UC Berkeley

This thesis consists of two parts. The first half concerns various foundational aspects ofthe theory of enriched ∞-categories. We develop the theory of adjunctions and weighted limits and colimits in enriched ∞-categories. We introduce theories of enriched ∞-props and operads, which provide a framework for the study of higher algebra in the enriched context. Finally, we study the theory of monads and monadic adjunctions in enriched (∞, 2)-categories, and prove an enriched generalization of the Barr-Beck-Lurie monadicity theorem. The second half of this thesis applies the results of the first half to the study of higher categorical sheaf theory in derived algebraic geometry. We introduce and study a theory of quasicoherent sheaves of presentable stable (∞, n)-categories on prestacks, generalizing the case n = 1 studied in [Gai15]. We prove a universal property for the (∞, n + 1)-category of correspondences, generalizing and providing a new approach to the case n = 1 from [GR17], and use it to show that our higher quasicoherent sheaves give rise to representations of the higher categories of correspondences of prestacks. We also introduce a notion of n-affineness for prestacks and provide a simple inductive criterion for checking n-affineness, which allows one to reduce affineness questions to the case n = 1 studied in [Gai15].