  Perverse schobers are conjectural categorical analogs of perverse sheaves. We
show that such structures appear naturally in Homological Minimal Model Program
which studies the effect of birational transformations such as flops, on the
coherent derived categories. More precisely, the flop data are analogous to
hyperbolic stalks of a perverse sheaf.
  In the first part of the paper we study schober-type diagrams of categories
corresponding to flops of relative dimension 1, in particular we determine the
categorical analogs of the (compactly supported) cohomology with coefficients
in such schobers.
  In the second part we consider the example of a "web of flops" provided by
the Grothendieck resolution associated to a reductive Lie algebra g and study
the corresponding schober-type diagram. For g=sl(3) we relate this diagram to
the classical space of complete triangles studied by Schubert, Semple and
others.
