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SKETCHES IN HIGHER CATEGORY THEORY

By

Rémy Tuyéras

A thesis submitted to Macquarie University for the degree of Doctor of Philosophy

Centre of Australian Category Theory (CoAct) Department of Mathematics

September 2015

ii

c© Rémy Tuyéras, 2016-2017. Typeset in LATEX 2ε.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

§1.1. Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §1.2. Conventions and usual theory . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 2. Vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

§2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 §2.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 §2.3. Theory of vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 §2.4. Examples of everyday vertebrae . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 3. Spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

§3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 §3.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 §3.3. Theory of spines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 §3.4. Examples of everyday spines . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Chapter 4. Vertebral and Spinal Categories . . . . . . . . . . . . . . . . . . . . . . . 155

§4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 §4.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 §4.3. Algebraic structures on vertebrae and spines . . . . . . . . . . . . . . . . . . 163 §4.4. Theory of spinal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 §4.5. Everyday examples of vertebral and spinal categories . . . . . . . . . . . . . 202

Chapter 5. Construct of Homotopy Theories . . . . . . . . . . . . . . . . . . . . . . . 205

§5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 §5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 §5.3. System of models for a croquis . . . . . . . . . . . . . . . . . . . . . . . . . . 214 §5.4. From narratives to combinatorial categories . . . . . . . . . . . . . . . . . . . 231 §5.5. From spinal categories to homotopy theories . . . . . . . . . . . . . . . . . . 262

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iv Contents

Chapter 6. Towards the Homotopy Hypothesis . . . . . . . . . . . . . . . . . . . . . 271

§6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 §6.2. Vertebral structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 §6.3. Spines and their functorial framings . . . . . . . . . . . . . . . . . . . . . . . 289 §6.4. Spinal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Chapter A. List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Abstract

A sketch, in the sense of Charles Ehresmann, provides the data needed to specify a type of mathematical structure. The category of structures for a given sketch has good properties assured by the existence of reflection functors from presheaf categories. For example, a Grothendieck site is an example of a sketch and the reflection functor assigns the sheaf associated to a presheaf.

The present thesis proposes a generalisation of sketch for higher categories. The motiva- tion for this comes from homotopy theory rather than universal algebra. For the requisite homotopy structure on a category, we introduce vertebral categories and spinal categories, rather than starting with a Quillen model category where the weak equivalences are part of the data. For us, the weak equivalences are defined from the vertebrae in much the same way as they are constructed from discs and spheres in topology. The categories of structures for our generalised sketches are categories of fibrant objects in the usual cases. Categories of stacks and spectra are examples. Moreover, our construct of the reflection functor is an extension of the small object argument of Quillen.

Finally, using our algorithm for constructing weak equivalences, we show that Grothen- dieck’s ∞-groupoids form a spinal category. By combining all the results developed in the present thesis, future work will aim at proving that the category of ∞-groupoids admits a Quillen model structure and satisfies the Homotopy Hypothesis (conjectured by Grothendieck in 1983 and still unproved).

v

vi Abstract

Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowl- edge, original and has not been submitted in whole or part for a degree in any university.

Rémy Tuyéras

Acknowledgements

I would like to thank the following people, who, in one way or another, made the writing of this thesis – and its content – possible. I would thus like to thank

Michael Batanin, my first supervisor, for his continuous support, interest and sugges- tions, his guidance and availability as well as his trust in my ideas even if they seemed strange in their very first form;

Dimitri Ara, Timothy Porter and Dorette Pronk for accepting to review my thesis and giving me useful advice regarding the ways it could be improved. I would particularly like to thank Timothy Porter for his impressive and thorough report regarding every section of the present thesis;

Steve Lack, my second supervisor, for his enthusiastic support and advice;

Ross Street, for his help, kindness, advice and great stories told around glasses of wine and cheese platers;

Camell Kachour, for his questions as well as the interesting and passionate discussions about all sorts of mathematics – I learnt a lot;

Dimitri Ara and Georges Malstiniotis, for their interest in my research, questions, remarks and discussions. I would particularly like to thank Georges Malstiniotis for his invitation to talk at the Working group Algèbre et Topologie Homotopiques of Paris VII in February 2014. I would also like to thank Dimitri Ara, Ieke Moerdjeck and all the team of Radboud University Nijmegen for inviting me to talk at their Algebraic topology seminar in December 2013;

André Joyal, for interesting discussions and communications, pertinent comments as well as good advice. Our discussions allowed me to have a broader point of view on my research and think more deeply about future research investigations;

Dorette Pronk and David White, for their discussions, questions, remarks and com- ments that lead me to enrich the content of this thesis;

Mathieu Anel and Clemens Berger, for their pertinent questions;

Benôıt Fresse, Paul-Andé Melliès and Clemens Berger without whom I would not have been able to do my Ph.D. at Macquarie University;

Anne-Laurence Bibost, for her warm and lively encouragements as well as the inter- esting discussions about Behavioural Ecology;

Poon Leung and Ramón Abud Alcalá, for their friendship, mathematical conversa- tions and amazing support during the writing of this thesis;

vii

viii Acknowledgements

Jim Andrianopoulos, Sherwin Bagheri, Leon Beale, Gaetan Bisson, Mitchell Buckley, Matthew Burke, Geoff Edington-Cheater, Benjamin Alarcón Heredia, Bryce Kerr, Daniel Lin, Audrey Markowskei, Brank Nicolić, Chris Nguyen, Alina Ostafe, Ringo, Nataliya Sokolovska and Daniel Sutantyo for all the mathematical conversations, various seminars, hanging out after work, entertaining moments and probably more, which made the writing of the thesis really enjoyable;

The CoAct team of Macquarie University (Michael Batanin, Richard Garner, Steve Lack, Ross Street, Dominic Verity, Mark Weber) for their interests, questions, remarks, mathematical discussions and letting me express my ideas at their seminar;

Finally, last but not least, I would like to thank my family for their eternal and uncondi- tional support.

Chapter 1

Introduction

1.1. Presentation

1.1.1. Goals of the thesis. Although there are many ways of telling a story, it is always useful to have a storyline in mind. I will therefore introduce the present work by giving several questions, which the reader may use as a breadcrumb trail to make sense of the different chapters and better appreciate the reading of this text. In this respect, the present thesis addresses the following three main questions related to homotopy theory and (higher) category theory.

1. Is there a systematic way of producing weak equivalences for general homotopy the- ories? The intended description should be intuitive.

2. Is there a systematic way of describing the colimits of a category of models for a sketch? The intended description should allo

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