N-spherical functors and categorification of Euler's continuants

TL;DR

This paper introduces N-spherical functors as higher analogs of spherical functors using categorified Euler's continuants, linking them to N-periodic semi-orthogonal decompositions in stable infinity-categories.

Contribution

It defines N-spherical functors via categorification of Euler's continuants, extending the classical spherical functor concept to higher orders and connecting them to categorical decompositions.

Findings

Defined N-spherical functors with vanishing twist and cotwist of order N-1

Characterized N-periodic semi-orthogonal decompositions via N-sphericity

Established analogy between continued fractions and iterated orthogonals

Abstract

Euler's continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely, coherently commutative cubes) of functors involving compositions of a given functor and its adjoints of various orders, with the differentials built out of units and counits of the adjunctions. In the stable infinity-categorical context these complexes/cubes can be assigned totalizations which are new functors serving as higher analogs of the spherical twist and cotwist. We define N-spherical functors by vanishing of the twist and cotwist of order N-1 in which case those of order N-2 are equivalences. The usual concept of a spherical functor corresponds to N=4. We characterize N-periodic semi-orthogonal decompositions of triangulated (stable infinity-) categories in terms of N-sphericity of their gluing functors. The procedure of forming iterated orthogonals turns out to be analogous to the procedure of forming a continued fraction.