Sphericalization and the Universal Spherical Adjunction

TL;DR

This paper introduces a procedure to invert twist and cotwist functors in adjunctions of stable $ abla$-categories, leading to explicit constructions of spherical adjunctions and their classification.

Contribution

It provides a simple method for inverting twist and cotwist functors and constructs adjoints to the inclusion of spherical adjunctions within all adjunctions.

Findings

Explicit construction of left and right adjoints to spherical adjunctions.

Description of the walking spherical adjunction as a classifying $( abla,2)$-category.

Proof that every spherical functor has infinitely many adjoints.

Abstract

For every adjunction of stable $\infty$-categories -- or more generally, in any locally stable $(\infty,2)$-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 $(\infty,2)$-category of spherical adjunctions of stable $\infty$-categories into all adjunctions. We utilize these adjoints to give a description of the walking spherical adjunction, a locally stable $(\infty,2)$-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.