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Homotopy theory of Grothendieck ∞-groupoids and ∞-categories

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posted on 28.03.2022, 00:46 authored by Edoardo Lanari
This work investigates the homotopy theory of globular models for higher categorical structures. In particular, we focus on weak ∞-groupoids, but most of the constructions can be performed also for weak ∞-categories, and we explicitly mention this when appropriate. Motivated by Grothendieck’s homotopy hypothesis, we study algebraic models of homotopy types in the form of ∞-groupoids, and we address the problem of constructing a path object for these structures, after having introduced their homotopy theory. In detail, we define (trivial) cofibrations and (trivial) fibrations of ∞-groupoids, and prove some basic facts about the induced factorization systems. The construction of a path-object endofunctor is a highly non-trivial task, and the first step we take is to characterize those globular theories whose category of models can be endowed with cofibrantly generated semi-model structure of a precise form, and we also give a sufficient condition for this to happen, based on the existence of a path object endofunctor. We then construct the underlying globular set of this path object based on the notion of cylinders, and show how to endow it with systems of structures, involving compositions, identities and inverses. Using the combinatorics of finite planar rooted trees we construct an approximation of the algebraic structure needed for the construction of the path object and we introduce modifications to “correct” this approximation in low dimensions, and we thus interpret all operations of dimension less than or equal to 2. Finally, we manage to complete this construction for the finite-dimensional case of Grothendieck 3-groupoids, thanks to the introduction of a bicategory of cylinders and modifications. We thus establish a semi-model structure on this category, which is conjectured to model homotopy 3-types.

History

Table of Contents

Chapter 1. Introduction -- Chapter 2. Globular theories and models -- Chapter 3. Basic homotopy theory of ∞-groupoids -- Chapter 4. Semi-model structures on categories of models of globular theories -- Chapter 5. Main constructions -- Chapter 6. Systems of structure on the path object -- Chapter 7. Elementary interpretation of operations on cylinders -- Chapter 8. Path object on Grothendieck 3-groupoids of type CW - Chapter 9. Perspectives -- Appendices -- Bibliography.

Notes

Empirical thesis. Bibliography: pages 120-121

Awarding Institution

Macquarie University

Degree Type

Thesis PhD

Degree

PhD, Macquarie University, Faculty of Science and Engineering, Department of Mathematics and Statistics

Department, Centre or School

Department of Mathematics and Statistics

Year of Award

2019

Principal Supervisor

Richard Garner

Additional Supervisor 1

Dominic Verity

Rights

Copyright Edoardo Lanari 2018. Copyright disclaimer: http://mq.edu.au/library/copyright

Language

English

Extent

1 online resource (x, 121 pages)

Former Identifiers

mq:70978 http://hdl.handle.net/1959.14/1269609