UCLA

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.