The main topic of this thesis is the stable homotopy theory of dendroidal sets. This topic
belongs to the area of mathematics called algebraic topology. Algebraic topology studies
the interaction between the algebraic and topological structures.
Examples of topological spaces with a very rich algebraic structure are (iterated) loop
spaces. Loop spaces carry an algebraic structure which is called an A∞-structure, while
infinite loop spaces carry an E∞-structure. These structures consist of an infinite sequence
of operations that satisfy various coherence laws. As it is difficult to grasp all these data,
one usually uses topological operads to efficiently describe this information. One can think
of operads as carrying “blueprints” for the algebraic structure which is realized in every
space with that structure. The characterization results for (iterated) loop spaces using
topological operads have been established in the early 1970’s by the work of P. May, M.
Boardman and R. Vogt. In the 1990’s it became evident that it is important to understand
the homotopy theory operads.
The theory of dendroidal sets provides a new context for studying operads up to homotopy. Dendroidal sets were introduced in 2007 by I. Moerdijk and I. Weiss. Subsequent
work of I. Moerdijk and D.-C. Cisinski shows that dendroidal sets indeed model topological/simplicial operads. An important advantage of dendroidal sets is that the theory is
built in a natural way as a generalization of the theory of simplicial sets. The study of
dendroidal sets is very combinatorial in its nature since it is based on the notion of trees
(graphs with no loops). Also, as a category of presheaves, the category of dendroidal sets
has nice categorical properties.
Simplicial sets provide combinatorial models for spaces (think of it in terms of triangulations of spaces given by simplicial approximations) and dendroidal sets provide combinatorial models for infinite loops spaces as spaces together with complicated algebraic
structure. In fact, the precise formulation of this idea is one of the main topics of this
thesis.
A precise formulation of our results is given in the language of Quillen’s model categories. Model categories provide a formalism to study and compare homotopy theories in
various contexts (topological spaces, chain complexes, simplicial sets, operads etc.) One
of the main results of this thesis is that the category of dendroidal sets admits a model
structure such that the underlying homotopy theory is equivalent to the homotopy theory
of infinite loop spaces (equivalently, of grouplike E∞-algebras or connective spectra). We
call this model structure the stable model structure on dendroidal sets.
Constructing a model structure is a tedious job. In our case it requires a great deal
of technical combinatorial results about dendroidal sets (i.e. about trees). In order to
simplify our arguments, in Chapter 4 we develop a combinatorial technique for proving
results about dendroidal anodyne extensions. This technique can be viewed as a result
in its own right as one might apply it also in different ways than it is used in the later
chapters of the thesis.
We give two constructions of the stable model model structure. The first construction is
more elementary and has an advantage of providing a characterization of fibrations between
fibrant objects. This construction is based on standard model-theoretical arguments and
it is given in Chapter 5.
The second construction, given in Chapter 6, is based on the work of G. Heuts. This
approach makes it possible to show that the stable model structure on dendroidal sets
is Quillen equivalent to a model structure on E∞-spaces with grouplike E∞-spaces as
fibrant objects. The equivalence to grouplike E∞-objects (i.e. connective spectra) might
be considered as a solution to the problem of geometric realization of dendroidal sets.
Also, these results open new possibilities to investigate the connective part of classical
stable homotopy theory.
The results of the thesis presented in Chapter 7 go in that direction. In that final
chapter we discuss homology groups of dendroidal sets. This homology theory generalizes
the well-known homology theory of simplicial sets (i.e. the singular homology of spaces).
The generalization is not straightforward because we work with non-planar trees, but we
want to use a certain sign-convention for planar trees. After giving the definition, we
establish that these homology groups are homotopy invariant and that they compute the
standard homology of the corresponding connective spectrum. The results of Chapters 6
and 7 are joint work with T. Nikolaus.
Author(s): Matija Basic
Series: PhD thesis at Radboud Universiteit Nijmegen
Year: 2015
Language: English
Commentary: Downloaded from https://web.math.pmf.unizg.hr/~mbasic/thesis.pdf
City: Nijmegen
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
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
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 A∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.7 Acyclicity argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Bibliography 159
Summary 163
Samenvatting 165
Curriculum vitae 167
Acknowledgments 169
