Spherical monadic adjunctions of stable infinity categories

TL;DR

This paper explores spherical adjunctions in stable infinity categories, proving key properties, providing examples from local systems on spheres, and characterizing when monadic adjunctions are spherical based on monad properties.

Contribution

It establishes a characterization of spherical monadic adjunctions in stable infinity categories using properties of the monad and twist functors, extending prior work on semiorthogonal decompositions.

Findings

Proves the 2/4 property of spherical adjunctions in stable $$-categories.

Provides examples of spherical adjunctions from local systems on spheres.

Characterizes spherical monadic adjunctions via the twist functor and monad properties.

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, 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.