  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, Shipman and by
Dyckerhoff, Kapranov, Schechtman, Soibelman. 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.
