory. Next, we wish to present the theory of dendroidal sets as a generalization of the theory of simplicial sets. Therefore, we will shortly review certain aspects of simplicial sets (that are probably familiar to the reader) with an aim of making the passage from simplicial sets to dendroidal sets natural. We will review the main results from the theory of dendroidal sets that have been known before writing this thesis and on which our work is based. In the final part of this chapter we will present our new results and explain how the rest of the thesis is organized. 1.1 Algebraic structures in topology 1.1.1 Homotopy invariants The subject of this thesis falls under the area of mathematics called algebraic topology. One of the main objectives of topology is to classify all topological spaces up to various equivalence relations. Two spaces are homeomorphic (or topologically equivalent) if there is a continuous bijection between them such that the inverse is also a continuous map. Although classifying spaces up to a homeomorphism is the most important question, we often consider coarser equivalence relations. For two continuous maps f; g : X ! Y we say that f is homotopic to g if there exists a continuous map H : X × [0; 1] ! Y such that H(x; 0) = f(x) and H(x; 1) = g(x) for all x 2 X. A map f : X ! Y is called a homotopy equivalence if there is a map g : Y ! X such that fg is homotopic to the identity 1Y and gf is homotopic to the identity 1X . In such cases, we say that g is a homotopy inverse of f and that the spaces X and Y are homotopy equivalent (or of the same homotopy type). With this definition at hand we 1 2 Chapter 1. Introduction may consider classification of all topological spaces up to homotopy equivalence. Obviously, every homeomorphism is a homotopy equivalence, but not vice versa. To show that two spaces are not of the same type we consider invariants shared by all spaces of the same type. An invariant might be a certain property of a space (e.g. compactness or connectedness) or a mathematical object assigned to it (e.g. the number of connected components, the Euler characteristic, the Betti numbers, the cohomology ring, the homotopy groups etc.). Typically, to each topological space we assign an object with a certain algebraic structure and to each continuous map we assign a morphism respecting that algebraic structure. For example, to a topological space X we might assign a group π(X) and to a continuous map f : X ! Y a group homomorphism π(f): π(X) ! π(Y ). The most important invariants in algebraic topology are functorial. This means that they are respecting the composition of maps, i.e. π(fg) = π(f)π(g). We say that π : Top ! Grp is a functor from the category Top of topological spaces and continuous maps to the category Grp of groups and group homomorphisms. If π : Top !C is a functor that sends homotopic maps to the same morphism in a category C, then π sends homotopy equivalences to isomorphisms. We say that π is a homotopy invariant. Now, consider the homotopy category Ho(Top). The objects of Ho(Top) are topological spaces and morphisms are homotopy equivalence classes of continuous maps. There is a functor γ : Top ! Ho(Top) sending each continuous map to its equivalence class. The pair (Ho(Top); γ) has the following universal property. Every homotopy invariant π : Top !C induces a functor π : Ho(Top) !C such that π = γπ. This illustrates that by studying homotopy invariants we study the homotopy category and functors defined on it. Category theory provides an efficient way to compare various invariants and to study various properties as we are passing from a topological to an algebraic context. The language of categories will be used throughout this thesis in an essential way. 1.1.2 Homotopy invariant algebraic structures Algebraic topology also studies topological spaces with an additional algebraic structure and the way the algebraic and the topological structures interact. Here is a basic example. Let X be a topological monoid and Y a space that is homotopy equivalent to X. One can transfer the multiplicative structure from X to Y via the homotopy equivalence. The transferred structure will not satisfy the strictly associative law (y1y2)y3 = y1(y2y3) for all y1; y2; y3 2 Y , but the maps F0;F1 : Y × Y × Y ! Y given by F0(y1; y2; y3) = (y1y2)y3 and F1(y1; y2; y3) = y1(y2y3) will be homotopic.