  In this paper we study derived categories of nodal singularities. We show
that for all nodal singularities there is a categorical resolution whose kernel
is generated by a $2$ or $3$-spherical object, depending on the dimension. We
apply this result to the case of nodal cubic fourfolds, where we describe the
kernel generator of the categorical resolution as an object in the bounded
derived category of the associated degree six K3 surface.
  This paper originated from one of the problem sessions at the Interactive
Workshop and Hausdorff School "Hyperk\"ahler Geometry", Bonn, September 6-10,
2021.
