  It has long been known that a key ingredient for a sheaf representation of a
universal algebra A consists in a distributive lattice of commuting congruences
on A. The sheaf representations of universal algebras (over stably compact
spaces) that arise in this manner have been recently characterised by Gehrke
and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of
the notion of softness.
  In this paper, we extend the scope of this theory by replacing varieties of
algebras with Barr-exact categories, thus encompassing a number of
"non-algebraic" examples. Our approach is based on the notion of K-sheaf:
intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined
on compact ones. Throughout, we consider sheaves on complete lattices rather
than spaces; this allows us to obtain point-free versions of sheaf
representations whereby spaces are replaced with frames.
  These results are used to construct sheaf representations for the dual of the
category of compact ordered spaces, and to recover Banaschewski and Vermeulen's
point-free sheaf representation of commutative Gelfand rings (Quaest. Math.,
2011).
