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Higher derivators as a foundation for ∞-category theory

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posted on 2025-01-29, 01:44 authored by Nicola Di Vittorio

Derivators were introduced by Grothendieck and Heller to formalise homotopy limits and colimits in a universal way by axiomatising what happens when we work with collections of homotopy categories of diagram model categories. Model categories can in turn be seen as a presentation for (∞, 1)-categories, a concept which has been described using a range of combinatorial and homotopy theoretic models. With a shift of perspective, Riehl and Verity realised that (∞, 1)-categories can be treated as a primitive concept within their theory of ∞-cosmoi. In this context, one can develop the theory of (∞, 1)-categories without ever having to define them explicitly, analogous to the case of formal category theory inside a 2-category. Some deep results ensure that much of the theory of an ∞-cosmos can be developed inside a quotient, its homotopy 2-category.

This thesis develops a higher dimensional version of the theory of derivators suitable to capture key properties of the ∞-cosmological approach to (∞, 1)-category theory. In particular, we prove coherence results for 2-dimensional pasting diagrams in Gray-categories needed for some of the axioms of a 2-derivator. Furthermore we introduce further axioms proving that they hold in a variety of cases. We also provide an account of monadicity within the setting of 2-derivators.

History

Table of Contents

Introduction -- 1 Background -- 2 Pasting results in Gray-categories -- 3 Higher derivator theory -- 4 Future work -- References

Awarding Institution

Macquarie University

Degree Type

Thesis PhD

Degree

Doctor of Philosophy

Department, Centre or School

School of Mathematical and Physical Sciences

Year of Award

2024

Principal Supervisor

Dominic Verity

Additional Supervisor 1

Stephen Lack

Rights

Copyright: The Author Copyright disclaimer: https://www.mq.edu.au/copyright-disclaimer

Language

English

Extent

141 pages

Former Identifiers

AMIS ID: 397945

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