We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable \({\infty }\) -categories. We study the relative Waldhausen S-construction \(S_\bullet (F)\) of the spherical functor F and show that it has a natural paracyclic structure (“rotation symmetry”). This fulfills a part of the general program of perverse schobers which are conjectural categorical upgrades of perverse sheaves. If we view a spherical functor as defining a schober on a disk, then each component \(S_n(F)\) of the S-construction gives a categorification of the cohomology of a perverse sheaf on a disk with support in a union of \((n+1)\) closed arcs in the boundary. In other words, \(S_n(F)\) can be interpreted as the Fukaya category of the disk with coefficients in the schober and with support (“stops”) at the boundary arcs. The importance of the paracyclic structure is that it allows us to naturally associate the above data to disks on oriented surfaces. The action of the paracyclic rotation is a categorical analog of the monodromy of a perverse sheaf.
