Spherical Monadic Adjunctions of Stable Infinity Categories

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].

Complexes of stable $\infty$-categories

“…The fact that the upper right square and the lower left square are biCartesian, i.e. both pullback and pushout, follows from the general properties of units and counit of spherical adjunctions, see for instance Remark 2.9 and Lemma 2.10 in [Chr22c]. By the pasting laws for biCartesian squares, we find that the horizontal middle composite is a natural equivalence if and only if the vertical middle composite is a natural equivalence.…”

“…6.4.13]. These Kan fibrations are furthermore spherical fibrations, meaning that their fibres are spheres, the sphericalness of the functors was thus shown in [Chr22c]. We note that in general, tot > pLocpAqq is not a spherical complex.…”

“…In particular, the composition of the spherical twists associated to a pair of spherical adjunctions with the same target, must again be a spherical twist (also cf. [Chr22c] for an 8-categorical generalization). The main result of [Bar20] provides an explicit description of a spherical adjunction that describes this composite twist in terms of the given spherical adjunctions.…”

Ginzburg algebras of triangulated surfaces and perverse schobers

“…Proof. The adjunction f * ⊣ f * is spherical with twist functor T RModR ≃ [1 − n], see [Chr22c], the same thus holds for f * ⊣ f * . By Proposition 5.7, the functor f * thus admits a weak right n-Calabi-Yau structure if Ind Fun(S n−1 , RMod perf R ) admits a weak right n-Calabi-Yau structure.…”

“…8.1]. One implication of this can be proven using [Chr22c,Prop. 4.5] and the fact that the iterated adjoints between proper, compactly generated ∞-categories can be obtained by compositions with powers of the Serre functors.…”