%%  Abstract for: Hereditary and modal closure operators
%%  with Gabriele Castellini and George E. Strecker

  We first investigate a symmetrized generalization of the notion of
  categorical closure operator on an arbitrary class  M  of morphisms.
  For this we introduce the concept of  Z - modality which presents a
  common generalization of hereditariness and modality.  Applications
  are indicated concerning the Cassidy-H\'ebert-Kelly Galois
  connection linking certain pre-factorizations with reflective
  subcategories of a finitely complete category that has arbitrary
  intersections of strong monos.  In the case that  M  is part of a
  factorizations system  <E,M>  for sinks and that  E  is sufficiently
  well-behaved, we present constructions for  Z - modal cores and
  hulls, and investigate how these constructions interact with closure
  operators that are weakly hereditary and with those that are
  idempotent.  If  E  is not well-behaved, but the category is  M -
  well-powered, it is shown that iterative methosd can be used to
  construct various hulls.