This thesis consists of three disjoint parts. The first part, Schemes relative to Actegories, develops Categorical Algebraic Geometry by bringing together Toën–Vaquié’s [2009] notion of a relative scheme (over a bicomplete closed symmetric monoidal category a.k.a. a Bénabou cosmos C) and the notion of vertical categorification of monoid actions, popularly known as “Actegories”. This is motivated by Baez–Dolan’s Microcosm Principle, which suggests that actegories are the right setups for internalizing monoid actions, just as (braided/symmetric) monoidal categories are the right setups to internalize (commutative) monoids. The proposed notion is that of a scheme relative to a C-actegory M. While generalizing relative schemes, the emphasis is on the ideas and techniques from category theory. Among other things, we present a theorem which investigates the behaviour of our notion of a scheme under base changes along an adjunction in the 2-category symMonCatlax (of symmetric monoidal categories, lax symmetric monoidal functors and monoidal transformations) together with a lax linear functor between actegories. We conclude with a generalization of a construction of Connes and Consani which combines classical Z-schemes and their notion of schemes over CMon_0 (the category of commutative monoids with absorbing elements). The proposed construction is speculative and extends Connes and Consani’s work by involving M-schemes for an actegory M over an arbitrary bicomplete closed symmetric monoidal category C with a zero object.

The second part, Heavy separability of the second kind, addresses Categorical Algebra by synthesizing two well-studied developments of separable functors: separable functors of the second kind (Caenepeel and Militaru [2003]) and heavily separable functors (Ardizzoni and Menini [2020]). The 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.

The third part, Construction of Eilenberg-Moore objects in the 2-category vDbl, applies the formal theory of monads to Virtual Double Category Theory. Virtual double categories, introduced by Burroni [1971] as “multicatégories” and later popularized by Leinster [2002] as “fc-multicategories”, are 2-dimensional categorical structures generalizing monoidal categories, multicategories, bicategories, and (pseudo) double categories. In 2010, Cruttwell–-Shulman defined monads on them to provide a unified framework for generalized multicategories. The main result in this brief part establishes that the 2-category vDbl (of virtual double categories) admits Eilenberg-Moore objects (à la Street [1972]), enabling the application of the formal theory of monads to virtual double categories.

In this talk, we will focus on the first part.