In this thesis, we study the 2-category of infinity-categories, largely

with attention to its relationships with the 2-category of prederivators.

We prove that the 2-category of infinity-categories admits a small set of objects detecting equivalences and

satisfies a Brown representability theorem, which we formulate using a new notion of compactly generated 2-category. We show that the canonical

2-functor from the 2-category of infinity-categories into the 2-category of prederivators detects equivalences and, under appropriate

size conditions, induces an equivalence on hom-categories.

We explain how to extend prederivators defined on the 2-category of

ordinary categories to the domain of all infinity-categories using the delocalization theorem. We use the

Brown representability theorem to give conditions under which a prederivator is representable by an infinity-category. We also show how to extend derivators defined on categories

and satisfying a mild size condition to derivators on infinity-categories, using an extension

of Cisinski's theorem on the universality of derivators of spaces. This extension allows us to give conditions under which the small sub-prederivators of quite general derivators are all representable by infinity-categories.

This thesis is concerned with two disparate results in the field of abstract homotopy theory, treated through the lens of derivators. In Chapter 2, we recall essential results in the theory of derivators, mostly drawn from [Groth, Algebr. Geom. Topol. 13 (2013), no. 1, 313–374], but provide new proofs in many cases. We do this in order to expand results on derivators to results on half derivators, a weaker notion that is required for many examples of interest. In Chapter 3, we revisit and improve Alex Heller's results on the stabilization of derivators in [Heller, J. Pure Appl. Algebra 115 (1997), no. 2, 113–130], recovering his results entirely. Along the way we give some details of the localization theory of derivators and prove some new results in that vein. We are able to give a 2-categorical statement of stabilization that should allow for future applications to derivator K-theory, which was the impetus for the study. In Chapter 4, we answer questions in derivator K-theory asked by [Muro-Raptis, Ann. K-Theory 2 (2017), no. 2, 303–340]. We define a new class of pointed derivators suitable for K-theory and prove some interesting properties about these per se. We then prove that derivator K-theory satisfies additivity and has the structure of an infinite loop space. We anticipate being able to prove properties of derivator K-theory using the theorems herein.

We consider separable ring objects in symmetric monoidal categories and investigate what it means for an extension of ring objects to be (quasi)-Galois. Reminiscent of field theory, we define splitting ring extensions and examine how they occur. We also establish a version of quasi-Galois-descent for ring objects.

Specializing to tensor-triangulated categories, we study how extension-of-scalars along a quasi-Galois ring object affects the Balmer spectrum. We define what it means for a separable ring to have constant degree, which turns out to be a necessary and sufficient condition for the existence of a quasi-Galois closure. Finally, we illustrate the above for separable rings occurring in modular representation theory.

For each object in a tensor triangulated category, we construct a natural continuous map from the object's support---a closed subset of the category's triangular spectrum---to the Zariski spectrum of a certain commutative ring of endomorphisms. When applied to the unit object this recovers a construction of P. Balmer. These maps provide an iterative approach for understanding the spectrum of a tensor triangulated category by starting with the comparison map for the unit object and iteratively analyzing the fibers of this map via "higher" comparison maps. We illustrate this approach for the stable homotopy category of finite spectra. In fact, the same underlying construction produces a whole collection of new comparison maps, including maps associated to (and defined on) each closed subset of the triangular spectrum. These latter maps provide an alternative strategy for analyzing the spectrum by iteratively building a filtration of closed subsets by pulling back filtrations of affine schemes.

We prove that the idempotent completion of a strong additive derivator~$\mathbb D$ is also a strong additive derivator which is moreover stable if~$\mathbb D$ is. This recovers the main results in~\cite{Balmer-idempotent} and~\cite{IdempotentCompletionpretriangulated} assuming that the (pre)triangulated categories we are working with are the base of a strong additive (stable) derivator (which happens in all examples one encounters in practice). We also show that given a separable cocontinuous monad on a triangulated derivator, the levelwise Eilenberg-Moore categories of modules glue together to a triangulated derivator. This allows us to give examples of derivators that are stable but not strong. Furthermore, we study localization of right derivators under classes that are closed under colimits and, as a special case, recover Franke's result~\cite{Franke} that we can form Verdier quotients of small triangulated derivators. Finally, we study compact objects in big triangulated derivators. We show that any compact object can be written as a retract of a finite direct colimit of a coherent diagram that is pointwise in a chosen generating set. We also show that levelwise and pointwise compact objects over small diagrams coincide. As an application, we extend the equivalence of~\cite{Neeman-Thomason} to an equivalence of~$\mathbf{Dir}_{\text f}$-derivators.

This dissertation investigates objects known as ``shifted-localized derivators'' through the lens of algebraic geometry by building affine and projective space objects over an arbitrary derivator. For affine space, we give a definition of $\bbA^n$ over a derivator $\bbD$, and then show a series of results identifying it as extending the $\bbA^n$-construction in algebraic geometry, including a universal property. We then note that this construction is not specific to the choice of $\bbA^n$ but can be used for any choice of abelian, unital monoid.

In order to tackle the projective space case, first we prove some technical results on two topics: compact generation of triangulated derivators, and a Day convolution structure on symmetric monoidal derivators shifted by symmetric monoidal categories. To construct $\bbP^n$, we first shift by a symmetric monoidal category $Q_n$ to achieve an analogue of graded modules over polynomial rings, and then localize by localizing to copies of $\bbA^n$. We prove generation and semiorthogonal decomposition results in $\bbP^n$ by using this formulation of localization, and the aforementioned technical results.