  Heaps are para-associative ternary operations bijectively exemplified by
groups via the operation $(x,y,z) \mapsto x y^{-1} z$. They are also ternary
self-distributive, and have a diagrammatic interpretation in terms of framed
links. Motivated by these properties, we define para-associative and heap
cohomology theories and also a ternary self-distributive cohomology theory with
abelian heap coefficients. We show that one of the heap cohomologies is related
to group cohomology via a long exact sequence. Moreover we construct maps
between second cohomology groups of normalized group cohomology and heap
cohomology, and show that the latter injects into the ternary self-distributive
second cohomology group. We proceed to study heap objects in symmetric monoidal
categories providing a characterization of pointed heaps as involutory Hopf
monoids in the given category. Finally we prove that heap objects are also
"categorically" self-distributive in an appropriate sense.
