Spherical Monadic Adjunctions of Stable Infinity Categories

Abstract

Abstract 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].