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Dendroidal Sets

Boomachtige Verzamelingen

(met een samenvatting in het Nederlands)

Proefschrift

ter verkrijging van de graad van doctor aan deUniversiteit Utrecht op gezag van de rector magnificus,

prof.dr. W.H. Gispen, ingevolge het besluit van hetcollege voor promoties in het openbaar te verdedigen

op dinsdag 18 september 2007 des middags te 2.30 uur

door

Ittay Weiss

geboren op 24 januari 1977 te Jeruzalem, Isral

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Promotor: Prof. I. Moerdijk

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Beoordelingscommissie: Prof. dr. E.D. FarjounProf. dr. K. Hess-BellwaldProf. dr. D. NotbohmProf. dr. J. Rosicky

ISBN 978-90-3934629-72000 Mathematics Subject Classification: 55P48, 55U10, 55U40

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It is a miracle that curiosity survives formal education

(Albert Einstein)

To Rahel

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Contents

Introduction 8Background 8Content and results 10

Preliminaries 120.1. Category theory 12

0.2. A formalism of trees 14

Chapter 1. Operads 191.1. Operads, functors, and natural transformations 191.2. Free operads and operads given by generators and relations 291.3. Limits and colimits in the category of operads 311.4. Yonedas lemma 331.5. Closed monoidal structure on the category of operads 351.6. Quillen model structure on the category of operads 381.7. Grothendieck construction for operads 431.8. Enriched operads 461.9. Comparison with the usual terminology 47

Chapter 2. Dendroidal sets 492.1. Motivation - simplicial sets and nerves of categories 492.2. An operadic definition of the dendroidal category 502.3. An algebraic definition of the dendroidal category 542.4. Dendroidal sets - basic definitions 632.5. Closed monoidal structure on the category of dendroidal sets 682.6. Skeletal filtration 73

Chapter 3. Operads and dendroidal sets 753.1. Nerves of operads 753.2. Inner Kan complexes 793.3. Anodyne extensions 823.4. Grafting in an inner Kan complex 833.5. Homotopy in an inner Kan complex 863.6. The exponential property 963.7. I nner Kan complex generated by a dendroidal set 99

Chapter 4. Enriched operads and dendroidal sets 1014.1. Case study: A-spaces 1014.2. The Boardman-Vogt W-construction 1034.3. The homotopy coherent nerve 1084.4. Algebras and the Grothendieck construction 112

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CONTENTS 7

4.5. Categories enriched in a dendroidal set 1154.6. Weak n-categories 1164.7. Quillen model structure on dSet 122

Bibliography 125

Index 127

Samenvatting 129

Acknowledgements 131

Curriculum vitae 132

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Introduction

In this thesis we introduce the new concept of dendroidal set which is an exten-sion of simplicial set. This notion is particularly useful in the study of operads andtheir algebras in the context of homotopy theory. We hope to convince the readerthat the theory presented below provides new tools to handle some of the difficultiesarising in the theory of up-to-homotopy algebras and supplies a uniform setting forthe weakening of algebraic structures in many contexts of abstract homotopy. The

thesis is based on, and expands [37, 38].

Background

An operad is a an algebraic gadget that can be used to describe sometimesvery involved algebraic structures on objects in various categories. The notionwas developed by May [36] in the theory of loop spaces. Indeed the complexityof the algebraic structure present on a loop space necessitates some machinery toeffectively handle that complexity, and operads do the job. The theory of operadsexperienced, in the mid 90s, the so called renaissance period [ 32], and consequentlythe importance of operad theory in many areas of mathematics became established.

Let us quickly explain in some more detail how operads are used in the contextof up-to-homotopy algebras. For simplicity let us only consider topological operads.

Loosely speaking, an up-to-homotopy algebraic structure is the structure present ona space Y that is weakly equivalent to a space X endowed with a certain algebraicstructure. So ifX is a topological monoid then Y will have the structure of an A-space. We say then that an A-space is the up-to-homotopy (or weak) version of atopological monoid. The way operads come into the picture is explained by the work[5] of Boardman and Vogt who construct for each operad P another operad WPsuch that WP-algebras correspond to weak P-algebras and the same constructioncan also be used to produce a notion of weak maps between WP-algebras.

It would appear that the problem of weak algebras is fully solved by operadsand by the Boardman-Vogt W construction. However, there is one difficulty thatarises, namely that the collection of weak algebras and their weak maps rarely formsa category. The reason is that the composition of weak maps, if at all defined, isin general not associative. Boardman and Vogt offer the following solution. Using

their W construction they produce for any operad P a simplicial set X in which X0is the set of weak P-algebras and X1 is the set of weak maps of weak P-algebras.An element ofX2 consists of three weak P-algebras A1, A2, A3 and three weak mapsf : A1 A2, g : A2 A3, and h : A1 A3 together with some extra structurethat can be thought of as exhibiting h as a possible composition of g with f.Similarly, Xn consists of chains of n weak maps and possible compositions of thesemaps, compositions of the compositions and so on. They show that this simplicialset satisfies what they call the restricted Kan condition. Joyal is studying such

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BACKGROUND 9

simplicial sets under the name quasi-categories, emphasizing that a quasi-categoryis a weakened notion of a category. Let us briefly outline some of the concepts ofquasi-categories.

Recall that a Kan complex is a simplicial set X such that every horn k[n] Xhas a filler [n] X. A simplicial set is a quasi-category if it is required that everyhorn k[n] X with 0 < k < n has a filler. Such horns are called inner horns.Recall also the nerve functor N : Cat sSet given by

N(C)n = HomCat([n], C).

It is easy to see that N(C) is a quasi-category for any category C and that thenerve functor N is fully-faithful. Moreover, those simplicial sets that are nerves ofcategories can be characterized as follows. Call a quasi-category X a strict quasi-

category if every inner horn

k

[n] X has a unique filler. It can then be shownthat a simplicial set is a strict quasi-category if, and only if, it is the nerve of acategory. In this way quasi-categories can be seen to extend categories. A quasi-category can be thought of as a special case of an -category, one in which all cellsof dimension bigger than 1 are invertible. As it turns out, much of the theory ofcategories can be extended to quasi-categories. Thus the notion of a quasi-categoryis a good replacement for categories particularly in cases such as weak algebras whenthe objects we wish to study do not form a category but do form a quasi-category.For instance in [23] Joyal lays the foundations of the theory of limits and colimitsin a quasi-category, so that it becomes meaningful for example to talk about limitsand colimits of weak P-algebras inside the quasi-category of such algebras.

Another, somewhat less common, approach to operads is as a generalization ofcategories. In a category every arrow has an object as its domain and an object

as its codomain. If instead of having just one object as domain we allow an arrowto have an ordered tuple of objects as domain (including the empty tuple) then weobtain the notion of an operad. We should immediately emphasize that from nowon by an operad we mean a symmetric coloured operad in Set, which is also knownas a symmetric multicategory. A category is then precisely an operad in which theonly operations present are of arity 1 (i.e, they only have 1-tuples as domains). Theobjects of the category are the colours of the operad and the arrows in the categoryare the operations in the operad. In this way the category of all small categoriesembeds in the category of all small operads (an operad is small if its colours andits operations form a set). Similarly, for a symmetric monoidal category E, thecategory of categories enriched in E embeds in the category of symmetric colouredoperads in E.

While this point of view is almost trivial, the development of operad theory

made it quite obscure. The reason, we suspect, is that originally operads weredefined in topological spaces and had just one object. Such operads were alreadycomplicated enough and, more importantly, they did the job they were designedfor (see [36]). One can say that early research of operad theory concerned itselfwith the sub-category of symmetric topological coloured operads spanned by thoseoperads with just one object. On the other hand, category theory was from theoutset concerned with categories with all possible objects and not just one-objectcategories (i.e., monoids) and enriched categories came later.

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