UCLA

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.