N-spherical functors and categorification of Euler’s continuants

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 \({∞ }\) -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 \({∞ }\) -) 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.