  Constructible sheaves of abelian groups on a stratified space can be
equivalently described in terms of representations of the exit-path category.
In this work, we provide a similar presentation of the abelian category of
perverse sheaves on a stratified surface in terms of representations of the
so-called paracyclic category of the surface. The category models a hybrid
exit-entrance behaviour with respect to chosen sectors of direction, placing it
"in between" exit and entrance path categories. In particular, this perspective
yields an intrinsic definition of perverse sheaves as an abelian category
without reference to derived categories and t-structures.
