Perverse Schobers

 We propose to develop a categorical analog of
the theory of perverse sheaves, when vector spaces are replaced by
triangulated categories. We call these hypothetical objects
perverse Schobers. So a perverse Schober should give a perverse
sheaf after passing to Grothendieck groups.
Many explcit "quiver" description of
categories of perverse sheaves admit immediate categorical
analogs thus providing ad hoc definitions of perverse Schobers
in various cases. In particular, the categorical analog of perverse
sheaves on the disk is provided by the concept of a spherical
functor. Joint work with V. Schechtman.