Higher derivators as a foundation for ∞-category theory

<p dir="ltr">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.</p><p dir="ltr">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.</p>