This paper concerns spherical adjunctions of stable ∞-categories and their relation to monadic adjunctions. We begin with a proof of the 2/4 property of spherical adjunctions in the setting of stable ∞-categories. The proof is based on the description of spherical adjunctions as 4-periodic semiorthogonal decompositions given by Halpern-Leistner and Shipman [HS16] and Dyckerhoff, Kapranov, Schechtman and Soibelman [DKSS19]. We then describe a class of examples of spherical adjunctions arising from local systems on spheres. The main result of this paper is a characterization of the sphericalness of a monadic adjunctions in terms of properties of the monad. Namely, a monadic adjunction is spherical if and only if the twist functor is an equivalence and commutes with the unit map of the monad. This characterization is inspired by work of Ed Segal [Seg18]. − → b if e is a coCartesian edge. Definition 1.3. Let p : M → ∆ 1 be an adjunction between A and B. An edge e : a → a ′ in A is called a unit map if there exists a diagram in Fun(∆ 2 , M) of the form b a a ′ e ! * with b ∈ B. An edge e ′ : b → b ′ in Fun(∆ 1 , B) is called a counit map if there exists a diagram in Fun(∆ 2 , M) of the form a b b ′ ! * e ′with a ∈ A.

This paper concerns spherical adjunctions of stable $\infty $-categories and their relation to monadic adjunctions. We begin with a proof of the 2/4 property of spherical adjunctions in the setting of stable $\infty $-categories. The proof is based on the description of spherical adjunctions as $4$-periodic semiorthogonal decompositions given by Halpern-Leistner and Shipman [4] and Dyckerhoff et al. [3]. We then describe a class of examples of spherical adjunctions arising from local systems on spheres. The main result of this paper is a characterization of the sphericalness of a monadic adjunctions in terms of properties of the monad. Namely, a monadic adjunction is spherical if and only if the twist functor is an equivalence and commutes with the unit map of the monad. This characterization is inspired by work of Segal [10].

We relate the derived category of a relative Ginzburg algebra of an n-angulated surface to the geometry of the underlying surface. Results include the description of a subset of the objects in the derived category in terms of curves in the surface and their Hom's in terms of intersection. By using the description of such a derived category as the global sections of perverse schober, we arrive at the geometric model through gluing local data. Nearly all results also hold for the perverse schober defined over any commutative ring spectrum.As a direct application of the geometric model, we categorify the extended mutation matrices of a class of cluster algebras with coefficients, associated to multi-laminated marked surfaces by Fomin-Thurston [FT18]. 7 The Jacobian gentle algebra 47 8 Derived equivalences associated to flips of the n-angulation 49

Ginzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the surface. Inspired by this perspective, we provide a description of the unbounded derived category in terms of a perverse schober. The main novelty is a gluing formalism describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a first application, we give a new construction of derived equivalences between these Ginzburg algebras associated to flips of an edge of the triangulation. Finally, we note that the perverse schober as well as the resulting gluing construction can also be defined over the sphere spectrum.

We study complexes of stable 8-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry, and differential topology. One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions. As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context. We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for CP 2 . Finally, we develop the concept of a lax additive p8, 2q-category and propose it as a suitable framework to formulate further aspects of categorified homological algebra.