The present thesis is organized around the structural theme that classical constructions–geometric, algebraic and categorical–support meaningful generalizations and specializations in settings endowed with additional structure, properties or higher dimensional features. Its three parts, although logically independent, are bound together by this general philosophy.

The first part develops Categorical Algebraic Geometry by bringing together the theme of categorification of monoid actions and Toën-Vaquié’s theory of a scheme relative to a Bénabou cosmos. Motivated by Baez-Dolan’s Microcosm principle which insists that actegories are the right setups to internalize monoid actions just as (braided/symmetric) monoidal categories are the right setups to internalize (commutative) monoids, we incorporate a suitably structured actegory into the symmetric monoidal setting and investigate the extent to which the foundational aspects of the theory persist in this broader context. In particular, we study the behaviour of our notion of a scheme relative to an actegory under base-change induced by a symmetric monoidal adjunction, together with a lax linear functor between actegories.

The second part is devoted to the analysis of separability phenomena for functors. We introduce and investigate a new class of functors, termed heavily separable functors of the second kind, which simultaneously generalize the heavily separable functors of Ardizzoni-Menini and specialize the separable functors of the second kind of Caenepeel-Militaru. This yields a unified framework in which multiple strands of separability theory can be treated simultaneously. Our main result is a Rafael-type theorem providing necessary and sufficient conditions for heavy separability of the second kind for adjoint functors, with applications to three classical contexts in separable functor theory : ringoids, (co)monads and entwined modules.

The third part addresses the existence of Street’s Eilenberg–Moore and Kleisli objects in the 2-category vDbl of virtual double categories. Virtual double categories are highly general 2-dimensional categorical structures that arise naturally in a variety of contexts. In 2010, Cruttwell–Shulman defined monads on them to formulate a unified framework for generalized multicategories. We show that usual constructions pertaining to ordinary monads on categories extend just as naturally to the virtual-double-categorical setting. This is followed by an application to the theory of monads in the 2-category Multicat of (plain) multicategories/coloured operads. One of the highlights of the work undertaken in this part is the use of change-of-shape arguments to establish 2-adjunctions connecting the 2-categories vDbl, Cat and Multicat.