Sphericalization and the Universal Spherical Adjunction

For every adjunction of stable $\mathrm{\infty }$-categories – or more generally, in any locally stable $(\mathrm{\infty },2)$-category – we give a simple procedure for inverting the twist and cotwist functors associated to this adjunction. As a consequence, we obtain an explicit construction for a left and right adjoint to the inclusion of the $(\mathrm{\infty },2)$-category of spherical adjunctions of stable $\mathrm{\infty }$-categories into all adjunctions. We utilize these adjoints to give a description of the walking spherical adjunction, a locally stable $(\mathrm{\infty },2)$-category which classifies spherical adjunctions, and to provide a synthetic proof of the fact that every spherical functor admits infinitely many left and right adjoints.