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posted on 2022-05-23, 05:02 authored by Nicola Di VittorioNicola Di Vittorio

Introduced independently by Grothendieck and Heller in the 1980s, derivators provide a formal way to study homotopy theories by working in some quotient category such as the homotopy category of a model category. One of the advantages of derivator theory is that they enable a calculus of homotopy Kan extensions that relies almost entirely on ordinary category theory (with a bit of 2-category theory). They can also be seen as an approximation of (1; 1)-categories, a concept which has been realized using a range of combinatorial and homotopy theoretic models. Quasi-categories are presumably the best developed between such models, and their theory has been established in the 2000s by Joyal and Lurie. In 2015 Riehl and Verity introduced 1-cosmoi, which are particular (1; 2)-categories where one can develop (1; 1)-category theory in a synthetic way. They noticed that much of the theory of 1-cosmoi can be developed inside a quotient, the homotopy 2-category. Inspired by this philosophy, we introduce a set of axioms that mirror key properties of the 1-cosmological approach to 1-category theory and demonstrate they hold in a variety of models, including common models related to 1-category theory. We also prove that these axioms are stable under a particular shift operation.


Table of Contents

Introduction -- 1 Background -- 2 Two-dimensional derivator theory


A thesis submitted to Macquarie University for the degree of Master of Research

Awarding Institution

Macquarie University

Degree Type

Thesis MRes


Thesis (MRes), Macquarie University, Faculty of Science and Engineering

Department, Centre or School

Department of Mathematics and Statistics

Year of Award


Principal Supervisor

Dominic Verity


Copyright disclaimer: Copyright Nicola Di Vittorio 2020




iv, 53 pages

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