  We develop the theory of semi-orthogonal decompositions and spherical
functors in the framework of stable $\infty$-categories. Building on this, we
study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical
functor $F$ and equip it with a natural paracyclic structure (``rotational
symmetry''). This fulfills a part of the general program to provide a rigorous
account of perverse schobers which are (thus far conjectural) categorifications
of perverse sheaves. Namely, in terms of our previous identification of
perverse sheaves on Riemann surfaces with Milnor sheaves, the relative
$S$-construction with its paracyclic symmetry amounts to a categorification of
the stalks of a Milnor sheaf at a singularity of the corresponding perverse
sheaf. The action of the paracyclic rotation is a categorical analog of the
monodromy on the vanishing cycles of a perverse sheaf. Having this local
categorification in mind, we may view the S-construction of a spherical functor
as defining a schober locally at a singularity. Each component $S_n(F)$ can be
interpreted as a partially wrapped Fukaya category of the disk with
coefficients in the schober and with $n+1$ stops at the boundary.
